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abstract: 'Let $X$ be a locally finite, connected graph without vertices of degree $1$. Non-backtracking random walk moves at each step with equal probability to one of the “forward” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of old results concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but *small cycles are dense in* $X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of [Grigorchuk]{} and [Cohen]{}.'
author:
- '**Ronald ORTNER and Wolfgang WOESS**'
date: 'June 27, 2003'
title: 'Non-backtracking random walks and cogrowth of graphs'
---
[^1]
Introduction and results {#intro}
========================
Let $X$ be the vertex set of a locally finite, connected graph, possibly with multiple edges and loops. We write $e(x,y)$ for the number of edges between the vertices $x$ and $y$, if $y \ne x$, while $e(x,x)$ is *twice* the number of loops at $x$ (see §\[cogrowth\].B for a discussion). The degree of a vertex $x \in X$ is $\deg(x)=\sum_y e(x,y)$. We assume that $\deg(x) \ge 2$ for all $x \in X$. *Non-backtracking (simple) random walk* (NBRW) is the following random process: at the beginning, the walker starts at some vertex $x$ and chooses with equal probablity one of the incident edges. He steps to the other end of that edge. At the later steps, the rule is the same, but the walker selects with equal probability only among those incident edges that are different from the one transversed at the previous step.
We write $q^{(n)}(x,y)$ for the probability that the random walker, starting at vertex $x$, is at vertex $y$ at the $n$-th step. Note that NBRW is *not* a Markov chain on $X$. The defining property of a Markov chain, that “the future depends only on the actual state and not on the past”, is violated, since the walker has to remember the edge along which he reached the actual state before moving on.
However, it is easy to turn NBRW into a Markov chain by changing the state space: with each edge, we associate two oppositely oriented edges $e, \check e$ (with $\check{\check e} = e$). We write $e^-$ and $e^+$ for the initial and terminal vertex of the edge $e$, so that $(\check e)^- = e^+$ and $(\check e)^+ = e^-$. (Note in particular, that for each a priori unoriented loop we get two oriented ones !) We now consider NBRW as a *Markov* process whose new state space is the set $E = E(X)$ of oriented edges, with transition matrix $Q_E = \bigl(q_E(e,f)\bigr)_{e,f\in E}$ given by $$q_E(e,f) = \begin{cases} \dfrac{1}{\deg(e^+)-1}\,,
&\mbox{if $e \to f$, that is, $f^- = e^+$ and $f \ne \check e$\,,}\\
0\,,&\mbox{otherwise.}
\end{cases}$$ Then there is the following obvious link between edge-NBRW and vertex-NBRW: for vertices $x, y \in X$, $$\label{qn}
q^{(n)}(x,y) = \frac{1}{\deg(x)}
\sum_{\scriptstyle e,f \in E : \atop \scriptstyle e^+=x,f^+=y}
q_E^{(n)}(e,f)\,,$$ where $q_E^{(n)}$ denotes the $n$-step transition probabilities, i.e., the elements of the matrix power $Q_E^n$, with $Q_E^0 = I_E$, the identity matrix over $E$. (Attention: $q^{(n)}(x,y)$ is *not* the $(x,y)$-element of an $n$-th matrix power over $X$!)
The following result is then a consequence of basic Markov chain theory.
\[limit1\] [(a)]{} If $X$ is finite, connected, with minimum degree $2$, then for all $x,y \in X$, $$\lim_{n \to \infty} \frac{1}{n}\Bigl( q^{(1)}(x,y) + q^{(2)}(x,y)
+ \dots + q^{(n)}(x,y)\Bigr) = \frac{\deg(y)}
{|E(X)|}\,.$$
[(b)]{} If in addition to the assumptions of [(a),]{} $X$ has minimum degree $3$, then for all $x,y \in X$, $$\begin{aligned}
\lim_{n \to \infty} q^{(n)}(x,y) &= \frac{\deg(y)}{|E(X)|}\,,\quad
\mbox{if $X$ is not bipartite,}\\
\lim_{n \to \infty} q^{(2n+\delta)}(x,y) &= \frac{2\deg(y)}{|E(X)|}\,,\quad
\mbox{if $X$ is bipartite,}
\end{aligned}$$ where $\delta=0$ (resp. $\delta=1$) according to whether $x$ and $y$ are at even (resp. odd) distance.
[(c)]{} If $X$ is infinite, connected, with minimum degree $2$, then for all $x, y \in X$, $$\lim_{n \to \infty} q^{(n)}(x,y) = 0\,.$$
In statements (a) and (b), note that $|E(X)|$ is twice the number of non-oriented edges.
As usual, the *distance* $d(x,y)$ between two vertices $x,y \in X$ is the minimum length of a path connecting the two. The *ball* of radius $R$ centred at $x$ is the subgraph $B(x,R) = \{ y \in X : d(y,x) \le R \}$ of $X$. Recall that a *cycle* of length $n$ in $X$ consists of a sequence $e_n = e_0, \dots, e_{n-1}$ of distinct edges whose initial vertices are all distinct, such that $e_{k-1} \to e_k$ for all $k=1, \dots, n$.
\[cycles\] We say that *small cycles are dense in* $X$, if there is $R > 0$ such that every ball $B(x,R)$ in $X$ contains a cycle.
Every finite, connected graph with minimum degree $2$ satisfies this condition.
The *automorphism group* of $X$ consists of all bijections $g: X \to X$ which satisfy $e(gx,gy) = e(x,y)$ for all $x,y \in X$. A graph is called *transitive*, resp. *almost transitive* if the automorphism group acts with one orbit, resp. finitely many orbits on $X$. Obviously, an infinite, almost transitive graph with minimum degree $2$ has dense small cycles unless it is a tree. (To be precise, we require of a tree that it does not have multiple edges.)
\[radius\] If small cycles are dense in $X$ then $$\rho(Q) = \limsup_{n \to \infty} q^{(n)}(x,y)^{1/n}$$ is independent of $x, y \in X$, and $0 < \rho(Q) \le 1$. (If $X$ is finite then $\rho(Q) = 1$.)
The following strengthens Theorem \[limit1\](c) for almost transitive graphs.
\[limit2\] If $X$ is infinite, connected, with minimum degree $2$, and almost transitive, then for all $x, y \in X$, $$\lim_{n \to \infty} q^{(n)}(x,y)/\rho(Q)^n = 0\,.$$
The *isoperimetric constant* $\iota(X)$ of a connected, locally finite graph $X$ is $$\iota(X) = \inf \left\{ \frac{{\mbox{\sl Area}}(F)}{{\mbox{\sl Vol}}(F)} :
F \subset X \;\mbox{finite} \right\}\,,$$ where ${\mbox{\sl Vol}}(F) = \sum_{x \in F} \deg(x)$ and ${\mbox{\sl Area}}(F)$ is the number of edges with one endpoint in $F$ and the other in $X \setminus F$. The graph is called *amenable* if $\iota(X) = 0$. Non-amenable graphs are also called (infinite) *expanders*.
Consider the Hilbert space $\ell^2(E)$ of all functions $F: E \to {\mathbb R}$ with $\langle F,F \rangle < \infty$, with the ordinary inner product $$\langle F,G \rangle = \sum_{e \in E} F(e)G(e).$$ Then $Q_E$ acts on this space by $Q_EF(e) = \sum_{f \in E} q_E(e,f)F(f)$. We denote by $\|Q_E\|$ the corresponding operator norm, and by $\rho_2(Q_E) = \lim_n \|Q_E^n\|^{1/n}$ its spectral radius. Note that $\rho(Q) \le \rho_2(Q_E) \le \|Q_E\|$ in general.
\[norm\] [(a)]{} One has always $\|Q_E\|=1$.\
[(b)]{} If small cycles are dense in $X$, then $\rho(Q) = \rho_2(Q_E)$.
\[amenable\] Suppose that $X$ is connected, that small cycles are dense, and that there is $M < \infty$ such that $2 \le \deg(x) \le M$ for all $x \in X$.
Then $X$ is amenable if and only if $\rho(Q) = 1$.
With these results and their proofs we aim principally at extending and explaining previous material regarding *cogrowth* of graphs and groups and at shedding new light on cogrowth by studying it in terms of NBRW on the oriented edges. We also think that NBRW on the (oriented) edge set of an arbitrary graph is an interesting random process in its own right.
In §\[cogrowth\], we first recall (ordinary) simple random walk on a graph and some of its basic properties in order to put our results on NBRW in the right perspective. We then consider cogrowth of graphs, which is best understood in terms of universal covering trees, and explain how Theorems \[limit1\], \[limit2\] and \[amenable\] apply. In §2 we also give various references.
§\[proofs\] is dedicated to the proofs of the results stated here.
Some additional remarks and observations can be found in §\[final\].
Simple random walk, and cogrowth of graphs {#cogrowth}
==========================================
[**A. Simple random walk**]{} (SRW) is mostly considered on graphs without multiple edges, and loops are usually counted only once for the degree of a vertex. Here, multiple edges are admitted, and we count each loop twice. SRW is the Markov chain on the (vertex set of the) graph $X$ with transition matrix $P = \bigl( p(x,y) \bigr)_{x,y \in X}$ given by $$p(x,y) = \frac{e(x,y)}{\deg(x)}.$$ Thus, contrary to NBRW, the walker does not remember from where he did come at the previous step, and chooses at random any one among the outgoing edges at the actual vertex. A possible interpretation for counting each loop twice is that topologically, the walker standing at a vertex $x$ sees two “ends” of each loop at $x$ among which he may choose. We write $p^{(n)}(x,y)$ for the $n$-step transition probability from $x$ to $y$.
The transition matrix $P$ acts by $Pg(x) = \sum_y p(x,y)g(y)$ on the Hilbert space $\ell^2(X,\deg)$ of all functions $g: X \to {\mathbb R}$ with $\langle g,g \rangle < \infty$, where the inner product is $$\langle g,h \rangle = \sum_{x \in X} g(x)h(x)\,\deg(x).$$ We denote by $\|P\|$ the norm of this operator.
Here is a list of well-known properties of SRW. (Recall once more that $E = E(X)$ is the set of oriented edges as in §\[intro\], so that $|E(X)|$ is twice the number of “ordinary” non-oriented edges.)
\[SRW\] Let $X$ be a connected, locally finite graph.
- If $X$ is finite and not bipartite, then for all $x,y \in X$, $$\lim_{n \to \infty} p^{(n)}(x,y) = \frac{\deg(x)}{|E(X)|}\,.$$ If $X$ is finite and bipartite, then for all $x,y \in X$, with $\delta \in \{0,1\}$ such that $d(x,y) \equiv \delta \mod 2$, $$\lim_{n \to \infty} p^{(2n+\delta)}(x,y) = 2\frac{\deg(x)}{|E(X)|}\,.$$
- If $X$ is infinite, then for all $x,y \in X$, $$\lim_{n \to \infty} p^{(n)}(x,y) = 0\,.$$
- The *spectral radius* $$\rho(P) = \limsup_{n \to \infty} p^{(n)}(x,y)^{1/n}$$ is independent of $x,y \in X$, and $\|P\| = \rho(P)$.
- If $X$ is infinite and almost transitive then $$\lim_{n \to \infty} p^{(n)}(x,y)/\rho(P)^n = 0\,.$$
- $X$ is amenable if and only if $\rho(P) =1\,.$
Statements (a) and (b) follow from basic Markov chain theory, see e.g. [Chung]{} [@Chu] or [Seneta]{} [@Sen]: the Markov chain given by $P$ is *irreducible* ($\forall\ x, y \in X \;\exists\;n=n(x,y)\ge 0$ such that $p^{(n)}(x,y) > 0$). Its *period* ${\mathfrak d}(P) = \gcd \{ n : p^{(n)}(x,x) > 0 \}$ is $=2$ when $X$ is bipartite, and $=1$, otherwise. Finally, $\mu(x) = \deg(x)$ defines an *invariant* measure. If $X$ is finite then $\mu(X)=|E(X)|$, and $\mu_0(x)=\mu(x)/|E(X)|$ is an invariant probability measure. Therefore, (a) follows from the basic convergence theorem, see [@Chu], Thm. 1 in §I.6 or [@Sen], Thm. 4.2. If $X$ is infinite then $\mu(X) = \infty$, whence the random walk cannot be *positive recurrent*, and (b) must hold. We shall encounter these notions in more detail in §\[proofs\].
For statement (c), see e.g. [Woess]{} [@Wbook], §10. In particular, the fact that $\rho(P) = \rho_2(P)$, the $\ell^2$-spectral radius of $P$, follows from self-adjointness of $P$ on $\ell^2(X,\deg)$.
Regarding statement (d), this is immediate when $\sum_n p^{(n)}(x,y)/\rho(P)^n < \infty\,$. If the series diverges then it follows from Theorem 7.8 in [@Wbook] (which is basically due to [Guivarc’h]{} [@Gui]) that $\rho(P)=1$, and we can apply (b).
Statement (e) has a long history, going back to [Kesten]{}’s amenability criterion for finitely generated groups [@Kes]. The version stated here is due to [Dodziuk and Kendall]{} [@Dod-Ken] based on a previous paper by [Dodziuk]{} [@Dod].
[**B. Cogrowth**]{} is a notion of asymptotic density of a graph. It is best understood in terms of the *universal cover* of the graph $X$. This is a (unique) *tree* $T$ together with a surjective mapping $\pi: T \to X$ which is a local homeomorphism, i.e., if $\tilde x, \tilde y$ are neighbours in $T$ then so are $\pi(\tilde x), \pi(\tilde y)$ in $X$, and $\deg_T(\tilde x) =
\deg_X\bigl(\pi(\tilde x)\bigr)$ for every vertex $\tilde x \in T$.
The covering tree can be constructed as follows: a non-backtracking walk of length $n\ge 0$ in $X$ is a sequence $e_1, \dots, e_n$ of edges such that $e_{k-1} \to e_{k}$ for $k=2, \dots, n$. Its initial and terminal vertices are $e_1^-$ and $e_n^+$, respectively. If $n=0$, we have an empty path, for which we have to specify its initial = terminal vertex. We now choose a root (reference vertex) $o \in X$, and define $T$ as the set of all non-backtracking paths $\tilde x$ starting at $o$, including the empty path. Two such paths are defined to be neighbours in $T$ if one of them extends the other by a single edge. The mapping $\pi$ assigns to each $\tilde x \in T$ its terminal vertex $x \in X$.
Now let $x, y \in X$, and choose $\tilde x \in T$ such that $\pi(\tilde x)=x$. Write $T(y) = \{ \tilde y \in T : \pi(\tilde y) = y\}$, and consider the sphere $S(\tilde x, n) = \{ \tilde v \in T : d_T(\tilde v,\tilde x) = n\}$, where $d_T(\cdot,\cdot)$ is the distance in $T$. Then *(ordinary) cogrowth* at $x, y \in X$ is the sequence $$\label{ordinary}
{\mbox{\sl cog}}_n(x,y) = \frac{|S(\tilde x, n) \cap T(y)|}{|S(\tilde x, n)|}\,,\quad
n \ge 0\,.$$ The graph $X$ being “small” corresponds to $\bigl( {\mbox{\sl cog}}_n(x,y) \bigr)_n$ being “large”. Besides finiteness, also amenability is a “smallness” condition, whence it is natural to look for a link between cogrowth and amenability.
Cogrowth was initially introduced by [Grigorchuk]{} [@Gri] and later [Cohen]{} [@Coh] for finitely generated *groups*. If ${\Gamma}$ is such a group, then we can represent it as a factor ${\mathbb F}_s/N$, where ${\mathbb F}_s$ is the *free group* on $s$ free generators $\tilde a_1, \dots, \tilde a_s$, and $N$ is a normal subgroup of ${\mathbb F}_s$. Let $\pi:{\mathbb F}_s \to {\Gamma}$ be the factor map. We write $\tilde a_{-i} = \tilde a_i^{-1}$ and set $\tilde S = \{ \tilde a_i : i = \pm 1, \dots, \pm s \}$. Then the Cayley graph of ${\mathbb F}_s$ with respect to $\tilde S$ is the $2s$-regular tree, which is the covering tree of the Cayley graph of ${\Gamma}$ with respect to the generators $a_i=\pi(\tilde a_i)$. It is best to consider immediately the oriented edges of that Cayley graph: every $x \in {\Gamma}$ is the inital point of an edge of type $\tilde a_i$, whose endpoint is $xa_i$; the associated “inverse” edge goes from $xa_i$ to $x$ and has type $\tilde a_{-i}$ ($i = \pm 1, \dots, \pm s$). Every pair of this type corresponds to one unoriented edge. Note that generators with $a_i=a_{-i} \ne {\mbox{\sl id}}$ give rise to multiple edges, and when $a_i=a_{-i} = {\mbox{\sl id}}$, we get loops. This also explains why loops should be counted twice for the degrees. Thus, the factor map $\pi$ becomes the covering map from the tree onto the Cayley graph.
Note that for groups, ${\mbox{\sl cog}}_n(x,x)$ is the same for all $x$. Amenability of a finitely generated group ${\Gamma}$ is equivalent with amenability of any of its (locally finite) Cayley graphs. The main result of [@Gri] and [@Coh], restated in our notation, was that $$\label{classic}
{\Gamma}\;\mbox{is amenable} \iff \limsup_{n\to\infty} {\mbox{\sl cog}}_n(x,x)^{1/n} = 1\,.$$
This has been generalized to regular graphs by [Northshield]{} [@No1], who was also the first to explain cogrowth in terms of covering trees. One of the basic tools for studying cogrowth of regular graphs is a functional equation between the generating functions $C(x,y|t) = \sum_n {\mbox{\sl cog}}_n(x,y)\,t^n$ of the cogrowth sequence and $G(x,y|z) = \sum_n p^{(n)}(x,y)\,z^n$ of the transition probabilites of SRW: if $X$ is $d$-regular then with our notation and normalizations, $$\label{functional}
C(x,y|t) = \frac{1}{d}\delta_x(y) +
\frac{(d-1)^2-t^2}{d(d-1+t^2)}G\bigl(x,y|z(t)\bigr)\,,\quad\mbox{where}\quad
z(t) = \frac{dt}{d-1+t^2}\,,$$ A first version of (\[functional\]) is contained in the Ph.D. thesis of [Grigorchuk]{}. Various proofs of that formula have appeared: [Woess]{} [@Wcog], [Szwarc]{} [@Szw] (both for groups), [Northshield]{} [@No1] (shortest), [Bartholdi]{} [@Bar] (more general). In spite of [@Bar], there is no satisfactory version of that formula for non-regular graphs. Nevertheless, [Northshield]{} [@No2] proves a clever extension of (\[classic\]) to *quasi-regular* graphs (non-regular graphs satisfying a certain uniform growth condition).
More generally, we can consider a sequence $\nu = (\nu_{\tilde x,n})_{\tilde x \in T, n \ge 0}$, where each $\nu_{\tilde x,n}$ is a probability measure concentrated on the sphere $S(\tilde x,n)$ of radius $n$ centred at $\tilde x$ in the covering tree $T$ of $X$, with $\pi(\tilde x) = x$. Note that there is a natural bijection between $S(\tilde x, n)$ and $S(\tilde x',n)$, when $\pi(\tilde x) = \pi(\tilde x')$. We require that in this case, $\nu_{\tilde x',n}$ is the image of $\nu_{\tilde x,n}$ under that bijection. Then we can define $$\label{sequence}
{\mbox{\sl cog}}^{\,\nu}_n(x,y) = \nu_{\tilde x,n}\bigl(T(y)\bigr)\,,\quad x, y \in X\,,
\;\pi(\tilde x) = x\,.$$ When each $\nu_{\tilde x,n}$ is equidistribution on $S(\tilde x,n)$, this is ordinary cogrowth.
Another choice is to define $$\nu_{\tilde x,n}(\tilde y) = \frac{1}{\deg(\tilde x)}
\frac{1}{\deg(\tilde x_1)-1} \cdots \frac{1}{\deg(\tilde x_{n-1})-1}\,,$$ where $\tilde x, \tilde x_1, \dots, \tilde x_{n-1},\tilde y$ are the consecutive vertices on the unique path in $T$ from $\tilde x$ to $\tilde y \in S(\tilde x,n)$. Cogrowth with respect to this choice of $\nu$ is the same as NBRW: $$\label{NBcog}
{\mbox{\sl cog}}^{\,\nu}_n(x,y) = q^{(n)}(x,y)$$ In the specific case of regular graphs, the two concepts coincide. Thus, besides ordinary cogrowth, non-backtracking random walk is another way to extend cogrowth from regular to arbitrary graphs.
Proofs
======
In this section, we always use the basic assumption that $X$ is a locally finite, connected graph with minimum degree $2$.
It may be best to think of edge-NBRW as simple random walk on the *oriented line graph* (OLG) of $X$. This is the digraph whose vertex set is $E=E(X)$, and there is an oriented (2nd order) edge from $e$ to $f$ ($e,f \in E$) if $e \to f$. Our Markov chain with transition matrix $Q_E$ is not symmetric, nor reversible like SRW on an unoriented graph. However, the counting measure ${\lambda}$, given by ${\lambda}(e) = 1$, is an invariant measure for $Q_E$, that is, $$\label{invariant}
\sum_{e \in E} {\lambda}(e)q_E(e,f) = {\lambda}(f) \quad \forall\ f \in E\,.$$ We now recall a few basic Markov chain notions. We write $e {\buildrel * \over \to}f$ if there is $n \ge 0$ such that $q_E^{(n)}(x,y) > 0$ (i.e., there is an oriented path from $e$ to $f$ in the OLG, a transitive relation), and $e {\buildrel * \over \leftrightarrow}f$ if $e {\buildrel * \over \to}f $ and $f {\buildrel * \over \to}e$. The equivalence classes with respect to the relation ${\buildrel * \over \leftrightarrow}$ are called *irreducible classes*. An *essential* class $V$ is an irreducible class with the property that $e \in V$ and $e {\buildrel * \over \to}f$ implies $f \in V$. Its elements are also called essential. The Markov chain and its transition matrix $Q_E$ are called irreducible if the state space $E$ forms a single irreducible class. (In graph theoretic terminology, this means that the OLG is *strongly connected*.)
\[irred\] If $X$ is finite then $Q_E$ is irreducible, unless $X$ is a cycle.
Assume that $X$ is not a cycle. Since $X$ is connected, for any pair of edges $e, f$, at least one of $e {\buildrel * \over \to}f$, $e {\buildrel * \over \to}\check f$, $\check e {\buildrel * \over \to}f$, or $\check e {\buildrel * \over \to}\check f$ must hold. Therefore it is sufficient to show that $e {\buildrel * \over \to}\check e$ for every $e \in E$.
Let us first assume that $e$ is not contained in any cycle of $X$. As $\deg(x) \ge 2 \;\forall x$ we can find inductively a sequence $e=e_0, e_1, e_2, \dots$ of edges such that $e_{k-1} \to e_k$. By finiteness of $X$, there must be a minimal index $m$ such that $e_m^+ = e_i^-$ for some $i \in \{ 1, \dots, m-1 \}$. The edges $e_i, \dots, e_m$ form a cycle $C_1$, so that $$e =e_0 {\buildrel * \over \to}e_m \to \check e_{i-1} {\buildrel * \over \to}\check e_0 = \check e\,.$$
Now assume that $e$ is contained in a cycle $C_1$ formed by edges $e=e_0, \dots, e_m$. Since we are assuming that $X$ is not a cycle, there is a vertex $e_i^-=:x$ in $C_1$ with $\deg(x) \ge 3$. Thus, there an edge $f$ with $f^-=x$ such that $f\notin\{\check e_{i-1},e_i\}$ (for $i=0$ we intend $e_{-1}=e_m$). If $f$ does not lie on any cycle in $X$, we have already seen that $f {\buildrel * \over \to}\check f$, whence $$e =e_0 {\buildrel * \over \to}e_{i-1} \to f {\buildrel * \over \to}\check f \to \check e_{i-1} {\buildrel * \over \to}\check e_0 = \check e\,.$$ On the other hand, assume that $f$ is contained in a cycle $C_2$ formed by edges $f=f_0, \dots, f_\ell$. Then there must be another edge $f_k$ ($k>0$) incident with some vertex in $C_1$. Let $j$ be the minimal index $\in \{ 0, \dots, m \}$ with $e_j^+=f_k^+$ for some $k \in \{ 1, \dots, \ell \}$. Then $$e {\buildrel * \over \to}e_{i-1} \to f=f_0 {\buildrel * \over \to}f_k \to \check e_j {\buildrel * \over \to}\check e\,. \qedhere$$
If $X$ is a finite cycle, then the OLG consists of two disjoint, oriented cycles of the same length, each of which constitutes an essential class of $Q_E$, on which $NBRW$ moves “forward” deterministically.
\[infinite\] If $X$ is infinite then for any edge $e \in E$ there are infinitely many edges $f\in E$ with $e {\buildrel * \over \to}f$.
Let $e \in E$ and $X'$ be the graph that results from $X$ by removing $e$ and $\check e$. If $X'$ is connected then by infiniteness, $e {\buildrel * \over \to}f$ for infinitely many $f \in E$. The same holds if $e$ is directed towards an infinite component. Thus, let us assume that $e$ is directed towards a finite component $X_1'$ of $X'$. By infiniteness of $X$, $\check e$ is directed to the other, infinite component, so that $\check e {\buildrel * \over \to}f$ for infinitely many $f \in E$. Applying the method of proof of Lemma \[irred\] to $X_1'$, we have $g {\buildrel * \over \to}\check g$ for some edge $g$ with $e \to g$ in $X$ (remember that we assumed that $\deg(e^+)\geq 2$). It follows that $e \to g {\buildrel * \over \to}\check g \to \check e$ and hence $e {\buildrel * \over \to}f$ for infinitely many $f \in E$.
In general, if $Q_E$ is irreducible, then we can define its *period* by $${\mathfrak d}= {\mathfrak d}(Q_E) = \gcd \{ n : q_E^{(n)}(e,e) > 0 \}\,,$$ which is independent of $e \in E$.
\[period\] Let $X$ be a finite, connected graph with $\deg(x)\ge 3$ for all $x\in X$. Then the period of the associated edge-NBRW is either 2 or 1, depending on whether $X$ is bipartite or not (respectively).
First we shall show that ${\mathfrak d}(Q_E)\in\{1,2\}$. Let $e,f,g$ be three edges with $e^-=f^-=g^-=:x$. By Lemma \[irred\] we have $e {\buildrel * \over \to}\check f$ and $e {\buildrel * \over \to}\check g$. That is, there are two non-backtracking closed paths at $x$ in $X$ both starting with $e$, one terminating with $\check f$, the other one with $\check g$. Since the starting edge in both these paths is not the reversed terminating edge, they can easily be turned into two cycles $C_1,C_2$ at $x$ formed by edges $e_1,\ldots,e_n$ and $f_1,\ldots,f_m$, respectively. Both $C_1$ and $C_2$ start with the same edge $e_1=f_1=e$. We claim that we may assume that the second edges $e_2$ and $f_2$ in $C_1$ and $C_2$ (resp.) do not coincide. Consider the case where $e_2=f_2$. By assumption, $\deg(e^+)\geq 3$ and there is another edge $g\neq e_2,\check e$ with $e^+=g^-$. Since initial vertices do not occur more than once in each cycle, neither $C_1$ nor $C_2$ contains $g$. As $\deg(\cdot) \ge 3$ we can find inductively a sequence $g=g_1, g_2, g_3, \dots$ of edges with distinct initial vertices such that $g_{i} \to g_{i+1}$. By finiteness of $X$, there must be a minimal index $k$ such that $g_k^+$ occurs as initial vertex of an edge in one of the cycles $C_1,C_2$. Let us assume that $g_k^+=e_\ell^-$ for some $\ell \in \{ 1, \dots, n \}$. Then we may replace the cycle $C_1$ by $e_1,g_1,\ldots,g_k,e_\ell,\ldots,e_n$ so that the two cycles in $X$ have the claimed property. A similar argument shows that we also may assume that $e_n\neq f_m$.
Thus we have two cycles of length $n$ and $m$, respectively. Since we assumed $e_2\neq f_2$ and $e_n\neq f_m$, we have $\check e_n {\buildrel * \over \to}\check e_n$ in $n+m-2$ steps via $$\check e_n \to \ldots\to \check e_2 \to f_2 \to \ldots \to f_m \to \check e_n.$$ Therefore, ${\mathfrak d}(Q_E)$ must be a factor of $n$, $m$ and $n+m-2$, whence ${\mathfrak d}(Q_E) \in \{1,2\}$.
It is now clear that we must have ${\mathfrak d}(Q_E) = 2$, if $X$ is bipartite. Otherwise, $X$ contains an odd cycle, so that $q_E^{(k)}(e,e) >0$ for some odd $k$. Thus, we cannot have ${\mathfrak d}(Q_E) = 2$, that is, ${\mathfrak d}(Q_E) = 1$.
(a+b) If $X$ is finite, but not a cycle, then we can use Lemma \[irred\]. Let $e, f \in E$ and $r \ge 0$ such that $q_E^{(r)}(e,f) > 0$. Then $q_E^{(n)}(e,f) > 0$ if and only if $n \equiv r \mod {\mathfrak d}$ and $n$ is sufficiently large (see [@Sen], Thm. 1.3). The fundamental convergence theorem (see [@Chu], Thm. 1 in §I.6 or [@Sen], Thm. 4.2) implies that $$\label{fund}
\lim_{n \to \infty} q_E^{(n{\mathfrak d}+ r)}(e,f) = {\mathfrak d}\,{\lambda}_0(f) = \frac{{\mathfrak d}}{|E(X)|},$$ where ${\lambda}_0$ is the unique invariant probability measure, that is, ${\lambda}_0(f) = \frac{1}{|E|}\,.$ In view of Lemma \[period\], this together with (\[qn\]) yields statement (b), when $\deg(x) \ge 3$ for all $x \in X$.
Otherwise, $$\lim_{n \to \infty}
\frac{1}{n} \Bigl( q_E^{(1)}(e,f) + \dots + q_E^{(n)}(e,f) \Bigr) =
\frac{1}{|E(X)|}\,,$$ and combining this with (\[qn\]), we obtain the limit proposed in statement (a) of Theorem \[limit1\].
In the case where $X$ is a cycle the $q^{(n)}(x,y)$ can be calculated explicitly, whence the claim of the Theorem follows. This is left as a simple exercise to the reader.
\(c) We distinguish two cases. First, if the edge-NBRW starting at $e \in E$ is *transient*, that is, the probability of returning to $e$ is $< 1$, then $\sum_n q^{(n)}(e,f) < \infty$ for every $f \in E$, see [@Chu], Thm. 4 in §I.6. Therefore, $q^{(n)}(e,f) \to 0$.
If the random walk starting at $e$ is *recurrent*, i.e., it returns to $e$ with probability $1$, then $e$ must be an essential state, see [@Chu], Thm. 4 in §I.4 or [@Sen], Lemma 5.2. Now by Lemma \[infinite\], there are infinitely many $f \in E$ such that $e {\buildrel * \over \to}f$. Therefore, the – essential – irreducible class $V$ of $e$ is infinite. Since the random walk starting at $e$ does not leave $V$, we can consider the restriction of $Q_E$ to $V$. It defines an irreducible, recurrent Markov chain with invariant measure ${\lambda}$, the counting measure. Recurrence yields that this is the unique invariant measure up to normalization. It has total mass ${\lambda}(V) = \infty\,$, the chain is *null recurrent*, see [@Chu], §I.6 or [@Sen], §§5.2–5.3. Therefore the convergence theorem for recurrent Markov chains yields that $q^{(n)}(e,f) \to 0$ for all $f \in V$. If $f \notin V$ then $q^{(n)}(e,f) = 0$ for all $n$. Since $X$ is by assumption locally finite, formula (\[qn\]) yields the result stated in (c).
[**Uniformly irreducible random walks and amenability.**]{} We now make a small detour regarding more general random walks on graphs, recalling and improving upon the material in [@Wbook], §10.B.
Let $X$ be a locally finite, connected graph with graph metric $d(\cdot,\cdot)$, and consider the transition matrix $P = \bigl(p(x,y)\bigr)_{x,y\in X}$ of an arbitrary random walk (Markov chain) on the set $X$. Then $P$ is called *uniformly irreducible* if there are constants $K, {\varepsilon}_0 > 0$ such that for any pair of neighbours $x, y$ there is some $k \le K$ such that $p^{(k)}(x,y) \ge {\varepsilon}_0$. Furthermore, $P$ is said to have *bounded range,* if there is $R > 0$ such that $p(x,y) > 0$ only if $d(x,y) \le R$. These two are conditions of adaptedness of $P$ to the graph structure.
If $P$ has an *invariant measure* $\nu$, then it acts on the Hilbert space $\ell^2(X,\nu)$ of all $F: X \to {\mathbb R}$ with $\langle F,F \rangle < \infty$, where $
\langle F, G \rangle = \sum_x F(x)G(x)\,\nu(x)\,.
$ The operator norm satisfies $\|P\| \le 1$, and its $\ell^2$-spectral radius is $\rho_2(P) = \lim_n \| P^n \|^{1/n}$. Note that for $\rho(P) = \limsup_n p^{(n)}(x,y)^{1/n}$ (independent of $x,y$ by irreducibility) one has $\rho(P) \le \rho_2(P)$, and equality does not hold in general. The adjoint (more precisely, $\nu$-adjoint) $P^*$ of $P$ on $\ell^2(X,\nu)$ has the stochastic kernel $p^*(x,y) = \nu(y)p(y,x)/\nu(x)$.
\[improved\] Suppose that $X$ is connected, with bounded vertex degrees, and that $P$ is uniformly irreducible with bounded range and has an invariant measure $\nu$ satisfying $C^{-1} \le \nu(\cdot) \le C$ for some $C \ge 1$.
Then $\rho_2(P) = 1$ if and only if the graph $X$ is amenable.
Theorem 10.6 in [@Wbook] states that under the given assumptions, $\rho(P) =1$ implies amenability of $X$. After the proof of that theorem, it is explained that the condition $\rho(P)=1$ may be replaced with $\rho_2(P)=1$.
Conversely, Theorem 10.8 in [@Wbook] states that amenability of $X$ implies $\|P\|=1$. Now, let $I$ be the identity operator (or matrix), and fix $n \ge 1$. Set $\bar P = \frac{1}{2}(I+P)$. Then $\bar P^n$ is uniformly irreducible, has bounded range and invariant measure $\nu$. If $X$ is amenable, then we get that $\|\bar P^n\| = 1$. This is true for every $n$. Consequently, $\rho_2(\bar P) = 1$. By basic spectral theory, also $\rho_2(P) = 1$.
More generally, the bounded range assumption can be replaced with tightness of the step length distributions of $P$ and $P^*$ as in [@Wbook], Thm. 10.8.
We want to apply Theorem \[improved\] not to random walks on our “original” graph $X$, but to edge-NBRW on the OLG. However, the latter is not a graph (with unoriented edges), but a digraph. Therefore, we symmetrize it by “removing the arrows” from its edges. (Recall that the latter are “second order” edges, connecting edges of the original graph $X$). The resulting SOLG (symmetrized oriented line graph) still has as its vertex set the set $E$ of *oriented* edges of the original graph $X$, but neighbourhood in the SOLG is given by $e \sim f$, if $e \to f$ or $f \to e$. We observe that in the SOLG, $q_E(e,f) > 0$ implies $e \sim f$, but *not conversely*.
\[uniform\] If $2 \le \deg(x) \le M$ for all $x \in X$, and small cycles are dense in $X$, then there is $L > 0$ such that for each $e \in E$, we have $e {\buildrel * \over \to}\check e$ in at most $L$ steps of edge-NBRW.\
In particular, $Q_E$ is uniformly irreducible on the symmetrized OLG.
We may suppose that $X$ is infinite. Observe that the first statement of the lemma implies uniform irreducibility. Indeed, let $f$ be a neighbour of $e$ in the OLG. Then either $e^+ = f^-$, in which case $q_E(e,f) \ge 1/(M-1)$, or $f^+ = e^-$, in which case $e {\buildrel * \over \to}\check e \to \check f {\buildrel * \over \to}f$ in $k \le 2L+1$ steps with probability $\ge 1/(M-1)^{2L+1}$.
Now let $R > 0$ be such that $B(x,R)$ contains a cycle for every $x \in X$. By Lemma \[infinite\] there are infinitely many edges $f$ with $e {\buildrel * \over \to}f$. Since the vertex degree in $X$ is bounded by $M$, the number of vertices in each $B(x,R)$ cannot exceed a certain constant $K=K(M,R)$. It follows that $e {\buildrel * \over \to}f$ for an edge $f$ with $f^+$ not contained in $B(e^-,R)$ in at most $K$ steps of the edge-NBRW. By assumption $B(f^+,R)$ contains a cycle $C_1$ formed by edges $e_1,\ldots,e_m$ ($m \leq K$). Since $d(e^-,f^+)>R$ neither $e$ nor $\check e$ are edges inside the ball $B(f^+,R)$ in $X$, and consequently neither of the two is among the edges $e_1,\ldots,e_m$ of $C_1$. Now, either $f {\buildrel * \over \to}e_i$ (case 1) or $\check f {\buildrel * \over \to}e_i$ (case 2) for some $i\in\{1,\ldots,m\}$ in at most $R$ steps. If $f$ or $\check f=e_i$ for $i\in\{1,\ldots,m\}$, then $$\begin{aligned}
&& e {\buildrel * \over \to}f=e_i \to e_{i+1} \to \ldots \to e_m \to e_1 \to \ldots
e_{i-1} {\buildrel * \over \to}\check e \\
\mbox{or } && e {\buildrel * \over \to}f=\check e_i \to \check e_{i+1} \to \ldots \to \check e_m \to \check e_1 \to \ldots
\check e_{i-1} {\buildrel * \over \to}\check e, \end{aligned}$$ respectively, in $\leq K+K+K=3K$ steps. Now let us assume that $f,\check f\neq e_i$ for $i\in\{1,\ldots,m\}$. Then we have in case 1 $$e {\buildrel * \over \to}f {\buildrel * \over \to}e_i \to e_{i+1} \to \ldots \to e_m \to e_1 \to \ldots
e_{i-1} {\buildrel * \over \to}\check f {\buildrel * \over \to}\check e$$ in $\leq K+R+K+R+K=2R+3K$ steps. In case 2 we have to turn off on the way to $f$ to arrive at the cycle $C_1$. More exactly, let $e=f_0,\ldots,f_n=f$ be a walk from $e$ to $f$ in $n\leq K$ steps. Now consider a walk from $\check f=\check f_n$ to $e_i$ in $\leq R$ steps. It contains some (at least one) of the edges $\check f_n, \check f_{n-1},\ldots,
\check f_{1}$. Let $\ell$ be the minimal index such that $\check f_\ell$ is not contained in the walk. Then we have $$e {\buildrel * \over \to}f_\ell {\buildrel * \over \to}e_i \to e_{i+1} \to \ldots \to e_m \to e_1 \to \ldots e_{i-1}
{\buildrel * \over \to}\check f_\ell {\buildrel * \over \to}\check e$$ again in $\leq K+R+K+R+K=2R+3K$ steps. Thus setting $L= 2R+3K$ we have $e {\buildrel * \over \to}\check e$ in $\leq L$ steps.
If $X$ is not a cycle then by Lemma \[irred\], $Q_E$ is irreducible and a standard argument (see e.g. [@Sen], Thm. 6.1, or [@Wbook], §1.B) yields that $$\label{rhoQ}
\rho(Q) = \rho(Q_E) = \limsup_n q_E^{(n)}(e,f)^{1/n}$$ is independent of $e, f \in E$. Now apply (\[qn\]). If $X$ is a cycle, $\limsup_n q^{(n)}(x,y)$ is constantly either $\frac{1}{2}$ or $1$.
The fact that $\rho(Q) = \rho(Q_E)$, as stated in (\[rhoQ\]), is immediate from (\[qn\]) and will be tacitly used several times.
If $X$ is a tree then for each pair $e, f \in E$ there is at most one $n$ such that $q_E^{(n)}(e,f) > 0$.
Otherwise, $X$ has a cycle, and since it is almost transitive, small cycles are dense in $X$. By Lemma \[radius\], $Q_E$ is irreducible, and the OLG of $X$ is connected. Therefore the series $\sum_n q_E^{(n)}(e,f)/\rho(Q)^n$ either converge for all $e,f$ or diverge for all $e, f \in E$, see e.g. [@Wbook], §1.B.
In the convergent case, $q_E^{(n)}(e,f)/\rho(Q)^n \to 0$.
In the divergent case, edge-NBRW is *$\rho$-recurrent*. The automorphism group ${\Gamma}$ of $X$ also acts with finitely many orbits on the OLG. Therefore we can apply an adaptation of a result of [Guivarc’h]{} [@Gui], see [@Wbook], Thm. 7.8 and its proof: it yields that there is a positive function $H$ on $E$ such that $Q_EH = \rho(Q)\cdot H$, and $$q_H(e,f) = \frac{q_E(e,f)H(f)}{\rho(Q)H(e)}$$ defines a new random walk which is ${\Gamma}$-invariant and recurrent. By Theorem 3.26 and Lemma 3.25 in [@Wbook], $Q_H$ has an invariant measure which is constant on each ${\Gamma}$-orbit, and consequently has infinite total mass. Therefore, $Q_H$ is null recurrent, and $q_H^{(n)}(e,f) \to 0$ for all $e, f$. Since $$q_H^{(n)}(e,f) = \frac{q_E^{(n)}(e,f)H(f)}{\rho(Q)^nH(e)}\,,$$ we find that $q_E^{(n)}(e,f)/\rho(Q)^n \to 0$.
A *rough isometry* between two metric spaces $(X,d), (X',d')$ is a mapping $\varphi: X \to X'$ with the following properties. $$\label{rough}
\begin{aligned}
A^{-1} d(x,y) - A^{-1}B &\le d'(\varphi x, \varphi y)
\le A\,d(x,y) + B \quad \forall\ x, y \in X,
\mbox{ and}\\
d'(x',\varphi X) &\le B
\quad \forall\ x' \in X',
\end{aligned}$$ where $A \ge 1$ and $B \ge 0$. In this case we say that the two spaces are *roughly isometric*.
\[OLGri\] If $X$ is a connected graph with $2 \le \deg(x) \le M$ that is not a cycle and has dense small cycles, then it is roughly isometric with its symmetrized oriented line graph.
Two finite connected graphs are always roughly isometric. Let us assume that $X$ is infinite, with edge set $E$. Throughout this proof, we write $d_X(\cdot,\cdot)$ for the graph distance in $X$, and $d_E(\cdot,\cdot)$ for the graph distance in the SOLG of $X$. Define the mapping $\varphi: E \to X$ by $\varphi e=e^-$. Evidently, $\varphi$ is surjective and hence $$\label{a}
d_X\bigl(x,\varphi(E)\bigr)=0 \quad \mbox{for all}\; x\in X.$$ Now given two vertices $x,y$ in $X$ with $d_X(x,y)=d$ it is clear that two arbitrary edges $e,f$ starting in $x$ and $y$, respectively, have distance at least $d$ in the SOLG of $X$. It follows that $$\label{b}
d_X(\varphi e,\varphi f)\le d_E(e,f)\,.$$ On the other hand, we obtain also an upper bound for $d_E(e,f)$. Clearly, if $e,f$ are oriented the “right way" we have $e {\buildrel * \over \to}f$ in $d_X(e^-,f^-)$ steps. If one of them is oriented the other way, by Lemma \[uniform\] it takes at most $L$ steps to turn around, i.e. to reach $\check e$ from $e$. Thus we have $e {\buildrel * \over \to}f$ in at most $2 L + d_X(e^-,f^-)$ steps, so that $$\label{c}
d_E(e,f) - 2L \le d_X(\varphi e,\varphi f)\,.$$ Now, setting $A=1$ and $B=2 L$ and combining (\[a\])–(\[c\]) yields (\[rough\]).
\(a) We have $\| Q_E \| = \| Q_E^*Q_E \|^{1/2}$, where the adjoint operator $Q_E^*$ has kernel $q_E^*(e,f) = q_E(f,e)$. Let $F: E \to {\mathbb R}$, and let $e \in E$. Then $$Q_E^*Q_E F(e) = \sum_{f \in E} \sum_{g \in E} q_E(g,e)q_E(g,f)\,F(f)\,.$$ Thus, $Q_E^*Q_E$ is a symmetric, stochastic operator that takes a weighted average of all values of $F$ on each of the finite sets $\{ f \in E : f^- = e^-\}$, where $e \in E$. Consequently, it has norm $1$.\
(b) Instead of $Q_E$ we shall use the new transition operator $\bar Q_E = \frac12(I_E+Q_E)$, where $I_E$ is the identity operator. Of course, its invariant measure is again the counting measure on $E$, and $\bar Q_E^* = \frac12(I_E+Q_E^*)$. If we fix $n$, then $\bar Q_E^{*\,n} \bar Q_E^n$ is again doubly stochastic, has finite range, and all its matrix elements are bounded below by those of $c_n Q_E$, where $c_n = n/4^{n}$. Since $Q_E$ is (uniformly) irreducible by Lemma \[uniform\], the same holds for $\bar Q_E^{*\,n} \bar Q_E^n$.
We shall now use the obvious, but crucial relation $$\label{reverse}
q_E^{(n)}(e,f) = q_E^{(n)}(\check f, \check e)\,,$$ which also holds for $\bar Q_E^n$ in the place of $Q_E^n$. Lemma \[uniform\] implies that for every $e \in E$, $$\bar q_E^{(L)}(e,\check e) \ge 1/C\,,\quad\mbox{where}\quad C=(2M)^L\,.$$ ($M$ is the upper bound on the vertex degrees.) Therefore, using (\[reverse\]), $$\bar q_E^{*\,(n)}(e,f) = \bar q_E^{(n)}(f, e)
= q_E^{(n)}(\check e, \check f)
\le C^2\, \bar q_E^{(L)}(e, \check e)\,
\bar q_E^{(n)}(\check e, \check f)\, \bar q_E^{(L)}(\check f, f)
\le C^2 \bar q_E^{(n+2L)}(e,f)\,.$$ In particular, we obtain that $\bar Q_E^{*\,n} \bar Q_E^n \le C^2\,
\bar Q_E^{2n+2L}$ matrix-elementwise.
Now, since $\bar Q_E^{*\,n} \bar Q_E^n$ is symmetric (self-adjoint) and irreducible, Lemma 10.1 in [@Wbook] implies that its norm satisfies $\| \bar Q_E^{*\,n}\bar Q_E^n\| = \rho(\bar Q_E^{*\,n} \bar Q_E^n)\,$, the latter number being defined in the same way as in (\[rhoQ\]), but for the powers of $\bar Q_E^{*\,n}\bar Q_E^n$. Thus, if we take $e \in E$, then $$\begin{aligned}
\rho(\bar Q_E^{*\,n} \bar Q_E^n) &= \lim_{m \to \infty}
\bigl\langle \bigl(\bar Q_E^{*\,n} \bar Q_E^n\bigr)^m\delta_e,\delta_e
\bigr\rangle^{1/m}\le
\lim_{m \to \infty}
C^2 \, \bigl\langle \bar Q_E^{(2n+2L)m}\delta_e,\delta_e\bigr\rangle^{1/m}\\[2pt]
&= \lim_{m \to \infty} C^2 \, \bar q_E^{((2n+2L)m)}(e,e)^{1/m}
\le C^2\,\rho(\bar Q_E)^{2n+2L}\,,
\end{aligned}$$ since $\bar q_E^{(k)}(e,e) \le \rho(\bar Q_E)^k$ for all $k \ge 0$ and $e \in E$, a well known fact, see e.g. [@Sen], §6.1 or [@Wbook], Lemma 1.9. We infer that $$\rho_2(\bar Q_E) = \lim_{n \to \infty} \| \bar Q_E^{*\,n}\bar Q_E^n \|^{1/2n}
\le \lim_{n \to \infty} \bigl(C^2\,\rho(\bar Q_E)^{2n+2L}\bigr)^{1/2n}
= \rho(\bar Q_E)\,.$$ Since $\rho(\bar Q_E) = \frac12\bigl(1+\rho(Q_E)\bigr)$ and $\rho_2(\bar Q_E) = \frac12\bigl(1+\rho_2(Q_E)\bigr)\,$, we conclude that $\rho_2(Q_E) \le \rho(Q_E)$. The reversed inequality is obvious.
It is by now a well established fact that for connected graphs with bounded vertex degrees, amenability is rough-isometry-invariant. See e.g. [@Wbook], Thm. 4.7 (the isoperimetric inequality $IS_{\infty}$ referred to there is the condition $\iota(X) > 0$, i.e., nonamenability), or also the book by [de la Harpe]{} [@Har]. Thus, in view of Proposition \[OLGri\], under the assumptions of Theorem \[amenable\] the graph $X$ is amenable if and only if its SOLG is amenable. By (\[invariant\]), edge-NBRW has the counting measure ${\lambda}$ on $E$ as an invariant measure, and by Lemma \[uniform\], it is uniformly irreducible. Therefore, we can apply Theorem \[improved\] to the SOLG, and Proposition \[norm\](b) allows us to replace the $\ell^2$-spectral radius with $\rho(Q)$.
Final remarks and observations {#final}
==============================
[**A.**]{} Regarding Theorem \[limit1\] (a+b), the condition $\deg(x)\geq 3$ in Lemma \[period\] is necessary for the stronger convergence result of (b), as the following example shows. Thus, if there are vertices of degree $\leq 2$ it is in general not true that for vertex-NBRW, one has convergence of $q^{(2n+\delta)}(x,y)$ ($\delta \in \{0,1\}$) or $q^{(n)}(x,y)$ according to whether $X$ is bipartite or not (respectively).
$$\beginpicture
\setcoordinatesystem units <.25cm,.25cm>
\setplotarea x from -5 to 5, y from -7 to -2
\plot 0 0 6.9 -4 6.9 4 0 0 -6.9 -4 -6.9 4 0 0 /
\multiput {\scriptsize $\bullet$} at 0 0 6.9 -4 6.9 4 -6.9 -4 -6.9 4 /
\put {$x$} at 0 -1
\put {$y$} at -6.9 -5
\put {$v$} at -6.9 5
\endpicture$$ Clearly, edge-NBRW has period ${\mathfrak d}= 3$. Write $e$ for the edge from $y$ to $x$ and $f$ for the edge from $v$ to $y$. We have $$q^{(3n)}(x,x) = 1 {\quad\mbox{and}\quad}q^{(3n+1)}(x,x) = q^{(3n+2)}(x,x) = 0
\quad\forall\ n\,.$$ For the edges terminating at $y$, we have $q_E^{(n)}(\check e, f) > 0$ only if $n \equiv 1 \mod 3$ and $q_E^{(n)}(f, \check e) > 0$ only if $n \equiv 2 \mod 3$, while $q_E^{(n)}(\check e, \check e)$ and $q_E^{(n)}(f, f)$ are $>0$ only if $n \equiv 0 \mod 3$. Therefore, using (\[qn\]) and (\[fund\]), $$\begin{aligned}
q^{(3n)}(y,y) &=
\frac12\bigl( q_E^{(3n)}(\check e, \check e) + q_E^{(3n)}(f,f)\bigr)
\to \frac14\,, \\
q^{(3n+1)}(y,y) &=
\frac12q_E^{(3n+1)}(\check e, f) \to \frac18\,,{\quad\mbox{and}\quad}\\
q^{(3n+2)}(y,y) &=
\frac12q_E^{(3n+2)}(f, \check e) \to \frac18\,,
\quad\mbox{as $n \to \infty\,$.}
\end{aligned}$$
[**B.**]{} For regular, almost transitive graphs, Lemma 3.9 of [Bartholdi]{} [@Bar] states what is Proposition \[SRW\] (a)+(b)+(d) and Theorems \[limit1\]+\[limit2\] here. (We remark that in Lemma 3.9 of [@Bar], the identity “$\lim \sup_n \frac{g_n}{\beta^n}=
\lim \sup_n \frac{f_n}{\alpha^n}=\ldots$" should read “$\lim \sup_n \frac{g_n}{\beta^n}=
\frac{d}{d-1}\lim \sup_n \frac{f_n}{\alpha^n}=\ldots$".) In [@Bar], a proof for SRW is suggested where one starts with the finite case, while for an infinite graph, one takes the sequence of balls $B(o,r)$ around a “root” vertex, applies the “finite” result to each ball, and lets the radius tend to infinity, thereby exchanging two limits. [@Bar] then suggests to use the same argument for cogrowth. This argument has also found its way into a recent paper of [Kapovich et al.]{} [@Kap], who state an extension to arbitrary regular graphs. However, the argument is problematic because a priori it is by no means clear that the two limits (for $n,r \to \infty$) may be exchanged.
As a matter of fact, this was the starting point for the present note, since several colleagues asked us how the mentioned argument can be made rigorous. When applied to regular graphs, our method provides a simple and rigorous proof of those statements for infinite graphs.
[**C.**]{} Theorems \[limit1\] and \[limit2\] extend the corresponding results for Cayley graphs of [@Wcog] to arbitrary graphs. At the same time, the functional equation (\[functional\]) is no more needed. The extension of the amenability criterion (Theorem \[amenable\]) required more work, since the functional equation (\[functional\]) can be used only in the regular case. Also, in the regular case, that criterion does not require denseness of small circles. However, our result *is* a full generalization of that amenability criterion for (Cayley graphs of) finitely generated groups. Indeed, according to our definition of the Cayley graph, small circles will always be dense in the latter unless the group is freely generated by the generating set that defines the Cayley graph. (Remember that when one of the generators satisfies $a_i=a_i^{-1} \ne {\mbox{\sl id}}$, it leads to double edges. But double edges give rise to circles of length $2$ according to our definition !)
[**Acknowledgement.**]{} The second author acknowledges discussions with G. Noskov that stand at the origin of the questions considered in this paper. We also acknowledge discussions with M. Neuhauser and a decisive hint of F. Lehner regarding the proof of Theorem \[amenable\].
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Grigorchuk, R. I.: *Symmetric random walks on discrete groups*, in *Multicomponent Random Systems* (eds. R. L. Dobrushin and Ya. G. Sinai), Nauka, Moscow 1978; English transl. in *Advances in Probability and Related Topics* **6** (eds. D. Griffeath and P. Ney), M. Dekker, New York 1980, pp. 132–152.
Guivarc’h, Y.: *Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire*, Astérisque **74** (1980), 47–98.
de la Harpe, P.: *Topics in Geometric Group Theory*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.
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Northshield, S.: *Cogrowth of regular graphs*, Proc. Amer. Math. Soc. **116** (1992), 203–205.
Northshield, S.: *Quasi-regular graphs, cogrowth, and amenability*, preprint, SUNY-Plattsburgh (2002).
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Szwarc, R.: *A short proof of the Grigorchuk-Cohen cogrowth theorem*, Proc. Amer. Math. Soc. **106** (1989), 663–665.
Woess, W.: *Cogrowth of groups and simple random walks*, Arch. Math. (Basel) **41** (1983), 363–370.
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[^1]: Supported by FWF (Austrian Science Fund) project P15577
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the fermionic zero modes of BPS semilocal magnetic vortices in N=2 supersymmetric QED with a Fayet-Iliopoulos term and two matter hypermultiplets of opposite charge. There is a one-parameter family of vortices with arbitrarily wide magnetic cores. Contrary to the situation in pure Nielsen–Olesen vortices, new zero modes are found which get their masses from Yukawa couplings to scalar fields that do not wind and are non-zero at the core. We clarify the relation between fermion mass and zero modes. The new zero modes have opposite chiralities and therefore do not affect the net counting (left minus right) of zero modes coming from index theorems but manage to evade other index theorems in the literature that count the total number (left plus right) of zero modes in simpler systems.'
author:
- 'A. Achúcarro'
- 'A.C. Davis'
- 'M. Pickles'
- 'J. Urrestilla'
title: 'Fermion zero modes in $N{\!\!=\!\!}2$ supervortices'
---
Ł[[L]{}]{}
Introduction
============
Stable magnetic flux tubes, or [*strings*]{}, occur in many high energy models. Fermionic zero modes on these strings can in some cases render them superconducting, and dramatically change their lifetime and interactions. In most of the examples that have been studied in a cosmological context the strings are topological, yet the effect of fermion zero modes on stable non–topological strings is equally powerful and interesting.
In this paper we investigate the fermionic zero modes on magnetic vortices in supersymmetric (SUSY) $N{\!\!=\!\!}2$ QED with two hypermultiplets of opposite charge. This model was proposed as a toy model for the low energy limit of type-II superstrings compactified on Calabi–Yau manifolds, where the possibility of magnetic vortices was first suggested in [@GMV96] (in the full low energy theory, there are sixteen hypermultiplets charged under fifteen U(1) groups, that is, fifteen copies of the model studied here).
In the absence of Fayet–Iliopoulos terms, it was proved in [@ARH98] that the strings appearing in that model are unstable. In [@ADPU01] the present authors studied the case where a (gauge) symmetry breaking Fayet-Iliopoulos (FI) term was added. In this case we found BPS string solutions. We also found that out of all the possible vacua, only certain choices can give rise to static strings, due to a [*vacuum selection*]{} effect [@PRTT96]. Even so, this effect was still not enough to obtain stable strings, unlike in the case of a comparable $N{\!\!=\!\!}1$ supersymmetric model with two chiral multiplets [@ADPU01; @PU02]. The strings formed are semilocal strings [@VA91], and in the BPS limit (equal scalar and gauge masses) they are only neutrally stable [@leese; @H92].
In the present work, a study of the possible fermionic zero modes arising in this model has been carried out, in order to see whether they stabilise the string by further augmenting the vacuum selection effect. If this were so, the semilocal strings could possibly turn into chiral cosmic strings [@CP99; @DKPS00], long-lived vortices in which the fermions move in one direction only. However we have found that this is not the case. The fermions respect the same vacuum selection effect as the bosons, meaning that the underlying string remains semilocal. Moreover, the fermions move in both directions along the string, and therefore they could mix and leave the string. It is known that the fermion back-reaction can modify the stability of the background string [@N95]. We argue that this is not the case in the present context, because, unlike the systems studied in [@N95], the Bogomol’nyi bound is protected by the remnant unbroken SUSY. The “topological charge” is identical for all the vortices in the family, and we do not expect any member of the family to be singled out by the fermion back-reaction.
Nevertheless, we have found that, due to the bosonic zero mode of the semilocal string, one of the fermions is coupled to a boson which is not zero at the core of the string, contrary to what might be expected from related assertions in the literature. To our knowledge this is the first time that the effect of charged, non–winding scalars has been considered on fermion zero modes, and it is interesting to point out that our model falls outside the scope of a number of index theorems for vortex backgrounds in scalar–fermion systems.
In section II we present a brief review of the bosonic sector of this model; we then obtain fermionic zero modes by SUSY transformations of the bosonic background in section III. We check that these in fact satisfy the fermionic equations of motion in section IV, and derive the mass matrix. The results are then discussed in section V.
Background
==========
We will investigate an $N{\!\!=\!\!}2$ supersymmetric (SUSY) model consisting of two $N{\!\!=\!\!}2$ hypermultiplets $(h_{ai},\psi_a,F_{ai})$, labelled by $a{\!\!=\!\!}1,2$, with opposite charges $q_a{\!\!=\!\!}\pm q$, coupled to an $N{\!\!=\!\!}2$ abelian vector multiplet $(A_\mu,M,N,\lambda^i,\vec{D})$ (in Wess-Zumino gauge), $i{\!\!=\!\!}1,2$, plus a Fayet-Iliopoulos term $\vec{k}
\cdot \vec{D}$, where $\vec{k}$ can be taken to be $(0,0,k)$ without loss of generality [@ADPU01]. This last term is responsible for breaking the $U(1)$ gauge symmetry, although it preserves the supersymmetry.
Using the conventions of [@sohnius], the Lagrangian we are interested in can be written as
Ł&=&D\^h\_a\^[ i]{} D\_h\_[ai]{} + i |\_a \^D\_\_a + |\_i\^\_\^i\
& &-F\^F\_ + i \_a h\_a\^[ i]{} |\_ i \_a- i \_a |\_a \^i h\_[ai]{}\
& &-(H\_1\^[ 1]{}-H\_2\^[ 2]{}-)\^2-(H\_2\^[ 1]{} + H\_1\^[ 2]{})\^2 - (i H\_2\^[ 1]{}-i H\_1\^[ 2]{})\^2, \[lagrangian\] where $\hat{q}_a=q_a/q$, $h_a^{\, i}=h^*_{aj}$, $H_j^{\,
i}=-(\hat{q}_a/2)h_a^{\, i}h_{aj}$, $D_\mu=\d_\mu + i \hat{q}_a A_\mu$ and $A_{\mu}$ is a U(1) gauge field. Auxiliary fields have been eliminated, all fields with zero vacuum expectation value (that is, $M$ and $N$), have been set to zero and suitable rescalings have been performed [@ADPU01].
We can use a Bogomol’nyi argument to express the energy of the bosonic part of a static, straight vortex configuration as
E&=&\^2 x\
& &-\^2 x B, where $B{\!\!=\!\!}\d_1 A_2-\d_2 A_1$, and the integral in the last term is the magnetic flux, which is quantised in units of 2$\pi$. The Bogomol’nyi equations can then be immediately read off
&(D\_1+iD\_2)h\_[11]{}=0; (D\_1-iD\_2)h\_[12]{}=0;& \[bogomolnyi\]\
&(D\_1+iD\_2)h\_[21]{}=0; (D\_1-iD\_2)h\_[22]{}=0;&\
&H\_1\^[ 2]{}=H\_2\^[ 1]{}=0; B+H\_1\^[ 1]{}-H\_2\^[ 2]{}+=0&.
The only solutions to these equations have $h_{12}{\!\!=\!\!}h_{21}{\!\!=\!\!}0$ (this was dubbed a “vacuum selection effect” in [@ADPU01]) ; the remaining equations are those of a semilocal string [@VA91; @H92] in $(h_{11},h^*_{22})$.
Up to global SU(2) transformations in the $(h_{11},h^*_{22})$ space, the unit winding, cylindrically symmetric semilocal string solution can be expressed as h\_[11]{}&=&f(r)e\^[i]{};\
h\_[22]{}&=&g(r);\
A\_&=&a(r). \[semi\] with boundary conditions $f(0){\!\!=\!\!}a(0){\!\!=\!\!}g'(0){\!\!=\!\!}0$ and $f(\infty){\!\!=\!\!}1$, $g(\infty){\!\!=\!\!}0$ and $a(\infty){\!\!=\!\!}-1$. $g(r)$ is given by g(r)=. \[g\] where the constant $\kappa$ essentially measures the width of the string, ranging from $\kappa{\!\!=\!\!}0$ –the Nielsen-Olesen [@NO73] string– to $\kappa{\!\!=\!\!}\infty$ –a $CP^1$ instanton [@H92]–. Note the lack of winding in $h_{22}$ and furthermore that $h_{22}$ does not necessarily have to be zero at $r{\!\!=\!\!}0$. Note also that $B$ is always negative with this choice of boundary condition, and its total flux does not depend on $\kappa$.
Fermionic zero modes
====================
We are now interested in studying the fermionic zero mode solutions to this system to see whether they influence this vacuum selection effect. These solutions could be obtained by solving the explicit fermionic equations of motion, but we can also use the SUSY transformation to get the zero modes directly [@DDT97]. This will give us static configurations in the plane perpendicular to the string, and we will subsequently introduce $t$ and $z$ dependence on the solutions.
By zero mode solutions, we mean infinitesimal changes to the background configuration that preserve the action (for static configurations, that amounts to leaving the energy unchanged) and satisfy their equations of motion. We know that a SUSY transformation of a given configuration leaves the energy unchanged. Moreover, as we started with a static solution to the bosonic equations of motion, the fermions produced by this transformation must automatically satisfy their equations of motion.
The fermionic content of our system consists of two higgsinos (two Dirac fermions $\psi_1$ and $\psi_2$) coming from the hypermultiplets, and two gauginos (two symplectic Majorana fermions $\lambda^1$ and $\lambda^2$) coming from the gauge vector multiplet. Recall that the symplectic Majorana spinors are 4-component $SU(2)$ covariant spinors defined from 2-component spinors $\lambda_{\alpha i}$ and $\bar\lambda^i_{\dot\alpha}\equiv(\lambda_{\alpha i})^\dagger$ as
\^i;|\_i=(\_i\^, i\_[ij]{}|\^j\_), \[sympl\] where $\varepsilon_{12}=\varepsilon^{12}=-\varepsilon_{\dot 1\dot 2}=-\varepsilon^{\dot 1\dot 2}=1$.
In $N{\!\!=\!\!}1$ SUSY language, the gauginos can be thought of as consisting of one gaugino coming from the $N{\!\!=\!\!}1$ gauge multiplet and one higgsino coming from a neutral (with respect to the gauge multiplet) chiral multiplet. The two SUSY generators ($\xi_{\alpha(1)}$, $\xi_{\alpha(2)}$) will be combined into two symplectic Majorana fermions $\epsilon^1$, $\epsilon^2$ (\[sympl\]).
Let us perform a SUSY transformation of the system in the background of the (bosonic) semilocal string obtained in the previous section. The bosonic fields do not transform. The higgsinos take the form
\_[(a)]{}=-i \^D\_\^[(i)]{} h\_[ai]{}, while the gauginos may be written as
\^[(i)]{}=-\^\^[(i)]{}F\_-i\^[(j)]{} \^[(i)]{}\_[(j)]{}.
Our conventions are [@sohnius] \^=, i=\^, where, $\sigma^\mu=(1,\mathbf{\sigma})$, $\bar\sigma^\mu=(1,-\mathbb{\sigma})$ and $\sigma^i$ are the Pauli matrices.
The higgsinos and the gauginos can be written more explicitly as \_[(1)]{}&=&-;\
\_[(2)]{}&=&-;\
\^[(1)]{}&=&\^1\^2\^[(1)]{} B- i (H\_1\^[ 1]{}-H\_2\^[ 2]{}+ )\^[(1)]{};\
\^[(2)]{}&=&\^1\^2\^[(2)]{} B + i (H\_1\^[ 1]{}-H\_2\^[ 2]{}+)\^[(2)]{}, where we have used the Bogomol’nyi equations (\[bogomolnyi\]).
We expect the strings to be $\half$BPS saturated [@WO78], so let us try to obtain the broken and unbroken part of the SUSY transformation, i.e., let us obtain the fermionic zero modes. In order to do so, we can use the following projectors
P\_(1i\^1\^2), which with our conventions, are given by P\_+(1,0,1,0),P\_-(0,1,0,1).
These projectors, besides $P_\pm ^2= P_\pm ^{\ \dagger}= P_\pm$ and $P_\pm P_\mp=0$, have the following properties:
& &\^1P\_=P\_\^1;\
& &\^2P\_=P\_\^2;\
& &P\_\^1=i P\_\^2. $i\gamma^1\gamma^2$ is essentially a two-dimensional version of $\gamma^5$, acting in the plane perpendicular to the string. Applying these projectors onto the fermions, we learn that
P\_+\_[(1)]{}&&\_[(1)+]{}=-i(D\_1-iD\_2)h\_[11]{}\^1 P\_-\^[(1)]{} =-2iD\_1 h\_[11]{}\^1P\_-\^[(1)]{};\
P\_-\_[(1)]{}&&\_[(1)-]{}=-i(D\_1+iD\_2)h\_[12]{}\^1 P\_+\^[(2)]{} =0;\
P\_+\_[(2)]{}&&\_[(2)+]{}=-i(D\_1-iD\_2)h\_[21]{}\^1 P\_-\^[(1)]{} =0;\
P\_-\_[(2)]{}&&\_[(2)-]{}=-i(D\_1+iD\_2)h\_[22]{}\^1 P\_+ \^[(2)]{} =-2iD\_1h\_[22]{}\^1P\_+ \^[(2)]{},
and
P\_+\^[(1)]{}&&\^[(1)]{}\_+=-i(B+H\_1\^[ 1]{}-H\_2\^[ 2]{}+1) P\_+\^[(1)]{}=0;\
P\_-\^[(1)]{}&&\^[(1)]{}\_-=i(B-H\_1\^[ 1]{}+H\_2\^[ 2]{}-1) P\_-\^[(1)]{}=2iBP\_-\^[(1)]{};\
P\_+\^[(2)]{}&&\^[(2)]{}\_+=-i(B-H\_1\^[ 1]{}+H\_2\^[ 2]{}-1) P\_+\^[(2)]{}=-2iBP\_+\^[(2)]{};\
P\_-\^[(2)]{}&&\^[(2)]{}\_-=i(B+H\_1\^[ 1]{}-H\_2\^[ 2]{}+1) P\_-\^[(2)]{}=0.
Note that $\delta\psi_{(1)-}$ and $\delta\psi_{(2)+}$ vanish on BPS states due to the vacuum selection effect $h_{12}=h_{21}=0$.
It is clear that $P_-\epsilon^{(1)}$ and $P_+\epsilon^{(2)}$ generate the fermionic zero modes, whereas $P_+\epsilon^{(1)}$ and $P_-\epsilon^{(2)}$ are the generators of the unbroken SUSY. Note that there are fermionic zero modes of both chiralities, since N=2 SUSY is non–chiral.
Fermionic equations of motion
=============================
In this section we investigate the structure of the fermionic zero modes by analysing the equations of motion directly, without reference to supersymmetry. We use two-spinor notation and define
\_[(1)]{} \_[(2)]{} \^[(1)]{}\^[(2)]{}
The projector operators can be defined in two spinor notation as \_+=;\_-=. \[twopro\]
In this notation the zero modes that we found previously are \_[(1)]{}&=& -2i D\_1 h\_[11]{} \^1 \_- |\^[(1)]{};\
|\^[(1)]{}&=&-2 D\_1 h\_[11]{}|\^1\_-\_[(2)]{};\
\_[(2)]{}&=& -2 iD\_1 h\_[22]{} \^1 \_+ |\^[(2)]{};\
|\^[(2)]{}&=&2 D\_1 h\_[22]{} |\^1 \_+ \_[(1)]{};\
\_[(1)]{}&=&-2iB\_+\_[(1)]{};\
\_[(2)]{}&=&2iB\_-|\_[(2)]{}, \[feomdef\] where $h_{11}, \ h_{22}$ and $B$ satisfy (\[bogomolnyi\]).
Consider the fermionic equations of motion derived from (\[lagrangian\]) in the bosonic background given by a solution to (\[bogomolnyi\]). Recall that $h_{12} = h_{21} = 0$, and all other fields are independent of $t$ and $z$:
(|\^) D\_\_[(1)]{} -|\^[(1)]{} h\_[11]{}&=&0;\
(\^) D\_|\_[(1)]{}+i \_[(2)]{} h\_[11]{}&=&0;\
(|\^) D\_\_[(2)]{} +|\^[(2)]{} h\_[22]{}&=&0;\
(\^) D\_|\_[(2)]{}+i \_[(1)]{} h\_[22]{}&=&0;\
(\^) \_|\^[(1)]{} +h\^\*\_[11]{}\_[(1)]{}+i\_[(2)]{} h\_[22]{}&=&0;\
(\^) \_|\^[(2)]{} -h\^\*\_[22]{}\_[(2)]{}+i\_[(1)]{} h\_[11]{}&=&0. \[feom\]
It can be checked that the static, $z$–independent configurations (\[feomdef\]) satisfy these equations. Including the $(t,z)$ dependence into (\[feom\]) we learn that three of the fermions are functions of $(t-z)$ and therefore move in the positive $z$ direction, while the other three move in the negative $z$ direction:
\_[(1)]{}, \_[(2)]{}, \_[(1)]{} && t+z;\
\_[(2)]{}, \_[(1)]{}, \_[(2)]{} && t-z.
The fermions move in opposite directions, as expected, since $N{\!\!=\!\!}2$ SUSY is intrinsically non-chiral and it has not been broken in this model. We are interested in the $(r, \theta)$ dependence of the zero modes.
The usual Nielsen-Olesen string is one of the possible configurations in the family of semilocal strings, and it corresponds to the narrowest string. One can characterise this string as having $h_{22}{\!\!=\!\!}0$, annihilating two out of the six Weyl fermions $\phi_{(2)}{\!\!=\!\!}\bar{\chi}_{(2)}{\!\!=\!\!}0$ [@hou]. Moreover, if we remove one SUSY generator out of the two, we recover the Nielsen-Olesen string, but with chiral fermions [@DDT97] moving in the same direction. This is a good check of our results.
The situation is different when $h_{22}\!\!\ne\!\!0$. The fermions $\phi_{(2)}$ and $\chi_{(2)}$ are coupled to a field which is not zero at $r{\!\!=\!\!}0$, which might seem surprising. In terms of the general string ansatz (\[semi\]), these fermions may be expressed in the form \_[(2)]{}&=&-2i\_r()e\^[i]{};\
\_[(2)]{}&=&-2\_r()e\^[-i]{}, and, as can be seen from figure \[fig\], they tend to $0$ at $r{\!\!=\!\!}0$. These two fermions are the only ones that wind.
![\[fig\] Profiles of the derivatives of the functions $f(r)$ and $g(r)$ given by equations (\[semi\]) and (\[g\]), for different values of the parameter $\kappa$.](fp.eps "fig:"){width="7cm"} ![\[fig\] Profiles of the derivatives of the functions $f(r)$ and $g(r)$ given by equations (\[semi\]) and (\[g\]), for different values of the parameter $\kappa$.](gpr.eps "fig:"){width="7cm"}\
In order to analyse the fermion zero modes, we convert the Dirac equations (\[feom\]) into second order equations by acting with the operators $\sigma\cdot D$ and $\bar\sigma\cdot D$:
& &(+\^3 B+|h\_[11]{}|\^2)\_[(1)]{}+ih\_[11]{}h\_[22]{}\_[(2)]{}-(D)h\_[11]{}|\^[(1)]{}=0;\
& &(+\^3 B+|h\_[22]{}|\^2)\_[(2)]{}-ih\_[11]{}\^\*h\_[22]{}\^\*\_[(1)]{}+ i(D)h\_[22]{}\^\*|\^[(1)]{}=0;\
& &(+|h\_[11]{}|\^2+|h\_[22]{}|\^2)|\^[(1)]{}+(|D)h\_[11]{}\^\*\_[(1)]{}+i(|D)h\_[22]{}\_[(2)]{}=0;\
& &(-\^3 B+|h\_[22]{}|\^2)\_[(2)]{}-ih\_[11]{}h\_[22]{}\_[(1)]{}+(D)h\_[22]{}|\^[(2)]{}=0;\
& &(-\^3 B+|h\_[11]{}|\^2)\_[(1)]{}+ih\_[11]{}\^\*h\_[22]{}\^\*\_[(2)]{}+ i(D)h\_[11]{}\^\*|\^[(2)]{}=0;\
& &(+|h\_[11]{}|\^2+|h\_[22]{}|\^2)|\^[(2)]{}-(|D)h\_[22]{}\^\*\_[(2)]{}+i(|D)h\_[11]{}\_[(1)]{}=0, \[box1\] where $\sigma\cdot D{\!\!=\!\!}\sigma^1D_1+\sigma^2D_2$ and $\bar\sigma^{1,2}=-\sigma^{1,2}$.
We will use two dimensional projectors also in this case; and we define the projected two-spinors $\Psi_\pm$ by $\Psi_\pm \equiv \sigma_\pm \Psi$, for any two-spinor $\Psi$, where $\sigma_\pm$ are given in equation (\[twopro\]).
Half of the equations (\[box1\]) correspond to the fermions coming from the unbroken SUSY symmetry: & &(-B+|h\_[11]{}|\^2)\_[(1)-]{}+ih\_[11]{}h\_[22]{}\_[(2)-]{}=0;\
& &(-B+|h\_[22]{}|\^2)\_[(2)-]{}-ih\_[11]{}\^\*h\_[22]{}\^\*\_[(1)-]{}=0;\
& &(+|h\_[11]{}|\^2+|h\_[22]{}|\^2)|\^[(1)]{}\_+=0;\
& &(-B+|h\_[22]{}|\^2)\_[(2)+]{}-ih\_[11]{}h\_[22]{}\_[(1)+]{}=0;\
& &(-B+|h\_[11]{}|\^2)\_[(1)+]{}+ih\_[11]{}\^\*h\_[22]{}\^\*\_[(2)+]{}=0;\
& &(+|h\_[11]{}|\^2+|h\_[22]{}|\^2)|\^[(2)]{}\_-=0. \[eq1\]
We can think of these equations as giving position–dependent [*masses*]{} (squared) for the fermions, in the sense that after diagonalisation, all six equations are of the form $\Box \Psi + {\it
M^2} \Psi = 0$, where
=[diag]{} ( -B +(|h\_[11]{}|\^2 + |h\_[22]{}|\^2), -B, |h\_[11]{}|\^2 + |h\_[22]{}|\^2, -B +(|h\_[11]{}|\^2 + |h\_[22]{}|\^2), -B, |h\_[11]{}|\^2 + |h\_[22]{}|\^2). \[m\]
Note that the matrix $M^2$ is not the same as the fermion mass matrix squared, due to the presence of derivative terms (in particular $M^2$ involves the magnetic field strength and is gauge invariant). Both matrices of course agree on constant bosonic backgrounds. As a simple check, far from the core we recover the fermion masses; indeed, $B \to 0$, $h_{11} \to 1$ and $h_{22} \to 0$ as $r \to
\infty$, and the diagonal terms become $1,0,1$, the correct masses for one higgsino, one goldstino and one gaugino of each chirality.
Recall that we are using signature $(+,-,-,-)$. Since $\Box = -
\nabla^2$, the laplacian in the plane perpendicular to the string, (\[eq1\]) is a set of coupled time–independent Schrödinger equations in $(r, \theta)$ and we are looking for the zero energy states. Obviously, if the “potential” is everywhere positive there are no zero energy eigenstates. The functions in ${\it M^2}$ (\[m\]) are all greater than zero everywhere, and thus there are no non-zero normalizable solutions to (\[eq1\]), as can be shown by simply multiplying by $\Psi^*$ and integrating by parts.
Thus [*all*]{} of these fermionic components are automatically zero because of the equations of motion, and this agrees with the form of the solution obtained from SUSY transformations (\[feom\]).
The other half of the equations reads & &(+B+|h\_[11]{}|\^2)\_[(1)+]{}+ih\_[11]{}h\_[22]{}\_[(2)+]{}-2\^2 |\^[(1)]{}\_-D\_2 h\_[11]{}=0;\
& &(+B+|h\_[22]{}|\^2)\_[(2)+]{}-ih\_[11]{}\^\*h\_[22]{}\^\*\_[(1)+]{}+2i\^2 |\^[(1)]{}\_- D\_2 h\_[22]{}\^\*=0;\
& &(+|h\_[11]{}|\^2+|h\_[22]{}|\^2)\^2 |\^[(1)]{}\_- - 2\_[(1)+]{} D\_2 h\_[11]{}\^\* - 2i \_[(2)+]{} D\_2 h\_[22]{} =0;\
& &(+B+|h\_[22]{}|\^2)\_[(2)-]{}-ih\_[11]{}h\_[22]{}\_[(1)-]{}+2 \^2 |\^[(2)]{}\_+ D\_2 h\_[22]{}=0;\
& &(+B+|h\_[11]{}|\^2)\_[(1)-]{}+ih\_[11]{}\^\*h\_[22]{}\^\*\_[(2)-]{}+2i\^2 |\^[(2)]{}\_+ D\_2 h\_[11]{}\^\* =0;\
& &(+|h\_[11]{}|\^2+|h\_[22]{}|\^2)\^2 |\^[(2)]{}\_+ + 2\_[(2)-]{} D\_2 h\_[22]{}\^\* -2i \_[(1)-]{} D\_2 h\_[11]{} =0, where, again, the Bogomol’nyi equations (\[bogomolnyi\]) have been used extensively.
These may be written as two sets of three coupled equations[^1] and we again seek to diagonalize the resulting mass matrices.
+ = 0 + = 0
For simplicity we just consider the first set of three equations. In the case of the Nielsen-Olesen string member of the semilocal family , for which $\kappa{\!\!=\!\!}0$, $h_{22}{\!\!=\!\!}0$, the spinors $\chi_{(2)+}$ and $\phi_{(2)-}$ do not couple to any spinors, and have mass squared B, which is negative in the core of the string, and zero at infinity. After diagonalisation, the other mass squared terms are (B + 2 |h\_[11]{}|\^2 ) which correspond to two (r-dependent) linear combinations of $\phi_{(1)+}$ and $\sigma^2 \bar\Lambda^{(1)}_-$. At $r{\!\!=\!\!}0$, the signs of these masses are $+,-$ respectively. At infinity these masses are 1, 1, and the mass states are simply the uncombined spinors. We can immediately see by the same reasoning as before that one combination of fermions is zero everywhere, since its mass squared is positive everywhere. The other one is a combination of the higgsino and the gaugino.
For the general case, $\kappa \neq 0$, we have to diagonalize two sets of $3\times 3$ matrices. The signs of the “eigenvalues” of the $M^2$ matrices are $+,-,-$ at $r{\!\!=\!\!}0$ and $+,0,+$ at infinity, the “eigenvectors” being a combination of the three fermions. Once again, the “eigenvector” whose mass–squared function is always positive is zero, and thus we are left with two non-zero “eigenvectors”.
In all cases, the fermions at infinity have masses $1,0,1,1,0,1$, which agree with the masses of the fields $h_{11},h_{12},A_\mu,h_{22},h_{21},M+iN$, as should be the case since supersymmetry is unbroken there.
Discussion
==========
We have studied the fermion zero modes arising in SUSY $N{\!\!=\!\!}2$ QED with two hypermultiplets and a FI term. It is known that in such a model there is a vacuum selection effect in the bosonic sector, and only some of all the possible vacua form a family of neutrally stable vortices [@ADPU01]. We showed that the fermion zero modes do not improve the stability of the vortices. This is due to the fact that fermions obey the same vacuum selection effect as the vortices. On the other hand, as the SUSY is $N{\!\!=\!\!}2$, fermions with both chiralities are present in the vortex.
We do not expect the stability to be altered by the back-reaction of the fermions either. A recent detailed calculation of quantum corrections for SUSY kinks shows that the Bogomol’nyi bound is preserved [@G]), even if the mass receives corrections.
The model analysed in this paper contains a family of vortices that have the same topological charge, and all saturate the Bogomol’nyi bound. SUSY is half broken for all members of the family, and the unbroken SUSY will preserve the Bogomol’nyi bound in all cases, since the multiplets are shortened. Thus the BPS condition holds for all members of the family, and as they all have the same topological charge, the energy of all of them is also the same. Thus, no member of the family will be singled out by fermionic back-reaction.
The fermions in this system are also interesting because some are coupled to a scalar field which is not zero at the core of the vortex. It is sometimes stated in the literature that the reason why vortices support fermion zero modes in their core is because the fermion masses, which come from coupling to the scalars, are zero there. We have shown here that this heuristic argument is incorrect by calculating explicitly the zero modes that couple to the non–zero scalar at the core ($h_{22}$, if the string is in $h_{11}$).
It is interesting to try to relate the zero modes derived from supersymmetry to the zero modes we see in the equations of motion. The two fermionic zero modes obtained directly from supersymmetry are related to translational zero modes in the bosonic sector.
We can see from the second order equations of motion that in fact there are two more fermionic zero modes: In the bosonic sector, there are zero modes corresponding to changes in the parameter $\kappa$ of the semilocal string, expanding the string core. This bosonic zero mode corresponds to the fermionic zero modes that have negative mass at zero, and which are massless at infinity. One way to see this is to take the case where there is just one hypermultiplet $(h_i,\psi,F_i)$. The string formed would then be an ordinary Nielsen-Olesen string, which would not possess the bosonic zero mode. Furthermore, the spinors $\phi_2$ and $\chi_2$ would disappear, and the remaining eigenstates would have signs $+,-,+,-$ for their masses squared at zero and $+,+,+,+$ at infinity, so that only two of the eigenstates are non-zero. When we re-introduce the second hypermultiplet, the extra fermionic zero modes correspond to the extra bosonic zero mode associated with $h_{22}$.
It is also possible to consider supersymmetric transformations of the bosonic zero mode to give the fermionic zero mode. We perturb the field $h_{22}$, adjusting the other fields so that we retain the Bogomol’nyi equations (\[bogomolnyi\]), and hence keep the same energy. In the limit $\kappa=0$, the semilocal string has $h_{22}=0$, and an infinitesimal perturbation of $h_{22}$ of the form $\delta
h_{22}= \alpha h_{11\,(background)}/r$ does not modify the other fields. Hence, a supersymmetry transformation of this zero mode gives fermionic zero modes in $\phi_{(2)}$ and $\chi_{(2)}$ only.
At infinity the bosonic zero mode corresponding to changing the string width is a goldstone boson, and this agrees with what we have discovered from the fermionic equations of motion: that the eigenstates that have mass zero at infinity are pure $\phi_{(2)}$ or $\chi_{(2)}$ in the $\kappa{\!\!=\!\!}0$ case.
For a general semilocal string $(\kappa\neq 0)$, all the bosonic fields will be affected by a perturbation in the string width, and so the corresponding fermionic zero modes are combinations of each set of three spinors.
We note that the index theorem of Davis, Davis and Perkins [@DDP97] does not apply in the case of the semilocal string due to the fact that one of the fields doesn’t wind in the core – an assumption made in the derivation of the index theorem. Hence we can’t use it to ascertain the number of zero modes in this case. In the case where the index theorem does apply, namely the Nielsen–Olesen string (i.e. one hypermultiplet), our results agree with the index theorem. For the semilocal string there are two extra zero modes corresponding to the change of the string width; the fermion zero modes are those fermions corresponding to the bosonic zero mode. This has been constructed explicitly in the Nielsen–Olesen limit of the semilocal string. The model also falls outside the scope of the index theorem of Ganoulis and Lazarides[@GL88], because $h_{22}$ is charged under $U(1)$ but goes to zero at infinity. Our results agree with Weinberg’s index theorem[@W81] (see also [@JR81]), since the new zero modes have opposite chiralities and therefore do not affect the counting of net (left minus right) fermion zero modes. This is in agreement with the fact that the magnetic flux measured at infinity is the same for all semilocal strings, including the Nielsen–Olesen string.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank F. Freire, T. Vachaspati and P. van Baal for very useful discussions. This work was partially supported by the ESF COSLAB programme (M.P.) and by grants AEN99-0315, FPA 2002-02037 and 9/UPV00172.310-14497/2002 (A.A., J.U.). J. Urrestilla and M. Pickles thank the University of Leiden for their hospitality.
[99]{}
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T. Vachaspati and A. Achúcarro, Phys. Rev. [**D44**]{}, 3067 (1991) M. Hindmarsh, Phys. Rev. Lett. [**68**]{}, 1263 (1992) R.A. Leese, Phys. Rev. [**D46**]{}, 4677 (1992) B. Carter, P. Peter, Phys. Lett. [**B466**]{}, 41 (1999) A.C. Davis, T.W.B. Kibble, M. Pickles, D.A. Steer, Phys. Rev. [**D62**]{}, 083516 (2000) S.G. Naculich, Phys. Rev. Lett. [**75**]{}, 998 (1995); H. Liu, T. Vachaspati, Nucl. Phys. [**B470**]{}, 176 (1996); M. Groves, W.B. Perkins, Nucl. Phys. [**B573**]{}, 449 (2000) M.F. Sohnius, Phys. Rept. [**128**]{}, 39 (1985) H. Nielsen, P. Olesen, Nucl. Phys. [**B61**]{}, 45 (1973) S.C. Davis, A.C. Davis, M. Trodden, Phys. Lett. [**B405**]{}, 257 (1997) E. Witten, D.I. Olive, Phys. Lett. [**B78**]{}, 97 (1978) X. Hou, Phys. Rev. [**D63**]{}, 045015 (2001) A. Rebhan, P. van Nieuwenhuizen, R. Wimmer, Nucl. Phys. [**B648**]{}, 174 (2003); A.S. Goldhaber, A. Rebhan, P. van Nieuwenhuizen, R. Wimmer “Quantum corrections to the mass and central charge of solitons in $1+1$ dimensions”, based on a talk given by P. van Nieuwenhuizen at the 3rd Sakharov Conference On Physics, June 2002, Moscow; [hep-th/0211087]{} S.C. Davis, A.C. Davis, W.B. Perkins, Phys. Lett. [**B408**]{}, 81 (1997) N. Ganoulis, G. Lazarides, Phys. Rev. [**D38**]{}, 547 (1988) E.J. Weinberg, Phys. Rev. [**D24**]{}, 2669 (1981) R. Jackiw, P. Rossi, Nucl. Phys [**B190**]{} 681 (1981).
[^1]: Note that each set of equations correspond to fermions of one given chirality and coming from one of the supersymmetry generators
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Enterprise application integration (EAI) solutions are the centrepiece of current enterprise IT architectures ([e.g., ]{}cloud and mobile computing, business networks), however, require the formalization of their building blocks, represented by integration patterns, verification and optimization.
This work serves as an instructive pattern formalization catalog that leads to the formalization of all currently known integration patterns. Therefore, we explain the classification of the underlying requirements of the pattern semantics and formalize representative patterns from the different categories, by realizing them in timed db-net. In this way, the catalog will allow for the addition of future patterns by assigning them to a category and applying the described formalism.
author:
-
-
-
bibliography:
- 'formal\_v2.bib'
title: Catalog of Formalized Application Integration Patterns
---
Introduction
============
The enterprise integration patterns (EIPs) from 2004 [@hohpe2004enterprise] denote messaging patterns that serve as building blocks, when implementing an enterprise application integration (EAI) system [@Linthicum:2000:EAI:328930]. While the EIPs are still practically relevant today [@DBLP:journals/software/ZimmermannPHW16; @Ritter201736], the emerging technological, social and business trends since then require pattern extensions, [e.g., ]{}for integration adapters and endpoints [@DBLP:conf/caise/0001H15], exception handling and fault tolerance [@DBLP:conf/edoc/RitterS14; @ritter2016exception], as well as information security among many other aspects [@DBLP:journals/corr/RitterR15; @Ritter201736]. With the totality of those patterns playing a major role in real-world application integration architectures, the lack of a comprehensive formalization of the single pattern semantics and their compositions (beyond the currently only attempt using plain coloured petri nets (CPN) [@DBLP:conf/caise/FahlandG13; @fahland2012using]) will be instrumental for the verification and optimization of the current and future EAI process modeling and architectural solutions [@Ritter201736].
While we found a suitable formal representation in the recent work on db-nets [@DBLP:journals/topnoc/MontaliR17] and extended them to timed db-nets for the formalization of EIPs, this work strives to collect and formalize all patterns from the aforementioned, currently known pattern catalogs, by being a catalog of formalized patterns itself. However, with a total amount of $166$ patterns and approximately $139$ that are due to formalization ([i.e., ]{}no meta concepts), a complete catalog of formalized patterns would not be practical and lead to many repetitive formalizations of the same underlying concepts.
Consequently, we categorized the underlying requirements for the patterns’ semantics ([i.e., ]{}data and control flow, external resources, transactions, complex message formats, and time) and discuss representative patterns from each of the categories ([e.g., ]{}data-control, data-time patterns) and their realizations in timed db-net in \[sec:formalization\]:
- Data, transact. resource
- Control, transacted resource, time
- Control-time
- Data, transact. Resource Time
With these representatives of each category, the others can be easily formalized due to the same underlying semantic concepts, making this work rather an instructive catalog manual, than a simple pattern reference. The new patterns identified after $2004$ denote a case on how to add and formalize potentially new patterns beyond this work, since they strengthened existing and added new categories.
Then in \[sec:testing\], we briefly discuss how the timed db-net formalism helps to experimentally test the correctness of the patterns–again by category and not for each pattern– and \[sec:conclusion\] concludes this work.
Formalized Patterns by Category {#sec:formalization}
===============================
In this section, we define selected patterns from the requirement categories discussed before. The subsequent categories have been chosen, since the go beyond the already existing formalisms and in their combination they allow for a complete representation of the known as well as future patterns.
Data, transact. resource patterns: Resequencer
----------------------------------------------
The stateful Resequencer is a pattern that guarantees the order of messages in (asynchronous) communication [@hohpe2004enterprise]. shows the resequencer in timed db-net representation.
![Resquencer pattern[]{data-label="fig:resequencer"}](resequencer_v2){width="1.0\columnwidth"}
The entering message `msg` contains sequence (`seq`) and order (`ord`) information and is persisted in the database, represented by a db-net view place $ch_p$. For the first message of a sequence the sequence will be created in view place `Message Sequences`, and for all subsequent message of that same sequence, the messages are stored. As soon as the sequence is complete, [i.e., ]{}all messages of that sequence arrived, the messages of that sequence are queried from the database in order by the `Reorder` transition. Eventually, the messages are forwarded in ascending order to $ch_{out}$.
Control, transacted resource, time patterns: Circuit Breaker
------------------------------------------------------------
The Circuit Breaker pattern [@Ritter201736] addresses failing or hang up remote communication, which impacts the control flow of a Request-Reply pattern [@hohpe2004enterprise] by using transacted access to external resources. shows a request-reply representation in timed db-net, extended by a circuit breaker “wrapper” (using db-net view places) that protect the remote call.
![Circuit breaker[]{data-label="fig:circuit_breaker"}](circuit_breaker_v2){width="1.0\columnwidth"}
At the beginning, the circuit is closed, [i.e., ]{}communication is enabled. In case of an exception (`exc`) during send or a timeout of 30 time units during receive, the number of failed attempts (`nexc`) is increased and stored into the `Enpoints` view place, which maintains a list of all endpoints (`epid`) and their consecutive failures. If the number of failures reaches a limit ([e.g., ]{}nexc $>5$) the circuit trips, and thus updates the `status` entry in view place `Circuits` to “open”, which let all subsequent messages immediately go to the exception place (`exc`). The circuit breaker can be closed by manually updating the status to “closed” to give the remote endpoint another chance. With additional logic, self-resetting mechanisms can be implemented.
Control-time patterns: Throttler, Delayer
-----------------------------------------
The following patterns mostly require control flow and time aspects, and are thus in timed CPNs with guards.
The Throttler pattern helps to ensure that a specific receiver does not get overloaded by regulating the number of transferred messages. shows the realization of a throttler that emits five messages per second to the receiving place $ch_3$.
$
\begin{array}{cc}
\subfigure[Throttler]{\label{fig:throttler}\includegraphics[width=0.5\columnwidth]{throttler_v2}} &
\subfigure[Delayer]{\label{fig:delayer}\includegraphics[width=0.5\columnwidth]{delayer_v2}}
\end{array}$
A slightly different pattern of this category is the delayer, as shown in \[fig:delayer\], which uses a timer to reduce the frequency of messages sent to the receiving place $ch_3$.
Data, transact. Resource Time patterns: Aggregator
--------------------------------------------------
The combination of data, transacted resources and time aspects in patterns makes them the semantically most complex ones. For example, \[fig:aggregator\] specifies the semantics of a commonly used stateful Aggregator [@hohpe2004enterprise] pattern.The aggregator persistently collects messages in a special timed db-net view place $ch_p$ and aggregates them in an `Aggregate` PN transition based on a completion condition ([e.g., ]{}sequence `isComplete`) or on timeout, depending on a sequence `seq`, represented as PN guards. For this an incoming message `msg` is correlated (cf. `correlate`) to an existing sequence based on its content. If the message is the first of a sequence, a new sequence and a message to sequence assignment is created in a persistent store called `Message Sequences`. If a message correlates to an existing sequence, which is aggregated due to a timeout `isExpired`, the update fails. Then a roll-back is executed (red reverse timed db-net arc) that puts the message back to the message channel $ch_{in}$ (PN Place). Now, this message matches `first msg` and a new sequence is created accordingly.
![Aggregator pattern.[]{data-label="fig:aggregator"}](aggregator_v8){width="1.0\columnwidth"}
Correctness Testing of timed db-net Patterns {#sec:testing}
============================================
The correctness of an integration pattern realization represented in timed db-net can be validated by evaluating the execution trace on the persistence layer. According to the timed db-net execution semantics, a pattern produces several B-snapshots $s_1,..,s_n$ during the execution of the pattern from an input snapshot $s_1= \langle I_1,m_1 \rangle$ with database instance $s_n= \langle I_n,m_n \rangle$ to a final snapshot $s_n$, denoted by $s_1[t,\sigma \rangle s_2$,..,$s_i[t,\sigma \rangle s_n$. In case of a database instance $I_j$ is not compliant with $\mathfrak{P}$, then the execution stops and leaves the timed db-net in an intermediate state $I_j= I_{i-1}$, otherwise $I_n$ is the final state. Hence, the control flow can be validated, by checking, whether the pattern produces a token to the correct final database instance. gives a schematic view of an timed db-net pattern, for which the inner workings are unknown and the data is exchanged through input places $ch_1,..,ch_i$, output places $ch_{n-m},..,ch_n$, and an intermediate place $ch_j$ in $\mathfrak{N}$ that subsumes all exceptional places, together with the corresponding database intances $I_1,..,I_i$, $I_{n-m},..,I_n$, and $I_j$. The input or newly created tokens eventually manifest in entries in the DB instances.
[**Data and (transacted) resouce-bound patterns**.]{} With respect to data, format , transacted resources and exceptional situations, for a given instance with test data $I_1$, either an expected final persistent database instance $I_n$ with the correct schema or an expected error state $I_j$ must be produced by the pattern. Otherwise the pattern is incorrect with respect to its definition for the requirements. [**Patterns with msg. channel order**.]{} Similarly, the channel execution order can be validated. In case of the content-based router, an initial instance $I_1$ will result a different output instance $I_n$, depending on the values in $I_1$ and the routing conditions. While this can be checked as for the first case, the balancer requires a sequence of input instances, which then have to produce data entries in the output instances that fit the probability values and distribution of the balancer ([e.g., ]{}Kolmogorov-Smirnov test [@1933sulla]).
![Throttler testing (schematic)[]{data-label="fig:throttler_testing"}](throttler_testing.png){width="0.6\columnwidth"}
[**Time-bound patterns**.]{} Finally, a timed pattern can be validated by assigning timestamps time to the database instances (as “on-insert timestamps” in actual databases). This allows for checking delays by comparing the insert timestamps $time(I_1)$, $time(I_n)$ of data to instance $I_1$ and those of $I_n$. With this, a numeric delay $d$ can be checked by $time(I_n)-time(I_1)>=d$ and a message per time ratio $m/time$ by counting the number of entries \#(e) inserted to $I_n$ within time buckets $b/unit$, making pattern valid, if $\#(e)_{I_n} <=$ expected.
For instance, \[fig:throttler\_testing\] shows a throttler pattern, provided with test messages into its input place $ch_1$ in $\mathfrak{N}$, bound to a database instance $I_1$ in $\mathfrak{P}$. The transition $t_1$ takes the messages from $ch_1$ one after the other ([i.e., ]{}capacity `guard` cap=1) and inserts them into $ch_2$ with instance $I_2$, where the messages are picked up by a timed transition `Timer` and moved to $ch_3$, and thus slowing down the processing. The data logic layer $\mathfrak{L}$ mediates by rewriting the database instances from the input $s_1[t,\sigma \rangle s_2$, to the output $s_2[t,\sigma \rangle s_3$. Thereby the average size of $I_2$ denotes the size of the time buckets $b/unit$.
[**Example: flawed pattern implementation**.]{}
In the absence of a tool that allows for the verification (model-checking) of a pattern, we test their correctness through simulation, by example of a “flawed” content-based router implementation. Therefore, we use our timed db-nets CPN Tools extension[^1].
A content-based router, is a pattern that takes one input message and passes it, read-only to exactly one receiver. This is done by evaluating a condition per recipient on the content of the message. shows one out of many router implementations, which look correct, however, violates this definition on the data and not the control level.
{width="1.0\linewidth"}
For the evaluation we use the aforementioned method for “data and (transacted) resource-bound patterns”, which is based on the reachability of a correct database state. Such a correct state would be a database instance with data in table `channel1` and an empty `channel2` table.
Now, let us explore the inner workings of this implementation using timed db-net. In \[fig:flawed\_router\_1\], transition T reads the token in place I and then conditionally inserts it to the two subsequent places.
{width="1.0\linewidth"}
Since the value of the token matches all conditions, both output places $O_1$ and $O_2$ receive a copy of the token, shown in \[fig:flawed\_router\_2\]. In terms of application integration, this could mean that two companies receive a payment request or a sales order that was actually meant for only one of them. In the net, the two subsequent transitions $push_1$ and $push_2$ are enabled and fire by executing the database inserts $ADD\_TO\_CHANNELx$, while $x$ being the respective database table of the receiver. From the net alone, the semantics seem to be correct. However, on the persistence layer, no correct state has been reached. This is illustrated by looking into the database instance after the tokens have been processed successfully on the control layer (cf. \[fig:channel1\], \[fig:channel2\]).
$
\begin{array}{cc}
\subfigure[\texttt{channel1}]{\label{fig:channel1}\includegraphics[width=0.5\linewidth]{flawed_content_based_router_db_1}} &
\subfigure[\texttt{channel2}]{\label{fig:channel2}\includegraphics[width=0.5\linewidth]{flawed_content_based_router_db_2}}
\end{array}$
In the persistence layer, both tables are filled with data, which is an invalid state according to the definition of the content-based router. Hence, the deep insight into the process and corresponding data aspects of timed db-net allow for detecting flaws in the pattern implementations as well as richer information for fixing it.
Conclusion {#sec:conclusion}
==========
In this work, we define and discuss the formalization of important integration patterns per category, and thus contribute an instructive catalog of pattern realizations and a description of testing their correctness for the current and future patterns to come. The power of the chosen timed db-net formalism becomes especially clear for complex patterns, since for the first time, all of their underlying data, transactional resource and time semantics become explicit now.
[^1]: Demonstrator available for download, 10/2018: <https://github.com/dritter-hd/db-net-eip-patterns>; containing a flawed implementation of a content-based router, and the non-flawed implementation of the aggregator pattern from \[fig:aggregator\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
This paper studies an infinite horizon optimal consumption problem under exponential utility, together with non-negativity constraint on consumption rate and a reference point to the past consumption peak. The performance is measured by the distance between the consumption rate and a fraction $0\leq\lambda\leq 1$ of the historical consumption maximum. To overcome its path-dependent nature, the consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional depending on the wealth variable $x$ and the reference variable $h$. The associated Hamilton-Jacobi-Bellman (HJB) equation is expressed in the piecewise manner across different regions to take into account constraints. By employing the dual transform and smooth-fit principle, the classical solution of the HJB equation is obtained in an analytical form, which in turn provides the feedback optimal investment and consumption. For $0<\lambda<1$, we are able to find four boundary curves $x_1(h)$, $\breve{x}(h)$, $x_2(h)$ and $x_3(h)$ for the wealth level $x$ that are nonlinear functions of $h$ such that the feedback optimal consumption satisfies: (i) $c^*(x,h)=0$ when $x\leq x_1(h)$; (ii) $0<c^*(x,h)<\lambda h$ when $x_1(h)<x<\breve{x}(h)$; (iii) $\lambda h\leq c^*(x,h)<h$ when $\breve{x}(h)\leq x<x_2(h)$; (iv) $c^*(x,h)=h$ but the running maximum process remains flat when $x_2(h)\leq x<x_3(h)$; (v) $c^*(x,h)=h$ and the running maximum process increases when $x=x_3(h)$. Similar conclusions can be made in a simpler fashion for two extreme cases $\lambda=0$ and $\lambda=1$. Numerical examples are also presented to illustrate some theoretical conclusions and financial insights. \
\
**Keywords**: Exponential utility, non-negative consumption, historical consumption maximum, path dependence, dual transform, free boundary.
author:
- 'Shuoqing DENG[^1]'
- 'Xun LI[^2]'
- 'Huyên PHAM[^3]'
- 'Xiang YU[^4]'
bibliography:
- 'consumption.bib'
title: Optimal Consumption with Reference to Past Spending Maximum
---
Introduction {#sec1}
============
The Merton problem, also known as continuous time optimal portfolio and consumption via utility maximization firstly studied in [@Mert1] and [@Mert2], has been one of the milestones in quantitative finance, which bridges the investment decision making and some advanced mathematical tools such as PDE theories and stochastic analysis. The celebrated dynamic programming principle enables one to solve the stochastic control problem by looking for the solution of the associated HJB equation. Isoelastic utility and exponential utility have attracted dominant attention in academic research as they enjoy the merits of homogeneity and scaling property. In abundant work on terminal wealth optimization, the value function can be conjectured in some convenient separation forms or the change of variables can be exercised, the dimension reduction can consequently be applied to simplify the HJB equation. When intermediate consumption is taken into account, the study of exponential utility becomes relatively rare in the literature due to its unnatural allowance of negative consumption behavior. To be precise, as the exponential utility is defined on the whole real line, the resulting optimal consumption from the first order condition can be negative in general. For technical convenience, some existing literature such as [@Mert1], [@Vayanos], [@LiuH] and many subsequent work simply ignore the constraint or interpret the negative consumption as the infusion of funds, i.e., the negative consumption control is assigned by different financial meanings so that the non-negativity constraint can be avoided in their mathematical problems.
The case of exponential utility with non-negative consumption has been examined before by [@CoxHuang] using the martingale method, in which the optimal consumption can be expressed in an integral form involving the state price density process. As illustrated in [@CoxHuang], the structure of the value function and the optimal consumption differ substantially from the case when the constraint is neglected. Some technical endeavors are actually required to fulfill the non-negativity constraint on the control process. In the present paper, we aim to revisit this problem under the exponential utility binding strictly with the constraint that the consumption rate must be non-negative. Moreover, unlike the time separable utility studied in [@CoxHuang], our paper further attempts to go beyond the conventional preference and investigate the consumption behavior when a reference point is combined in the utility as well. In particular, our new preference essentially concerns how far the investor is away from the past consumption maximum level, and this intermediate gap is chosen as the metric to generate the utility of the investor in a dynamic way. Due to the consumption running maximum process inside the utility function, the martingale method developed in [@CoxHuang] can no longer handle our path-dependent optimization problem because it turns to be difficult to conjecture the valid dual processes and the associated dual problem.
Our problem formulation is mainly motivated by the psychological viewpoint that the consumer’s satisfaction level and risk tolerance sometimes depend on recent changes instead of the absolute rate. Some large amount of expenditures, such as purchasing a car, a house or some luxury goods, not only spur some long term continuing spending for maintenance and repair, but also lift up the investor’s standard of living gradually. A striking decline in future consumption plan may result in intolerable disappointment and discomfort. To depict the quantitative influence of the relative change towards the investor’s preference, it makes good sense to introduce the utility to measure the distance between the consumption control and a proportion of the past consumption peak. On the other hand, during some economic recession periods such as recent global economy battered by Covid-19, it is unrealistic to mandate that the investor needs to catch up with the past spending maximum all the time. To capture the possibility that the investor may strategically decrease the consumption budget to fall below the benchmark so that more wealth can be accumulated from the financial market to meet future higher consumption plan, we choose to work with the exponential utility instead of Isoelastic utility that is defined on the positive real line. As a direct consequence, the investor can bear a negative gap between the consumption control and the reference level. The flexibility to compromise the consumption plan below the reference point from time to time makes the model suitable to accommodate more versatile market environments and mathematically unique and interesting.
Utility maximization with a reference point has become an important topic in the research of prospect theory and behavioral finance, see [@tvekah92], [@He1], [@He2] and [@He3] on portfolio management with either a fixed or an adaptive reference level. Our paper differs from the previous work as we do not distinguish the utility on gain and loss separately and our reference level has path-dependent nature and is dynamically updated by the control itself. The impact of the reference to the past consumption maximum becomes highly implicit in our setting, which makes the problem appealing and challenging. On the other hand, our formulation is closely related to the so-called consumption habit formation preference, which measures the deviation of the consumption from the standard of living conventionally defined as the average of the accumulative consumption. See some previous work on addictive consumption habit formation in [@constantinides1990habit], [@detemple1992optimal], [@schroder2002isomorphism], [@munk2008portfolio], [@englezos2009utility], [@yu2015utility], [@yu2017], [@YYu] and non-addictive consumption habit formation in [@DepKart]. Recently, there are also some emerging research on the combination of the reference point and the consumption habit formation, see for instance [@Curatola17] and [@Bilsen17], in which the reference point is generated by the habit formation process and different utility functions are equipped when the consumption is above the habit and when the consumption is below the habit. It will be an interesting future work for us to also consider this S-shaped utility defined on the difference between the consumption and the consumption peak reference level and investigate the structure of the optimal consumption. Among the aforementioned work, it is worth noting that [@DepKart] considers the utility defined on the whole real line and also permits the admissible consumption to fall below the habit level from time to time, namely the consumption habit is not addictive. [@DepKart] extends the martingale method in [@CoxHuang] by using the adjusted state price density process, which produces a nice construction of the optimal consumption in the complete market model. However, our running maximum process in the utility function differs substantially from [@DepKart] and the duality approach is again not applicable.
One of the main contributions of the present paper is to show that our path-dependent control problem with consumption constraint can be solved under the umbrella of dynamic programming and PDE approach. The optimal consumption and portfolio can be obtained in piecewise feedback forms across different regions. Furthermore, all free boundary curves to separate these regions, albeit complicated, can be fully explicitly characterized. Comparing with Merton problem with exponential utility, our value function and feedback optimal controls have distinctive and more interesting features. On the other hand, in terms of the control problem and the associated HJB equation, it is worth noting that [@Arun], [@GHR] and [@BAY] are technically close to the present paper. However, [@GHR] studies the optimal consumption under a Cobb-Douglas utility that is defined on the ratio of the consumption rate and the consumption running maximum, and [@Arun] and [@BAY] considers an optimal consumption and dividend control problem respectively with a standard power utility and the drawdown constraint is only mandated on the control and does not appear in the utility. As opposed to [@GHR] and [@BAY], our utility measures the difference between the control and its running maximum and the non-negativity constraint on consumption is actively imposed under the exponential utility function. Mathematically speaking, the change of variable and dimension reduction in [@GHR] and [@BAY] can not be exercised in the present framework and we confront a two dimensional value function and its associated nonlinear PDE problem. Despite of its complex structure and the non-negativity constraint on optimal control, it is revealed in the present paper that the existence of the classical solution to the associated HJB equation can be obtained in the analytic form with the aid of the dual transform, the smooth-fit principle and other novel arguments.
In summary, by noting that the consumption control is restricted between 0 and the peak level, we first heuristically derive the HJB equation in different forms based on the decomposition of the domain $\{(x,h)\in \mathbb{R}_+\times \mathbb{R}_+\}$ into disjoint regions of $(x,h)$ such that the feedback optimal consumption satisfies (i) $c^*(x,h)=0$; (ii) $0<c^*(x,h)<h$; (iii) $c^*(x,h)=h$. To overcome the obstacle from nonlinearity, we apply the dual transformation only with respect to the state variable $x$ and treat $h$ as the parameter that is involved in some free boundary conditions. The linearized dual PDE in different regions can be handled as ODE problem with the parameter $h$. By using smooth-fit principle and some intrinsic boundary conditions from the nature of the problem, we successfully obtain the explicit solution of the dual ODE problem that eventually enables us to express the value function, the feedback optimal investment and consumption in terms of the primal variables after the inverse transform. Unlike [@Arun], [@GHR] and [@BAY], taking the weight parameter $0<\lambda<1$ for instance, we can explicitly characterize the thresholds $x_1(h)$, $\breve{x}(h)$, $x_2(h)$ and $x_3(h)$ for the wealth variable $x$ as sophisticated nonlinear functions of the variable $h$ such that we can provide the feedback optimal consumption in the way that: (i) $c^*(x,h)=0$ when $x\leq x_1(h)$; (ii) $0<c^*(x,h)<\lambda h$ when $x_1(h)<x<\breve{x}(h)$; (iii) $\lambda h\leq c^*(x,h)<h$ when $\breve{x}(h)\leq x<x_2(h)$; (iv) $c^*(x,h)=h$ but $h$ is a previously attained maximum level when $x_2(h)\leq x<x_3(h)$; (v) $c^*(x,h)=h$ and the instant $c^*(x,h)$ creates a new historical maximum level when $x=x_3(h)$. Two extreme cases $\lambda=0$ and $\lambda=1$ are also discussed separately. In particular, $\lambda=0$ corresponds to Merton problem with non-negativity constraint and we recover the result in [@CoxHuang] using PDE approach. When $\lambda=1$, it is interesting to observe that the value function is not strictly concave any more so that we need to apply the dual transform in a restricted domain. Moreover, we reveal an interesting observation that there is no need to consider the singular consumption that increases its running maximum process, which differs from the case $0<\lambda<1$. At last, the complete proof of the verification theorem is rigorously established.
Building upon the explicit value function and the feedback optimal controls, some quantitative properties and numerical examples are presented. The impacts of the variable $h$ and the reference weight parameter on the boundary curves $x_1(h)$, $\breve{x}(h)$, $x_2(h)$ and $x_3(h)$ can be numerically illustrated and the financial insights are observed. We also perform some sensitivity analysis on the value function, the optimal consumption and portfolio with respect to the reference weight parameter, the drift of the risky asset, the volatility of the risky asset and the risk aversion parameter respectively and conclude some interesting financial implications.
The remainder of the paper is organized as follows. Section \[sec:introduc\] introduces the market model and formulates the stochastic control problem under the utility with the reference to consumption peak. Section \[sec:solveHJB\] presents the associated HJB equation and our technical computations to obtain the fully explicit solution using dual transform, smooth-fit principle and some intrinsic boundary conditions. Some numerical sensitivity analysis are presented in Section \[sec:numerics\]. At last, Section \[sec:proofs\] provides the rigorous proof of the verification theorem and other main results in the previous sections.
Market Model and Problem Formulation {#sec:introduc}
====================================
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, in which $\mathbb{F}=(\mathcal{F}_t)_{t\geq 0}$ satisfies the usual conditions. We consider a financial market consisting of one riskless asset and one risky asset. The riskless asset price satisfies $dB_t=rB_tdt$ where $r\geq 0$ represents the constant interest rate. The risky asset price follows the dynamics $$dS_t=S_t\mu dt+S_t\sigma dW_t,\nonumber$$ where $W$ is an $\mathbb{F}$-adapted Brownian motion and both the mean return $\mu$ and volatility $\sigma>0$ are given constants. The sharp ratio parameter is denoted by $\kappa:= \frac{\mu-r}{\sigma}$. It is worth noting that our mathematical arguments and all conclusions can be readily generalized to the model with multiple risky assets as long as the market is complete. For the sake of simple presentation, we shall only focus on the model with a single risky asset. It is assumed that $\kappa>0$ from this point onwards, i.e. $\mu>r$ that the return of the risky asset is higher than the interest rate.
Let $(\pi_t)_{t\geq 0}$ represent the dynamic amount that the investor allocates in the risky asset and $(c_t)_{t\geq 0}$ denote the dynamic consumption rate of the investor. The resulting self-financing wealth process $(X_t)_{t\geq 0}$ satisfies $$dX_t=rX_tdt+ \pi_t(\mu -r)dt+\pi_t\sigma dW_t-c_tdt,\ \ \ t\geq 0,$$ with the initial wealth $X_0=x\geq 0$.
The consumption-portfolio pair $(c, \pi)$ is said to be *admissible*, denoted by $(c, \pi)$ $\in$ $\mathcal{A}(x)$, if the consumption rate maintains non-negative, i.e. $c_t\geq 0$ a.s. for all $t\geq 0$ and both $c$ and $\pi$ are $\mathbb{F}$-progressively measurable and satisfy the integrability condition $\int_0^{\infty} (c_t+\pi_t^2)dt<\infty$ a.s. Moreover, no bankruptcy of the investor is allowed in the sense that $X_t\geq 0$ a.s. for $t\geq 0$.
Let us focus on the exponential utility preference $U(x)=-\frac{1}{\beta}e^{-\beta x}$ in the present paper with $\beta>0$, $x$ $\in$ $\mathbb{R}$. We are interested in the following infinite time utility maximization defined on the difference between the current consumption rate and its historical running maximum that $$\begin{aligned}
\label{primalvalue}
u(x, h)=\sup_{(\pi, c)\in\mathcal{A}(x)}\mathbb{E}\left[\int_0^{\infty} e^{-\rho t}U(c_t-\lambda H_t)dt\right],\end{aligned}$$ where we define $$H_t=\max{\{h,\ \ \sup_{s\leq t} c_s\}},\ \ H_0=h\geq 0,\nonumber$$ and the proportional constant $0\leq \lambda\leq 1$ depicts the intensity towards the reference level that the investor adheres to the past spending pattern. Moreover, as we are working in an infinite horizon framework with constant interest rate, it is assumed in this paper that the discount rate $\rho=r$ to simplify some future computations. Here, $H_0=h\geq 0$ describes the reference level of the consumption that the individual aims to surpass at the initial time.
It is worth noting that the exponential utility function that we choose is defined on the whole real line, which implies that the current consumption rate is actually allowed to fall below the reference process $H$. To achieve the value function, it is not necessary for the optimal consumption control to exceed the reference level at any time. The lifetime average of the outperformance between the consumption and the reference level plays the key role. Meanwhile, as $c_t-\lambda H_t$ can be negative sometimes, the non-negativity constraint $c_t\geq 0$ a.s. is actively enforced for all time $t\geq 0$ so that $c_t$ represents the consumption rate in the conventional sense. This control constraint spurs some new mathematical challenges when we handle the associated HJB equation using dynamic programming arguments in subsequent sections.
\[comparepref\] The problem stems from some psychological consumption behavior that the investor sometimes can be very sensitive to the deviation from the past consumption pattern. In particular, the investor may have a strong memory of the past large amount of expenditures such as to purchase a large house or a fancy car, which may overturn the investor’s living environment and future budget plan. In response to the psychological consistency on consumption stream, the investor can be prone to aggressively catch up with the past consumption peak to some extent, which motivates us to consider the preference that measures the distance between the current consumption choice and a proportion of the historical maximum level.
The problem in the extreme case $\lambda=0$ is reduced to the standard Merton problem on consumption rate under exponential utility with non-negativity constraint that has been studied in [@CoxHuang] in the complete market model. In particular, to handle the constraint that the consumption rate is non-negative, [@CoxHuang] applied the martingale method and formulated the control problem with constraint into a relaxed form by introducing the Lagrange multipliers. By using the dual representation, the optimal non-negative consumption can be expressed in a technical integral form involving the unique state price density process. In the same complete market framework, this martingale method has been further refined by [@DepKart] to study the optimal consumption under non-addictive habit formation when the utility is generated by the difference between the non-negative consumption rate and the accumulative integral of past consumption control. By introducing the adjusted state price density process and the stochastic Lagrange multiplier process, the duality gap can be closed and the optimal non-negative consumption can be constructed and verified using the dual representation. Contrary to [@CoxHuang] and [@DepKart], the presence of consumption running maximum inside the utility invalidates the martingale method because it becomes very complicated to construct the adjusted dual process and dual problem.
On the other hand, the problem in the extreme case $\lambda=1$ is related to the so-called ratcheting consumption behavior studied in the seminal paper [@Dyb] and several subsequent work. In [@Dyb] with power utility function, the ratcheting constraint that the consumption rate is non-decreasing, i.e. $c_t\geq\sup_{s\leq t} c_s$ is mandated purely in the definition of admissible strategies while its utility function is defined on the consumption rate in a conventional way. In the same framework with power utility, [@Arun] further generalizes the ratcheting constraint in [@Dyb] to a drawdown type constraint on consumption in the sense that $c_t\geq \lambda\sup_{s\leq t} c_s$ for some $\lambda\in [0,1]$. Our present formulation differs substantially from [@Dyb] and [@Arun]. The outperformed difference between the current consumption rate and a fraction of the benchmark process is now chosen as the metric to measure the satisfaction of the investor. Moreover, we choose to work with exponential utility instead of power utility so that the ratcheting or drawdown constraint is no longer strictly enforced. The investor can strategically suppress the current consumption to some subsistence level that is below the reference level sometimes, which in turn may benefit the investor to attain a larger future consumption rate that is beyond the reference level with a higher probability and longer time periods.
Main Results {#sec:solveHJB}
============
Heuristic derivation of the HJB equation
----------------------------------------
To embed the control problem into the Markovian framework and derive the associated HJB equation using dynamic programming arguments, we treat both $X_t$ and $H_t$ as the controlled state processes given the control policy $(c,\pi)$. The value function $u(x,h)$ becomes two dimensional depending on variables $x\geq 0$ and $h\geq 0$, namely the initial wealth and the initial reference level for consumption. Let us consider the process $$\label{supmartpro}
\Gamma_t:=e^{-r t} u(X_t, H_t)+\int_0^t e^{-r s}U(c_s-\lambda H_s)ds.$$ The martingale optimality principle implies that $(\Gamma_t)_{t\geq 0}$ is a local supermartingale under all admissible controls and $(\Gamma_t)_{t\geq 0}$ is a local martingale given the optimal control (if it exists).
If the function $u(x,h)$ is smooth enough, by applying Itô’s formula to the process $(\Gamma_t)_{t\geq 0}$, we can derive that $$\begin{aligned}
e^{r t}d\Gamma_t=\left[ -r u+u_x(rX_t+\pi(\mu-r)-c_t)+\frac{1}{2}\sigma^2\pi^2 u_{xx}+U(c_t- \lambda H_t) \right]dt+u_hdH_t+u_x\pi \sigma dW_t,\end{aligned}$$ which heuristically leads to the associated HJB variational inequality $$\label{HJB_eqn}
\left\{
\begin{array}{rcl}
\underset{c\in [0,h], \pi\in\mathbb{R}}{\sup}\left[ -r u+u_x(rx+\pi(\mu-r)-c)+\frac{1}{2}\sigma^2\pi^2 u_{xx}-\frac{1}{\beta}e^{\beta(\lambda h-c)}\right] &=& 0, \\
u_h(x,h) & \leq & 0,
\end{array}
\right.$$ for $x\geq 0$, $h \geq 0$. To guarantee the local martingale property of $u(X^*_t, H^*_t)$ under the optimal portfolio $\pi^*_t$ and consumption control $c^*_t$, we have to require that $u_h(X^*_t, H^*_t)=0$ whenever $H_t^*$ increases for some $\omega$, i.e., the current consumption rate $c_t^*$ creates the new historical maximum level that $H_t^*=c_t^*$ and $c_t^*>H_s^*$ for $s<t$. This motivates us to mandate an important free boundary condition that $u_h(x,h)=0$ on some set of $(x,h)$ that will be determined explicitly later in in the section when we derive and analyze the associated HJB equation.
In the present paper, we aim to find some deterministic functions $\pi^{\ast}(x,h)$ and $c^{\ast}(x,h)$ to provide the feedback form of the optimal portfolio and consumption strategy. To this end, if $u(x,\cdot)$ is $C^2$ w.r.t the variable $x$, the first order condition gives the optimal portfolio in a feedback form by $\pi^{\ast}(x,h)=-\frac{\mu-r}{\sigma^2}\frac{u_x}{u_{xx}}$. The previous HJB variational inequality can first be written as $$\label{ODE}
\sup_{c\in [0,h]} \left[ -\frac{1}{\beta}e^{\beta(\lambda h-c)} - c u_x \right] -r u + rx u_x -\frac{\kappa^2}{2}\frac{u_x^2}{u_{xx}} = 0,\ \ \text{and}\ \ u_h\leq 0,\ \ \forall x\geq0,h\geq 0,$$ together with the free boundary condition $u_h=0$ on some set of $(x,h)\in\mathbb{R}_+\times\mathbb{R}_+$ that will be characterized later. To handle the control constraint $0\leq c\leq h$, we consider two extreme cases that $\lambda=0$ and $\lambda=1$ and the more interesting case $0<\lambda<1$ respectively in the subsequent subsections.
The case $\lambda=0$
--------------------
To tackle the HJB equation , let us first consider the extreme case without the reference to its historical maximum, i.e. $\lambda=0$. Recall that this case corresponds to the standard optimal consumption under exponential utility with non-negativity constraint that has been studied in [@CoxHuang]. Instead of using the martingale method as in [@CoxHuang], we provide the solution in a more explicit manner based on the analysis of the HJB equation.
In this case, the value function $u$ actually does not depend on $h$, and we can simply write it as $u(x)$. The free boundary condition can be ignored and some results in this extreme case will be used later in the problem when $\lambda>0$.
As $\lambda=0$, the HJB variational inequality can be simplified into a standard ODE problem without worrying about $u_h(x,h)\leq 0$. The first order condition without the non-negativity constraint gives the auxiliary feedback control $\hat{c}(x):=- \frac{1}{\beta}\ln u_x$, and we need to distinguish two cases based on the value of $\hat{c}(x)$ as below.\
\
*Region I:* on the set $\{x\in \mathbb{R}_+: u_x(x) \geq 1 \}$, we have $\hat{c}(x)\leq 0$. The optimal consumption is therefore $c^*(x)=0$ and the ODE is simplified to $$\label{ODE1}
-\frac{1}{\beta}-r u + rx u_x - \frac{\kappa^2}{2}\frac{u_x^2}{u_{xx}} = 0.$$ \
*Region II:* on the set $\{x\in \mathbb{R}_+: u_x(x)< 1 \}$, we have $\hat{c}(x)>0$. The optimal consumption is then $c^*(x)=-\frac{1}{\beta}\ln u_x> 0$ and the ODE is written as $$\label{ODE2}
-\frac{1}{\beta}u_x+ \frac{1}{\beta}u_x \ln u_x - r u + rx u_x -\frac{\kappa^2}{2}\frac{u_x^2}{u_{xx}} = 0.$$
To guarantee the global regularity of the solution, we need to impose the smooth-fit condition along the free boundary $\{x\in \mathbb{R}_+: u_x(x)=1\}$. Moreover, with the aid of some boundary conditions at $x=0$, we can actually determine its solution explicitly. To be precise, we observe that as the wealth level $x$ declines to zero, the consumption rate $c$ will first turn to zero at some point $x^*$(to be determined later), then when $x$ continues to tend to $0$, the optimal investment $\pi$ should also go to $0$. Otherwise, we will confront the risk of bankruptcy by keeping trading with the nearly $0$ wealth. Using the optimal portfolio $\pi^{\ast}(x)=-\frac{\mu-r}{\sigma^2}\frac{u_x}{u_{xx}}$, the boundary condition becomes $$\begin{aligned}
\label{bound0-1}
\lim_{x\rightarrow 0}\frac{u_x(x)}{u_{xx}(x)}=0.\end{aligned}$$ In addition, note that if we start with $0$ initial wealth, the wealth level will never change as there is no trading according to the previous condition, and the consumption rate should be $0$ all the time consequently. Therefore, we can conclude that $$\begin{aligned}
\label{bound0-2}
\lim_{x\rightarrow 0}u(x)= \int_0^{+\infty} - \frac{1}{\beta}e^0 e^{-r t} dt =-\frac{1}{r\beta}.\end{aligned}$$ On the other hand, as the wealth tends to infinite, one can consume as much as possible that leads to infinitely large admissible consumption rate and also a small variation in the wealth has the negligible effect on the change of the value function. It thus follows that $$\begin{aligned}
\label{boundinf}
\lim_{x\rightarrow +\infty}u(x)=0\ \ \text{and}\ \ \lim_{x\rightarrow +\infty}u_x(x)= 0.\end{aligned}$$
To handle the nonlinear terms in the HJB equation and , we employ the dual transform of the function $u(x)$ that is defined by $v(y) := \sup_{x \geq 0} (u(x)-xy)$, $y>0$. For the given $x$, we consider the variable $y:=u_x(x)$ and it holds that $u(x)=v(y)+xy$. We can further deduce that $$x = - v_y(y),\ \ u(x)=v(y)-yv_y(y)\ \ \text{and}\ \ u_{xx}(x) = - \frac{1}{v_{yy}(y)}.$$ The nonlinear ODE and can be linearized as $$\label{EDPtildeU1}
\frac{\kappa^2}{2} y^2 v_{yy} - r v =\left\{
\begin{aligned}
&\frac{1}{\beta}, & & \mbox{if } y \geq 1,\\
&\frac{1}{\beta}y -\frac{1}{\beta} y \ln y, & & \mbox{if } y < 1,
\end{aligned}
\right.$$ and the free boundary condition is transformed to the point $y=1$. Note that $y \geq 1$ corresponds to the primal region $c^*(x)=0$ and $y <1$ corresponds to the primal region $c^*(x)>0$.
Based on the dual transform, we can translate the boundary condition to $$\begin{aligned}
\label{dualcond-1}
\lim_{y\rightarrow 0}v_y(y)= -\infty\ \ \text{and}\ \ \lim_{y \rightarrow 0} (v(y)-yv_y(y))=0.\end{aligned}$$
Using the duality transform again, the boundary conditions and at $x=0$ can be reformulated into free boundary conditions that $$\begin{aligned}
\label{dualcond-2}
yv_{yy}(y) \rightarrow 0\ \text{and}\ \ v(y)-yv_y(y) \rightarrow -\frac{1}{r\beta}\ \ \text{as}\ \ v_y(y) \rightarrow 0.\end{aligned}$$
The next result gives the explicit solution to the dual ODE problem and its proof is given in Section \[otherproofs\].
\[0lambda\] Given the boundary conditions in , the free boundary conditions in and also the smooth-fit condition at $y=1$, The ODE admits the unique solution given explicitly by $$\begin{aligned}
v(y) =\left\{
\begin{aligned}
&C_2 y^{r_2} - \frac{1}{r\beta}, & & \mbox{if } y \geq1,\\
&C_3 y^{r_1} + \frac{y}{r\beta}(\ln y + \frac{\kappa^2}{2r} - 1), & & \mbox{if } y < 1,
\end{aligned}
\right.\end{aligned}$$ where constants $C_2$ and $C_3$ are given by $$\begin{aligned}
\label{c14}
\left\{
\begin{aligned}
&C_2 := \frac{r_1-1}{r_1-r_2}\frac{\kappa^2}{2r^2\beta}>0,\\
&C_3 := \frac{r_2-1}{r_1-r_2}\frac{\kappa^2}{2r^2\beta}<0,
\end{aligned}
\right.\end{aligned}$$ in which the constants $r_1>1$ and $r_2<0$ are two roots of the algebraic equation $$z^2 - z - \frac{2r}{\kappa^2} =0,$$ which are given by $$\begin{aligned}
\label{def-r12}
r_{1,2}=\frac{1}{2} \Big( 1 \pm \sqrt{1+\frac{8 r}{\kappa^2}} ~ \Big).\end{aligned}$$
By using the dual value function $v(y)$ in Proposition \[0lambda\], the optimal consumption and investment $c^*$ and $\pi^*$ can be expressed in terms of the dual value function and dual variable in the feedback form for $y>0$ in the next main result.
\[verify-0\] Let $x\geq 0$ be the initial wealth. We consider the process $Y_t:=y^*e^{r t} M_t$, where $M_t:= e^{-(r+\frac{\kappa^2}{2})t - \kappa W_t}$ is the discounted state price density process, where $y^*=y^*(x)$ is the unique solution to the budget constraint $\mathbb{E}[\int_0^{\infty}c^*(Y_t)M_tdt]=x$. The optimal consumption $c^*_t=c^*(Y_t)$ and portfolio $\pi^*_t=\pi^*(Y_t)$ in the problem for $\lambda=0$ are given by $$\begin{aligned}
\pi^*(y)&=\frac{\mu-r}{\sigma^2}yv_{yy}(y)=\frac{\mu-r}{\sigma^2} \left\{
\begin{aligned}
&r_2(r_2-1)C_2y^{r_2-1}=\frac{2r}{\kappa^2}C_2y^{r_2-1}, & & \mbox{if } y \geq 1,\\
&r_1(r_1-1)C_3y^{r_1-1}+\frac{1}{r\beta}=\frac{2r}{\kappa^2}C_3y^{r_1-1}+\frac{1}{r\beta}, & & \mbox{if } y< 1,
\end{aligned}
\right.
\ \\
c^*(y)&=\left\{
\begin{aligned}
&0, & & \mbox{if } y \geq 1, \\
&-\frac{1}{\beta} \ln y, & & \mbox{if } y< 1,
\end{aligned}
\right.\end{aligned}$$ where we used the fact that $r_1(r_1-1)=r_2(r_2-1)=\frac{2r}{\kappa^2}$ from . Note that as it is assumed that $\mu>r$, we always have $\pi^*(y)>0$ be the definition of $C_2$ and $C_3$.
We actually can further rewrite the optimal controls in terms of the primal variables using the inverse transform. To this end, let us denote $k(x):=u'(x)$ and the duality relationship implies that $u(x)=v(k(x))+xk(x)$. Moreover, we know that $k(x)$ will have two different expressions $k_1(x)$ and $k_2(x)$ depending on $x$. We therefore can obtain that $$\begin{aligned}
x=\left\{
\begin{aligned}
&-C_2r_2(k_2(x))^{r_2-1}, & & \mbox{if } k_2(x) \geq 1, \\
&-C_3r_1(k_1(x))^{r_1-1}-\frac{1}{r\beta}\left(\ln k_1(x)+\frac{\kappa^2}{2r} \right), & & \mbox{if } k_1(x)< 1.
\end{aligned}
\right.\end{aligned}$$ Therefore, we can obtain the free boundary point $x^*$ that $$x^* = -C_2 r_2>0,$$ with $C_2$ given in . For $x>-C_2r_2$, the function $k_1(x)$ is uniquely defined by the implicit equation that $x=-C_3r_1(k_1(x))^{r_1-1}-\frac{1}{r\beta}\left(\ln k_1(x)+\frac{\kappa^2}{2r}\right)$ because the function $G(y)=-C_3r_1y^{r_1-1}-\frac{1}{r\beta}\left(\ln y+\frac{\kappa^2}{2r}\right)$ is decreasing and $\lim_{y\rightarrow 0}G(y)=+\infty$ and $\lim_{y\rightarrow 1}G(y)=-C_3r_1-\frac{\kappa^2}{2r^2\beta}=-C_2r_2$. When $x\leq -C_2r_2$, we obtain that $k_2(x)=\left(-\frac{x}{C_2r_2}\right)^{\frac{1}{r_2-1}}$. The next result follows directly from Theorem \[verify-0\] and the arguments above.
\[veri-2-0\] For the initial wealth $x\geq 0$, we can express the value function and feedback optimal consumption and portfolio by: $$\begin{aligned}
\label{primal0lambda}
u(x)=\left\{
\begin{aligned}
&C_2\left(-\frac{x}{C_2r_2}\right)^{\frac{r_2}{r_2-1}}-\frac{1}{r\beta}+x\left(-\frac{x}{C_2r_2}\right)^{\frac{1}{r_2-1}}, & & \mbox{if } x \leq -C_2 r_2, \\
&C_3(k_1(x))^{r_1}+\frac{k_1(x)}{r\beta}\left[\ln k_1(x)+\frac{\kappa^2}{2r}-1+xr\beta\right], & & \mbox{if } x > -C_2 r_2.
\end{aligned}
\right.\end{aligned}$$ The optimal strategy $c^*$ and $\pi^*$ can therefore be written in the feedback form using $x\geq 0$ by $$\begin{aligned}
\pi^*(x)&=\frac{\mu-r}{\sigma^2} \left\{
\begin{aligned}
&(1-r_2)x,& & \mbox{if } x \leq -C_2 r_2, \\
&\frac{2r}{\kappa^2}C_3k_1^{r_1-1}(x)+\frac{1}{r\beta},& & \mbox{if } x > -C_2 r_2,\label{zeropi}
\end{aligned}
\right.\ \\
c^*(x)&=\left\{
\begin{aligned}
&0, & & \mbox{if } x \leq -C_2 r_2, \\
&-\frac{1}{\beta}\ln k_1(x), & & \mbox{if } x > -C_2 r_2.\label{zerocon}
\end{aligned}
\right.\end{aligned}$$
Here, we choose market parameters $r = 0.05$, $\mu = 0.1$, $\sigma = 0.25$ and $\beta=1$ and graph the value function $u(x)$ in Figure-1, the optimal consumption $c^*(x)$ in Figure-2, and the optimal investment $\pi^*(x)$ in Figure-3. In particular, we use the vertical dot line to highlight the free boundary point $x=-C_2r_2$ in all figures to separate the domain of $x$.
$$\begin{array}{ccc}
\hspace{-0.3in}
\includegraphics[height=1.75in]{Figure01.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure02.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure03.eps} \\
\mbox{\footnotesize{Figure 1}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 2}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 3}}
\end{array}$$
The case $0<\lambda<1$
----------------------
We next consider the original control problem binding with the reference to the historical consumption peak when $0<\lambda<1$ and the non-negativity constraint $c_t\geq 0$ is enforced. In view of the constraints that $0\leq c_t\leq H_t$, we first need to decompose the domain $(x,h)\in\mathbb{R}_+\times \mathbb{R}_+$ into three different regions such that the feedback optimal consumption strategy satisfies: (1) $c^*(x,h)=0$; (2) $0<c^*(x,h)<h$; (3) $c^*(x,h)=h$. Let us denote the auxiliary control $\hat c(x,h):= - \frac{1}{\beta}\ln u_x + \lambda h$, which is simply derived by the first order condition in the HJB equation . We need to separate the following regions:
*Region I*: on the set $\mathcal{R}_1:=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+:u_x(x,h)\geq e^{\lambda \beta h} \right\}$, we have $\hat c(x,h) \leq 0$, and therefore the optimal consumption rate is $c^*(x,h)=0$ and the HJB variational inequality becomes $$\label{veq-1}
-\frac{1}{\beta}e^{\lambda \beta h} -r u+ rx u_x - \frac{\kappa^2 u_x^2}{2 u_{xx}} = 0, \ \text{and}\ \ u_h \leq 0.$$ *Region II*: on the set $\mathcal{R}_2:=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+:e^{-(1-\lambda)\beta h} < u_x(x,h)< e^{\lambda \beta h} \right\}$, we have that $0 < \hat c(x,h) < h$, and therefore the optimal consumption rate is $c^* = -\frac{1}{\beta} \ln u_x + \lambda h$. The HJB variational inequality can be written as $$\label{veq-2}
-\frac{1}{\beta}u_x+ u_x (\frac{1}{\beta}\ln u_x - \lambda h) -r u + rx u_x -\frac{\kappa^2 u_x^2}{2 u_{xx}} = 0,\ \text{and}\ \ u_h \leq 0.$$
\[comparec-lambdah\] As pointed out in Remark \[comparepref\], the main reason for us to consider the exponential utility resides in the flexibility that the optimal consumption $c^*$ can fall below the reference level $\lambda H^*$, which matches better with the real life situation that the investor can bear unfulfilling consumption during the economic recession periods. Based on the feedback form of the optimal consumption $c^* = -\frac{1}{\beta} \ln u_x + \lambda h$ in Region II, we can characterize the domain of $(x,h)$ such that the investor lowers the consumption rate below the reference, i.e. $c_t^*<\lambda H_t^*$ if and only if $(x,h)$ is in the subset $\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+:1<u_x(x,h)< e^{\lambda \beta h}\right\}$. This subset will be further characterized explicitly in Remark \[discussthresh\] as a threshold (depending on $h$) of the wealth level $x$.
*Region III*: on the set $\mathcal{R}_3:=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+:u_x(x,h) \leq e^{-(1-\lambda)\beta h} \right\}$, we have $\hat c(x,h) \geq h$ and the optimal consumption rate is $c^*(x,h)= h$ that implies the instant consumption rate $c_t^*$ coincides with the running maximum process $H_t^*$. However, two subtle cases may occur that motivate us to split this region further.
- In a certain region (to be determined), the historical maximum level is already attained at some previous time $s$ before time $t$ and the current optimal consumption rate is either to revisit this maximum level from below or to sit on the same maximum level. This is the case that the running maximum process $H_t$ keeps flat from time $s$ to time $t$, and the feedback form of $c_t^*=H_t^*=c_s^*$ for some time $s<t$. In this case, it is very natural to treat $H_t^*$ as the state process and plug it to the feedback form $c^*(x,h)=h$.
- In the complementary region, the optimal consumption rate creates a new record of the maximum level that is strictly larger than its past consumption, and the running maximum process $H_t$ is strictly increasing at that instant time $t$. This corresponds to the case that $c_t^*=H_t^*$ is a singular control and $c_t^*>H_s^*$ for $s<t$ and we have to mandate the free boundary condition $u_h(x,h)=0$ from the martingale optimality condition. In this region, the feedback form $c^*(x,h)=h$ is useless because $H_t^*$ is updated by $c_t^*$ itself, which can not provide any effective information.
Restricted to the set $\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+:u_x(x,h)> e^{\lambda \beta h} \right\}$, the case $(ii)$ that $H_t^*$ increases and is updated by the singular control $c_t^*$ suggests us to treat the $H_t^*=c^*_t$ as a singular control instead of the state process. That is, the dimension of the problem can be reduced and we can first substitute $h=c$ in and then apply the first order condition to $-\frac{1}{\beta}e^{\beta(\lambda c - c)} - c u_x$ with respect to $c$. Under the condition that $\lambda<1$, we can obtain the auxiliary singular control $\hat{c}(x):=\frac{1}{\beta(\lambda-1)} \ln (\frac{u_x}{1-\lambda})$, which is the feedback form depending only on $X_t$. It then becomes convenient to see that $c_t^*$ can update $H_t^*$ to a new level if and only if the feedback control $c_t^*=\hat{c}(X_t^*)\geq H_t^*$ so that $H_t^*$ is instantly increasing. We can then separate Region III into three subsets:\
\
*Region III-(i)*: on the set $\mathcal{D}_1:=\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: (1-\lambda) e^{-(1-\lambda)\beta h}<u_x(x,h) \leq e^{-(1-\lambda)\beta h} \}$, we have a contradiction that $\hat{c}(x)<h$, and therefore $c^*_t$ is not a singular control. We still need to follow the previous feedback form $c^*(x,h) = h$, in which $h$ is a previously attained maximum level. The corresponding running maximum process remains flat at the instant time. In this region of $(x,h)$, we only know that $u_h(x,h)\leq 0$ as we have $dH_t=0$. The HJB variational inequality is written as $$\label{veq-3}
-\frac{1}{\beta}e^{\beta(\lambda h-h)} - h u_x -r u + rx u_x -\frac{\kappa^2 u_x^2}{2 u_{xx}} = 0,\ \text{and}\ \ u_h \leq 0.$$ \
*Region III-(ii)*: on the set $\mathcal{D}_2:=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: u_x(x,h)=(1-\lambda) e^{-(1-\lambda)\beta h}\right\}$, we get $\hat{c}(x)=h$ and the feedback optimal consumption is $c^*(x,h)=\frac{1}{\beta(\lambda-1)} \ln (\frac{u_x}{1-\lambda})=h$. This corresponds to the singular control $c_t^*$ that creates a new peak for the whole path and $H_t^*=c_t^*=\frac{1}{\beta(\lambda-1)} \ln (\frac{u_x(X_t^*,H_t^*)}{1-\lambda})$ is strictly increasing at the instant time so that $H_t^*>H_s^*$ for $s<t$ and we must require the following free boundary condition that $$\label{martingale_condition}
u_h(x,h)=0\ \ \text{on}\ \left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: u_x(x,h)=(1-\lambda) e^{-(1-\lambda)\beta h}\right\}.$$ In this region, it is noted that $c^*(x,h)=h=\frac{1}{\beta(\lambda-1)} \ln (\frac{u_x}{1-\lambda})$. Therefore, the HJB equation follows the same PDE but together with the new free boundary condition .\
\
*Region III-(iii)*: on the set $\mathcal{D}_3:=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: u_x(x,h)<(1-\lambda) e^{-(1-\lambda)\beta h}\right\}$, we get $\hat{c}(x)>h$. This indicates that the initial reference level $h$ is below the feedback control $\hat{c}(x)$, and the optimal consumption is again a singular control $c^*(x,h)=\frac{1}{\beta(\lambda-1)} \ln (\frac{u_x}{1-\lambda})$, which creates a new consumption peak. As the running maximum process $H_t^*$ is updated immediately by $c_t^*$, the feedback optimal consumption pulls the associated $H_{t-}^*$ upward from its original value to the new value $\frac{1}{\beta(\lambda-1)} \ln (\frac{u_x(X_t^*, H_t^*)} {1-\lambda})$ in the direction of $h$ and $X_t^*$ remains the same, in which $u(x,h)$ is the solution of the HJB equation on the set $\mathcal{D}_2$. This suggests that for any given initial value $(x,h)$ in the set $\mathcal{D}_3$, the feedback control $c^*(x,h)$ pushes the value function jumping immediately to the point $(x, \hat{h})$ on the boundary set $\mathcal{D}_2$ where $\hat{h}=\frac{1}{\beta(\lambda-1)} \ln (\frac{u_x(x,\hat{h})}{1-\lambda})$ for the given level of $x$. Therefore, it is sufficient for us to only concentrate $(x,h)$ on the *effective domain* of the original stochastic control problem that $$\begin{aligned}
\label{primedomain}
\mathcal{C}:=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: u_x(x,h)\geq (1-\lambda) e^{-(1-\lambda)\beta h}\right\},\end{aligned}$$ equivalently $\mathcal{C}=\mathcal{R}_1\cup\mathcal{R}_2\cup\mathcal{D}_1\cup\mathcal{D}_2\subset \mathbb{R}_+^2$. The only possibility for $(x,h)\in \mathcal{D}_3=\mathcal{C}^c$ occurs at the initial time $t=0$, and the value function is just equivalent to the value function of $(x,\hat{h})$ on the boundary $\mathcal{D}_2$ with the same $x$. In other words, if the controlled process $(X_0^*, H_0^*)$ starts from $(x,h)$ in the region $\mathcal{C}$, then $(X_t^*, H_t^*)$ will always stay inside the region $\mathcal{C}$ and will either reflect at the boundary or move along the boundary $\mathcal{D}_2$ whenever it hits the boundary $\mathcal{D}_2$ (but will never go across the boundary). On the other hand, if the process $(X_0^*, H_0^*)$ starts from the value $(x,h)$ inside the region $\mathcal{D}_3$, the optimal control enforces an instant jump (and the only jump) of the process $H$ from $H_{0-}=h$ to $H_0=\hat{h}$ on the boundary $\mathcal{D}_2$ and both processes $X_t$ and $H_t$ become continuous processes diffusing inside the effective domain $\mathcal{C}$ afterwards for $t>0$.
In addition, to ensure the desired global regularity of the solution, we also need to impose the smooth-fit conditions along two free boundaries of $(x,h)$ such that $u_x(x,h)= e^{\lambda \beta h}$, $u_x(x,h) = e^{-(1-\lambda)\beta h}$, which separate the different regions that we discussed above.
Similar to the case when $\lambda=0$, we can again employ the dual transform of the value function to linearize the HJB equation. In particular, we choose the dual transform only with respect to the variable $x$ and treat the variable $h$ as a parameter. Let $h\geq 0$ be fixed, we consider $x\geq 0$ such that $(x,h)\in\mathcal{C}$ and define the dual function on the domain $y\geq (1-\lambda) e^{-(1-\lambda)\beta h} $ that $$\begin{aligned}
v(y,h) & := \sup_{\substack{(x,h)\in\mathcal{C},\\ x\geq 0}} [ u(x,h) - xy],\ \ y\geq (1-\lambda) e^{-(1-\lambda)\beta h} .\end{aligned}$$
For the given $(x,h)$, let us define $\hat{y}(x,h):=u_x(x,h)$ (short as $\hat{y}$), the dual representation implies $u(x,h)=v(\hat{y}, h)+x\hat{y}$ as well as $v_y(\hat{y}, h)=-x$. We then have $$\begin{aligned}
u_h(x,h)=\frac{\partial}{\partial h}(v(\hat{y},h)+x\hat{y})=v_h(\hat{y},h)+(v_y(\hat{y},h)+x)\frac{d\hat{y}}{dh}=v_h(\hat{y},h).\end{aligned}$$ In view of the free boundary condition , we obtain the boundary condition $$\begin{aligned}
\label{freedul}
v_h(y,h)=0\ \ \text{on the set}\ \left\{(y,h)\in(0,+\infty)\times \mathbb{R}_+: y=(1-\lambda)e^{(\lambda-1)\beta h}\right\}.\end{aligned}$$
To align with nonlinear HJB variational inequality , , in three different regions, the transformed dual variational inequality can be written as $$\label{EDPtildeU2}
\frac{\kappa^2}{2} y^2 v_{yy} - r v =\left\{
\begin{aligned}
& \frac{1}{\beta}e^{\lambda \beta h}, & & \mbox{if } y \geq e^{\lambda \beta h}, \\
& \frac{1}{\beta}y - y \left(\frac{1}{\beta} \ln y - \lambda h\right), & & \mbox{if } e^{(\lambda-1)\beta h} < y < e^{\lambda \beta h}, \\
& \frac{1}{\beta}e^{(\lambda-1)\beta h} + hy, & & \mbox{if } (1-\lambda)e^{(\lambda-1)\beta h} \leq y \leq e^{(\lambda-1)\beta h},
\end{aligned}
\right.$$ together with the free boundary condition . As we regard $h$ as a parameter from this point onwards, we can fix $h$ and study the above equation as the ODE problem of the variable $y$.
Similar to the case when $\lambda=0$, after the dual transform, the boundary condition gives $$\begin{aligned}
\label{yhcond-1}
\lim_{y\rightarrow 0}v_y(y,h)= -\infty\ \ \text{and}\ \ \lim_{y \rightarrow 0} (v(y,h)-yv_y(y,h))=0,\end{aligned}$$ and the boundary conditions and at $x=0$ is equivalent to $$\begin{aligned}
\label{yhcond-2}
yv_{yy}(y,h) \rightarrow 0\ \text{and}\ \ v(y,h)-yv_y(y,h) \rightarrow -\frac{1}{r\beta}e^{-\lambda \beta h}\ \ \text{as}\ \ v_y(y,h) \rightarrow 0.\end{aligned}$$
By using the previous conditions, we can solve the dual ODE fully explicitly and its proof is provided in Section \[otherproofs\].
\[dual\_value\_function\] Let $h\geq 0$ be a given parameter. Given the boundary conditions in , free boundary conditions and free boundary condition , the smooth-fit conditions with respect to $y$ at free boundary points $y=e^{\lambda\beta h }$ and $y=e^{(\lambda-1)\beta h}$, the ODE in the domain $y\geq (1-\lambda)e^{(\lambda-1)\beta h}$ admits the unique solution given explicitly by $$\begin{aligned}
\label{part2-sol}
v(y,h) =\left\{
\begin{aligned}
& C_2(h) y^{r_2} - \frac{1}{r\beta} e^{\lambda \beta h}, & & \mbox{if } y \geq e^{\lambda \beta h}, \\
& C_3(h) y^{r_1} + C_4(h) y^{r_2}-\frac{y}{r\beta}+\frac{y}{r\beta } \left(\ln y-\lambda\beta h+\frac{\kappa^2}{2r}\right), & & \mbox{if } e^{(\lambda-1)\beta h} < y < e^{\lambda \beta h}, \\
& C_5(h) y^{r_1} + C_6(h) y^{r_2} -\frac{1}{r}hy-\frac{1}{r\beta} e^{(\lambda-1)\beta h}, & & \mbox{if } (1-\lambda)e^{(\lambda-1)\beta h} \leq y \leq e^{(\lambda-1)\beta h},
\end{aligned}
\right.\end{aligned}$$ where functions $C_2(h)$, $C_3(h)$, $C_4(h)$, $C_5(h)$ and $C_6(h)$ are given explicitly in , , , and respectively that $$\begin{aligned}
C_2(h) := & \frac{(1-\lambda)^{r_1-r_2}(r_2-1)\kappa^2}{2(r_1-r_2)\beta r^2} \left[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h} -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \right]\notag\\
&+ \frac{(1-r_1)\kappa^2}{2(r_1-r_2)\beta r^2}\left[e^{(\lambda-1)(1-r_2)\beta h}-e^{\lambda(1-r_2)\beta h} \right];\label{C2h}\\
C_3(h):=&\frac{(r_2-1) \kappa^2}{2(r_1-r_2)\beta r^2}e^{\lambda(1-r_1)\beta h}\label{C3h};\\
C_4(h) := & \frac{(1-\lambda)^{r_1-r_2}(r_2-1)\kappa^2}{2(r_1-r_2)\beta r^2} \left[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h} -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \right]\notag\\
&+ \frac{(1-r_1)\kappa^2}{2(r_1-r_2)\beta r^2}e^{(\lambda-1)(1-r_2)\beta h}\label{C4h};\\
C_5(h) :=& \frac{(1-r_2) \kappa^2}{2(r_1-r_2)\beta r^2} \left[e^{(\lambda-1)(1-r_1)\beta h}- e^{\lambda (1-r_1)\beta h}\right];\label{C5h}\\
C_6(h):= & \frac{(1-\lambda)^{r_1-r_2}(r_2-1)\kappa^2}{2(r_1-r_2)\beta r^2} \left[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h} -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \right].\label{C6h}\end{aligned}$$ Here, constants $r_{1,2}$ are given previously in .
\[remarkord\] Based on explicit forms in , and , let us note the following asymptotic results of the coefficients that $$\begin{aligned}
& C_2(h) = O \left( e^{(\lambda-1)(1-r_2)\beta h} \right) + O \left( e^{[\lambda(1-r_2)-(r_1-r_2)]\beta h} \right) + O\left( e^{\lambda (1-r_2)\beta h} \right),\\
& C_3(h) = O \left( e^{\lambda\beta h (1-r_1)} \right), \\
& C_4(h) = O \left( e^{(\lambda-1)(1-r_2)\beta h} \right) + O \left( e^{[\lambda(1-r_2)-(r_1-r_2)]\beta h} \right),
\end{aligned}$$ which will be used in later proofs.
We can now present the main result of this paper, which provides the optimal investment and consumption in the feedback form explicitly using the dual variables for $0<\lambda<1$. The complete proof is deferred to Section \[sec:verification\].
\[verthm\] Let $(x,h)\in\mathcal{C}$ and $0<\lambda<1$, where $x$ is the initial wealth and $h\geq 0$ is the initial reference level and $\mathcal{C}$ stands for the effective domain . We consider the process $Y_t:=y^*e^{r t} M_t$, where $M_t:= e^{-(r+\frac{\kappa^2}{2})t - \kappa W_t}$ is the discounted state price density process and $H_t^*=h\lor \sup_{s \leq t} c^*(Y_s,H_s^*)$ is the reference process under the optimal control, and the constant $y^*=y^*(x,h)$ is the unique solution to the budget constraint $\E \left[\int_0^{\infty} c^*(Y_t,H_t^*) M_t dt \right]=x$. The value function $u(x,h)$ can be attained by employing the optimal consumption and portfolio strategies in the feedback form that $c_t^*=c^*(Y_t,H_t^*)$ and $\pi_t^*=\pi^*(Y_t,H_t^*)$, $t\geq 0$, which are given by:\
\
$$\begin{aligned}
\label{feedbackcp}
c^*(y,h) =\left\{
\begin{aligned}
&0, & & \mbox{if } y \geq e^{\lambda \beta h}, \\
&- \frac{1}{\beta}\ln y + \lambda h, & & \mbox{if } e^{(\lambda-1)\beta h} < y < e^{\lambda \beta h}, \\
&h, & & \mbox{if } (1-\lambda)e^{(\lambda-1)\beta h} <y \leq e^{(\lambda-1)\beta h}, \\
&\frac{1}{(\lambda-1)\beta} \ln \Big(\frac{1}{1-\lambda} y\Big), & & \mbox{if } y = (1-\lambda)e^{(\lambda-1)\beta h},
\end{aligned}
\right.\end{aligned}$$
$$\begin{aligned}
\label{feedbackpi}
\begin{aligned}
&\pi^*(y,h) = \frac{\mu-r}{\sigma^2}yv_{yy}(y,h) \\
= & \frac{\mu-r}{\sigma^2}\left\{
\begin{aligned}
& \frac{2r}{\kappa^2}C_2(h) y^{r_2-1}, & & \mbox{if } y \geq e^{\lambda \beta h}, \\
& \frac{2r}{\kappa^2}C_3(h) y^{r_1-1} + \frac{2r}{\kappa^2}C_4(h) y^{r_2-1}+ \frac{1}{r\beta}, & & \mbox{if } e^{(\lambda-1)\beta h} < y < e^{\lambda \beta h}, \\
& \frac{2r}{\kappa^2}C_5(h) y^{r_1-1} + \frac{2r}{\kappa^2}C_6(h) y^{r_2-1}, & & \mbox{if } (1-\lambda)e^{(\lambda-1)\beta h} \leq y \leq e^{(\lambda-1)\beta h},
\end{aligned}
\right.
\end{aligned}\end{aligned}$$
In particular, the running maximum process $H_t^*$ is strictly increasing such that $H_t^*=c_t^*>c_s^*$ for any time $s<t$ if and only if $Y_t=(1-\lambda)e^{(\lambda-1)\beta H_t^*}$ and its feedback optimal consumption is $c_t^*=\frac{1}{(\lambda-1)\beta} \ln \Big(\frac{1}{1-\lambda}Y_t\Big)$. If we have $y^*(x,h)<(1-\lambda)e^{(\lambda-1)\beta h}$ at the initial time, the optimal consumption creates a new peak and brings $H_{0-}^*=h$ jumping immediately to a higher level $H_0^*=\frac{1}{(\lambda-1)\beta} \ln \Big(\frac{1}{1-\lambda} y^*(x,h)\Big)$ such that $t=0$ becomes the only jump time of $H^*_t$. Moreover, for any initial data $(X_0^*, H_0^*) = (x, h) \in \mathcal{C}$, the stochastic differential equation $$\begin{aligned}
\label{wealthSDE}
d X_t^* = r X_t^* dt+ \pi^*_t(\mu -r)dt+ \pi^*_t\sigma dW_t- c_t^*dt\end{aligned}$$ has a unique strong solution given the optimal feedback control $(c^*, \pi^*)$ as above.
In Theorem \[verthm\], the feedback forms of the optimal investment and consumption are given explicitly in terms of the dual value function and the dual variables. We can also conduct the inverse dual transformation and express the primal value function $u(x,h)$ and the feedback controls in terms of $x$ and $h$, albeit in more complicated forms. In the main body of the proof of Theorem \[verthm\], we will take full advantage of the simplicity in the dual feedback formulas and verify the optimality of the feedback controls using the duality relationship and some estimations based on the dual process $Y_t=y^*e^{r t} M_t$. However, to show the existence of a unique strong solution of SDE , we have to derive the feedback controls in terms of $X_t^*$ and $H_t^*$ and the step of inverse dual transform becomes necessary, which will be established as follows.
By using the dual relationship between $u$ and $v$, we have that the optimal $$\label{sol_for_x}
x= g(\cdot, h) := -v_y(\cdot,h).$$ Defining $f(\cdot,h)$ as the inverse of $g(\cdot,h)$, we have that $$\label{dualrelationship}
u(x,h) = v \circ \left( f(x,h), h \right) + x f(x,h).$$ Note that $v$ has different expressions in the regions $c =0$, $0 < c < h$ and $c = h$, the function $f$ should also have three different expressions in these regions and we denote them respectively $f_1$, $f_2$ and $f_3$.
By the definition of $g$ in , the invertibility of the map $x \mapsto g(x,h)$ is guaranteed by the following important result and its proof is deferred to Section \[otherproofs\].
\[vyy\_positive\] In all three regions, we have that $v_{yy}(y, h) > 0$, $\forall h >0$ and the inverse Legendre transform $u(x,h)=\inf_{y\geq(1-\lambda)e^{-(1-\lambda)\beta h}} [v(y,h)+xy]$ is well defined. Moreover, it implies that the feedback optimal portfolio $\pi^*(y,h)>0$ always holds.
Using and Proposition \[dual\_value\_function\], the function $f$ is implicitly determined in different regions by the following equations:
- If $f_1(x,h) \geq e^{\lambda \beta h}$, $f_1(x,h)$ can be determined by $$\begin{aligned}
\label{f1-equt}
x=-C_2(h) r_2 (f_1(x,h))^{r_2-1}. \end{aligned}$$
- If $e^{(\lambda-1)\beta h} < f_2(x,h) < e^{\lambda \beta h}$, Lemma \[vyy\_positive\] implies that $v_y(y,h)$ is strictly increasing in $y$ and $f_2(x,h)$ can be uniquely determined by $$\begin{aligned}
\label{f2-equt}
x=-C_3(h) r_1 (f_2(x,h))^{r_1-1} - C_4(h) r_2 (f_2(x,h))^{r_2-1} - \frac{1}{r\beta } \left(\ln f_2(x,h)-\lambda\beta h +\frac{\kappa^2}{2r}\right).\end{aligned}$$
- If $(1-\lambda)e^{(\lambda-1)\beta h} \leq f_3(x,h) \leq e^{(\lambda-1)\beta h}$, Lemma \[vyy\_positive\] implies that $v_y(y,h)$ is strictly increasing in $y$ and $f_3(x,h)$ can be uniquely determined by $$\begin{aligned}
\label{f3-equt}
x= -C_5(h) r_1 (f_3(x,h))^{r_1-1} - C_6(h) r_2 (f_3(x,h))^{r_2-1} + \frac{h}{r}.\end{aligned}$$.
In region $\mathcal{R}_1$, we can obtain the explicit form of $f_1(x,h)=\left( \frac{-x}{C_2(h) r_2} \right)^{\frac{1}{r_2-1}}$. The condition $f_1(x,h) \geq e^{\lambda \beta h}$ gives us that this is valid when $x \leq x_1(h)$, where we define the free boundary by $$\begin{aligned}
\label{x1-def}
x_1(h):=-e^{\lambda \beta h (r_2-1)} C_2(h)r_2.\end{aligned}$$
In region $\mathcal{R}_2$, the function $f_2$ is uniquely determined implicitly by when $x_1(h) < x < x_2(h)$, where $x_2(h)$ is the solution of $$f_2(x,h) = e^{(\lambda-1)\beta h}.$$ In view of the definition of $f_2(x,h)$ in , we can obtain the free boundary point explicitly as $$\begin{aligned}
\label{x2-def}
x_2(h)=-C_3(h) r_1 e^{(\lambda-1)(r_1-1)\beta h} - C_4(h) r_2 e^{(\lambda-1)(r_2-1)\beta h} + \frac{h}{r}-\frac{\kappa^2}{2r^2\beta}.\end{aligned}$$
\[discussthresh\] In addition, as in Remark \[comparec-lambdah\], we know that the optimal consumption falls below the reference level if and only if $1<f_2(x,h)<e^{\lambda\beta h}$. Using again, we can determine the critical point $\breve{x}(h)$ by $$\begin{aligned}
\label{xbreve-def}
\breve{x}(h):=-C_3(h) r_1 - C_4(h) r_2 + \lambda\frac{h}{r}-\frac{\kappa^2}{2r^2\beta}.\end{aligned}$$ It then follows that if and only if the wealth level $x$ is sufficiently small that satisfies $x_1(h)<x< \breve{x}(h)$, the optimal consumption rate meets the compromised plan $0<c_t^*(x,h)<\lambda H_t^*(x,h)$.
In region $\mathcal{D}_1\cup\mathcal{D}_2$, the expression of $f_3$ is uniquely defined implicitly by the equation . This expression of $f_3$ holds when $x_2(h) \leq x \leq x_3(h)$, where $x_3$ is the solution of $$\begin{aligned}
f_3(x,h) = (1-\lambda) e^{(\lambda-1) \beta h}.\end{aligned}$$ It follows from that the free boundary point $x_3(h)$ is explicitly given by $$\begin{aligned}
\label{x3-def}
x_3(h):=-C_5(h)r_1(1-\lambda)^{r_1-1}e^{(\lambda-1)(r_1-1)\beta h}-C_6(h)r_2(1-\lambda)^{r_2-1}e^{(\lambda-1)(r_2-1)\beta h}+\frac{h}{r}.\end{aligned}$$ Moreover, in view of definitions of $C_5(h)$ and $C_6(h)$ in and , one can check that $x_3(h)$ is strictly increasing in $h$ and hence we can define the inverse function $$\begin{aligned}
\label{inverseh}
\tilde{h}(x):=(x_3)^{-1}(x),\ \ x\geq 0.\end{aligned}$$ Therefore, along the free boundary $x=x_3(h)$, we can write the feedback form of the optimal consumption in for $y=(1-\lambda) e^{(\lambda-1) \beta h}$ by $c^*(x)=\frac{1}{(\lambda-1)\beta} \ln \Big(\frac{1}{1-\lambda} f_3(x,\tilde{h}(x))\Big)$ only depending on the variable $x$. That is, the optimal consumption can be determined by the current wealth process $X_t^*$ and the associated running maximum process $H_t^*$ is instantly increasing.
In what follows, for some parameters $r = 0.05$, $\mu = 0.1$, $\sigma = 0.25$, $\beta=1$, $\lambda=0.5$, we graph all free boundary curves $x_1(h)$, $\breve{x}(h)$, $x_2(h)$ and $x_3(h)$ as functions of $h\geq 0$ in Figure 4 on the left panel. On the right panel, we choose the same market parameters and fix the variable $h=1$ and plot all boundary curves in terms of the parameter $\lambda\in[0.01,0.98]$ (recall that each function $C_i(h;\lambda)$ depends on $\lambda$). $$\begin{array}{cc}
\includegraphics[height=2.2in]{Figure04.eps}\hspace{-0.1in}\includegraphics[height=2.2in]{Figure06.eps}\\
\mbox{\footnotesize{Figure 4}}\hspace{2in}\mbox{\footnotesize{Figure 5}}
\end{array}$$
Although $x_1(h)$, $\breve{x}(h)$, $x_2(h)$ and $x_3(h)$ are all complicated nonlinear functions of $h$, from Figure 4, we note that all free boundary curves are increasing in the variable $h$ with the given parameters. The graphs are consistent with the intuition that if the past reference level is higher, the investor would expect larger wealth thresholds to trigger the change of consumption from 0 to $c^*>0$ and from $c^*<H^*$ to the historical maximum $c^*=H^*$. We also recall that we only consider the effective domain that is the region below the boundary curve $x_3(h)$ (including the boundary curve $x_3(h)$). It is interesting to observe from Figure 5 that $x_1(h;\lambda)$ and $x_2(h;\lambda)$ are both decreasing in $0.01\leq \lambda\leq 0.98$, while $\breve{x}(h;\lambda)$ and $x_3(h)$ are both increasing in $0.01\leq\lambda\leq 0.98$. That is, if the investor clings to a larger proportion of the past spending maximum, it is more likely that the investor will switch from zero consumption to positive consumption and from a lower consumption $c^*<H_t^*$ to the past maximum level $H^*_t$, which match with the real life situations. On the other hand, with a higher proportion $\lambda$ towards the consumption peak, the investor needs to accumulate larger wealth to consume at the reference level $c^*=\lambda H_t^*$ or consume at the peak to create a new historical maximum record that $H_t^*=c_t^*>H_s^*$ for $s<t$.
In particular, Figure 5 illustrates that $\breve{x}(h;\lambda)$ is increasing in terms of $\lambda$, which indicates that if the investor adheres more to the past consumption peak with a larger proportion $\lambda$, it is more likely that the investor will suppress the optimal consumption rate $c_t^*$ below $\lambda H_t^*$ due to the larger threshold $\breve{x}(h;\lambda)$ for the wealth level. That is, the more the investor cares about the past consumption peak $H_t^*$, the more conservative the investor will become by comparing $c_t^*$ and $\lambda H_t^*$. This observation can partially explain the real life situations that the constantly aggressive consumption behavior may not lead to a long term happiness. A high consumption plan also creates a high level of psychological competition with the past pattern such that this aggressive consumption behavior may not be sustainable for the whole lifetime. A wise investor who takes into account the past reference will strategically lower the consumption rate from time to time (triggered by a wealth threshold) below the target reference such that the reference process can be maintained at a reasonable level and the overall lifetime performance can eventually become a win.
Plugging all difference pieces of $f$ back into equation , we can readily get the following result, in which the primal value function $u$ and optimal feedback controls are all given in terms of the primal variables $x$ and $h$.
\[value\_function\_primal\] For $(x,h)\in\mathcal{C}$ and $0<\lambda<1$, the value function $u(x,h)$ of the control problem in can be explicitly expressed in a piecewise manner by $$\begin{aligned}
\label{explicit_primal}
& u(x,h) \nonumber \\
=&\left\{
\begin{aligned}
& C_2(h) f_1(x,h)^{r_2} - \frac{1}{r\beta} e^{\lambda \beta h} + x f_1(x,h), & & \mbox{if } x \leq x_1(h), \\
& C_3(h) (f_2(x,h))^{r_1} + C_4(h) (f_2(x,h))^{r_2} & & \\
& +\frac{f_2(x,h)}{r\beta } \left[\ln f_2(x,h)-\lambda\beta h+\frac{\kappa^2}{2r} -1+xr\beta \right], & & \mbox{if } x_1(h) < x < x_2(h), \\
& C_5(h) (f_3(x,h))^{r_1} + C_6(h) (f_3(x,h))^{r_2} -\frac{1}{r}h f_3(x,h) & &\\
&-\frac{1}{r\beta} e^{(\lambda-1)\beta h} + x f_3(x,h), & & \mbox{if } x_2(h) \leq x \leq x_3(h),\\
\end{aligned}
\right.\end{aligned}$$ where the free boundaries $x_1(h)$, $x_2(h)$ and $x_3(h)$ are given explicitly in , and respectively. Moreover, the feedback optimal consumption and portfolio can also be given in terms of primal variables $(x,h)$ accordingly: $$\begin{aligned}
\label{consumption:primal}
c^*(x,h) =\left\{
\begin{aligned}
&0, & & \mbox{if } x \leq x_1(h), \\
&- \frac{1}{\beta}\ln f_2(x,h) + \lambda h, & & \mbox{if } x_1(h) < x < x_2(h), \\
&h, & & \mbox{if } x_2(h) \leq x < x_3(h), \\
&\frac{1}{(\lambda-1)\beta} \ln \Big(\frac{1}{1-\lambda} f_3(x,\tilde{h}(x))\Big), & & \mbox{if } x= x_3(h),
\end{aligned}
\right.\end{aligned}$$ where $\tilde{h}(x)$ is given in , and $$\begin{aligned}
\label{invest:primal}
& \pi^*(x,h) \nonumber \\
=&\frac{\mu-r}{\sigma^2} \left\{
\begin{aligned}
&(1-r_2)x, & & \mbox{if } x \leq x_1(h), \\
&\frac{2r}{\kappa^2}C_3(h)f_2^{r_1-1}(x,h)+\frac{2r}{\kappa^2}C_4(h)f_2^{r_2-1}(x,h)+\frac{1}{r\beta}, & & \mbox{if } x_1(h) < x < x_2(h), \\
& \frac{2r}{\kappa^2}C_5(h) f_3^{r_1-1}(x,h) + \frac{2r}{\kappa^2}C_6(h) f_3^{r_2-1}(x,h), & & \mbox{if } x_2(h) \leq x \leq x_3(h).
\end{aligned}
\right.\end{aligned}$$ We also have that $0<c_t^*(x,h)<\lambda H_t^*(x,h)$ if and only if $x_1(h)< x< \breve{x}(h)$ where the threshold $\breve{x}(h)$ is given by .
In all referred work [@Arun], [@GHR] and [@BAY], the domain of $(x,h)$ is split into several regions by linear free boundaries such as $\nu_1\leq \frac{x}{h}\leq \nu_2$ for some constants $\nu_{1,2}$, in which different optimal consumption policies (or dividends) need to be followed. On the contrary, our free boundary curves $x_1(h)$, $x_2(h)$ and $x_3(h)$ can be explicitly characterized by , and (see graphs in Figure 4), which are nonlinear functions of the variable $h$. The sophisticated and more interesting decomposition of the domain results from both the exponential utility function with non-negativity constraint and the presence of the consumption running maximum inside the utility function.
The case $\lambda=1$
--------------------
At last, we present some main results for the extreme case $\lambda=1$. We separate this subsection from the previous case $0<\lambda<1$ because there are some new and distinct features in the optimal feedback controls when $\lambda=1$. Solving the HJB equation essentially follows the same arguments in the case $0<\lambda<1$. However, the effective domain $\mathcal{C}$ defined in needs to be modified to $$\begin{aligned}
\label{C-2}
\mathcal{C}:=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: u_x(x,h)\geq 0\right\}.\end{aligned}$$ Equivalently, $\mathcal{C}=\mathcal{R}_1\cup\mathcal{R}_2\cup\mathcal{R}_3=\mathbb{R}_+^2$, where $\mathcal{R}_1$, $\mathcal{R}_2$ are defined the same as in the previous subsection for $0<\lambda<1$.
In particular, we recall that $\mathcal{R}_1=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: u_x(x,h)\geq e^{\beta h}\right\}$, in which the optimal consumption $c_t^*=0$. We also have $\mathcal{R}_2=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: 1<u_x(x,h)<e^{\beta h}\right\}$, in which the feedback optimal consumption $c_t^*=-\frac{1}{\beta}\ln u_x+h$. As opposed to the case $0<\lambda<1$, we now consider $\mathcal{R}_3=\left\{(x,h)\in\mathbb{R}_+\times \mathbb{R}_+: 0\leq u_x(x,h)\leq 1\right\}$ and note that the previous auxiliary singular control $\hat{c}(x)=\frac{1}{\beta(\lambda-1)}\ln(\frac{u_x}{1-\lambda})$ to further split the region $\mathcal{R}_3$ for the case $0<\lambda<1$ is no longer well defined when we have $\lambda=1$.
In fact, in the extreme case $\lambda=1$, there is no need to consider the singular optimal consumption that excesses the previous maximum level $h$. In the whole region $\mathcal{R}_3$, the optimal consumption in no longer unique, but one feedback optimal consumption $c^*(x,h)=h$ is to follow the previously attained maximum level, which is the initial level $H_0^*=h$ and $c^*(x,h)\leq h$ for any $x\geq 0$. Comparing with other work [@Arun], [@GHR] and [@BAY], this unique and interesting phenomenon that we can only focus on the optimal control such that $H_t^*$ will never increase for $\lambda=1$ results from the nature of the formulation $U(c_t-H_t)$ where the utility is defined on the difference. For the case $0<\lambda<1$, the utility $U(c_t-\lambda H_t)$ allows the investor to gain positive outperformance $c_t-\lambda H_t>0$ if he chooses a large $c_t$ to increase $H_t$. On the other hand, for the case $\lambda=1$, the investor can only obtain $0=c_t-H_t$ by choosing to consume more than the past maximum. However, the investor can also easily achieve the same goal of zero difference $c_t-H_t$ by following the previously attained maximum level without creating any new record. Therefore, to achieve the largest gap $c_t-H_t=0$, one equivalent optimal way is to sit on the previous consumption peak and the investor has no incentives to switch to a singular control to increase the reference process $H_t$ at any time. Even if the initial wealth is sufficiently large, the investor will choose to consume the constant initial level $h=H_0$ such that $c_t=H_t$ as long as it is sustainable but never to excess this reference level during the life time. Consequently, in this subsection, we shall only adopt the feedback control $c^*(x,h)=h$ in the region $\mathcal{R}_3$.
Based on the observations above, if the wealth $x$ is larger than or equal to the subsistence level $x^*=\frac{h}{r}$, the investor can always choose to invest zero amount $\pi^*_t\equiv 0$ in the risky asset and consume the initial reference level $c^*_t=H_0=h$ constantly so that $c^*_t-H^*_t=0$ for $t\geq 0$. As a consequence, the value function defined in attains its maximum value $u(x,h)=-\frac{1}{r\beta}$ for $x\geq \frac{h}{r}$. That is, the primal value function $u(x,h)$ for $\lambda=1$ is no longer strictly concave and $u(x,h)$ remains constant (and $u_x(x,h)=0$) for $x\geq\frac{h}{r}$, which differs substantially from the case $0<\lambda<1$. Therefore, we have the asymptotic conditions that $$\begin{aligned}
\label{asymp1}
\lim_{x\rightarrow \frac{h}{r}}u_x(x,h)=0,\ \ \text{and}\ \ \lim_{x\rightarrow\frac{h}{r}}u(x,h)=-\frac{1}{r\beta}.\end{aligned}$$ For each $h\geq 0$, we expect that the value function $x\mapsto u(x,h)$ is strictly concave for $0\leq x<\frac{h}{r}$ and the dual transform method in the previous sections can still be applied on this interval $[0,\frac{h}{r})$. In view of the set $\mathcal{C}$ when $\lambda=1$, we will now consider $y>0$ for the dual problem and define $$\begin{aligned}
v(y,h):=\sup_{0\leq x<\frac{h}{r}} [u(x,h)-xy],\ \ y>0,\end{aligned}$$
As a consequence of , we have the asymptotic conditions that $$\begin{aligned}
\label{yto0cond}
\lim_{y\rightarrow 0}v_y(y,h)= -\frac{h}{r}\ \ \text{and}\ \ \lim_{y \rightarrow 0} (v(y,h)-yv_y(y,h))=-\frac{1}{r\beta},\end{aligned}$$ which are completely different from the boundary condition for $0<\lambda<1$.
Based on the same analysis in the case $0<\lambda<1$, we can write down the linear dual ODE for the case $\lambda=1$ as $$\label{EDPtildeU3}
\frac{\kappa^2}{2} y^2 v_{yy} - r v =\left\{
\begin{aligned}
& \frac{1}{\beta}e^{\beta h}, & & \mbox{if } y \geq e^{\beta h}, \\
& \frac{1}{\beta}y - y \left(\frac{1}{\beta} \ln y - h\right), & & \mbox{if } 1< y < e^{\beta h}, \\
& \frac{1}{\beta} + hy, & & \mbox{if } 0<y \leq 1,
\end{aligned}
\right.$$
By following the arguments of Proposition \[dual\_value\_function\], and replacing the free boundary condition now by the new boundary condition as $y\rightarrow 0$ in the third region, we can establish the next result.
Let $h\geq 0$ be a given parameter, the ODE admits the unique solution explicitly by $$\begin{aligned}
\label{part2-sol}
v(y,h) =\left\{
\begin{aligned}
& C_2(h) y^{r_2} - \frac{1}{r\beta} e^{\beta h}, & & \mbox{if } y \geq e^{\beta h}, \\
& C_3(h) y^{r_1} + C_4(h) y^{r_2}-\frac{y}{r\beta}+\frac{y}{r\beta } \left(\ln y-\beta h+\frac{\kappa^2}{2r}\right), & & \mbox{if }1 < y < e^{\beta h}, \\
& C_5(h)y^{r_1} -\frac{1}{r}hy-\frac{1}{r\beta}, & & \mbox{if } 0< y \leq 1,
\end{aligned}
\right.\end{aligned}$$ where $C_i(h)$, $i=2,3,4,5$ are defined in , , and $\eqref{C5h}$ in Proposition \[dual\_value\_function\] by setting $\lambda=1$.
We can similarly present the result of the verification theorem when $\lambda=1$ as below.
\[verificationthm-3\] Let $(x,h)\in\mathbb{R}_+^2$. We consider the process $Y_t:=y^*e^{r t} M_t$, where $M_t:= e^{-(r+\frac{\kappa^2}{2})t - \kappa W_t}$ is the discounted state price density process and $H_t^*\equiv H_0^*=h$ is the constant reference process under the optimal control, and the constant $y^*=y^*(x,h)$ is the unique solution to the budget constraint $\E \left[\int_0^{\infty} c^*(Y_t,H_t^*) M_t dt \right]=x$. The value function $u(x,h)$ can be attained by employing the optimal consumption and portfolio strategies in the feedback form that $c_t^*=c^*(Y_t,H_t^*)$ and $\pi_t^*=\pi^*(Y_t,H_t^*)$, $t\geq 0$, which are given by:\
\
$$\begin{aligned}
\label{feedbackcp}
c^*(y,h) =\left\{
\begin{aligned}
&0, & & \mbox{if } y \geq e^{\beta h}, \\
&- \frac{1}{\beta}\ln y + h, & & \mbox{if } 1 < y < e^{\beta h}, \\
&h, & & \mbox{if } 0 <y \leq 1,
\end{aligned}
\right.\end{aligned}$$
$$\begin{aligned}
\label{feedbackpi}
\begin{aligned}
&\pi^*(y,h) = \frac{\mu-r}{\sigma^2}\left\{
\begin{aligned}
& \frac{2r}{\kappa^2}C_2(h) y^{r_2-1}, & & \mbox{if } y \geq e^{\beta h}, \\
& \frac{2r}{\kappa^2}C_3(h) y^{r_1-1} + \frac{2r}{\kappa^2}C_4(h) y^{r_2-1}+ \frac{1}{r\beta}, & & \mbox{if } 1 < y < e^{ \beta h}, \\
& \frac{2r}{\kappa^2}C_5(h) y^{r_1-1}, & & \mbox{if } 0< y \leq 1,
\end{aligned}
\right.
\end{aligned}\end{aligned}$$
Following the same inverse dual transform arguments in the previous subsection that $u(x,h)=\inf_{y>0}[v(y,h)+xy]$ for $0\leq x<\frac{h}{r}$ and $u(x,h)=-\frac{1}{r\beta}$ for $x\geq\frac{h}{r}$, we can also obtain the functions $\bar{f}_i(x)=u_x(x,h)$ in different regions that:\
\
(i) $\bar{f}_1(x,h)=\left( \frac{-x}{C_2(h) r_2} \right)^{\frac{1}{r_2-1}}$ for $0\leq x\leq \bar{x}_1(h)$, where we define $$\begin{aligned}
\label{barx-1}
\bar{x}_1(h):=-e^{\beta h(r_2-1)}C_2(h)r_2.\end{aligned}$$ (ii) $\bar{f}_2(x,h)$ that is uniquely determined by $$\begin{aligned}
\label{barf-2}
x=-C_3(h) r_1 (\bar{f}_2(x,h))^{r_1-1} -C_4(h) r_2 (\bar{f}_2(x,h))^{r_2-1} - \frac{1}{r\beta } \left(\ln \bar{f}_2(x,h)-\beta h +\frac{\kappa^2}{2r}\right),\end{aligned}$$ for $\bar{x}_1(h)<x<\bar{x}_2(h)$ where $$\begin{aligned}
\label{barx-2}
\bar{x}_2(h):=-C_3(h) r_1 -C_4(h) r_2+ \frac{h}{r}-\frac{\kappa^2}{2r^2\beta}.\end{aligned}$$ (iii) $\bar{f}_3(x,h)=\left(\frac{\frac{h}{r}-x}{C_5(h)r_1}\right)^{\frac{1}{r_1-1}}$ for $\bar{x}_2(h)\leq x<\frac{h}{r}$.\
\
We can conclude the corollary below on the value function and feedback optimal controls in the whole domain.
\[value\_function\_3\] For $(x,h)\in\mathbb{R}_+^2$ and $\lambda=1$, the value function $u(x,h)$ of the control problem in can be explicitly expressed in a piecewise manner by $$\begin{aligned}
& u(x,h) \nonumber \\
=&\left\{
\begin{aligned}
& C_2(h) \left( \frac{-x}{C_2(h) r_2} \right)^{\frac{r_2}{r_2-1}} - \frac{1}{r\beta} e^{\beta h} + x \left( \frac{-x}{C_2(h) r_2} \right)^{\frac{1}{r_2-1}}, & & \mbox{if } x \leq \bar{x}_1(h), \\
& C_3(h) (\bar{f}_2(x,h))^{r_1} + C_4(h) (\bar{f}_2(x,h))^{r_2} & & \\
& +\frac{\bar{f}_2(x,h)}{r\beta } \left[\ln \bar{f}_2(x,h)-\beta h+\frac{\kappa^2}{2r} -1+xr\beta \right], & & \mbox{if } \bar{x}_1(h) < x < \bar{x}_2(h), \\
& C_5(h) \left(\frac{\frac{h}{r}-x}{C_5(h)r_1}\right)^{\frac{r_1}{r_1-1}} -\frac{1}{r}h \left(\frac{\frac{h}{r}-x}{C_5(h)r_1}\right)^{\frac{1}{r_1-1}} & &\\
&-\frac{1}{r\beta} + x \left(\frac{\frac{h}{r}-x}{C_5(h)r_1}\right)^{\frac{1}{r_1-1}} & & \mbox{if } \bar{x}_2(h) \leq x< \frac{h}{r},\\
&-\frac{1}{r\beta}& & \mbox{if } \frac{h}{r}\leq x,
\end{aligned}
\right.\end{aligned}$$ where the free boundaries $\bar{x}_1(h)$ and $\bar{x}_2(h)$ are given explicitly in and respectively and $\bar{f}_2(x,h)$ is given implicitly by . The feedback optimal consumption and portfolio are given by: $$\begin{aligned}
\label{consumption:primal}
c^*(x,h) =\left\{
\begin{aligned}
&0, & & \mbox{if } x \leq \bar{x}_1(h), \\
&- \frac{1}{\beta}\ln \bar{f}_2(x,h) + h, & & \mbox{if } \bar{x}_1(h) < x < \bar{x}_2(h), \\
&h, & & \mbox{if } \bar{x}_2(h) \leq x,
\end{aligned}
\right.\end{aligned}$$ and $$\begin{aligned}
\label{invest:primal}
& \pi^*(x,h) \nonumber \\
=&\frac{\mu-r}{\sigma^2} \left\{
\begin{aligned}
&(1-r_2)x, & & \mbox{if } x \leq \bar{x}_1(h), \\
&\frac{2r}{\kappa^2}C_3(h)\bar{f}_2^{r_1-1}(x,h)+\frac{2r}{\kappa^2}C_4(h)\bar{f}_2^{r_2-1}(x,h)+\frac{1}{r\beta}, & & \mbox{if } \bar{x}_1(h) < x < \bar{x}_2(h), \\
& \frac{2r}{\kappa^2r_1}\left(\frac{h}{r}-x\right), & & \mbox{if } \bar{x}_2(h) \leq x< \frac{h}{r},\\
& 0, & & \mbox{if } \frac{h}{r} \leq x,\\
\end{aligned}
\right.\end{aligned}$$ and the resulting consumption running maximum process is constant that $H_t^*=H_0^*=h$ for $t>0$.
At last, based on Corollaries \[veri-2-0\], \[value\_function\_primal\] and \[value\_function\_3\], we present the result of the asymptotic behavior of the optimal consumption-wealth ratio $\frac{c_t^*}{X_t^*}$ and the investment amount $\pi^*_t$ when the wealth is sufficiently large and its proof is given in Section \[otherproofs\].
\[propasymp\] For $\lambda=0$, we have $\lim_{x\rightarrow+\infty}\frac{c^*(x)}{x}=r$ and $\lim_{x\rightarrow+\infty}\pi^*(x)=\frac{\mu-r}{r\beta\sigma^2}$. For $0<\lambda<1$, as we have $x\leq x_3(h)$, we consider the asymptotic behavior along the boundary curve $x_3(h)$ as $x,h\rightarrow+\infty$, and we have $$\begin{aligned}
\lim_{\substack{x\rightarrow+\infty,\\(x,h)\in x_3(h)}}\frac{c^*(x,h)}{x}=r,\ \ \ \lim_{\substack{x\rightarrow+\infty,\\ (x,h)\in x_3(h)}}\pi^*(x,h)=\frac{(\mu-r)(1-\lambda)^{r_1-1}}{r\beta\sigma^2}.\end{aligned}$$ As the wealth level gets sufficiently large, the optimal consumption is asymptotically proportional to the wealth level that $c_t^*\approx rX_t^*$ and the optimal investment converges to a constant level that $\pi^*_t\approx \frac{(1-\lambda)^{r_1-1}}{r\beta}$ for $(0\leq \lambda<1)$. That is, the investor will only allocate constant amount of wealth into the risky asset and save most of the wealth into the bank account. For $\lambda=1$, the investor will stop the investment in the risky asset when the wealth exceeds the constant level $\frac{h}{r}$ and one optimal consumption is to constantly spend the initial reference amount $c_t^*=h$ for $t\geq 0$.
Numerical Examples and Sensitivity Analysis {#sec:numerics}
===========================================
This section reports some numerical examples of sensitivity analysis using the previous explicit value function and feedback optimal controls in Corollary \[value\_function\_primal\].
We first present the 3-dimensional graphs of the value function $u(x,h)$, the optimal consumption $c^*(x,h)$ and optimal portfolio $\pi^*(x,h)$ in the next three figures. In particular, we choose the market parameters that $r = 0.05$, $\mu = 0.1$, $\sigma = 0.25$, $\beta=1$, $\lambda = 0.5$ and we plot the graphs for the variable $h \in [0.5, 1.5]$ and $x \in [0, 25]$. $$\begin{array}{ccc}
\hspace{-0.3in}
\includegraphics[height=1.75in]{Figure07.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure08.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure09.eps} \\
\mbox{\footnotesize{Figure 6}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 7}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 8}}
\end{array}$$
We first perform the sensitivity analysis by plotting graphs of the value function, the feedback optimal consumption and the feedback optimal portfolio for different values of the reference weight parameter $\lambda = 0.1, 0.2, ..., 0.8, 0.9$. Here, we choose the market parameters that $r = 0.05$, $\mu = 0.1$, $\sigma = 0.25$, $\beta=1$ and fix the variable $h = 1$ and plot all graphs as functions of $x$ for $0\leq x\leq x_3(1)$. $$\begin{array}{ccc}
\hspace{-0.3in}
\includegraphics[height=1.75in]{Figure10.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure11.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure12.eps} \\
\mbox{\footnotesize{Figure 9}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 10}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 11}}
\end{array}$$ \
From Figure 10, for $x_1(1)<x<x_2(1)$, we can see that the feedback consumption $c^*(x,1)$ is increasing in $x$. More importantly, for the fixed $0<x<x_3(1)$, the feedback optimal consumption $c^*(x,1;\lambda)$ is increasing in the parameter $\lambda\in (0,1]$, which is consistent with the intuition: the stronger that the investor adheres to the past consumption maximum level, the more likely that the investor will consume more during the life-cycle. We can observe from Figure 11 (see also Figure 14, Figure 17 and Figure 20) that for $x\leq x_1(1)$, the feedback $\pi^*(x,1)$ is increasing and linear in $x$; and for $x_1(1)<x<x_2(1)$, the feedback $\pi^*(x,1)$ is increasing and concave in $x$; and for $x_2(1)\leq x\leq x_3(1)$, the feedback $\pi^*(x,1)$ is increasing and convex in $x$, Moreover, for the fixed $0<x<x_3(1)$, $\pi^*(x,1;\lambda)$ is decreasing in the parameter $\lambda\in(0,1]$, which coincides with the observation in Figure $10$ that the optimal consumption level is lifted up by a larger value of $\lambda$. The investor may strategically invest less in the market to save enough cash for higher consumption plan influenced by $\lambda$. From Figure 9, for each $0<x<x_3(1)$, the graphs illustrate that the value function $u(x,h;\lambda)$ is decreasing in $\lambda\in (0,1]$. This suggests that even the optimal consumption rate increases because of the increase in $\lambda$, it does not necessarily imply that the value function also increases. In our preference formulation, the utility function is defined on the difference between the consumption rate $c_t^*$ and the reference process $\lambda H_t^*$. As both $\lambda$ and $c^*_t$ increase, the reference process $\lambda H^*_t$ increases as well. From Figure 9, we can see that $\lambda H^*_t$ actually increases faster than the consumption rate $c_t$ when $\lambda$ increases, which leads to a drop of the difference $c^*_t-\lambda H_t^*$ so that the resulting value function actually decreases.
We next present the impact of the drift parameter $\mu$ on the value function, the feedback optimal consumption and the feedback optimal portfolio by considering $\mu = 0.1, 0.12, 0.14, 0.16, 0.18$. We again fix marker parameters $r = 0.05$, $\sigma = 0.25$, $\lambda = 0.5$, $\beta=1$ and the maximum reference variable $h = 1$ and plot the graphs as functions of $x$ for $0\leq x\leq x_3(1)$. $$\begin{array}{ccc}
\hspace{-0.3in}
\includegraphics[height=1.75in]{Figure13.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure14.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure15.eps} \\
\mbox{\footnotesize{Figure 12}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 13}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 14}}
\end{array}$$ \
From Figure $13$, we can see that the feedback function of the optimal consumption is increasing in terms of the drift parameter $\mu$. This implies that if the market performance is getting better as the return of the risky asset increases, the investor can accumulate more wealth from the financial market to support a higher consumption plan. Likewise, Figure $14$ illustrates that the investor’s optimal portfolio in the financial market increases as the stock return increases. Figure $12$ shows that the primal value function is increasing with respect to the drift parameter $\mu$. It illustrates that when the return parameter $\mu$ increases, the increase in optimal consumption rate $c^*_t$ dominates the increase in the running maximum process $H^*_t$ so that the life time value function is lifted up.
Similarly, for the market parameters $r = 0.05$, $\mu = 0.1$, $\beta=1$, $\lambda=0.5$ and the fixed variable $h=1$, we continue to present the sensitivity analysis of the value function $u(x,1)$, the feedback controls $c^*(x,1)$ and $\pi^*(x,1)$ with respect to different volatility parameters $\sigma=0.1, 0.2, 0.4, 0.6, 0.8$. $$\begin{array}{ccc}
\hspace{-0,3in}
\includegraphics[height=1.75in]{Figure16.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure17.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure18.eps} \\
\mbox{\footnotesize{Figure 15}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 16}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 17}}
\end{array}$$ From Figure 16, we observe that the monotonicity of the optimal consumption $c^*(x,1)$ on the volatility $\sigma$ is not guaranteed and the dependence becomes much subtle and complicated. We can also see that the consumption $c^*(x,1)$ is not simply concave or convex in the variable $x$ for the region $x_1(1)<x<x_2(1)$, which depends on the volatility $\sigma$ and other market parameters. In this example, we can observe that the threshold $x_2(1;\sigma)$ is increasing in $\sigma$ but the dependence of the threshold $x_1(1;\sigma)$ on $\sigma$ is unclear. Figure $15$ and Figure $17$ show that both the value function and the optimal portfolio are decreasing in the volatility $\sigma$. These graphs are consistent with the real life situation and are similar to some classical models such as the Merton problem that if the risky asset has a higher volatility, the investor will allocate less wealth in the risky asset and the value function on consumption also becomes lower.
At last, for the market parameters $r = 0.05$, $\mu = 0.1$, $\sigma=0.25$, $\lambda=0.5$ and the fixed $h=1$, we plot the sensitivity analysis of the value function $u(x,1;\beta)$, the feedback controls $c^*(x,1;\beta)$ and $\pi^*(x,1;\beta)$ with respect to the risk aversion parameter $\beta=0.2,0.5, 1, 2, 5$ in the following figures for $0\leq x\leq x_3(1)$. We can see from Figure 19 that the threshold $x_1(1;\beta)$ is decreasing in the risk aversion parameter $\beta$ and $x_2(h)$ is increasing in $\beta$. That is, for a more risk averse investor, it is more difficult to start a positive consumption $c^*>0$ but it becomes much easier to consume at the maximum level. However, the optimal consumption $c^*(x,1;\beta)$ has a very complicated dependence on the parameter $\beta$. Within this numerical example, the optimal portfolio $\pi^*(x,1;\beta)$ is still decreasing in $\beta$ from Figure $20$, but the impact of $\beta$ on the value function $u(x,1;\beta)$ is no longer monotone. Only when the wealth is sufficiently large, the value function behaves increasing with higher risk aversion $\beta$.
$$\begin{array}{ccc}
\hspace{-0.3in}
\includegraphics[height=1.75in]{Figure19.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure20.eps}\hspace{-0.2in}
\includegraphics[height=1.75in]{Figure21.eps} \\
\mbox{\footnotesize{Figure 18}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 19}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mbox{\footnotesize{Figure 20}}
\end{array}$$
\
Proofs of Main Results {#sec:proofs}
======================
Proof of the verification theorem {#sec:verification}
---------------------------------
In this subsection, we only provide the complete proof of Theorem \[verthm\] for the case $0<\lambda<1$, which exhibits more structures in the feedback optimal controls. One can follow similar arguments to prove Theorem \[verify-0\] and Theorem \[verificationthm-3\] and their proofs are omitted for the sake of length.
The proof of the verification theorem boils down to show that the solution of the PDE indeed coincides with the value function, i.e. there exists $(\pi^*, c^*) \in \mathcal{A}(x)$ such that $$u(x, h) = \E_{}\left[\int_0^{\infty} e^{- r t} U({c_t^* - \lambda H_t^*}) dt\right].$$ Taking into account the definition of ${H_t^*=h \lor \sup_{s \leq t} c_s^*}$, let us further define $$\begin{aligned}
\label{defhatH}
{\hat H_t(y)} := h \lor \left( \frac{1}{(\lambda-1)\beta} \ln\left[\frac{1}{1-\lambda} \inf_{s \leq t} Y_s(y) \right] \right), \end{aligned}$$ where $Y_t(y) \; = \; y e^{r t} M_t$ is the discounted martingale measure density process. For any admissible strategy $(\pi,c) \in \mathcal{A}(x)$, similar to the standard proof of Lemma 1 in [@Arun], we have $$\label{eq:SdfInequality}
\E \left[\int_0^{\infty} c_t M_t dt \right] \leq x.$$ Regarding $(\lambda,h)$ as fixed parameter, let us consider the dual transform of $U$ with respect to $c$ in the constrained domain that
- when $\lambda$ $=$ $0$: $$\begin{aligned}
V(y,h) &:= \sup_{c \geq 0} [U(c) - cy] \; = \;
\left\{
\begin{aligned}
& - \frac{1}{\beta}, & & \mbox{if } y \geq 1, \\
& -\frac{1}{\beta}y + \frac{1}{\beta}y \ln y, & & \mbox{if } y < 1,
\end{aligned}
\right.\end{aligned}$$
- when $\lambda$ $>$ $0$, $$\begin{aligned}
V(y,h) &:= \sup_{0\leq c \leq h} [U(c-\lambda h ) - cy] \; = \;
\left\{
\begin{aligned}
& - \frac{1}{\beta} e^{\lambda \beta h}, & & \mbox{if } y \geq e^{\lambda \beta h}, \\
& -\frac{1}{\beta}y + y ( \frac{1}{\beta} \ln y - \lambda h), & & \mbox{if } e^{(\lambda-1)\beta h} \leq y < e^{\lambda \beta h}, \\
& - \frac{1}{\beta}e^{(\lambda-1) \beta h} - hy, & & \mbox{if } (1-\lambda)e^{(\lambda-1)\beta h}\leq y < e^{(\lambda-1) \beta h}.
\end{aligned}
\right.\end{aligned}$$
We remark that when $\lambda=0$, $V(y,h)$ is independent of $h$. Moreover, $V(y,h)$ can be attained by the construction of the feedback function $c^*(y,h)$ given in .
For any admissible $(\pi,c)$ $\in$ $\mathcal{A}(x)$, recall its resulting reference process $H_t$ $=$ $h \lor \sup_{s \leq t} c_s$, and for all $y > 0$, we see that $$\begin{aligned}
\E_{} \left[\int_0^{\infty} e^{- r t} U({c_t - \lambda H_t}) dt \right] =& \E_{}\left[\int_0^{\infty} e^{- r t} (U(c_t - \lambda H_t) - Y_t(y) c_t ) dt \right]
+ y \E_{} \left[\int_0^{\infty} c_t M_t dt \right] \nonumber \\
\leq & \E_{} \left[\int_0^{\infty} e^{- r t} V(Y_t(y), {H_t^*}) dt \right] + yx \label{ineg} \\
= & \E_{} \left[\int_0^{\infty} e^{- r t} V(Y_t(y), {\hat H_t(y)} ) dt \right] + yx \nonumber \\
= & v (y, h) +y x. \nonumber \end{aligned}$$ where the second line follows by Lemma \[lemma:IneqAttained\], the third line holds thanks to Lemma \[lemma:ReplaceHat\] below, and the last line is consequent on Lemma \[lemma:DualVerification\]. In addition, in view of Lemma \[lemma:IneqAttained\], the inequality becomes equality with the choice of $c^*_t=c_t^*(Y_t(y^*), H_t^*(y^*))$, in which $y^*$ is the unique solution to the equation $\E \left[\int_0^{\infty} c^*(Y_t(y^*),H_t^*(y^*)) M_t dt \right]=x$ for the given $x>0$ and $h\geq 0$.
In conclusion, we arrive at $$\sup_{(\pi,c) \in \mathcal{A}(x)} \E_{} \Big[\int_0^{\infty} e^{- r t} U({c_t - \lambda H_t}) dt \Big] = \inf_{y > 0} ( v (y, h) +yx)= u(x, h),$$ which completes the proof of verification theorem.
We then proceed to prove some auxiliary results that have been used to support the previous proof of the main theorem.
\[lemma:DualVerification\] $$v(y, h) = \E_{} \left[\int_0^{\infty} e^{- r t} V(Y_t(y), \hat{H}_t(y)) dt \right].$$
Note that the martingale measure density process $M_t$ satisfies the equation $$d M_t = M_t (-r dt - \kappa d W_t).$$ By and , $v(y,h)$ satisfies the ODE $$\frac{\kappa^2}{2} y^2 v_{yy} - r v + V(y, h) = 0.$$ By Itô’s formula, we have that $$\begin{aligned}
d \Big(e^{- r t} v(Y_t(y), \hat{H}_t (y)) \Big) = &- e^{- r t} V(Y_t(y), \hat{H}_t (y)) dt - \kappa e^{-r t} v_y(Y_t(y), \hat{H}_t (y))Y_t(y) d W_t \\
&+ e^{-r t} v_h(Y_t(y), \hat{H}_t (y)) d \hat{H}_t(y). \end{aligned}$$ By defining the stopping time $$\begin{aligned}
\tau_n:= \inf \left\{ t \geq 0\Big | Y_t(y) \geq n,~ \hat{H}_t(y) \geq \frac{1}{(\lambda-1)\beta} \ln \frac{1}{(1-\lambda)n} \right\}\end{aligned}$$ and integrating the above equation from $0$ to $T \wedge \tau_n$, we have that $$\begin{aligned}
\label{refito}
v(y, h) = \E_{}\left[\int_0^{T \wedge \tau_n} e^{-r t} V(Y_t(y) \hat{H}_t (y)) dt\right]
+ \E_{}\left[ e^{-r (T \wedge \tau_n)} v(Y_{T \wedge \tau_n}(y), \hat{H}_{T \wedge \tau_n} (y))\right].\end{aligned}$$ To wit, the integral term with respect to $d \hat{H}_t(y)$ vanishes as $\hat{H}_t(y)$ increases only if $c^*_t(y)=\hat{H}_t(y)$ and we have $v_h(Y_t(y), \hat{H}_t (y))=0$ by the free boundary condition. The expectation of the integral of $d W_t$ also vanishes as the local martingale $$\int_0^{T \wedge \tau_n} \kappa v_y(Y_t(y), \hat{H}_t (y)) y M_t d W_t$$ becomes a true martingale thanks to the definition of $\tau_n$ and the fact that $v$ is of class $C^2$.
By passing to the limit as $n\rightarrow+\infty$, the first term in tends to $\E_{}\left[\int_0^{T} e^{-r t} V(Y_t(y) \hat H_t (y)) dt\right]$ by the monotone convergence theorem. Moreover, the second term in can be written as $$\begin{aligned}
\label{twoexpt}
&\E_{} \left[e^{-r (T \wedge \tau_n)} v(Y_{T \wedge \tau_n}(y), \hat H_{T \wedge \tau_n} (y))\right] \\ \nonumber
= &\E_{}\left[e^{-r T} v(Y_{T}(y), \hat H_{T} (y))\mathbf{1}_{\{T \leq \tau_n\}} \right] + \E_{}\left[ e^{-r \tau_n} v(Y_{\tau_n}(y), \hat H_{\tau_n} (y)) \mathbf{1}_{\{T > \tau_n\}}\right].
\end{aligned}$$ As $n\rightarrow+\infty$, the first term in clearly converges to $\E_{}\left[e^{-r T} v(Y_{T}(y), \hat H_{T} (y)) \right]$. We will further show that the transversality condition holds in the sense that $\E_{}\left[e^{-r T} v(Y_{T}(y), \hat H_{T} (y)) \right]$ converges to $0$ as $T\rightarrow+\infty$ in Lemma \[lemm:Transversality\].
We then claim that the second term in also converges to $0$ as $T\rightarrow+\infty$. To see this, it follows by the definition of $\tau_n$ that for all $t \leq \tau_n$, we have $\inf_{s \leq t}Y_s(y) \geq \frac{1}{n}$ and $Y_t(y) \leq n$. Using the fact that when $y$ is sufficiently large, $v(y, h)$ is of order $C_2(h)y^{r_2}$ and Remark \[remarkord\] gives that $$C_2(h)= O \left( e^{(\lambda-1)(1-r_2)\beta h} \right) + O \left( e^{[\lambda(1-r_2)-(r_1-r_2)]\beta h} \right) + O\left( e^{\lambda (1-r_2)\beta h} \right).$$ We can then compare the order of $v(Y_{\tau_n}(y), \hat H_{\tau_n} (y))$ accordingly for the fixed $\tau_n$. First of all, we have $Y_t(y)^{r_2} \leq (\frac{1}{n})^{r_2} = n^{-r_2}$. Secondly, it is easy to see that $O \left( e^{(\lambda-1)(1-r_2)\beta h} \right) = O \left( n^{r_2-1} \right)$, $O \left( e^{[\lambda(1-r_2)-(r_1-r_2)]\beta h} \right) = O \left( n^{\frac{\lambda(1-r_2) - (r_1-r_2)}{1-\lambda}} \right)$ as well as $O \left( e^{\lambda(1-r_2)\beta h} \right) = O \left( n^{\frac{\lambda}{\lambda-1}}(1-r_2) \right)$. Note that all these three terms have an order smaller than $O\left( 1 \right)$. Thirdly, similar to the proof of (A.25) in [@GHR], we have that $$\E[ \mathbf{1}_{\{\tau_n \leq T\}} ] \leq n^{-2 \kappa} (1 + y^{2 \kappa}) e^{CT},$$ for any $\kappa \geq 1$. Putting all pieces together, the desired claim holds that $$\lim_{T\rightarrow+\infty}\E_{}\left[ e^{-r \tau_n} v(Y_{\tau_n}(y), \hat H_{\tau_n} (y)) \mathbf{1}_{\{T > \tau_n\}}\right]=0.$$
\[lemma:ReplaceHat\] $$\E_{} \left[\int_0^{\infty} e^{- r t} V(Y_t(y), {H_t^*}) dt \right]
= \E_{} \left[\int_0^{\infty} e^{- r t} V(Y_t(y), \hat H_t(y) ) dt \right] .$$
The proof is similar to [@GHR]. For the sake of completeness, we present the argument in sketch. Suppose that $H_t^*$ is strictly increasing at $t$, the fact that $H_t^*=c_t^*$ implies that the optimal consumption is given by $c_t^*$ $=$ $\frac{1}{(\lambda-1)\beta} \ln (\frac{1}{1-\lambda} Y_t(y)) $.
Define $$\begin{aligned}
\mathcal{I}_t:= \{ s \leq t: H^*\ \text{is strictly increasing at}\ s \}.\end{aligned}$$ Then, for any $s \notin \mathcal{I}_t$, using the condition that $Y_s(y) \leq (1-\lambda)e^{(\lambda-1)\beta H_s^*} $ and the formula that $c_s^*=- \frac{1}{\beta}\ln Y_s(y) + \lambda H_s^*$ or $c_s^*=H_s^*$, we have that $c_s^* \leq H_s^*$. Thus, we derive that $$\begin{aligned}
H_t^* &= h \vee \sup_{s \in \mathcal{I}_t} c_s^* = h \vee \sup_{s \in \mathcal{I}_t} \frac{1}{(\lambda-1) \beta} \ln \left(\frac{1}{1-\lambda} Y_s(y)\right) \\
& = h \vee \sup_{s \leq t} \frac{1}{(\lambda-1)\beta} \ln \left(\frac{1}{1-\lambda} Y_s(y)\right) = \hat{H}_t (y).\end{aligned}$$
\[lemma:IneqAttained\] The inequality becomes an equality with the consumption control $c_t^*=c^*(Y_t(y^*),\hat H_t(y^*))$, $t$ $\geq$ $0$, with $y^*=y^*(x,h)$ as the unique solution to $\E \left[\int_0^{\infty} c^*(Y_t(y^*),\hat H_t(y^*)) M_t dt \right]=x$.
The definition of $V$ implies that for all $(\pi, c) \in \mathcal{A}(x)$, $U({c_t - \lambda H_t})- Y_t(y) c_t \leq V(Y_t(y), {H_t})$. The inequality holds as an equality with the control $ c_t^*$. In other words, for any admissible $(c_t)_{0 \leq t \leq T}$, we have that for all $t \in [0,T]$, $$U({c_t - \lambda H_t})- Y_t(y) c_t \leq U({c_t^* - \lambda H_t})- Y_t(y) c_t^* = V(Y_t(y), {H_t}).$$ Multiplying both sides by $e^{- r t}$ and integrating from $0$ to $T$, we have that $$\int_0^{\infty} e^{- r t} (U(c_t - \lambda H_t) - Y_t(y) c_t ) dt \leq \int_0^{\infty} e^{- r t} V(Y_t(y), {H_t^*}) dt.$$
To turn into an equality, the equality in needs to be attained with some $y$ to be determined later, and $$\label{eq:LegendreEquality}
U(c_t - \lambda H_t) - Y_t(y) c_t = V(Y_t(y), H_t)$$ also needs to hold. Hence, we choose to employ $$c_t^*(y) \; = \; c^*(Y_t(y),\hat H_t(y)) =: \hat H_t(y) F_t(y,Y_t(y)),$$ where we define $$F_t(y,z):=\mathbf{1}_{\{ (1-\lambda) e^{-(1-\lambda) \beta \hat H_t(y)} \leq z \leq e^{-(1-\lambda) \beta \hat H_t(y)} \}}
+ (-\frac{1}{\beta}\mbox{ln} z + \lambda \hat H_t(y)) \mathbf{1}_{ \{ e^{-(1-\lambda) \beta \hat H_t(y)} \leq z \leq e^{\lambda \beta \hat H_t(y)} \} }$$ Note that the construction of $c_t^*(y)$ guarantees the validity of the equality .
In view of the definition of $\hat{H}_t(y)$ in , one can obtain that: (i) If $y \downarrow 0$, then $\hat H_t(y) \uparrow +\infty$ and $F_t(y,Y_t(y)) > 0$, it yields that $\E_{} \left[\int_0^{\infty} M_t c^*_t(y) dt\right] \uparrow +\infty$; (ii) If $y \uparrow +\infty$, then $\hat H_t(y) \downarrow h$ and $F_t(y,Y_t(y)) \downarrow 0$, it yields that $\E_{} \left[\int_0^{\infty} M_t c^*_t(y) dt\right] \downarrow 0$.The existence of $y^*$ satisfying the budget constraint can be verified from the previous asymptotic behavior of $ \hat H_t(y)$ and $F_t(y,Y_t(y))$ by passing to the limit $y\rightarrow 0$ and $y\rightarrow+\infty$ and the fact that $\E_{} \left[\int_0^{\infty} M_t c^*_t(y) dt\right]$ is continuous in the variable $y$.
We then prove the transversality condition, which is a key step in the proof of Lemma \[lemma:DualVerification\]:
\[lemm:Transversality\] The following transversality condition holds that for all $y$ $>$ $0$, $$\lim_{T \rightarrow + \infty} \E_{}\left[ e^{-r T} v(Y_T(y), \hat H_{T} (y))\right] = 0.$$
Let us first recall that $$\hat H_t(y) = h \lor \left( \frac{1}{(\lambda-1)\beta} \ln\left[\frac{1}{1-\lambda} \inf_{s \leq t} Y_s(y)\right] \right).$$ As $T\rightarrow +\infty$, we can concentrate on the interval $[e^{(\lambda-1) \beta h}, + \infty)$ due to the fact that $e^{(\lambda-1) \beta \hat H_{T} (y)} < Y_T(y)$ a.s.
From Section \[sec:solveHJB\], in the interval $e^{(\lambda-1) \beta h} < y < e^{\lambda \beta h}$, which corresponds to the case $0 < c_t < H_t$, we have $$v(y,h)=C_3(h) y^{r_1} + C_4(h) y^{r_2} -\frac{y}{r \beta}+\frac{y}{r \beta} \left(\ln y-\lambda\beta h+\frac{\kappa^2}{2r}\right).$$
In the interval $y \geq e^{\lambda \beta h}$, which corresponds to the case $c_t=0$, we have $$v(y,h)=C_2(h) y^{r_2} - \frac{1}{r \beta} e^{\lambda \beta h} ,$$
We first deal with the case $0<c_t< H_t$ and check the asymptotic behavior of the expectation of the following process $$\mathbb{E}\left[e^{-r T} \left( C_3(\hat H_T(y)) (Y_T(y))^{r_1} + C_4(\hat H_T(y)) (Y_T(y))^{r_2}-\frac{ Y_T(y)}{r \beta}+\frac{Y_T(y) }{r \beta} \left(\ln Y_T(y)-\lambda \beta \hat H_T(y)+\frac{\kappa^2}{2r}\right) \right)\right].$$
We will consider its asymptotic behavior term by term.
\(i) *Step 1*: Let us start by considering the asymptotic behavior of the third and fourth terms. For the third term, it is easy to see that $$\begin{aligned}
\label{thirdconv}
\E_{}\left[y e^{-(r+\frac{\kappa^2}{2})T-\kappa W_T} \frac{1}{r \beta}\right]= \frac{y}{r \beta} e^{-(r+\frac{\kappa^2}{2})T} \E_{}\left[e^{- \kappa W_T}\right]=\frac{y}{r \beta} e^{-r T},\end{aligned}$$ which converges to $0$ as $T \rightarrow +\infty$.
For the fourth term, we have that $$\frac{y M_T}{r \beta} \left(\ln Y_T(y)-\lambda \beta \hat H_T(y)+\frac{\kappa^2}{2r}\right) = \frac{1}{r \beta} \left( y M_T \left(r T + \ln y +\frac{\kappa^2}{2r}\right) + y M_T \ln M_T - y M_T \lambda \beta \hat H_T(y) \right).$$ Similar to , we can show that $\mathbb{E}[y M_T (r T + \ln y - \ln p)]$ converges to $0$ and moreover we have $$\E_{} [ y M_T \ln M_T ] = -y e^{-(r+\frac{\kappa^2}{2})T} \left( \E_{}\left[ \kappa W_T e^{-\kappa W_T} \right] + \left(r+\frac{\kappa^2}{2}\right) \E_{}\left[e^{- \kappa W_T}\right] \right)
=-y e^{-rT} \left(r-\frac{\kappa^2}{2}\right),$$ which also converges to $0$ as $T\rightarrow+\infty$. Furthermore, we can deduce that $$\begin{aligned}
\E_{} [ y M_T \hat H_T(y) ] &\leq y M_T + y M_T \frac{1}{\lambda-1} \ln \left[ \frac{1}{1-\lambda} y \inf (e^{r s} M_s)\right] \\
&= O \left( y M_T \right) + O \left( y M_T \ln (y M_T e^{r T}) \right)\\
&=O \left( y M_T \right) + O \left( y M_T \ln (y M_T)\right)+ O(yM_Tr T),\end{aligned}$$ in which each term vanishes as $T\rightarrow+\infty$ by following similar computations as in .
\(ii) *Step 2*: Let us continue to consider the term with $C_3(h)$. In view of the constraint $Y_T(y) < e^{\lambda \beta \hat H_T}$, we have $$\lambda \beta \hat H_T (1-r_1) < (1-r_1) \ln ( Y_T(y))$$ and it follows that $$\begin{aligned}
& \ \E_{}\left[ e^{-r T} C_3(\hat H_T(y)) (Y_T(y))^{r_1} \right] \\
&= O\left( \E_{}\left[ e^{\lambda \beta \hat H_T(y) (1-r_1)} e^{-r T} (Y_T(y))^{r_1} \right] \right) \\
&\leq O\left( \E_{}\left[ (Y_T(y))^{1-r_1} e^{-r T} (Y_T(y))^{r_1} \right] \right) \\
&=O\left( \E_{}\left[ y M_T \right] \right) \; =O\left( e^{-rT} \right) \\
\end{aligned}$$ It is thus verified that the third term converges to $0$ as $T\rightarrow+\infty$.
\(iii) *Step 3*: Now let us work with the term $C_4(\hat H_T(y)) e^{-r T} (Y_T(y))^{r_2}$. Remark \[remarkord\] asserts that $$C_4(h) = O \left( e^{(\lambda-1)(1-r_2)\beta h} \right) + O \left( e^{[\lambda(1-r_2)-(r_1-r_2)]\beta h} \right).$$
Let us define the set $$\begin{aligned}
A:=\left\{\frac{1}{\lambda-1} \ln\left[\frac{1}{1-\lambda} y \inf_{s \leq T} (e^{r s} M_s) \right] \geq h \right\}=\left\{\inf_{s\leq T} (e^{r s} M_s) \leq(1-\lambda) e^{(\lambda-1)h}\frac{1}{y} \right\},\end{aligned}$$ and two auxiliary processes $$\begin{aligned}
G_t^1:= &\left(\frac{y}{1-\lambda}\right)^{(1-r_2)\beta} \inf_{s \leq T} \left(e^{r s} M_s \right)^{(1-r_2)\beta},\\
G_t^2:= &\left(\frac{y}{1-\lambda}\right)^{\frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta} \inf_{s \leq T} \left(e^{r s} M_s \right)^{\frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta}.\end{aligned}$$
Using the formula of $\hat{H}_t$ defined in , we have that $$\begin{aligned}
&\E\left[ e^{-r T} C_4(\hat H_T(y)) (Y_T(y))^{r_2} \right] \\
=& O\left( \E_{}\left[ e^{(\lambda-1) (1-r_2) \beta \hat H_T(y) } e^{-r T} (Y_T(y))^{r_2} + e^{[\lambda(1-r_2)-(r_1-r_2)] \beta \hat H_T(y)} e^{-r T} (Y_T(y))^{r_2} \right] \right) \\
=& O \Big( \E_{} \Big[ G_t^1 (e^{r T} M_T)^{r_2} e^{-r T} \mathbf{1}_{A} \Big] \Big) \vee O \Big( \E_{} \Big[ G_t^2 (e^{r T} M_T)^{r_2} e^{-r T} \mathbf{1}_{A} \Big] \Big) \vee O \Big( \E_{} \Big[ e^{-r T} (Y_T(y))^{r_2} \mathbf{1}_{A^c} \Big] \Big)\\
=& O(\Upsilon_1(T)) \vee O(\Upsilon_2(T)) \vee O \Big( \E_{} \Big[ e^{-r T} (Y_T(y))^{r_2} \mathbf{1}_{A^c} \Big] \Big) ,
\end{aligned}$$ in which we define $$\begin{aligned}
\Upsilon_1(T):=& \E_{} \left[ \left(\frac{y}{1-\lambda}\right)^{(1-r_2)\beta} \left[ \inf_{s \leq T} \left(e^{r s} M_s \right) \right]^{(1-r_2)\beta} (e^{r T} M_T)^{r_2} e^{-r T} \mathbf{1}_{A} \right], \\
\Upsilon_2(T):=& \E_{} \left[ \left(\frac{y}{1-\lambda}\right)^{\frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta} \left[ \inf_{s \leq T} \left(e^{r s} M_s \right) \right]^{\frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta} (e^{r T} M_T)^{r_2} e^{-r T} \mathbf{1}_{A} \right],\end{aligned}$$ and the third terms comes from $\bar{h}$ in the definition of $\hat H_t(y)$ with $A^c$ being the complementary set of $A$.
We then proceed to show that [all three terms $\Upsilon_1(T)$ and $\Upsilon_2(T)$ and $\E_{} \Big[ e^{-r T} (Y_T(y))^{r_2} \mathbf{1}_{A^c} \Big]$ ]{}converge to $0$ as $T\rightarrow+\infty$.
Thanks to Corollary A.7 of [@GHR] with $a_1=-\kappa r_2$, $b_1=-\kappa (1-r_2)\beta$, $b_2=-\kappa \frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta$, $\zeta=\frac{\kappa}{2}$, we obtain some upper bounds of the limit by $$\begin{aligned}
\begin{aligned}
&\lim_{T \rightarrow \infty}\frac{1}{T}\log \Upsilon_1(T)\\
=&\lim_{T \rightarrow \infty}\frac{1}{T}\log \left\{ \E_{} \left[ \left(\frac{y}{1-\lambda}\right)^{(1-r_2)\beta} \left[ \inf_{s \leq T} \left(e^{r s} M_s \right) \right]^{(1-r_2)\beta} (e^{r T} M_T)^{r_2} e^{-r T} \mathbf{1}_{A} \right] \right\} \\
\leq& \max \left\{ \frac{a_1(a_1+2\zeta)}{2}-r, \frac{(a_1+b_1)(a_1+b_1+2\zeta)}{2}-r, -\frac{\zeta^2}{2}-r \right\}
\end{aligned}\end{aligned}$$ Similarly, we have that $$\begin{aligned}
\begin{aligned}
&\lim_{T \rightarrow \infty}\frac{1}{T}\log \Upsilon_2(T)\\
=&\lim_{T \rightarrow \infty}\frac{1}{T}\log \left\{ \E_{} \left[ \left(\frac{y}{1-\lambda}\right)^{\frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta} \left[ \inf_{s \leq T} \left(e^{r s} M_s \right) \right]^{\frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta} (e^{r T} M_T)^{r_2} e^{-r T} \mathbf{1}_{A} \right] \right\} \\
\leq& \max \left\{ \frac{a_1(a_1+2\zeta)}{2}-r, \frac{(a_1+b_2)(a_1+b_2+2\zeta)}{2}-r, -\frac{\zeta^2}{2}-r \right\}.
\end{aligned}\end{aligned}$$
We now show that the above bounds are either negative or not attainable. For the first bound $\frac{a_1(a_1+2\zeta)}{2}-r$, direct calculations lead to $$\frac{a_1(a_1+2\zeta)}{2} -r = -\frac{1}{2}\kappa^2 r_1 (1-r_1) -r = 0.$$ However, this zero bound can not be reached as the corresponding condition to attain the bound is $a_1 +\zeta < 0$. However, we have instead that $$a_1 +\zeta = -\kappa r_2 + \frac{\kappa}{2}
= \frac{1}{\kappa} \sqrt{(-\frac{\kappa^2}{2})^2+2 r \kappa^2} > 0.$$
We can show that the second upper bound $\frac{(a_1+b_1)(a_1+b_1+2 \zeta)}{2}-r$ is strictly negative. From Corollary A.7 of [@GHR], the conditions under which this bound can be attained are $a_1+b_1+\zeta>0$, and $2 a_1 + b_1 + 2 \zeta>0$. Recall that $\kappa>0$, we have that $$\begin{aligned}
& 2 a_1 + b_1 + 2 \zeta>0 \\
\Longleftrightarrow & ~ - 2 \kappa r_2 - \kappa(1-r_2) \beta + \kappa >0 \\
\Longleftrightarrow & ~ \beta < \frac{1-2r_2}{1-r_2}. \ \
\end{aligned}$$
Now direct computations yield that $$\begin{aligned}
& (a_1+b_1)(a_1+b_1+2 \zeta) - 2r \\
&= \kappa^2 [r_2 + (1-r_2)\beta] [r_2 + (1-r_2) \beta - 1] - 2r \\
&= \kappa^2 \left\{ r_2^2 + (1-r_2)^2 \beta^2 + 2 r_2 (1-r_2) \beta - r_2 - (1-r_2) \beta \right\} - 2r \\
&= \kappa^2 \left\{ (1-r_2)^2 \beta^2 + 2 r_2 (1-r_2) \beta - (1-r_2) \beta \right\} \\
&= \kappa^2 (1-r_2) \beta \left[ (1-r_2)\beta + 2 r_2 -1 \right] < 0.\ \
\end{aligned}$$
We can also derive that the upper bound $\frac{(a_1+b_2)(a_1+b_2+2\zeta)}{2}-r$ is strictly negative. Once again from Corollary A.7 of [@GHR], the conditions under which this bound can be attained are $a_1+b_2+\zeta>0$, and $2 a_1 + b_2 + 2 \zeta>0$. As $\kappa>0$, we equivalently need $$\begin{aligned}
& 2 a_1 + b_2 + 2 \zeta>0 \\
\Longleftrightarrow & ~ - 2 \kappa r_2 -\kappa \frac{\lambda(1-r_2)-(r_1-r_2)}{\lambda-1}\beta + \kappa >0 \\
\Longleftrightarrow & ~ \beta < \frac{1-2r_2}{\lambda^*}, \ \
\end{aligned}$$ where we define $\lambda^* := \frac{(r_1-r_2) - \lambda (1-r_2)}{1-\lambda}$. Noting that $\lambda^* > 0$, we can show by straightforward computations that $$\begin{aligned}
& (a_1+b_2)(a_1+b_2+2 \zeta) - 2r \\
&= \kappa^2 (r_2 + \lambda^*\beta) (r_2 + \lambda^* \beta - 1) - 2r \\
&= \kappa^2 ( r_2^2 + \lambda^{*2} \beta^2 + 2 r_2 \lambda^* \beta - r_2 - \lambda^* \beta ) - 2r \\
&= \kappa^2 ( \lambda^{*2} \beta^2 + 2 r_2 \lambda^* \beta - \lambda^* \beta ) \\
&= \kappa^2 \lambda^* \beta \left( \lambda^* \beta +2 r_2 - 1 \right) < 0.
\end{aligned}$$
Putting all the pieces together, we conclude that $\Upsilon_1(T)\rightarrow 0$ and $\Upsilon_2(T)\rightarrow 0$ as $T\rightarrow +\infty$.
For the last term, we have that $\lim_{T \rightarrow + \infty} \E_{} \Big[ e^{-r T} (Y_T(y))^{r_2} \mathbf{1}_{A^c} \Big]=0$ from Lemma \[lemma:BrownianMax\] below with $a= - \kappa r_2, b=0, \eta=\frac{r}{\kappa r_2} + \frac{1}{2} \kappa$.
Let us finally deal with the case $c_t=0$. It is clear that the associated term takes the order $O\left( e^{\lambda \beta h (1-r_2)} \right)$. By virtue of the condition $e^{\lambda \beta \hat{H}_T(y)} < e^{r T} y M_T$, it holds that $$e^{\lambda \beta \hat{H}_T(y) (1-r_2)} < (e^{r T} y M_T)^{1-r_2}.$$ It follows that $$e^{- r T} e^{\lambda \beta \hat{H}_T(y) (1-r_2)} (e^{r T} y M_T)^{r_2} < y M_T.$$ It has been shown that the expectation of the last term converges to $0$ as $T\rightarrow +\infty$, which completes the whole proof.
The following result has been used in the previous proof, which is essentially similar to Corollary A.7 of [@GHR]. We present it here for the completeness.
\[lemma:BrownianMax\] Let $B_t^{(\zeta)}=B_t+\zeta t$, where $B$ is a standard Brownian motion, $\left( B_t^{(\zeta)} \right)^*$ be the running maximum of $B_t^{(\zeta)}$. Then for any constant $a,b,k$ with $2a+b+2 \zeta \neq 0$, $k \geq 0$, we have $$\begin{aligned}
& \E \left[ e^{a B^{(\zeta)}_{T} + b \left( B^{(\zeta)}_{T} \right)^*} \mathbf{1}_{\left\{ \left( B^{(\zeta)}_{T} \right)^* \leq k \right\}} \right] \\
=& ~ \frac{2(a+b+c)}{2a +b +\zeta} \exp\left\{ \frac{(a+b)(a+b+2\zeta)}{2}T \right\} \left[ \Phi\left( (a+b+\zeta)\sqrt{T}\right) - \Phi\left( (a+b+\zeta)\sqrt{T} -\frac{k}{\sqrt{T}} \right) \right]\\
&+\frac{2(a+\zeta)}{2a+b+2 \zeta} \Bigg[ \exp \left\{ \frac{a(a+2\zeta)}{2}T \right\} \Phi\left( -(a+\zeta)\sqrt{T} \right)\\
&-\exp \left\{ (2a+b+2\zeta)k + \frac{a(a+2\zeta)}{2}T \right\} \Phi\left( -(a+\zeta)\sqrt{T} -\frac{k}{\sqrt{T}} \right) \Bigg].
\end{aligned}$$ In particular, we have that $$\lim_{T \rightarrow +\infty} \E \left[ e^{a B^{(\zeta)}_{T} + b \left( B^{(\zeta)}_{T} \right)^*} \mathbf{1}_{\left\{ \left( B^{(\zeta)}_{T} \right)^* \leq k \right\}} \right] = 0.$$
At last, we turn to prove the existence of the unique strong solution to the SDE for $X_t^*$. First, we need to establish the following results concerning the regularity of the feedback functions $c^*(x,h)$ and $\pi^*(x,h)$.
By the definition of $g$ in and the fact that $f(\cdot, h)$ is the inverse of $g(\cdot, h)$, we have the following results of the function $f$.
\[f\_regularity\] The function $f$ is $C^1$ within each of the subsets of $\mathbb{R}_+^2$: $x \leq x_1(h)$, $x_1(h) < x < x_2(h)$ and $x_2(h) \leq x \leq x_3(h)$, and it is continuous at the boundary of $x=x_2(h)$ and $x=x_3(h)$. Moreover, we have that: $$\begin{aligned}
\label{f_derivative_x}
&f_x(x,h) = \frac{1}{g_y(f,h)} \nonumber \\
=& \left\{
\begin{aligned}
&\left(-C_2(h) r_2 (r_2-1) (f_1(x,h))^{r_2-2} \right)^{-1} , & & \mbox{if } x \leq x_1(h), \\
&\Bigg(-C_3(h) r_1 (r_1-1) (f_2(x,h))^{r_1-2} - C_4(h) r_2 (r_2-1) (f_2(x,h))^{r_2-2} & &\\
&\ \ \ -\frac{1}{r\beta f_2(x,h) } \Bigg)^{-1} , & & \mbox{if } x_1(h) < x < x_2(h), \\
&\left(-C_5(h) r_1 (f_3(x,h))^{r_1-1} - C_6(h) r_2 (f_3(x,h))^{r_2-1} + \frac{1}{r}h \right)^{-1}, & & \mbox{if } x_2(h) \leq x \leq x_3(h),\\
\end{aligned}
\right.\end{aligned}$$ and $$\label{f_derivative_z}
f_h(x,h)= -g_h(f(x,h),h) \cdot f_x(x,h).$$
The proof the lemma is similar to [@ElieTouzi]. As the inverse of $g$, the function $f$ satisfies that $$\label{relation_g_f}
g(f(x,h),h)=x, ~~~\mathrm{for}~(x,h) \in \mathbb{R}_+^2.$$ By definition, the function $g(\cdot,h)$ and its inverse $f(\cdot,h)$ are $C^1$ and decreasing, for any $h > 0$. Direct computation leads to . From the definition of $g$ in , we can calculate the partial derivative $g_h$ explicitly. [As $g_h$ is clearly a continuous function in each of the closed intervals, it is bounded, ]{} i.e. $\exists$ a constant $\alpha > 0$, such that $g_h(x,h) \leq \alpha, \forall (x,h) \in \mathbb{R}^2_+$. Now in order to prove that $f$ is $C^1$ within each of the intervals $x \leq x_1(h)$, $x_1(h) < x < x_2(h)$ and $x_2(h) \leq x \leq x_3(h)$, we can verify that $f$ is differentiable in each variable with continuous partial derivative.
First, let us prove that $f \in C^0$ in each of the closed intervals, which implies that $f_x \in C^0$ in each of the closed intervals. Indeed, for a pair $(x,h)$ belonging to one of the intervals and a $l_2$ small enough, we have that $$g(f(x, h+l_2), h) -x = g(f(x, h+l_2), h) - g(f(x, h+l_2), h+l_2) \leq \alpha l_2 \underset{l_2 \rightarrow 0}{\longrightarrow} 0.$$ Now using the continuity of $f(\cdot, h)$, we obtain $$f(x,h+l_2) - f(x,h) = f(g(f(x, h+l_2),h),h) - f(x,h) \underset{l_2 \rightarrow 0}{\longrightarrow} 0.$$ Finally, for sufficiently small $l_1$, we have that $$f(x+l_1, h+l_2) - f(x,h) = f_x(x_l, h+l_2) l_1 + f(x, h+l_2) -f(x,h),$$ which will tend to $0$ when $l_1, l_2$ tend to $0$, and this shows that $f$ is continuous at an arbitrary point $(x,h)$.
Secondly, let us show that $f$ is differentiable with respect to $h$ with continuous partial derivatives. Let the pair $(x,h)$ in a certain interval and $l$ small enough such that $(x, h+l)$ is in the same interval. We have that $$\begin{aligned}
\frac{1}{l} \{ f(x, h+l) - f(x,h) \} &= \frac{1}{l} \{ f(x, h+l) - f(g(f(x,h),h+l),h+l) \} \\
&= f_x(x_l, h+l) \frac{1}{l} \{ g(f(x,h),h) - g(f(x,h), h+l) \},
\end{aligned}$$ for some $x_l \in [x, x+ g(f(x,h),h+l)]$. Since $f_x \in C^0$ and $g_h(f(x,h),\cdot)$ is continuous, we obtain $$\frac{1}{l} \{ f(x, h+l) - f(x,h) \} \underset{l \rightarrow 0}{\longrightarrow} - f_x(x,h) g_h(f(x,h),h),$$ which indicates . Now the continuity of $f_h$ follows from and the continuity of $f$.
\[Lipschitz\_c\_pi\] The functions $c^*$ and $\pi^*$ are Lipschitz on $\mathcal{C}$.
By and the dual fransformation relationship, we can express $c^*(x,h)$ in terms of the primal variables as in . Using Lemma \[f\_regularity\] which implies the $C^1$ regularity of $f$ and , together with the continuity of $f$ at the boundary between the three regions, we can draw the conclusion that $c^*(x,h)$ is Lipschitz on $\mathcal{C}$.
Recall that from Proposition \[dual\_value\_function\], the coefficients $C_2(h)$, $C_3(h)$, $C_4(h)$, $C_5(h)$ and $C_6(h)$ are $C^1$ in closed intervals and hence are Lipschitz. From Lemma \[f\_regularity\], we get the Lipschitz property of $f$ on $\mathcal{C}$. Now using in which $\pi^*(x,h)$ is expressed in terms of the primal variables, we can conclude that $\pi^*(x,h)$ is Lipschitz on $\mathcal{C}$.
\
We can proceed to prove the existence of strong solution with the optimal feedback.
The SDE has a unique strong solution $(X_t^*, H_t^*)$ for any initial condition $(x,h) \in \mathcal{C}$.
Let us introduce the functional $$G(t, x(t), h(t)) : = r x(t) + \pi^*_t(x(t), h(t)) (\mu -r) - c^*_t(x(t), h(t)),$$ and $$H(t, x(t), h(t)) := \pi^*_t(x(t), h(t)).$$ By Lemma \[Lipschitz\_c\_pi\] which implies the Lipschitz property of $c^*$ and $\pi^*$, we can easily derive that both $G$ and $H$ are Lipschitz functions. This justifies the existence of strong solution for the SDE .
Proofs of other results in Section \[sec:solveHJB\] {#otherproofs}
----------------------------------------------------
It is easy to verify that the general solution of the equation is given by $$\begin{aligned}
v(y) =\left\{
\begin{aligned}
&C_1 y^{r_1} + C_2 y^{r_2} - \frac{1}{r\beta}, & & \mbox{if } y \geq 1,\label{gensl-1}\\
&C_3 y^{r_1} + C_4 y^{r_2} + \frac{y}{r\beta}(\ln y + \frac{\kappa^2}{2r} - 1), & & \mbox{if } y < 1,
\end{aligned}
\right.\end{aligned}$$ where $C_1, C_2, C_3, C_4$ are some constants to be determined and $r_{1,2}$ are given in . Note that we have $r_1 > 1$ by a straightforward calculation.
In view of the general solution in , the boundary condition $v_y(y) \rightarrow -\infty$ as $y\rightarrow 0$ first implies that $C_4 \leq 0$. Then $v(y)-yv_y(y) \rightarrow 0$ as $y\rightarrow 0$ further gives that $C_4\equiv 0$.
Suppose the convergence $v_y(y) \rightarrow 0$ holds as $y \rightarrow y_0$ for some $y_0$. It is obvious that $y_0 > 1$ as $x=0$ belongs to the region $c^*(x)=0$. It thus follows that $$\left\{
\begin{aligned}
C_1 r_1 y_0^{r_1-1} & = - C_2 r_2 y_0^{r_2-1} , \\
C_1 (1-r_1) y_0^{r_1} & = - C_2 (1-r_2) y_0^{r_2} , \\
C_1 r_1(r_1-1)y_0^{r_1-1} & = - C_2 r_2 (r_2-1) y_0^{r_2-1} .
\end{aligned}
\right.$$ If $C_1 \neq 0$ and $C_2 \neq 0$, the first equation and the third equation lead to a obvious contradiction. If $C_1=0$ and $C_2=0$, we get another contradiction from the zero-order and first-order smooth-fitting condition along the boundary $y=1$. Noting $r_1>1$ and $r_2<0$, the only possible choice is $y_0= + \infty$ and $C_1\equiv 0$. By taking the inverse transform, we note that $y_0= + \infty$ and $C_1=0$ actually imply the hidden boundary condition that $u_x \rightarrow + \infty$ as $x \rightarrow 0$.
Substituting $C_1=0$ and $C_4=0$ back into the general solution and using the smooth-fitting condition at $y=1$, we get equations $$\left\{
\begin{aligned}
&C_2 - \frac{1}{r\beta} = C_3 - \frac{1}{r\beta} + \frac{\kappa^2}{2r^2\beta}, \\
&C_2 r_2 = C_3 r_1 + \frac{\kappa^2}{2r^2\beta},
\end{aligned}
\right.$$ which give constants $C_2$ and $C_3$ in .
We can first obtain the special solution $v(y,h)=-\frac{1}{r\beta} e^{\lambda\beta h}$ for the first equation, $v(y,h)=-\frac{y}{r\beta}+\frac{y}{r\beta} (\ln y-\lambda\beta h+\frac{\kappa^2}{2r} )$ for the second equation, and $v(y,h)=-\frac{1}{r}hy-\frac{1}{r\beta} e^{(\lambda-1)\beta h}$ for the third equation in . Therefore, we can summarize the general solution of the ODE by $$\begin{aligned}
\label{explicitdual}
v(y,h) =\left\{
\begin{aligned}
& C_1(h) y^{r_1} + C_2(h) y^{r_2} - \frac{1}{r\beta} e^{\lambda \beta h}, & & \mbox{if } y \geq e^{\lambda \beta h}, \\
& C_3(h) y^{r_1} + C_4(h) y^{r_2}-\frac{y}{r\beta}+\frac{y}{r\beta } \left(\ln y-\lambda\beta h+\frac{\kappa^2}{2r} \right), & & \mbox{if } e^{(\lambda-1)\beta h} < y < e^{\lambda \beta h}, \\
& C_5(h) y^{r_1} + C_6(h) y^{r_2} -\frac{1}{r}hy-\frac{1}{r\beta} e^{(\lambda-1)\beta h}, & & \mbox{if } (1-\lambda)e^{(\lambda-1)\beta h} \leq y \leq e^{(\lambda-1)\beta h},\\
\end{aligned}
\right.\end{aligned}$$ in which $C_i(h)$, $i=1,...,6,$ are functions of $h$ to be determined.
By virtue of the form of $v(y,h)$ in along the free boundary $y=(1-\lambda)e^{(\lambda-1)\beta h}$, the condition $v_h(\hat{y},h)=0$ in implies that $$\label{eq:boundary}
C_5'(h) (1-\lambda)^{r_1} e^{(\lambda-1)\beta hr_1} + C_6'(h) (1-\lambda)^{r_2} e^{(\lambda-1) \beta hr_2} = \Big(\frac{1}{r}-\frac{1}{r\beta}\Big)(1-\lambda) e^{(\lambda-1)\beta h}.$$
Similar to the case when $\lambda=0$, the free boundary condition $v_y(y,h)\rightarrow 0$ in implies that $y\rightarrow +\infty$. Together with free boundary conditions in and the formula of $v(y,h)$ in the region $y\geq e^{\lambda\beta h}$, we deduce that $C_1(h)\equiv 0$. Moreover, it is easy to see that as $h\rightarrow+\infty$, we get $y\rightarrow 0$ in the third region $(1-\lambda)e^{(\lambda-1)\beta h} \leq y \leq e^{(\lambda-1)\beta h}$ and therefore the boundary conditions in also implies the asymptotic condition that $C_6(h) \rightarrow 0$ as $h\rightarrow+\infty$.
To determine the left parameters, we apply the smooth-fit conditions with respect to the variable $y$ at three boundary points $y=e^{\lambda\beta h}$ and $y=e^{(\lambda-1)\beta h}$. After simple manipulations, we can deduce the system of equations: $$\left\{
\begin{aligned}
&C_2(h) e^{\lambda \beta h r_2} - \frac{1}{r\beta} e^{\lambda\beta h}=C_3(h) e^{\lambda\beta h r_1} + C_4(h) e^{\lambda \beta h r_2} -\frac{1}{r\beta}e^{\lambda \beta h}+\frac{1}{2r^2\beta}e^{\lambda\beta h} \kappa^2, \\
&C_2(h) r_2 e^{\lambda \beta h r_2} =C_3(h) r_1 e^{\lambda\beta h r_1} + C_4(h) r_2 e^{\lambda\beta h r_2} +\frac{1}{2r^2\beta}e^{\lambda\beta h} \kappa^2 , \\
&C_3(h) e^{(\lambda-1)\beta h r_1} + C_4(h) e^{(\lambda-1)\beta h r_2} +\frac{1}{2r^2\beta}e^{(\lambda-1)\beta h} \kappa^2 \\
&= C_5(h) e^{(\lambda-1) \beta h r_1} + C_6(h) e^{(\lambda-1)\beta h r_2} - \frac{1}{r\beta}e^{(\lambda-1)\beta h} , \\
& C_3(h) r_1 e^{(\lambda-1) \beta h r_1} + C_4(h) r_2 e^{(\lambda-1) \beta h r_2} +\frac{1}{2r^2\beta}e^{(\lambda-1)\beta h} \kappa^2 \\
&=C_5(h) r_1 e^{(\lambda-1) \beta h r_1} + C_6(h) r_2 e^{(\lambda-1) \beta h r_2}.
\end{aligned}
\right.$$ The system of equations above can be solved fully explicitly. To this end, the linear system can be regarded as linear equations in terms of variables $C_3(h)$, $C_2(h)-C_4(h)$, $C_4(h)-C_6(h)$ and $C_3(h)-C_5(h)$. We can solve the first two equations and obtain $C_3(h)$ explicitly in and $C_2(h)-C_4(h)$. By solving the last two equations, we also get $C_3(h)-C_5(h)$, which yields $C_5(h)$ in by substituting the function $C_3(h)$.
Plugging the derivative $C_5'(h)$ back into the boundary condition , we obtain that $$\begin{aligned}
C_6'(h) (1-\lambda)^{r_2} e^{(\lambda-1)\beta h r_2} =& (1-\lambda)^{r_1} e^{(\lambda-1)\beta h r_1} \frac{(r_2-1) \kappa^2}{2(r_1-r_2)\beta r^2} \\
&\times \left[(\lambda-1)(1-r_1) e^{(\lambda-1)(1-r_1)\beta h}-\lambda (1-r_1) e^{\lambda (1-r_1)\beta h}\right].
\end{aligned}$$ By using the asymptotic condition that $C_6(h) \rightarrow 0$ when $h \rightarrow + \infty$ and $\lambda(1-r_2)-(r_1-r_2) < 0$, we can integrate the equation above on both sides, and get $C_6(h)$ explicitly in .
Substituting $C_6(h)$ back to $$(r_1-r_2)(C_6(h)-C_4(h)) e^{(\lambda-1)\beta hr_2}= (r_1-1) \frac{e^{(\lambda-1)\beta h}}{2r^2\beta} \kappa^2,$$ we can get $C_4(h)$ in . Substituting $C_4(h)$ to the equation that $$(r_1-r_2)(C_2(h)-C_4(h)) e^{\lambda \beta h r_2} = (r_1-1) \frac{e^{\lambda \beta h}}{2r^2\beta} \kappa^2,$$ we can at last obtain $C_2(h)$ in .
\
We shall analyze each region separately.
\(i) In the region $y \geq e^{\lambda \beta h}$, $v_{yy}(y,h) = r_2(r_2-1)C_2(h) y^{r_2-2}$, as $r_2(r_2-1)=\frac{2r}{\kappa^2}>0$ and $C_2(h)>0$ from its expression , we draw the conclusion easily.
\(ii) In the region $(1-\lambda)e^{(\lambda-1)\beta h} \leq y \leq e^{(\lambda-1)\beta h}$, $v_{yy}(y,h) = r_1(r_1-1)C_5(h) y^{r_1-2} + r_2(r_2-1)C_6(h) y^{r_2-2}$, the conclusion follows by the fact that $C_5(h)>0$, $C_6(h)>0$ and the identity that $r_1(r_1-1)=r_2(r_2-1)= \frac{2r}{\kappa^2}>0$.
\(iii) In the region $e^{(\lambda-1)\beta h} < y < e^{\lambda \beta h}$, we can proceed in the following two steps:
*Step 1*: We can first equivalently check that $y v_{yy}(y,h)>0$ at the two endpoints $e^{(\lambda-1)\beta h}$ and $e^{\lambda \beta h}$, i.e. $$\frac{2r}{\kappa^2} y^{r_2-1} [ C_3(h) y^{r_1-r_2} + C_4(h) ] + \frac{1}{\beta r} > 0.$$ Using the expression of $C_3(h)$ and $C_4(h)$, at the point $e^{\lambda \beta h}$ this boils down to prove that $$\begin{aligned}
& e^{\lambda \beta h (r_2-1)} \frac{1}{(r_1-r_2) \beta} \Bigg\{ e^{\lambda \beta h (1-r_2)} \frac{r_2-1}{r} + e^{(\lambda-1) \beta h (1-r_2)} \frac{1-r_1}{r} + \frac{(r_2-1) (1-\lambda)^{r_1-r_2}}{r} \\
& \times \left[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h} -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \right] \Bigg\}+\frac{1}{r \beta} > 0.\end{aligned}$$ Using the fact that $e^{\lambda \beta h (1-r_2)} > e^{(\lambda-1) \beta h (1-r_2)}$, the above is larger than $$\begin{aligned}
& e^{\lambda \beta h (r_2-1)} \frac{1}{(r_1-r_2) \beta} \Big\{ e^{\lambda \beta h (1-r_2)} \frac{r_2-1}{r}+ e^{\lambda \beta h (1-r_2)} \frac{1-r_1}{r} \Big\} +\frac{1}{r \beta} - e^{\lambda \beta h (r_2-1)} \\
&\times \frac{(1-r_2) (1-\lambda)^{r_1-r_2}}{(r_1-r_2) \beta r} \left[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h} -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \right]\\
&=- e^{\lambda \beta h (r_2-1)} \frac{(1-r_2) (1-\lambda)^{r_1-r_2}}{(r_1-r_2) \beta r} \Bigg[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h}\\
&\quad -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \Bigg],\end{aligned}$$ which is strictly positive. Hence $y v_{yy}(y,h)>0$ at the point $e^{\lambda \beta h}$.
For the point $e^{(\lambda-1)\beta h}$, similar as before, it is enough to show that $$\begin{aligned}
&e^{(\lambda-1) \beta h (r_2-1)} \frac{1}{(r_1-r_2) \beta} \Bigg\{ e^{[\lambda(1-r_2)-(r_1-r_2)] \beta h } \frac{r_2-1}{r} + e^{(\lambda-1) \beta h (1-r_2)} \frac{1-r_1}{r} + \frac{(r_2-1) (1-\lambda)^{r_1-r_2}}{r} \\
& \times \left[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h} -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \right] \Bigg\}+\frac{1}{r \beta} > 0.\end{aligned}$$ Using $e^{[\lambda(1-r_2)-(r_1-r_2)] \beta h } < e^{(\lambda-1) \beta h (1-r_2)}$, similar calculation as at the point $e^{\lambda \beta h}$ shows that the above term is also strictly larger than $$- e^{\lambda \beta h (r_2-1)} \frac{(1-r_2) (1-\lambda)^{r_1-r_2}}{(r_1-r_2) \beta r} \left[ \frac{1-r_1}{1-r_2} e^{(\lambda-1)(1-r_2)\beta h} -\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)} e^{\left[ \lambda(1-r_2)-(r_1-r_2) \right]\beta h} \right]>0,$$ and hence is strictly positive.
*Step 2*: In this step, we show that the function $$\gamma(y):=y v_{yy}(y,h)= \frac{2r}{\kappa^2} C_3(h) y^{r_1-1} + \frac{2r}{\kappa^2} C_4(h) y^{r_2-1} + \frac{1}{r \beta}$$ is either monotone or first increasing then decreasing. Combining with Step 1, this guarantees the statement of the lemma. Indeed, the extreme point $y^*$ of $\gamma(y)$ should satisfy the first order condition $\gamma'(y^*)=0$, i.e. $$C_3(h) (r_1-1) (y^*)^{r_1-r_2} + C_4(h) (r_2-1) = 0.$$ We remark that $C_3(h) < 0$, while $C_4(h)$ can be negative or positive. If $C_4(h) \leq 0$, there is no solution for $y^*$, hence $\gamma(y)$ is monotone. If $C_4(h) >0$, there exists a unique real solution to the above equation $$y^*=\left( \frac{C_4(h)(1-r_2)}{C_3(h) (r_1-1)} \right)^{\frac{1}{r_1-r_2}},$$ which might fall into the interval $[e^{(\lambda-1)\beta h}, e^{\lambda \beta h}]$. Noticing that $C_3(h) < 0$, and $$\gamma'(y)=\frac{2r}{\kappa^2} y^{r_2-2} \left( C_3(h) (r_1-1) (y)^{r_1-r_2} + C_4(h) (r_2-1) \right),$$ it follows that when $y \leq y^*$, $\gamma'(y) \geq 0$; when $y \geq y^*$, $\gamma'(y) \leq 0$. Hence $\gamma(y)$ increases before reaching $y^*$, then decreases after passing $y^*$.
\
We first verify the conclusion for $\lambda=0$. In view of $c^*(x)$ in , it is sufficient to check $\lim_{x\rightarrow+\infty}\frac{\ln g_1(x)}{x}$. As $g_1(x)$ satisfies the equation $x=-C_3r_1(g_1(x))^{r_1-1}-\frac{1}{r\beta}\left(\ln g_1(x)+\frac{\kappa^2}{2r}\right)$ and $\lim_{x\rightarrow+\infty}g_1(x)=0$, it is easy to see that $\lim_{x\rightarrow+\infty}\frac{\ln g_1(x)}{x}=-r\beta$ and hence $\lim_{x\rightarrow+\infty}\frac{c^*(x)}{x}=r$ follows. By $\lim_{x\rightarrow+\infty}g_1(x)=0$ again, we also get $\lim_{x\rightarrow+\infty}\pi^*(x)=\frac{\mu-r}{r\beta\sigma^2}$ using the feedback form in .
For the case $\lambda>0$, as we consider the asymptotic behavior along the boundary $x_3(h)$, we first have $$\begin{aligned}
\lim_{\substack{x\rightarrow+\infty,\\(x,h)\in x_3(h)}}\frac{c^*(x,h)}{x}=\lim_{h\rightarrow+\infty}\frac{h}{x_3(h)}.\end{aligned}$$ Taking into account the explicit form of $x_3(h)$ in , we need to compute two limits $$\begin{aligned}
\lim_{h\rightarrow+\infty} \frac{-C_5(h)r_1(1-\lambda)^{r_1-1}e^{(\lambda-1)(r_1-1)\beta h}}{h}=\lim_{h\rightarrow+\infty}\frac{\frac{-r_1(1-\lambda)^{r_1-1}(1-r_2)\kappa^2}{2(r_1-r_2)\beta r^2}[1-e^{(1-r_1)\beta h}]}{h}=0,\end{aligned}$$ and $$\begin{aligned}
&\lim_{h\rightarrow+\infty} \frac{-C_6(h)r_2(1-\lambda)^{r_2-1}e^{(\lambda-1)(r_2-1)\beta h}}{h}\\
=&\lim_{h\rightarrow+\infty}\frac{\frac{-r_2(1-\lambda)^{r_1-1}(r_2-1)\kappa^2}{2(r_1-r_2)\beta r^2}\left[\frac{1-r_1}{1-r_2}-\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)}e^{(1-r_1)\beta h}\right]}{h}=0.\end{aligned}$$ Therefore, we obtain that $$\lim_{\substack{x\rightarrow+\infty,\\(x,h)\in x_3(h)}}\frac{c^*(x,h)}{x}=r.$$ Similarly, thanks to the explicit form of $\pi^*(x,h)$ in , we need to compute two limits along $x_3(h)$ that $$\begin{aligned}
\lim_{h\rightarrow+\infty}\frac{2r}{\kappa^2}C_5(h)(1-\lambda)^{r_1-1}e^{(\lambda-1)\beta h (r_1-1)}&=\lim_{h\rightarrow+\infty}\frac{(1-\lambda)^{r_1-1}(1-r_2)}{(r_1-r_2)\beta r}[1-e^{(1-r_1)\beta h}]\\
&=\frac{(1-\lambda)^{r_1-1}(1-r_2)}{(r_1-r_2)\beta r},\end{aligned}$$ and $$\begin{aligned}
&\lim_{h\rightarrow+\infty}\frac{2r}{\kappa^2}C_6(h)(1-\lambda)^{r_2-1}e^{(\lambda-1)(r_2-1)\beta h}\\
=&\lim_{h\rightarrow+\infty}\frac{(1-\lambda)^{r_1-1}(r_2-1)}{(r_1-r_2)\beta r}\left[\frac{1-r_1}{1-r_2}-\frac{\lambda(1-r_1)}{\lambda(1-r_2)-(r_1-r_2)}e^{(1-r_1)\beta h}\right]
=\frac{(1-\lambda)^{r_1-1}(r_1-1)}{(r_1-r_2)\beta r}.\end{aligned}$$ Therefore, we conclude that $$\begin{aligned}
\lim_{\substack{x\rightarrow+\infty,\\ (x,h)\in x_3(h)}}\pi^*(x,h)=\frac{\mu-r}{\sigma^2}\left(\frac{(1-\lambda)^{r_1-1}(1-r_2)}{(r_1-r_2)\beta r}+ \frac{(1-\lambda)^{r_1-1}(r_1-1)}{(r_1-r_2)\beta r}\right) =\frac{(\mu-r)(1-\lambda)^{r_1-1}}{r\beta\sigma^2}.\end{aligned}$$
\
\
**Acknowledgements**: H. Pham and X. Yu appreciate the financial support by the PROCORE-France/Hong Kong Joint Research Scheme under no. F-PolyU501/17. X. Yu is partially supported by the Hong Kong Early Career Scheme under grant no. 25302116. X. Li is partially supported by the Hong Kong General Research Fund under grant no. 15213218 and no. 15215319. \
\
[^1]: Department of Mathematics, University of Michigan, Ann Arbor, USA. Email:`[email protected]`
[^2]: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. Email:`[email protected]`
[^3]: LPSM, Université de Paris and CREST-ENSAE, Paris, France. Email:`[email protected]`
[^4]: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. Email:`[email protected]`
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $\chi(g)\neq 0$ for all irreducible characters $\chi$ of $G$. Such an element is said to be *non-vanishing in* $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.
**Keywords** Finite groups $\cdot$ Normal subgroups $\cdot$ Irreducible characters $\cdot$ Conjugacy classes
**2010 MSC** 20C15 $\cdot$ 20E45
author:
- 'M. J. Felipe $^{*}$ $\cdot$ N. Grittini $^{\ddagger}$ $\cdot$ V. Sotomayor [^1]'
title: '**On zeros of irreducible characters lying in a normal subgroup**'
---
Introduction
============
In the sequel, all groups considered are finite. Within character theory, a classical theorem of Burnside asserts that a non-linear irreducible character of a finite group always vanishes on some element. It is not difficult to see that the converse is also true, so the rows of the character table of a group that contain a zero entry are completely characterised. However, the “dual” situation for conjugacy classes fails in general: a column that corresponds to a non-central conjugacy class may not contain a zero. This fact somehow violates the standard duality that in many cases arises between irreducible characters and conjugacy classes of a group. Therefore, for a group $G$, an element $g$ is said to be *non-vanishing* in $G$ if $\chi(g)\neq 0$ for every irreducible character $\chi$ of $G$.
An immediate corollary to the aforementioned Burnside’s result is that a group is abelian if and only if every element is non-vanishing. I.M. Isaacs, G. Navarro and T.R. Wolf obtained in [@INW] elegant results about the location of non-vanishing elements in certain groups. For example, for a nilpotent group $G$, an element is non-vanishing if and only if it lies in the centre of $G$. They also proved that if $G$ is soluble, then $g{\pmb{{\operatorname}{F}}(G)}$ is a $2$-element for a non-vanishing element $g$ of $G$. Consequently, if $g$ is of odd order, then $x$ lies in ${\pmb{{\operatorname}{F}}(G)}$. These authors conjectured that every non-vanishing element of a soluble group $G$ lies in ${\pmb{{\operatorname}{F}}(G)}$, and it is still an open problem. In this paper, we prove the following result which provides further evidence for this conjecture.
\[teoA\] Let $N$ be a normal subgroup of a group $G$, and let $p$ be a prime. If $\chi(x)\neq 0$ for every $p$-element $x\in N$ and for all $\chi\in{\operatorname}{Irr}(G)$, then $N$ has a normal Sylow $p$-subgroup.
In particular, if $\chi(x)\neq 0$ for every prime power order element $x\in N$ and for all $\chi\in{\operatorname}{Irr}(G)$, then $N$ is nilpotent.
Therefore, the arithmetical properties of the non-vanishing elements of $G$ that lie in a normal subgroup $N$ control the structure of $N$. This is interesting since, although we cannot construct the character table of $N$ from the one of $G$, normal subgroups and vanishing elements of $G$ can be easily read from its character table.
Regarding the first claim of Theorem \[teoA\] when $N=G$, we provide extra information on the structure of a $p$-complement of $G$ in Corollary \[cor-dpss\], which extends [@DPSS Theorem A]. Concerning the second assertion in Theorem \[teoA\], when $N=G$ it holds that the group is abelian (see Theorem \[carac\]). However, this fact might not happen for the case of a normal subgroup as Example \[suz\] shows.
Actually, we prove Theorem \[teoA\] as a consequence of the next result. We will denote by $1_G$ the trivial character of a group $G$.
\[TEOB\] Let $N$ be a normal subgroup of a group $G$, and let $P$ be a Sylow $p$-subgroup of $G$ for some prime $p$. Let $P_0=P\cap N$ and $\beta\in{{{\operatorname}{Irr}}(P/P_0)}$. Then the following conditions are pairwise equivalent:
*(i)* $P_0$ is a normal Sylow $p$-subgroup of $N$.
*(ii)* $\chi(x)\neq 0$ for all irreducible constituents $\chi$ of $(1_{P_0})^G$ and all $x\in P_0$.
*(iii)* $\chi(x)\neq 0$ for all irreducible constituents $\chi$ of $\beta^G$ and all $x\in P_0$.
Indeed Theorem \[TEOB\] generalises [@MN Theorem B] when $N=G$ (see Theorem \[MN\]). Notice that, by Theorem \[MN\] (i)-(ii), $P_0$ is normal in $N$ if and only if $\eta(x)\neq 0$ for all irreducible constituents $\eta$ of $(1_{P_0})^N$ and all $x\in P_0$; however this fact does not directly imply (ii) of Theorem \[TEOB\], nor vice versa. Further, the following equivalence, which is related to Theorem \[MN\] (i)-(iii), is not true: $P_0$ is normal in $N$ if and only if $p$ does not divide $\chi(1)$ for all irreducible constituents of $(1_{P_0})^G$; it is enough to observe Example \[degree\] (2).
As a consequence of Theorem \[teoA\], some features of a normal subgroup $N$ of a group $G$ can be obtained through the analysis of its $G$-conjugacy class sizes of elements which are zeros of some irreducible character of $G$; such an element is said to be *vanishing* in $G$. Let $\pi$ be any set of primes, and let $x^G$ be the conjugacy class of $x$ in $G$. If ${\ensuremath{\left| x^G \right|}}$ is a $\pi'$-number for every prime power order $\pi$-element $x$ in $N$, then by the main result of A. Beltrán et al. (see [@BFM Theorem B]), it is known that $N$ has a nilpotent Hall $\pi$-subgroup. In the next result, we show that we do not need to assume this condition for all prime power order $\pi$-elements in $N$, but for those which are vanishing in $G$. However, we need to assume the $\pi$-separabilty of $N$ to get that result.
\[teoC\] Let $N$ be a normal subgroup of a group $G$, and let $\pi$ be any set of prime numbers.
1. Suppose that ${\ensuremath{\left| x^G \right|}}$ is a $\pi'$-number for every prime power order $\pi$-element $x \in N$ which is vanishing in $G$. If $N$ is $\pi$-separable, then $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$ has a nilpotent normal Hall $\pi$-subgroup. In particular, the Hall $\pi$-subgroups of $N$ are nilpotent.
2. Suppose that ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number for every prime power order $\pi$-element $x \in N$ which is vanishing in $G$. If ${{\operatorname}{Hall}_{\pi}\left(N\right)}\neq \emptyset$, then $N$ has a normal Hall $\pi$-subgroup.
Additionally, if all ${\ensuremath{\left| x^G \right|}}$ are also $\pi$-numbers for the prime power order $\pi'$-elements $x\in N$ that are vanishing in $G$, then the Hall $\pi'$-subgroups of $N$ are nilpotent.
We do not know whether the $\pi$-separability condition on $N$ in (1) can be weakened in order to obtain the nilpotency of its Hall $\pi$-subgroups. What is certainly true is that this condition is necessary for the normality of the Hall $\pi$-subgroup of $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$, as Example \[pi-separa\] shows. Additionally, the statement (2) above extends for a set of primes the following result proved in [@BQ]: If a prime $p$ does not divide any conjugacy class size of a vanishing $p'$-element $x$ of prime power order of a group $G$, then $G$ has a normal $p$-complement. We do not know whether the assumption ${{\operatorname}{Hall}_{\pi}\left(N\right)}\neq \emptyset$ in (2) can be avoided.
Finally, we investigate in Theorem \[teoE\] the structure of $N$ when the $G$-conjugacy class lenghts of the considered vanishing elements in $G$ are prime powers. As a consequence of this study, when $N=G$ we obtain the next result.
\[cor\_nilpotent\] Let $G$ be a group. Assume that ${\ensuremath{\left| x^G \right|}}$ is a prime power for every vanishing element $x$ of $G$ of prime power order. Then $G'$ is nilpotent.
Other new interesting consequences arise from our contributions in the trivial case $N=G$ (see Section \[secCor\]).
Preliminaries
=============
The notation and terminology here is as follows. In the sequel, $p$ will be always a prime, and $\pi$ will denote a set of primes. The set of prime divisors of the order of $G$ is $\pi(G)$. As usual, the set of all Sylow $p$-subgroups of $G$ is denoted by ${{\operatorname}{Syl}_{p}\left(G\right)}$, and ${{\operatorname}{Hall}_{\pi}\left(G\right)}$ is the set of all Hall $\pi$-subgroups of $G$. We write ${\operatorname}{Irr}(G)$ for the set of all irreducible complex characters of $G$. The set of vanishing elements of a group $G$ will be denoted by ${{{\operatorname}{Van}}(G)}$. CFSG means classification of finite simple groups. The remaining notation and terminology is standard in the framework of finite group theory, and we refer to the book [@ISA] for details about character theory.
We gather some significant results for locating vanishing elements in a given group. As mentioned in the Introduction, it is elementary to see that a group is abelian if and only if every element is non-vanishing. In fact, this characterisation remains true, via the CFSG, when only prime power order elements are involved.
\[carac\] A group $G$ is abelian if and only if every prime power order element is non-vanishing in $G$.
This is a direct application of [@MNO Theorem B], which asserts that a non-linear irreducible character vanishes on some prime power order element.
\[suz\] Concerning the above theorem it is worth noting that, in general, it is not true that a normal subgroup $N$ is abelian if and only if every element of $N$ is non-vanishing in $G$, i.e. if $N\cap {{{\operatorname}{Van}}(G)}=\emptyset$:
On the one hand, if $G=Q_8$ is a quaternion group of $8$ elements and $N$ is a normal subgroup of $G$ isomorphic to a cyclic group of order $4$, then $N$ is abelian and $N\cap{{{\operatorname}{Van}}(G)}\neq \emptyset$. On the other hand, by [@INW Theorem 5.1], for any prime $p$ there exists a group $G$ having a normal non-abelian Sylow $p$-subgroup, and every $p$-element of $G$ is non-vanishing.
*[@INW Theorem B]* \[nilp\] $G\smallsetminus{\pmb{{\operatorname}{Z}}(G)}= {{{\operatorname}{Van}}(G)}$ for any nilpotent group $G$.
Observe that if a normal subgroup $N$ is nilpotent, then $N\smallsetminus{\pmb{{\operatorname}{Z}}(G)}$ may not coincide with ${{{\operatorname}{Van}}(G)}\cap N$. For instance, one can consider as $G$ the normaliser in a Suzuki group of degree 8 of a Sylow $2$-subgroup of it, and $N$ the Sylow $2$-subgroup. It holds that ${{{\operatorname}{Van}}(G)}\cap N=\emptyset$ although clearly $N\smallsetminus{\pmb{{\operatorname}{Z}}(G)}\neq \emptyset$.
*[@G Corollary 1.3]* \[gru\] Let $H$ be a subgroup of a group $G$. Assume that $G=H{\pmb{{\operatorname}{C}}_{G}(x)}$ for some $x\in H$. Then $x\in {{{\operatorname}{Van}}(G)}$ if and only if $x\in {{{\operatorname}{Van}}(H)}$.
The four lemmas below are crucial for proving Theorem \[teoB\], and the last two use the CFSG.
*[@BCLP Lemma 5]* \[BianchiLemma\] Let $N$ be a minimal normal subgroup of $G$ so that $N = S_1 \times \cdots \times S_t$, where $S_i$ is isomorphic to $S$, a non-abelian simple group. If $\sigma\in{{{\operatorname}{Irr}}(S)}$ extends to ${\operatorname}{Aut}(S)$, then $\sigma \times \cdots \times \sigma \in {{{\operatorname}{Irr}}(N)}$ extends to $G$.
*[@MN Lemma 2.2]* \[2.2MN\] Let $G$ be a finite group, $p$ a prime, and $P\in{{\operatorname}{Syl}_{p}\left(G\right)}$. If $\chi\in{{{\operatorname}{Irr}}(G)}$ has $p$-defect zero, then $\chi$ is a constituent of $(1_P)^G$ and vanishes on the non-trivial $p$-elements of $G$.
*[@MN Theorem 2.1]* \[2.1MN\] Let $S$ be a finite non-abelian simple group, $p$ a prime, and $P\in{{\operatorname}{Syl}_{p}\left(S\right)}$. Then either $S$ has a $p$-defect zero character, or there exists a constituent $\theta \in {{{\operatorname}{Irr}}(S)}$ of the permutation character $(1_P)^S$ such that $\theta$ extends to ${\operatorname}{Aut}(S)$ and $\theta(x)=0$ for some $p$-element $x$ of $S$.
*[@DPSS Lemma 2.8]* \[contradiction\_lemma\] Let $A$ be an abelian group that acts coprimely and faithfully by automorphisms on a group $M$. If $M$ is characteristically simple, then there exists $\theta \in {\operatorname}{Irr}(M)$ such that $I_A(\theta)=1$.
We also collect some preliminary results regarding conjugacy class sizes. We start with the next elementary properties which are frequently used, sometimes with no comment.
Let $N$ be a normal subgroup of a group $G$, and let $p$ be a prime. We have:
*(a)* ${\ensuremath{\left| x^N \right|}}$ divides ${\ensuremath{\left| x^G \right|}}$, for any $x\in N$.
*(b)* ${\ensuremath{\left| (xN)^{G/N} \right|}}$ divides ${\ensuremath{\left| x^G \right|}}$, for any $x\in G$.
*(c)* If $xN\in G/N$ is a $p$-element, then $xN=yN$ for some $p$-element $y\in G$.
\[wielandt\] Let $N$ be a normal subgroup of a group $G$, and let $H\in{{\operatorname}{Hall}_{\pi}\left(N\right)}$ for a set of primes $\pi$. If $x\in H$ is such that ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number, then $x$ lies in ${\pmb{{\operatorname}{O}}_{\pi}(N)}$.
Since ${\ensuremath{\left| x^N \right|}}$ divides ${\ensuremath{\left| x^G \right|}}$, then $({\ensuremath{\left| x^N \right|}}, {\ensuremath{\left| N:H \right|}})=1$. It follows $N=H{\pmb{{\operatorname}{C}}_{N}(x)}$ and so $\langle x^N\rangle \leqslant {\pmb{{\operatorname}{O}}_{\pi}(N)}$.
Next we recall a generalisation of the above lemma when $N=G$ and $\pi=\{p\}$.
*[@BK Lemma 3]* \[berkokazarin\] Let $x\in G$. If ${\ensuremath{\left| x^G \right|}}$ is a power of a prime $p$, then $[x^G, x^G]$ is a $p$-group.
We end this section with the main result of [@Br], which will be necessary for proving Theorem \[teoE\]. We present here an adapted version for our context of vanishing $G$-conjugacy classes.
\[brough\] Let $G$ be a group which contains a non-trivial normal $p$-subgroup $N$, for a given prime $p$. Then ${\ensuremath{\left| x^G \right|}}$ is a multiple of $p$ for each $x\in N\cap {{{\operatorname}{Van}}(G)}$.
Proof of Theorems \[teoA\] and \[TEOB\]
=======================================
Certainly, Theorem \[teoA\] is a direct application of Theorem \[TEOB\], so we focus on the proof of this last result. The next key proposition, which makes use of the CFSG, is inspired by the proof of [@MN Theorem B].
\[lemma:minimalsubgroup\] Let $M$ be a non-abelian minimal normal subgroup of a group $G$, and let $p$ be a prime divisor of ${\ensuremath{\left| M \right|}}$. Let $H$ be a subgroup of $G$ such that $H \cap M\in{{\operatorname}{Syl}_{p}\left(M\right)}$. Let $\beta\in{{{\operatorname}{Irr}}(H/H\cap M)}$ Then, there exists $\chi \in {\operatorname}{Irr}(G)$ such that $\chi$ is a constituent of $\beta^G$ and it vanishes on some $p$-element of $M$.
In particular, if $H=P\in{{\operatorname}{Syl}_{p}\left(G\right)}$, then there exists $\chi\in {\operatorname}{Irr}(G)$ such that $\chi$ is a constituent of $(1_P)^G$ and it vanishes on some $p$-element of $M$.
We have $M=S_1 \times \cdots \times S_k$, where all $S_i$ are isomorphic to a non-abelian simple group $S$ with $p\in\pi(S)$. If $\theta\in{{{\operatorname}{Irr}}(S)}$ is of $p$-defect zero, then $\eta=\theta \times \cdots \times \theta\in{{{\operatorname}{Irr}}(M)}$ and $\eta$ is also of $p$-defect zero. By Lemma \[2.2MN\] applied to $M$ we have $[\eta, (1_{H\cap M})^M]\neq 0$ and $\eta$ vanishes on the non-trivial $p$-elements of $M$.
Since $\beta\in{{{\operatorname}{Irr}}(H/H\cap M)}$, we have $[\beta_{H\cap M}, 1_{H\cap M}]\neq 0$. Then $(\beta^{HM})_M=(\beta_{H\cap M})^M=\beta(1)(1_{H\cap M})^M$ and $[\eta, (\beta^{HM})_M]=[\eta^{HM}, \beta^{HM}]\neq 0$. Hence there exists $\tau\in{{{\operatorname}{Irr}}(HM)}$ such that $[\tau, \eta^{HM}]\neq 0 \neq [\tau, \beta^{HM}]$. Let $\chi\in{{{\operatorname}{Irr}}(G)}$ over $\tau$. Then $\chi_M$ is sum of $G$-conjugate characters of $\eta$. Therefore $\chi$ vanishes on the non-trivial $p$-elements of $M$ and $[\chi, \beta^G]=[\chi_H, \beta]\neq 0$.
Suppose now that $S$ does not have a character of $p$-defect zero. By Lemma \[2.1MN\], there exists $\theta\in{{{\operatorname}{Irr}}(S)}$ such that $[\theta, (1_{H\cap S})^S]\neq 0$ (note $H\cap S\in{{\operatorname}{Syl}_{p}\left(S\right)}$) which extends to ${\operatorname}{Aut}(S)$, and there exists a $p$-element $x\in S$ such that $\theta(x)=0$. Thus $1\neq y=(x, \ldots, x)\in M$ is a $p$-element and $\eta=\theta\times\cdots\times\theta$ vanishes on $y$, and certainly $[\eta_{H\cap M}, 1_{H\cap M}]\neq 0$. Since $[\beta_{H\cap M}, 1_{H\cap M}]\neq 0$, arguing as in the previous paragraph, we may affirm that there exists $\tau \in{{{\operatorname}{Irr}}(HM)}$ over $\eta$ and over $\beta$. Let $\chi\in{{{\operatorname}{Irr}}(G)}$ be over $\eta$, so $[\chi, \beta^G]\neq 0$. By Lemma \[BianchiLemma\], $\eta$ extends to $G$. Let $\hat{\eta}$ be an extension of $\eta$. By Gallagher, $\chi=\hat{\eta}\rho$ for some $\rho\in{{{\operatorname}{Irr}}(G/M)}$. Therefore, $\chi$ lies over $\beta$ and $\chi(y)=\eta(y)\rho(1)=0$.
\[teoB\] Let $N$ be a normal subgroup of a group $G$, and let $P_0$ be a Sylow $p$-subgroup of $N$ for some prime $p$. Let $H$ be a subgroup of $G$ such that $H\cap N=P_0$, and let $\beta \in {{{\operatorname}{Irr}}(H/P_0)}$. Then, $P_0$ is normal in $N$ (and therefore in $G$) if and only if all irreducible constituents of $\beta^G$ do not vanish on any $p$-element of $N$.
Suppose $P_0\unlhd N$. Let $\chi$ be a constituent of $\beta^G$ with $\beta\in{{{\operatorname}{Irr}}(H/H\cap N)}$. We have $[\beta_{P_0}, 1_{P_0}]\neq 0$, so $[\chi_{P_0},1_{P_0}]\neq 0$. Since $P_0\unlhd G$, then $\chi(x)\neq 0$ for all $p$-elements $x\in N$. Conversely, we consider that all irreducible constituents of $\beta^G$, where $\beta\in{{{\operatorname}{Irr}}(H/H\cap N)}$, do not vanish on any $p$-element of $N$, and we claim that $P_0$ is normal in $N$.
Suppose that the claim is false, and let us consider a counterexample which minimises ${\ensuremath{\left| G \right|}}$. Let $M$ be a minimal normal subgroup of $G$ such that $M\leqslant N$. We check that the hypotheses are inherited by $\overline{G}=G/M$. Certainly $\overline{H}\cap \overline{N}=N/M\cap HM/M=(H\cap N)M/M\in{{\operatorname}{Syl}_{p}\left(N/M\right)}$. Since $\beta\in{{{\operatorname}{Irr}}(H/H\cap N)}$, then $\beta \in{{{\operatorname}{Irr}}(H/H\cap M)}$ so $\overline{\beta}\in{{{\operatorname}{Irr}}(HM/M)}$. Besides $H\cap N\leqslant {\operatorname}{ker}{\beta}$ so $\overline{H\cap N}\leqslant {\operatorname}{ker}{\overline{\beta}}$. Let $\overline{\chi}\in{{{\operatorname}{Irr}}(\overline{G})}$ be an irreducible constituent of $\overline{\beta}^{\overline{G}}$ and $\overline{x}\in\overline{N}$ a $p$-element. Then we may assume that $x\in N\smallsetminus M$ is a $p$-element and, since $[\overline{\chi}, \overline{\beta}^{\overline{G}}]\neq 0$, then it is easy to see that $[\chi_H, \beta]\neq 0$ and $\overline{\chi}(\overline{x})=\chi(x)\neq 0$. By minimality, we get $\overline{P_0}\unlhd \overline{G}$, so $P_0M\unlhd G$.
Let us assume that $p$ divides the order of $M$. If $M$ is a $p$-group, then $M\leqslant P_0$ and $P_0=P_0M\unlhd G$, a contradiction. Hence $M$ is non-abelian. Since $\beta\in{{{\operatorname}{Irr}}(H/H\cap M)}$, in virtue of Lemma \[lemma:minimalsubgroup\] there exists $\chi\in{{{\operatorname}{Irr}}(G)}$ such that $[\chi, \beta^G]\neq 0$ and $\chi(x)=0$ for some $p$-element $x\in M\leqslant N$, a contradiction again.
Thus $p$ does not divide the order of $M$ and ${\pmb{{\operatorname}{O}}_{p}(N)}=1$. Let $K/M$ be a chief factor of $G$ such that $K \leq P_0M\unlhd G$, so $K/M$ is an abelian $p$-group. Note $K=M(K\cap P_0)$ and $K\cap P_0\in{{\operatorname}{Syl}_{p}\left(K\right)}$ is abelian. By Frattini’s argument, $G=K{\pmb{{\operatorname}{N}}_{G}(K\cap P_0)}=M{\pmb{{\operatorname}{N}}_{G}(K\cap P_0)}$, so ${\pmb{{\operatorname}{C}}_{K\cap P_0}(M)}\unlhd G$ and ${\pmb{{\operatorname}{C}}_{K\cap P_0}(M)}\leqslant{\pmb{{\operatorname}{O}}_{p}(N)}=1$. Therefore $K\cap P_0$ is an abelian $p$-group which acts coprimely and faithfully on $M$, and $M$ is characteristically simple. By Lemma \[contradiction\_lemma\] and Clifford theory, there exists $\theta \in {\operatorname}{Irr}(M)$ such that $\eta =\theta^K$ is irreducible. In particular, $\eta$ and all its conjugates vanish on $K \setminus M$. Therefore, if we prove that there exists $\chi \in {\operatorname}{Irr}(G)$ which lies over both $\eta$ and $\beta$ we will reach the final contradiction.
Let $T$ be the inertia subgroup for $\theta$ in $P_0M\unlhd G$. Since $({\ensuremath{\left| T/M \right|}},{\ensuremath{\left| M \right|}})=1$ we have that $\theta$ extends to $\hat{\theta} \in {\operatorname}{Irr}(T)$ by [@ISA Corollary 6.28]. Further, $p$ does not divide $\hat{\theta}(1)$ so $\hat{\theta}_{P_0 \cap T}$ has at least one linear constituent $\lambda$. As $T=M(P_0\cap T)$, then $P_0 \cap T \cong T/M$ and we can see $\lambda$ also as a character of $T/M$. By Gallagher, $\nu = \bar{\lambda}\hat{\theta}$ is an irreducible character of $T$, where $\bar{\lambda}$ is the complex conjugate of $\lambda$. Moreover, $\nu_M=\theta$ and by Clifford correspondence $\nu^{P_0M}\in{{{\operatorname}{Irr}}(P_0M)}$. Hence $0\neq [1_{P_0\cap T}, \overline{\lambda}_{P_0\cap T}\hat{\theta}_{P_0\cap T}]=[1_{P_0\cap T}, \nu_{P_0\cap T}]=[(\nu_{P_0\cap T})^{P_0}, 1_{P_0}]=[(\nu^{P_0T})_{P_0}, 1_{P_0}]=[(\nu^{P_0M})_{P_0}, 1_{P_0}]=[\nu^{P_0M}, (1_{P_0})^{P_0M}]$. On the other hand, $(\beta^{HN})_N = \beta(1)(1_{P_0})^N=\beta(1)((1_{P_0})^{P_0M})^N$, so $[(\beta^{HN})_N, (\nu^{P_0M})^N]=[(\beta^{HN})_N, \nu^N]=[\beta^{HN}, \nu^{HN}]\neq 0$. Therefore there exists $\tau\in{{{\operatorname}{Irr}}(HN)}$ over $\beta$ and over $\nu$. Let $\chi\in{{{\operatorname}{Irr}}(G)}$ over $\tau$, so $[\chi, \beta^G]\neq 0$. Moreover, $\chi$ lies over $\theta$, and then $\chi$ lies over $\eta =\hat{\theta}$. Thus $\chi_K$ is a sum of $G$-conjugate characters of $\eta$. Hence $\chi(x)=0$ for all $x\in K\cap P_0$ and this is a final contradiction.
Theorem \[TEOB\] in the Introduction is now a corollary of the above result when we take $H$ a Sylow $p$-subgroup of $G$ (for Theorem \[TEOB\] (iii)) and $H=P_0$ (for Theorem \[TEOB\] (ii)). Moreover, when $N=G$ in Theorem \[TEOB\], then we obtain the characterisation (i)-(ii) below.
*[@MN Theorem B]* \[MN\] Let $G$ be a group, $p$ a prime number, and $P$ a Sylow $p$-subgroup of $G$. Then the following conditions are equivalent:
*(i)* $P$ is normal in $G$.
*(ii)* $\chi(x)\neq 0$ for all irreducible constituents $\chi$ of $(1_P)^G$ and all $x\in P$.
*(iii)* $p$ does not divide $\chi(1)$ for all irreducible constituents $\chi$ of $(1_P)^G$.
\[degree\] (1) Note that in Theorem \[TEOB\] we can have $\beta\in{{{\operatorname}{Irr}}(P/P_0)}$ distinct from $1_{P}$, in contrast to Theorem \[MN\]: Let $G$ be a symmetric group of degree $4$ and let $N$ be an alternating group of degree $4$. Take $P\in{{\operatorname}{Syl}_{2}\left(G\right)}$. Then there exists a non-trivial irreducible character $ \beta\in{{{\operatorname}{Irr}}(P)}$ with $P_0=P\cap N\leqslant {\operatorname}{ker}{\beta}$. Additionally, the irreducible constituents of $\beta^G$ do not vanish on the $p$-elements of $N$, so the hypotheses in Theorem \[TEOB\] (iii) are fulfilled.
\(2) The following equivalence, similar to Theorem \[MN\] (i)-(iii), is not true: $P_0$ is a normal Sylow $p$-subgroup of $N$ if and only if $p$ does not divide $\chi(1)$ for all irreducible constituents of $(1_{P_0})^G$: Consider $G$ and $N$ as above. Then $(1_{P_0})^G$ has three distinct irreducible constituents, being one of them of degree $2$.
Both examples have been checked using the software `GAP` [@GAP].
Let consider now a set of primes $\pi$ instead of a single prime $p$. As a consequence of Theorem \[teoA\], we give in the following proposition extra information on the structure of a $\pi$-complement of $G$ when $N$ contains a Hall $\pi$-subgroup of it.
\[nilphall\] Let $N$ be a normal subgroup of a group $G$ such that every prime power order $\pi$-element of $N$ is non-vanishing in $G$, for a set of primes $\pi$. Then $N$ has a nilpotent normal Hall $\pi$-subgroup.
Further, if ${\ensuremath{\left| G:N \right|}}$ is a $\pi'$-number, then any $\pi$-complement $F$ of $G$ verifies that $F{\pmb{{\operatorname}{Z}}(G)}$ is self-normalising.
Certainly, in virtue of Theorem \[teoA\] we have that $N$ has a nilpotent normal Hall $\pi$-subgroup, say $H$. In fact, if ${\ensuremath{\left| G:N \right|}}$ is not divisible by any prime in $\pi$, then $H$ is a normal Hall $\pi$-subgroup of $G$. Let $F$ be a $\pi$-complement of $H$ in $G$, so $G=HF$. We aim to show that $F{\pmb{{\operatorname}{Z}}(G)}={\pmb{{\operatorname}{N}}_{G}(F{\pmb{{\operatorname}{Z}}(G)})}$. Take a prime power order element $x\in{\pmb{{\operatorname}{N}}_{H}(F{\pmb{{\operatorname}{Z}}(G)})}$. Then $F^x{\pmb{{\operatorname}{Z}}(G)}=(F{\pmb{{\operatorname}{Z}}(G)})^x=F{\pmb{{\operatorname}{Z}}(G)}$, so there exists some $y\in F{\pmb{{\operatorname}{Z}}(G)}$ such that $F^x=F^y=F$. Thus $x\in {\pmb{{\operatorname}{N}}_{H}(F)}\leqslant {\pmb{{\operatorname}{C}}_{H}(F)}$ because $[{\pmb{{\operatorname}{N}}_{H}(F)}, F]\leqslant H\cap F=1$. Therefore $G=HF=H{\pmb{{\operatorname}{C}}_{G}(x)}$. Since $x\notin {{{\operatorname}{Van}}(G)}$ by assumption, then Lemma \[gru\] yields that $x\notin {{{\operatorname}{Van}}(H)}$. Now Proposition \[nilp\] applies because $H$ is nilpotent, so $x\in {\pmb{{\operatorname}{Z}}(H)}\cap{\pmb{{\operatorname}{C}}_{G}(F)}\leqslant{\pmb{{\operatorname}{Z}}(G)}$. As this argument is valid for every prime power order element in ${\pmb{{\operatorname}{N}}_{H}(F{\pmb{{\operatorname}{Z}}(G)})}$, then ${\pmb{{\operatorname}{N}}_{H}(F{\pmb{{\operatorname}{Z}}(G)})}\leqslant {\pmb{{\operatorname}{Z}}(G)}$. Finally, note that ${\pmb{{\operatorname}{N}}_{G}(F{\pmb{{\operatorname}{Z}}(G)})}= {\pmb{{\operatorname}{N}}_{G}(F{\pmb{{\operatorname}{Z}}(G)})} \cap HF = F({\pmb{{\operatorname}{N}}_{H}(F{\pmb{{\operatorname}{Z}}(G)})})=F{\pmb{{\operatorname}{Z}}(G)}$, as wanted.
\[cor-dpss\] Let $G$ be a group such that all the $p$-elements are non-vanishing. Then $G$ has a normal Sylow $p$-subgroup, and $F{\pmb{{\operatorname}{Z}}(G)}$ is self-normalising for any $p$-complement $F$ of $G$.
Lenghts of G-conjugacy classes of vanishing elements
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We start by showing an extension of Lemma \[berkokazarin\] for a set of primes $\pi$ and a $G$-conjugacy class. The proof is inspired by [@BF Theorem C] under the weaker hypothesis of the $\pi$-separability of the normal subgroup $N$.
\[in\_fitting\] Let $N$ be a normal $\pi$-separable subgroup of a group $G$. If $x \in N$ is such that ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number, then $[x^G,x^G] \leqslant {\pmb{{\operatorname}{O}}_{\pi}(N)}$. In particular, $x{\pmb{{\operatorname}{O}}_{\pi}(N)}/{\pmb{{\operatorname}{O}}_{\pi}(N)}\in{\pmb{{\operatorname}{Z}}({\pmb{{\operatorname}{F}}(N/{\pmb{{\operatorname}{O}}_{\pi}(N)})})}$.
Indeed, if $\pi$ consists of a single prime $p$, then the same statement is valid even if $N$ is not $p$-soluble.
In order to prove the first claim, let us consider a counterexample which minimises ${\ensuremath{\left| G \right|}}+{\ensuremath{\left| N \right|}}$. One can clearly assume ${\pmb{{\operatorname}{O}}_{\pi}(N)}=1$, so we aim to get the contradiction $[x^G,x^G]=1$. Let us suppose firstly that $\langle x \rangle$ is subnormal in $G$. Then $x \in {\pmb{{\operatorname}{F}}(G)}$. As ${\pmb{{\operatorname}{F}}(G)}$ is a $\pi'$-group and ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number, then clearly $x \in {\pmb{{\operatorname}{Z}}({\pmb{{\operatorname}{F}}(G)})}$ and $\langle x^G\rangle \leqslant {\pmb{{\operatorname}{Z}}({\pmb{{\operatorname}{F}}(G)})}$, so $[x^G,x^G]=1$.
Next we assume that the normal subgroup $M:=\langle x^G\rangle$ is proper in $N$. Then by minimality we obtain $[x^M,x^M]=1$, and it follows that $x \in {\pmb{{\operatorname}{Z}}(\langle x^M \rangle)}$. In particular, $\langle x \rangle$ is subnormal in $M$, and therefore in $G$, which contradicts the previous paragraph. Hence $M=N$.
Let $K:={\pmb{{\operatorname}{O}}_{\pi'}(N)}$. Since $N$ is $\pi$-separable, then $K$ is non-trivial. It follows from the class size hypothesis that $K$ centralises $x^G$, so $K$ is central in $N=\langle x^G\rangle$. As $[x^G,x^G]K/K \leqslant {\pmb{{\operatorname}{O}}_{\pi}(N/K)}$ by minimality, and ${\pmb{{\operatorname}{O}}_{\pi}(N/K)}={\pmb{{\operatorname}{O}}_{\pi}(N)}K/K$ because $K$ is central $N$, we deduce $[x^G,x^G]=N' \leqslant K\leqslant {\pmb{{\operatorname}{Z}}(N)}$. Therefore $N$ is a nilpotent $\pi'$-group. Since ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number, we obtain $x\in {\pmb{{\operatorname}{Z}}(N)}$ and $[x^G,x^G]=1$.
Next we concentrate on the second assertion. Let $\overline{G}:=G/{\pmb{{\operatorname}{O}}_{\pi}(N)}$. Then, $[\overline{x}^{\overline{G}},\overline{x}^{\overline{G}}]=1$ by the first claim. It follows that $\langle \overline{x}\rangle \unlhd {\pmb{{\operatorname}{Z}}(\langle \overline{x}^{\overline{G}} \rangle)} \unlhd \overline{G}$, so $\langle \overline{x}\rangle\leqslant {\pmb{{\operatorname}{F}}(\overline{G})} \cap \overline{N} \leqslant {\pmb{{\operatorname}{F}}(\overline{N})}$. As ${\pmb{{\operatorname}{F}}(\overline{N})}$ is a normal $\pi'$-subgroup of $\overline{G}$ and $|\overline{x}^{\overline{G}}|$ is a $\pi$-number, then necessarily $\overline{x}\in {\pmb{{\operatorname}{Z}}({\pmb{{\operatorname}{F}}(\overline{N})})}$.
Finally, observe that the last statement follows from Lemma \[berkokazarin\], since $[x^G, x^G]\leqslant {\pmb{{\operatorname}{O}}_{p}(G)}\cap N\leqslant {\pmb{{\operatorname}{O}}_{p}(N)}$.
Note that the $\pi$-separability assumption in the previous result cannot be removed, even when $N=G$: Consider any non-trivial element in the centre of a Sylow $p$-subgroup of a non-abelian simple group and $\pi=p'$, for a prime divisor $p$ of its order.
For a normal subgroup $N$ of a group $G$, note that if $xN$ is a vanishing (prime power order) element of $G/N$, then we can assume that $x$ is also a vanishing (prime power order) element of $G$. This is because there exists a bijection between ${\operatorname}{Irr}(G/N)$ and the set of all characters in ${\operatorname}{Irr}(G)$ containing $N$ in their kernel. This fact will be used in the sequel with no reference.
As an application of the above proposition and mainly Theorem \[teoA\], we prove Theorem \[teoC\] in the Introduction.
\(1) Assume that $N$ is $\pi$-separable, and that ${\ensuremath{\left| x^G \right|}}$ is a $\pi'$-number for every prime power order $\pi$-element $x \in {{{\operatorname}{Van}}(G)} \cap N$. Let us prove that $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$ has a normal Sylow $p$-subgroup for each prime $p\in\pi$. Certainly, whenever ${\pmb{{\operatorname}{O}}_{\pi'}(N)}\neq 1$, the assertion follows by induction, considering the groups $G/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$ and $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$. Therefore we may assume that ${\pmb{{\operatorname}{O}}_{\pi'}(N)}=1$. Let $Z_p:={\pmb{{\operatorname}{Z}}({\pmb{{\operatorname}{O}}_{p}(N)})}$. In virtue of Proposition \[in\_fitting\], we have that all the $p$-elements of ${{{\operatorname}{Van}}(G)} \cap N$ lie in ${\pmb{{\operatorname}{Z}}({\pmb{{\operatorname}{F}}(N)})}$, and thus in $Z_p$. Therefore, if we denote $\overline{G}:=G/Z_p$, then it follows that no prime power order $p$-element of $\overline{N}$ is vanishing in $\overline{G}$. Now Theorem \[teoA\] yields that $\overline{N}$ has a normal Sylow $p$-subgroup $\overline{P}$, where $P\in{{\operatorname}{Syl}_{p}\left(N\right)}$. Since $Z_p$ is a $p$-group, then $P$ is normal in $N$ clearly and we get the claim. As this is valid for each prime $p\in \pi$, then $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$ has a nilpotent normal Hall $\pi$-subgroup, as wanted.
\(2) Assume that $N$ has Hall $\pi$-subgroups, and that ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number for every prime power order $\pi$-element $x \in {{{\operatorname}{Van}}(G)} \cap N$. We claim that $N$ has a normal Hall $\pi$-subgroup. Clearly we may assume ${\pmb{{\operatorname}{O}}_{\pi}(N)}=1$. Let $H\in{{\operatorname}{Hall}_{\pi}\left(N\right)}$, and let $p\in \pi$. If $x\in N\cap{{{\operatorname}{Van}}(G)}$ is a $p$-element, then $x\in P\in{{\operatorname}{Syl}_{p}\left(N\right)}$. Hence there exists $g\in N$ such that $x^g \in P^g\in{{\operatorname}{Syl}_{p}\left(H\right)}$. Now Lemma \[wielandt\] yields $x^g\in {\pmb{{\operatorname}{O}}_{\pi}(N)}=1$. Thus there are no $p$-elements in $N\cap {\operatorname}{Van}(G)$, and by Theorem \[teoA\] we get that $N$ has a normal Sylow $p$-subgroup. Since this is valid for every prime $p\in \pi$, then $N$ has a (nilpotent) normal Hall $\pi$-subgroup, as desired.
Next we show that $N$ has nilpotent Hall $\pi'$-subgroups under the additional assumption that the prime power order $\pi'$-elements in $N\cap {{{\operatorname}{Van}}(G)}$ have also $G$-class sizes not divisible by any prime in $\pi'$. Note that $N$ is $\pi$-separable because it has a normal Hall $\pi$-subgroup, say $H$. If we take any prime power order element $xH\in (N/H) \cap {{{\operatorname}{Van}}(G/H)}$, then we may suppose that $x\in N\cap {{{\operatorname}{Van}}(G)}$ is a prime power order element, so by assumptions ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number. Thus ${\ensuremath{\left| (xH)^{G/H} \right|}}$ is also a $\pi$-number. Therefore every ${\ensuremath{\left| (xH)^{G/H} \right|}}$ is a $\pi$-number for each prime power order $\pi'$-element $xH\in (N/H) \cap {{{\operatorname}{Van}}(G/H)}$, so by assertion (1) the $\pi'$-group $N/H$ is nilpotent. Since $N/H$ is isomorphic to a Hall $\pi'$-subgroup of $N$, the proof is completed.
\[pi-separa\] We remark that the $\pi$-separability assumption in Theorem \[teoC\] (1) is necessary for the first claim. Let $G$ be a symmetric group of degree $5$, and let $N$ be an alternating group of degree $5$. Consider $\pi=\{3\}$. Then all the $3$-elements in $N\cap {{{\operatorname}{Van}}(G)}$ have conjugacy class size equal to $20$. Nevertheless, $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}=N$ does not have a normal Sylow $3$-subgroup.
It is not difficult to find groups satisfying the assumptions of Theorem \[teoC\]. For instance, let $G=A\Gamma(2^3)$ be an affine semilinear group of order $168$, and let $N$ be the Hall $3'$-subgroup of $G$. If we consider $\pi=\{7\}$, then the pair $(N, G)$ satisfies the hypotheses of Theorem \[teoC\] (1). Concerning Theorem \[teoC\] (2), if $\pi$ is any set of prime numbers, $G={\pmb{{\operatorname}{O}}_{\pi}(G)}\times{\pmb{{\operatorname}{O}}_{\pi'}(G)}$ and $N={\pmb{{\operatorname}{O}}_{\pi}(G)}$, then the pair $(N, G)$ certainly holds the hypotheses.
The next theorem combines the arithmetical conditions of Theorem \[teoC\] on the vanishing $G$-class sizes.
\[pi-pi’\] Let $N$ be a normal $\pi$-separable subgroup of a group $G$. Assume that ${\ensuremath{\left| x^G \right|}}$ is either a $\pi$-number or a $\pi'$-number for every prime power order $\pi$-element $x \in {{{\operatorname}{Van}}(G)} \cap N$. Then $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$ has a normal Hall $\pi$-subgroup. Thus $N$ has $\pi$-length at most 1.
First, we claim that $O:={\pmb{{\operatorname}{O}}_{\pi, \pi'}(N)}$ contains a Sylow $p$-subgroup of $N$, for a prime $p\in \pi$. Let $x \in {{{\operatorname}{Van}}(G)} \cap N$ be a $p$-element. If ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number, then $x$ lies in ${\pmb{{\operatorname}{O}}_{\pi}(N)}$ because of Lemma \[wielandt\], so clearly $x\in O$. If ${\ensuremath{\left| x^G \right|}}$ is a $\pi'$-number, then by Proposition \[in\_fitting\] we get $x{\pmb{{\operatorname}{O}}_{\pi'}(N)}\in {\pmb{{\operatorname}{F}}(N/{\pmb{{\operatorname}{O}}_{\pi'}(N)})}$, and again $x$ lies in $O$. It follows that $\overline{N}:=N/O$ contains no vanishing $p$-element of $G/O$, so $\overline{N}$ has a normal Sylow $p$-subgroup $\overline{P}$ in virtue of Theorem \[teoA\]. Since $p \in \pi$ and clearly ${\pmb{{\operatorname}{O}}_{\pi}(\overline{N})}=1$, thus $\overline{P}=1$.
Therefore $O$ contains a Sylow $p$-subgroup of $N$ for every $p\in\pi$, and thus $O/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$ is a Hall $\pi$-subgroup of $N/{\pmb{{\operatorname}{O}}_{\pi'}(N)}$.
The main theorem of [@BF] examines groups such that all their $\pi$-elements have prime power class sizes. The next result is a “vanishing version” of that theorem for prime power order elements and in the context of $G$-conjugacy classes.
\[teoE\] Let $N$ be a normal subgroup of a group $G$. Assume that ${\ensuremath{\left| x^G \right|}}$ is a prime power for each prime power order $\pi$-element $x \in N$ that is vanishing in $G$. Then $N/{\pmb{{\operatorname}{O}}_{\pi'}({\pmb{{\operatorname}{F}}(N)})}$ has a normal Hall $\pi$-subgroup.
In particular, if $\pi$ is the set of prime divisors of ${\ensuremath{\left| N \right|}}$, then $N/{\pmb{{\operatorname}{F}}(N)}$ is nilpotent.
We claim that $\overline{N}:=N/{\pmb{{\operatorname}{F}}(N)}$ has a normal Hall $\pi$-subgroup, and therefore $N/{\pmb{{\operatorname}{O}}_{\pi'}({\pmb{{\operatorname}{F}}(N)})}$ so does because ${\pmb{{\operatorname}{F}}(N)}/{\pmb{{\operatorname}{O}}_{\pi'}({\pmb{{\operatorname}{F}}(N)})}$ is a $\pi$-group. Arguing by contradiction, and in virtue of Proposition \[nilphall\], we may assume that $\overline{N}\cap {{{\operatorname}{Van}}(\overline{G})}$ contains a non-trivial $q$-element for some prime $q\in \pi$, say $\overline{x}$. Hence we may suppose that $x\in (N\cap {{{\operatorname}{Van}}(G)})\smallsetminus {\pmb{{\operatorname}{F}}(N)}$ is a $q$-element. By assumptions, we have that ${\ensuremath{\left| x^G \right|}}$ is a power of some prime $p$. Observe that, since $x\notin{\pmb{{\operatorname}{F}}(N)}$, then $q\neq p$ due to Lemma \[wielandt\]. Now the last statement of Proposition \[in\_fitting\] yields $(\langle x^G \rangle)'\leqslant {\pmb{{\operatorname}{O}}_{p}(N)}\leqslant {\pmb{{\operatorname}{F}}(N)}$, so $\overline{\langle x \rangle}$ is a subnormal nilpotent subgroup of $\overline{N}$. It follows that $\overline{x}\in{\pmb{{\operatorname}{F}}(\overline{N})}$, and as $\overline{x}$ is a $q$-element, then $\overline{x}\in {\pmb{{\operatorname}{O}}_{q}(\overline{N})}$. Now ${\ensuremath{\left| \overline{x}^{\overline{G}} \right|}}$ is a multiple of $q$ by Proposition \[brough\], and then ${\ensuremath{\left| x^G \right|}}$ so is, a contradiction.
Finally, if $\pi=\pi(N)$, then with a similar argument we deduce that there is no prime power order element in $N/{\pmb{{\operatorname}{F}}(N)}$ vanishing in $G/{\pmb{{\operatorname}{F}}(N)}$. Hence Theorem \[teoA\] applies and $N/{\pmb{{\operatorname}{F}}(N)}$ is nilpotent.
Some consequences on vanishing conjugacy classes {#secCor}
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New interesting contributions on the lengths of vanishing classes of a group $G$ emerge from Theorem \[teoC\], Theorem \[pi-pi’\] and Theorem \[teoE\] when $N=G$.
\[cor\_van\_pi’\] Let $G$ be a $\pi$-separable group. If ${\ensuremath{\left| x^G \right|}}$ is a $\pi'$-number for every prime power order $\pi$-element $x \in {{{\operatorname}{Van}}(G)}$, then $G/{\pmb{{\operatorname}{O}}_{\pi'}(G)}$ has a nilpotent normal Hall $\pi$-subgroup. Therefore, $G$ has nilpotent Hall $\pi$-subgroups, and its $\pi$-length is at most 1.
\[cor\_van\_pi\] Let $G$ be a finite group such that ${{\operatorname}{Hall}_{\pi}\left(G\right)}\neq \emptyset$. Assume ${\ensuremath{\left| x^G \right|}}$ is a $\pi$-number for every prime power order $\pi$-element $x\in {\operatorname}{Van}(G)$. Then $G$ has a normal Hall $\pi$-subgroup.
Further, if the prime power order $\pi'$-elements in ${{{\operatorname}{Van}}(G)}$ have also class size a $\pi$-number, then the Hall $\pi'$-subgroups of $G$ are nilpotent.
\[corE\] Let $G$ be a group. Suppose that ${\ensuremath{\left| x^G \right|}}$ is either a $\pi$-number or a $\pi'$-number for every prime power order $\pi$-element $x \in {{{\operatorname}{Van}}(G)}$. Then $G/{\pmb{{\operatorname}{O}}_{\pi'}({\pmb{{\operatorname}{F}}(G)})}$ has a normal Hall $\pi$-subgroup. In particular, $G$ has $\pi$-length at most 1.
Arguing as in the proof of Theorem \[teoE\] we can see that $G/{\pmb{{\operatorname}{F}}(G)}$ has no prime power order vanishing elements. Thus, Theorem \[carac\] applies and $G/{\pmb{{\operatorname}{F}}(G)}$ is abelian, so $G'$ is nilpotent.
**Acknowledgements:** This research has been carried out during a stay of the second author at the Instituto Universitario de Matemática Pura y Aplicada (IUMPA-UPV) of the Universitat Politècnica de València. He wishes to thank the members of the IUMPA for their hospitality. The authors would like to thank G. Navarro for useful conversations during the preparation of the paper.
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The GAP Group:
GAP – Groups, Algorithms, and Programming, <http://www.gap-system.org>, Version 4.10.0, 2018.
[^1]: Instituto Universitario de Matemática Pura y Aplicada (IUMPA-UPV), Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain. $^{\ddagger}$Dipartimento di Matematica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy.: `[email protected]`, `[email protected]`, `[email protected]`
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The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain). The research of the second author is partially funded by the Istituto Nazionale di Alta Matematica - INdAM. The third author acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain). The first and third authors are also supported by Proyecto PGC2018-096872-B-I00, Ministerio de Ciencia, Innovación y Universidades (Spain).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the phase transition dynamics of a quasi-2D antiferromagnetic spin-1 Bose-Einstein condensate from the easy-axis polar phase to the easy-plane polar phase, which is initiated by suddenly changing the sign of the quadratic Zeeman energy $q$. We observe the emergence and decay of spin turbulence and the formation of half-quantum vortices (HQVs) in the quenched condensate. The characteristic time and length scales of the turbulence generation dynamics are proportional to $|q|^{-1/2}$ as inherited from the dynamic instability of the initial state. In the evolution of the spin turbulence, spin wave excitations develop from large to small length scales, suggesting a direct energy cascade, and the spin population for the axial polar domains exhibit a nonexponential decay. The final equilibrated condensate contains HQVs, and the number is found to increase and saturate with increasing $|q|$. Our results demonstrate the time-space scaling properties of the phase transition dynamics near the critical point and the peculiarities of the spin turbulence state of the antiferromagnetic spinor condensate.'
author:
- Seji Kang
- Sang Won Seo
- Joon Hyun Kim
- 'Y. Shin'
title: 'Emergence and scaling of spin turbulence in quenched antiferromagnetic spinor Bose-Einstein condensates'
---
Introduction
============
The far-from-equilibrium dynamics of many-body quantum systems is a challenging subject to study in modern physics [@Review] and relevant to several areas from cosmology [@Traschen90] to condensed matter physics [@Fausti11]. Ultracold atomic gases provide a highly controllable platform for studying many-body physics [@Bloch_rmp08] and a quench protocol is typically employed to explore nonequilibrium dynamics [@Polkovnikov_rmp], where a system is prepared in a well-defined initial state and then its evolution is precisely examined after the system’s Hamiltonian is rapidly changed. One of the key topics in current research activities is quantum phase transition dynamics, which addresses the important question of how a many-body system evolves into a newly ordered quantum state. Recently, it was theoretically proposed that scaling behavior occurs in the phase transition dynamics near quantum critical points [@Lamacraft_prl; @Uhlmann_prl07; @Damski_prl07; @Rossini09; @DallaTorre13; @Karl_scirep] and the existence of scaling and universality was indeed demonstrated in various experiments of quantum phase transitions [@Braun_pnas; @Nicklas_prl; @Anquez16; @Clark_sci].
In this paper, we report on an experimental study of a quantum phase transition of a quasi-2D antiferromagnetic spin-1 Bose-Einstein condensate (BEC). The ground state of the antiferromagnetic BEC is a polar state with $\langle \textbf{F} \rangle =0$, where $\textbf{F}=(F_x,F_y,F_z)$ is the hyperfine spin operator of the atoms [@Kawaguchi_rev; @StamperK_rev]. The spin state is parametrized with a unit vector $\vec{d}=(d_x,d_y,d_z)$, called a spin nematic director, such that the system is in the $|m_F=0\rangle$ state for the quantization axis along $\vec{d}$. In an external magnetic field $\vec{B}$ (e.g., along the $z$ direction), an uniaxial spin anisotropy is imposed on the system due to the quadratic Zeeman energy $E_Z=q\langle F_z^2\rangle=q(1-d_z^2)$ and the BEC has two ground-state phases depending on the sign of $q$: the easy-axis polar (EAP) phase with $\vec{d}\parallel \hat{z}$ for $q>0$ and the easy-plane polar (EPP) phase with $\vec{d}\perp \hat{z}$ for $q<0$. Here, we investigate the EAP-to-EPP phase transition dynamics by preparing a quasi-2D BEC with a uniform spin texture with $\vec{d}\parallel \hat{z}$ for positive $q$ \[Fig. 1(a)\] and then suddenly changing the sign of $q$ to negative. In the transition to the EPP phase, the continuous spin-rotation symmetry in the $xy$ plane is spontaneously broken and topological point defects, which are half-quantum vortices (HQVs) in the EPP phase [@Zhou_prl01; @Zhou_IJMPB03; @Seo_prl15; @Seo_prl16], can be created in the spatially extended 2D system \[Fig. 1(b)\].
The dynamic instability of the initial EAP state for $q<0$ was demonstrated in experiments with elongated BECs [@Bookjans_prl; @Vinit_arxiv]. The Bogoliubov analysis of the EAP state gives two degenerate magnon modes with energy spectra of $E_k=\sqrt{(\epsilon_k +q)(\epsilon_k +q +2 c_2 n)}$ [@Kawaguchi_rev], where $\epsilon_k=\hbar^2 k^2 /(2m)$ is the single-particle spectrum ($m$ is the atomic mass), $c_2$ is the spin interaction coefficient, and $n$ is the atomic density. For small negative $q$ such that $|q|\ll c_2n$, the magnon modes with $k <k_q\equiv\sqrt{2|q|m}/\hbar$ have imaginary frequencies for $E_k^2 <0$, which means that small fluctuations in the transverse magnetization would be exponentially amplified in the EAP state. The dynamic instability rate is given by the maximum magnitude of the imaginary frequencies, $\Gamma_q=2|E_{k=0}|/\hbar \approx \sqrt{8|q|c_2 n}/\hbar$. Thus, we can define the time and length scales of $t_q = 2\pi/\Gamma_q\propto |q|^{-1/2}$ and $l_q= 1/k_q\propto |q|^{-1/2}$ to characterize the instability of the initial EAP state after the quench. The primary focus of this work is to determine how the intrinsic time-space scaling of the initial state near the critical point at $q=0$ is inherited and transformed in the subsequent phase transition dynamics.
{width="17.0cm"}
We investigate the scaling properties of the phase transition dynamics by measuring the spin texture evolution of the quenched BEC. We observe that spin turbulence emerges, which is featured by an irregular spin texture involving both a spatially disordered pattern of $\vec{d}$ and ferromagnetic spin excitations. We show that the turbulence generation dynamics are characterized by the time-space scaling of $\sim |q|^{-1/2}$ predicted for the initial EAP state. Furthermore, in the evolution of the spin turbulence, we observe that its spatial structure develops from large to small length scales, suggesting a direct energy cascade of spin wave excitations, and that the spin population for the axial polar domains exhibits a nonexponential decay. Finally, we observe that HQVs are created in the phase transition dynamics and find that the HQV number of the final equilibriated sample increases and saturates as $|q|$ increases.
The paper is organized as follows. In Sec. II, we describe the experimental procedures for sample preparation and spin texture imaging. In Sec. III, we present the characterization of the emergence and decay of spin turbulence in the quenched BEC and discuss the scaling properties of the phase transition dynamics. A summary and outlook are provided in Sec. IV.
Experiments
===========
Our experiment begins by preparing a BEC of $^{23}$Na atoms in the $|F=1, m_F=0\rangle$ state [@Seo_prl15]. The condensate is confined in an optical potential with trapping frequencies of $(\omega_x, \omega_y, \omega_z)=2\pi\times(3.8, 5.5, 400)$ Hz. The condensate contains $N^c\approx 8.0\times 10^6$ atoms and its Thomas-Fermi radii are $(R_x, R_y, R_z)\approx(232, 160, 2.2)~\mu$m. The sample preparation is carried out in a magnetic field of $B_z=0.5$ G. Initially, the sample only has the $m_z=0$ spin component and its thermal fraction is less than 10%. To initiate the transition dynamics to the EPP phase, we first adiabatically ramp $B_z$ to 33 mG for 0.2 s and then, we suddenly turn on a microwave field to change the quadratic Zeeman energy to a negative $q$ value [@Gerbier_pra; @Zhao_pra]. The residual field gradient was measured to be less than 0.1 mG/cm.
In our experiment, $q/h$ ranges from $-1.4$ Hz to $-20$ Hz. The excitation energy of the initial state after the quench is given by $|q|$ with respect to the EPP ground state, which is much smaller than the condensate chemical potential $\mu= h\times880$ Hz. Therefore, the subsequent evolution of the condensate can be approximated as pure spin dynamics, and does not involve density excitations. For the peak atomic density, $n$, of the sample, the spin interaction energy is $c_2 n/h= 14$ Hz [@Black_prl07] and the spin healing length is $\xi_s=\hbar/\sqrt{2m c_2 n}\approx 4.0~\mu$m. Because the condensate thickness $R_z$ is smaller than $\xi_s$, the spin dynamics in the oblate condensate are effectively 2D.
The spin texture of the condensate is examined by taking an absorption image after the Stern-Gerlach (SG) spin separation. After releasing the trapping potential, we apply a magnetic field gradient pulse along the $x$ direction and let the three $m_z=1,0,-1$ spin components be spatially separated for a 24-ms time of flight. The image reveals the density distributions of the individual spin components. The applied field gradient was slightly inhomogeneous and the expansions of the $m_z=\pm1$ spin components are not perfectly identical. We used $F=1$ imaging and calibrated the absorption coefficients for each spin component [@Kim16].
Spin-sensitive [*[in-situ]{}*]{} phase-contrast imaging is also employed to obtain further information on the spatial magnetization structure of the condensate [@Seo_prl15]. The probe beam frequency is tuned to give a signal proportional to the axial magnetization, $M_z$, i.e., the density difference of the $m_z=\pm 1$ spin components. The contribution of the $m_z=0$ spin component to the imaging signal is not significant and we interpret the phase-contrast image as the axial magnetization distribution $M_z(x,y)$ of the condensate.
Results
=======
Emergence of spin turbulence
----------------------------
Figure 1 displays two image data sequences of the quenched BEC for the two different $q$ values of $-5.4$ Hz and $-20$ Hz. After a short hold time, an irregular spin texture begins to appear in the condensate. It is clearly shown that the $m_z=\pm1$ components are spatially separated from the $m_z=0$ component \[Figs. 1(c) and 1(e)\], which results from the immiscibility of the $m_z=\pm1$ components with the $m_z=0$ component [@Stenger_nat]. The spin domains formed by an equal mixture of the $m_z=\pm1$ components have $\vec{d}\perp \hat{z}$, where the azimuthal direction of $\vec{d}$ is determined by the relative phase of the two spin components. It is observed that the irregular spin texture first emerges in the center region of the condensate and expands over the whole condensate. We attribute this result to the inhomogeneous density distribution of the trapped condensate because $\Gamma_q \propto \sqrt{n}$.
The appearance of an irregular spin texture is also observed in the magnetization image \[Figs. 1(d) and 1(f)\]. In the spin-exchange process where two $m_z=0$ atoms are scattered into a pair of $m_z=+1$ and $-1$ atoms, the quadratic Zeeman energy is converted into the kinetic energy of the $m_z=\pm1$ atoms, imparting opposite momenta to the pair. Thus, spin currents are generated in the $m_z=\pm 1$ spin domains and axial magnetization develops at the domain boundaries. The irregular structure in the $M_z$ image constitutes an observation of spin turbulence that has a complex spin current pattern.
As the hold time $t$ increases, the $m_z=0$ spin component is continuously depleted, and the spin texture becomes more complex. In particualr, the length scale of the spin texture decreases, implying a direct energy cascade in the spin turbulence. The condensate eventually relaxes into the EPP phase with the $m_z=0$ component vanising. In the final state, HQVs are observed as magnetized point defects in the $M_z$ image [@Seo_prl15] and are identified with the density-depleted holes in the SG image.
![Spin wave excitations for $q/h=-1.4$ Hz. SG images (left) and [*in-situ*]{} magnetization images (right) for various hold times. A ring-shaped oscillating spin texture appears and shortly becomes dismantled.[]{data-label="fig2"}](Fig2){width="6.5cm"}
It is apparent in the comparison between the two image data sets in Fig. 1 for $q/h=-5.4$ Hz and $-20$ Hz that when the system is closer to the critical point, the time and length scales of the quench dynamics become slower and larger, respectively, which is consistent with the theoretical anticipation based on the dynamic instability of the initial state. The length scale of the spin texture becomes even larger with lower $|q|$. For $|q/h|< 2$ Hz, we observed that the incipient spin texture shows a large ring-shaped pattern, which propagates toward the boundary and shortly becomes dismantled (Fig. 2). The ring-shaped pattern has the same ellipticiy as the trapped condensate and we believe that it corresponds to long-wavelength spin wave excitations induced by the trapping geometry of the finite-size sample [@Klempt_prl; @Scherer_prl].
Characterization of the quench dynamics
---------------------------------------
### Time evolutions of $\eta$ and $\delta M_z^2$
We first characterize the quench dynamics of the condensate by measuring the time evolutions of the fractional population, $\eta$, of the $m_z=0$ component and the magnetization variance, $\langle \delta M_z^2 \rangle$, of the spin texture (Fig. 3). Here $\eta= N^c_0/N^c$ and $ N^c=\sum_i N^c_i$, where $N^c_i$ is the $m_z=i$ atom number of the condensate ($i=1,0$, and $-1$) and is determined from the SG absorption image. The thermal cloud contribution is subtracted using a Gaussian fit to the outer thermal wing. For the measurement of $\langle \delta M_z^2 \rangle$, we set the central $206~\mu \textrm{m}\times206~\mu\textrm{m}$ region of the condensate as the region of interest.
{width="7.0cm"}
The quench evolution starts with a delayed, rapid decrease of $\eta$ as expected from the exponential growth of the dynamically unstable magnon modes. The early evolution of $\eta$ is described as $\eta(t)=1-b \exp(-\Gamma_q t)$, where $b$ is a constant determined by the magnitude of the magnetization fluctuations in the system. In the inset of Fig. 2(a), we display the time $t_1$ measured for $\eta(t_1)=0.8$ as a function of $|q|$. A power-law fit to the experimental data gives an exponent of $-0.53\pm 0.01$, which is in quantitatively good agreement with the predicted scaling of $\Gamma_q \sim |q|^{1/2}$. The measured $t_1$ values give $b\approx 4.6 \times 10^{-6}$, which is slightly higher than the value of $b\approx 3.3\times 10^{-6}$ estimated for quantum fluctuations at our peak atomic density [@Mele_pra13], indicating thermal enhancement in the experiment.
The rapid decay of $\eta$ is halted at a certain threshold value $\eta_{th}$ and after a short loitering period of a few tens of ms, $\eta$ resumes its decay. As $\langle \delta M_z^2\rangle$ rapidly increases when $\eta$ undergoes this change, it is reasonable to infer that the condensate enters a qualitatively different phase of the quench evolution, where the role of the generated spin turbulence becomes significant. The threshold value $\eta_{th}$ monotonically increases from $\approx 0.4$ for $q/h=-1.4$ Hz to $\approx 0.75$ for $q/h=-20$ Hz \[Fig. 6(c)\].
The spin turbulence develops further with maximizing $\langle \delta M_z^2 \rangle$ and then gradually relaxes with decreasing $\langle \delta M_z^2 \rangle$. After a long hold time, $t>5$ s, the system is equilibrated with $\eta\simeq 0$ and stationary $\langle \delta M_z^2 \rangle$. In our experiment, the equilibrium value of $\langle \delta M_z^2 \rangle$ was insensitive to $q$ because the final sample temperature is mainly determined by the heating from the microwave field dressing and the evaporation cooling due to the finite trap depth. Note that for $q/h=-1.4$ Hz, $\langle \delta M_z^2 \rangle$ monotonically increases and saturates to the equilibrium value over time. At the equilibrium, the thermal fraction was approximately 30% and the sample temperature was estimated to be $T\approx 100$ nK. $k_B T \ll |q|$ and the thermal cloud was an equal mixture of the three spin components [@Erhard_pra].
### Power spectrum of axial magnetization
To investigate the spatial structure of the generated spin turbulence, we measure the power spectrum of the magnetization distribution, $P(\vec{k})=|\int dr^2 e^{i\vec{k}\cdot\vec{r}}M_z(\vec{r})|^2$. In the measurement, a reference image was first obtained by averaging over ten images that were taken for the same experiment and we subtracted the reference image from the individual images to remove systematic fringes not related with the spin texture of the sample. Then, the power spectrum $P(\vec{k})$ was obtained by averaging the squared Fourier transforms of the subtracted images and subtracting the photon shot noise level. We introduce a relative spectrum $\tilde{P}(\vec{k})=P(\vec{k})/P_\textrm{eq}(\vec{k})$, where $P_\textrm{eq}(\vec{k})$ is the average spectrum of samples at thermal equilibrium in the EPP phase. Because $P_\textrm{eq}$ is measured with the same imaging system, the relative spectrum $\tilde{P}$ provides spectral information free from the systematic modifications of the imaging system. For the determination of the equilibrium spectrum $P_\textrm{eq}$, we selected samples for $q/h=-1.4$ Hz at $t=5$ s, in particular, without HQVs because HQVs can affect the spectrum due to their magnetized cores and the magnon excitations generated by their collisional motions [@Seo_prl16]. $\tilde{P}(\vec{k})$ was isotropic and we obtained a 1D spectrum, $\tilde{P}(k)$, by azimuthally averaging it over $|\vec{k}|=k$.
![Relative power spectra, $\tilde{P}(k)$, of the axial magnetizaion, $M_z(x,y)$, of the quenched BEC. Evolution of $\tilde{P}(k)$ for $q/h=-10$ Hz when $\langle \delta M_z^2\rangle$ increases (a) and decreases (b) \[Fig. 3(b)\]. (c) $\tilde{P}(k)$ at the time when $\langle \delta M_z^2\rangle$ is maximum and (d) at $t=5$ s when the quenched BEC is thermally equilibrated. The reference spectrum $\tilde{P}(k)=1$ was obtained from the thermal equilibrium samples at $T\approx 100$ nK, having no HQVs (see text).[]{data-label="fig2"}](Fig4){width="8.4cm"}
Figures 4(a) and 4(b) show the evolution of $\tilde{P}(k)$ for $q/h=-10$ Hz when $\langle \delta M_z^2\rangle$ increases and decreases, respectively. As observed in the visual examination of Figs. 1(d) and 1(f), the $\tilde{P}$ measurement results show that the spin turbulence develops from low to high wave numbers $k$, i.e., from large to small length scales \[Fig. 4(a)\]. When $\langle \delta M_z^2\rangle$ decreases, the power spectrum decays towards the equilibrium level, $\tilde{P}_\textrm{eq}=1$, where the spectral strength subsides more rapidly in the lower-$k$ region \[Fig. 4(b)\]. Over the entire growth and decay evolution, the spectral center-of-mass of $\tilde{P}(k)$ continues moving towards high $k$, which suggests a direct energy cascade of the spin wave excitations in the spin turbulence.
In Fig. 4(a), it is noted that when the spin turbulence is generated, the spectral slope is formed in $\tilde{P}(k)$ and propagates to the high-$k$ regions in a self-similar manner. In Fig. 4(c), we display $\tilde{P}(k)$ at the time when $\langle\delta M_z^2\rangle$ is maximum for various $q$ values and observe that the spectral slopes are almost identical, whereas the characteristic wave number increases with increasing $|q|$. The power-law scaling in the spin turbulence of antiferromagnetic BECs was predicted by Fujimoto *et al.* [@Fujimoto_pra12; @Fujimoto_pra13], but further analysis of our experimental data is limited by a lack of quantitative understanding of $P_\textrm{eq}(k)$. In Ref. [@Symes_pra], Symes [*et al.*]{} calculated the static structure factor of the antiferromagnetic BEC in the EPP phase at low temperatures. However, our sample temperature of $k_B T/(c_2 n)\approx 150$ is too high to be extrapolated from their results.
In Fig. 4(d), we display $\tilde{P}(k)$ at a long hold time, $t= 5$ s, when the sample is thermally equilibrated. For high $|q|$, the spectral strength in $k\leq 1/\xi_s$ is still noticeably higher than the equilibrium level of $\tilde{P}_\textrm{eq}=1$. This result is due to the presence of HQVs, which we confirmed by correlating the deviation magnitude in the low-$k$ region with the HQV number (Fig. 7).
![Time-space scaling in the spin turbulence generation. (a) Evolution of $\langle\delta M_z^2\rangle$ as a function of the rescaled time $\tilde{t}\equiv t/t_q$ with $t_q=h/\sqrt{8|q|c_2 n}$. The vertical dotted line denotes the time when $\eta=0.8$ and the grey region indicates the time window for $\langle \delta M_z^2\rangle$ to be maximized. (c) Relative spectra $\tilde{P}(k)$ with maximum $\langle \delta M_z^2\rangle$ \[Fig. 4(c)\] as functions of the rescaled wave number $\tilde{k}\equiv k/k_q$ with $k_q=\sqrt{2|q|m}/\hbar$.[]{data-label="fig3"}](Fig5){width="6.5cm"}
Scaling behavior
----------------
### Spin turbulence generation
The spin turbulence generation is seeded by the amplification of the dynamically unstable magnon modes in the initial EAP state, in which the characteristic time and length scales are given by $t_q\sim |q|^{-1/2}$ and $l_q\sim|q|^{-1/2}$. To examine the scaling properties of the subsequent development of the spin turbulence, Fig. 5(a) displays the time evolution data of $\langle\delta M_z^2\rangle$ as functions of the rescaled time $\tilde{t}\equiv t/t_q$. For all $q$, $\langle\delta M_z^2\rangle$ begins its rapid increase at $\tilde{t}_1\approx 1.7$ and becomes maximized in the range of $5<\tilde{t}<10$. In Fig. 5(b), we display the relative power spectra $\tilde{P}(k)$ for the maximum $\langle\delta M_z^2\rangle$ as functions of the rescaled wave number $\tilde{k}\equiv k/k_q$. All the spectra collapse into a single line for $1.5<\tilde{k}<4$. This observation demonstrates that the length scale of the initial state, $l_q\sim |q|^{1/2}$, is preserved in the subsequent turbulence generation dynamics.
![Decay of $\eta$ in the turbulence phase. $\eta(t)$ for (a) $q/h=-5.4$ Hz and (b) $-20$ Hz, and the threshold values $\eta_{th}$ are indicated by the right arrows. (c) $\eta_{th}$ versus $|q|$. (d) Decay curves of $\eta$ as functions of $t_d=t-t_2$, where $t_2$ is the starting time of the turbulence phase. The inset shows the same data with adjusted time offsets and the solid line is a power-law fit to the data. (e) Relaxation of the thermal clouds. The solid symbols denote the thermal fraction of the sample and the open symbols denote the $m_z=0$ spin fraction of the thermal cloud. Each data point is obtained by averaging about five measurements and its error bar denotes the standard deviation of the measurements.[]{data-label="fig4"}](Fig6){width="7.4cm"}
### Spin population relaxation
As the spin turbulence develops, $\eta$ exhibits adifferent decay behavior for $\eta<\eta_{th}$ \[Figs. 6(a) and 6(b)\]. In Fig. 6(d), we display the decay curves of $\eta$ for various $q$ values as functions of $t_d=t-t_2$, where $t_2$ is the starting time of the turbulence phase. $\eta$ shows a nonexponential decay, where the relative decay rate $\gamma=-\frac{1}{\eta}\frac{d \eta}{dt}$ decreases as $\eta$ decreases. In the turbulence phase, the $m_z=0$ spin component, being spatially separated from the $m_z=\pm1$ components, forms axial polar domains in the condensate (Fig. 1). As the density of the $m_z=0$ atoms in the spin domains is regulated by the condensate chemical potential, a local two-body decay process would result in the constant decay of $\eta$, which cannot explain the observed nonexponential decay. The inset of Fig. 6(d) shows a log-log plot of the same data with adjusted time offsets. The determination of the exponent from a power-law fit to the data is not reliable due to its high sensitivity to the time offsets.
In our experiments, the decay rate $\gamma$ is found to be insensitive to $q$, which is different from the general expectation of higher $\gamma$ for a higher excitation energy, $|q|$. It is speculated that the small domain size for the high $|q|$ proportionally suppresses $\gamma$, or this result may be due to the high thermal energy of the system, $k_B T \gg |q|$ [@Miesner99; @Shin04]. In Fig. 6(e), the relaxation of the thermal cloud during the turbulence phase is characterized. The $m_z=0$ spin fraction of the thermal cloud relaxes within 0.2 s to the equilibrium level of one third, and the thermal fraction of the sample increases from $\approx 20\%$ to $\approx 30\%$. The relaxation time is compatible to the value of $1/\gamma$ at the initial decay but the decrease in $\gamma$ does not correspond with the increase of the sample temperature. The spin turbulence relaxation dynamics including the coupling to the thermal cloud [@McGuirk_prl03] merits further investigation in future experiment.
![HQV number, $N_v$, versus $|q|$. $N_v$ was measured by counting density-depleted holes in the SG images of the $m_z=\pm 1$ spin components. Each data point is the average of fifteen measurements for the same experiment and its error bars indicates the standard error of the mean. The blue dashed line denotes a power-law fit to the data in the range of $|q|/h \leq 10$ Hz for $t=3$ s.[]{data-label="supple2"}](Fig7){width="6.8cm"}
### Creation of HQVs
At long hold times of $t\geq 3$ s, when the $m_z=0$ spin component nearly vanishes in the condensate, HQVs are unambiguously identified with the density-depleted holes in the SG images (Fig. 1). We measure the HQV number, $N_v$, and find that it increases with increasing $|q|$ and saturates for $|q|/h>10$ Hz (Fig. 7). When the spatial size of the spin domains in the quenched condensate scales with $l_q$, the number of point defects created in the 2D system is expected to scale as $N_v\propto l_q^{-2}$ [@Bray_adv94]. A power-law fits to the data for $t=3$ s in the range of $|q|/h\leq 10$ Hz gives an exponent of $1.0\pm 0.2$, which is consistent with the length scale, $l_q\sim |q|^{1/2}$, observed in $\tilde{P}(k)$. The saturation of $N_v$ for high $|q|$ might indicate another length scale involved in the defect creation, such as the spin healing length $\xi_s$ [@Tomasz_prl13], but we note that its effect was absent in our measurements of $\tilde{P}(k)$. As HQVs can be pair-annihilated in the turbulence relaxation process, leading to a non-exponential decrease of $N_v$ [@Kwon14], the HQV number measured at long hold times may not linearly reflect the initial $N_v$ of the quenched BEC.
Summary and outlook
===================
We have investigated the phase transition dynamics of quasi-2D antiferromagnetic spin-1 BECs quenched from the EAP phase to the EPP phase. We observed the emergence and decay of the spin turbulence in the quenched condensate and presented the time and space scaling properties of the phase transition dynamics near the quantum critical point.
We can extend this work to a deeper quench regime with $|q|>2c_2 n$. In this case, the dynamic instability of the initial EAP state is driven by the magnon modes with finite wave numbers centered at $k_q=\sqrt{2m(|q|-c_2 n)}/\hbar$ [@Matuszewski_prl10]. It would be of great interest to examine how the energy injected at finite wave numbers flows in the subsequent evolution of spin turbulence [@Fujimoto_pra16]. Because the length scale $\sim 1/k_q$ becomes comparable to and even smaller than the spin healing length $\xi_s$, it is speculated that qualitatively different turbulence states would emerge in this high $|q|$ regime. Another extension of this work is to study the effects of the quench rate across the critical point [@Anquez16; @Saito_pra07; @Tomasz_prl13]. In particular, the power-law scaling of the HQV number after the quench is reminiscent of the Kibble-Zurek (KZ) mechanism [@KZ]. We note that in the mean-field theory, the EAP-to-EPP phase transition is described as first-order; thus, the conventional KZ mechanism, which involves a continuous phase transition, cannot be applied directly to our system. A generalization of the KZ mechanism could be studied with this system.
The turbulence of the spinor BECs represents a unique turbulence state where the mass and spin superflows are entangled. Previous turbulence studies with spinor BECs were mostly focused on the case with ferromagnetic spin interactions both experimentally [@Sadler_nat06; @Vengalattore_prl08; @Guzman_pra11; @De_pra14] and theoretically [@Fujimoto_pra122; @Villasenor_pra14; @Williamson_prl16]. Our work demonstrates the peculiarities of the spin turbuelence in an antiferromagnetic BEC, enabling a comparative study between the ferromagnetic and antiferromagnetic cases of the spin-1 BEC system [@Fujimoto_pra12; @Fujimoto_pra13].
This work was supported by Samsung Science and Technology Foundation Under Project Number SSTF-BA1601-06.
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---
abstract: 'Entrainment by a pacemaker, representing an element with a higher frequency, is numerically investigated for several classes of random networks which consist of identical phase oscillators. We find that the entrainment frequency window of a network decreases exponentially with its depth, defined as the mean forward distance of the elements from the pacemaker. Effectively, only shallow networks can thus exhibit frequency-locking to the pacemaker. The exponential dependence is also derived analytically as an approximation for large random asymmetric networks.'
author:
- Hiroshi Kori
- 'Alexander S. Mikhailov'
date: '20.FEB.2004'
title: Entrainment of randomly coupled oscillator networks by a pacemaker
---
Pacemakers are wave sources in distributed oscillatory systems typically associated with a local group of elements having a higher oscillation frequency. Target patterns, generated by pacemakers, were the first complex wave patterns observed in the Belousov-Zhabotinsky system [@zhab]. Pacemakers play an important role in functioning of the heart [@winfree] and in the collective behavior of [*Dictyostelium discoideum*]{} [@Lee]. They are also observed in large-scale ecosystems [@Blasius]. In addition to pacemakers produced by local heterogeneities in the medium [@Fife], self-organized pacemakers in uniform birhythmic media have been theoretically studied [@Stich]. While the majority of related investigations have so far been performed for systems with local diffusive coupling between the elements, pacemakers can also operate in oscillator networks with complex connection topologies. The circadian rhythm in mammals is a daily variation of 24 hours that regulates basic physiological processes in such animals [@moore97]. It is produced by a complex network of neurons forming the so-called suprachiasmatic nucleus (SCN) [@abrahamson01]. As recently shown, this oscillator network undergoes spontaneous synchronization in absence of any environmental input, but its intrinsic synchronization period is then significantly longer than 24 hours [@yamaguchi03]. Therefore, the actual shorter rhythm results from the environmental entrainment and must be externally imposed. The entrainment is mediated by direct photic inputs from eyes into the SCN, which undergo periodic daily variation. However, it is known that only a distinct subset of neurons in this network is directly influenced by photic inputs [@kuhlman03]. Hence, functioning of this particular neural system is crucially dependent on the ability of the entire complex network to become entrained by an external pacemaker. Analogous behavior can also be expected, for example, in heterogeneous arrays of globally coupled electrochemical oscillators where synchronization and entrainment have been experimentally demonstrated [@Hudson].
To understand operation of pacemakers in networks with complex connection topologies, action of a pacemaker in a random oscillator network should first be investigated. In this Letter, networks of identical phase oscillators with random connections are considered. A pacemaker is introduced as a special element whose oscillations have a higher frequency and are not influenced by the rest of the system. Depending on the pacemaker frequency and the strength of coupling, the pacemaker can entrain the entire network, so that the frequencies of all its elements become equal to that of the pacemaker. We find that the entrainment window decreases exponentially with the depth of a network, defined as the mean forward distance of its elements from a pacemaker, and thus only shallow networks can effectively be entrained. This result is confirmed in numerical simulations for several different classes of random networks, including small-world graphs. It is further analytically derived as an approximation for random networks with asymmetric connections.
We consider a system of $N+1$ phase oscillators, one of them being a pacemaker. The model is given by a set of evolution equations [@kuramoto84] for the oscillator phases $\phi _{i}$ and the pacemaker phase $%
\phi _{0}$, $$\begin{aligned}
\dot{\phi}_{i} &=&\omega -\frac{\kappa }{pN}\sum_{j=1}^{N}A_{ij}\sin (\phi
_{i}-\phi _{j})-\mu B_{i}\sin (\phi _{i}-\phi _{0}), \nonumber \\
\dot{\phi}_{0} &=&\omega +\Delta \omega . \label{model}\end{aligned}$$ The topology of network connections is determined by the adjacency matrix ${\mathbf A}$ whose elements $A_{ij}$ are either $1$ or $0$. The element with $i=0$ is special and represents a pacemaker. Its frequency is increased by $\Delta \omega$ with respect to the frequency $\omega$ of all other oscillators [^1]. The pacemaker is acting on a randomly chosen subset of $N_{1}$ elements, specified by $B_{i}$ taking values $1$ or $0.$ The total number of connections to the pacemaker, $N_{1}=\sum_{i}B_{i}$, is fixed. The coupling between elements inside the network is characterized by strength $\kappa$. The strength of coupling from the pacemaker to the network elements is determined by the parameter $\mu $. In absence of a pacemaker, such networks undergo autonomous phase synchronization at the natural frequency $\omega $. Without loss of generality, we put $\omega =0$. Moreover, we rescale time as $t^{\prime }=t$ $\Delta \omega $ and introduce rescaled coupling strengths $\kappa ^{\prime }=\kappa /\Delta
\omega ,\mu ^{\prime }=\mu /\Delta \omega$. After such rescaling, the model takes the form of Eqs. (1) with $\Delta \omega =1$ and $\omega =0$ (we drop primes in the notations for the rescaled couplings). In terms of the original model (1), increasing the rescaled coupling between the elements is equivalent either to an increase of coupling $\kappa$ or to a decrease of the relative pacemaker frequency $\Delta \omega $.
The presence of a pacemaker imposes hierarchical organization. For any node $i$, its distance $h$ with respect to the pacemaker is given by the length of the minimum forward path separating this node from the pacemaker. All $N_{1}$ elements in the group directly connected to the pacemaker have distances $h=1$, the next elements which are connected to the elements from this group have distances $h=2$, etc. Thus, the whole network is divided into a set of shells [@dorogovtsev03], each characterized by a certain forward distance $h$ from the pacemaker. The set of numbers $N_{h}$ is an important property of a network. The depth $L$ of a given network, which is the mean distance from the pacemaker to the entire network, is introduced as $L=(1/N)\sum_{h}hN_{h}$. It should be noticed that such ordering of network nodes is based solely on the forward connections down the hierarchy and does not depend on the distribution of reverse (upward) connections in the system.
First, we investigated [*standard random asymmetric networks*]{}, where independently for all connections $A_{ij}=1$ with probability $p$ and $
A_{ij}=0$ otherwise. Only sparse random networks with relatively low mean connectivity $p$ and a small number $N_{1}$ of elements directly connected to the pacemaker were considered. Numerical simulations were performed for the networks of size $N=100$ starting with random initial conditions for the phases of all oscillators. For each oscillator, its effective long-time frequency $\omega _{i}$ was computed as $\omega
_{i}=T^{-1}\left[ \phi _{i}(t_{0}+T)-\phi_i (t_{0})\right]$ with sufficiently large $T$ and $t_{0}$. The simulations show that the response of a network to the introduction of a pacemaker depends on the strength $\kappa $ of coupling between the oscillators. When this coupling is sufficiently large (and coupling $\mu $ to the pacemaker is also sufficiently strong as assumed below), the pacemaker entrains the whole network (i.e., $\omega _{i}=1$ for all elements $i$). The frozen relative phases $\psi_{i}\equiv \phi _{i}-\phi _{0}$ are displayed in Fig. 1. Here, the elements are sorted according to their hierarchical shells. Despite random variations, there is a clear correlation between phases of oscillators and their positions in the hierarchy. Generally, the phase decreases for deeper shells, and the phase difference between the neighboring shells rapidly becomes smaller as deeper shells are considered. As the coupling strength $\kappa $ is decreased, the entrainment breaks down at a certain threshold value $\kappa _{{\rm
cr}}$. Our simulations show that synchronization between the first and the second shells was almost always the first to break down, and the frequencies of the second and deeper shells remained equal in most cases for the considered random networks.
Figure 2 displays in the logarithmic scale the thresholds $\kappa_{\rm
cr}$ for a large set of networks with different depths and different numbers of elements in the first shell. Each group with a certain $N_{1}$ is displayed by using its own symbol. Every such group generates a cluster of data points. Correlation between the entrainment threshold and the network depth is apparent. The distributions inside each cluster and the accumulation of the clusters yield the dependence $\kappa_{{\rm
cr}}(L)$ of the entrainment threshold on the network depth. Note that the statistical variation of the data becomes larger for deeper networks with larger $L$ and for smaller $pN$. Similar dependence was found for the networks with different mean connectivity $p$ (see inset). Remarkably, the observed dependences could be well numerically approximated by the exponential dependence $$\kappa_{{\rm cr}} \propto (1+pN)^{L}. \label{exponential}$$
As the second class, [*asymmetric small-world networks*]{} [@watts98] were considered. To generate them, we first constructed a one-dimensional lattice of $N$ elements where each element had incoming connections from up to its $k$th neighbor (the degree was thus $2k$). Then a randomly chosen link in the lattice was eliminated and a distant connection between two independently randomly chosen elements was introduced. This construction was repeated $qN$ times, with the parameter $q$ specifying the randomness of a network. When $q$ was small, the network was close to a lattice and, in this case, we have seen that stable wave solutions with different winding numbers were possible, depending on initial conditions (cf. [@kuramoto84; @manrubia]). To avoid this, we chose almost synchronized states as initial conditions. The entrainment thresholds for such small-world networks are displayed in Fig. 3 and again show a clear correlation between $\kappa _{{\rm cr}}$ and $L$. The dependence on the depth is approximately linear in lattices ($q=0$), but it becomes strongly nonlinear even when small randomness is introduced. For $q=0.1$, the dependence is already approximately exponential, though the dispersion of data is strong. As randomness $q$ is increased, the dependence approaches that of the standard random networks with $pN=2k$.
We have also investigated [*asymmetric scale-free random networks*]{} [@newman01], [*asymmetric regular random networks*]{} (where every element has exactly the same number of either incoming or outgoing connections), and [*symmetric standard random networks*]{}. For all of them, approximately exponential dependences of the entrainment threshold on the network depth were observed in a large parameter region.
The exponential dependence (\[exponential\]) can be approximately derived for asymmetric random networks with large $N$ and $pN$. In the large-size limit, random graphs have locally a tree-like structure [@dorogovtsev03]. The global tree approximation has previously been used for determining statistical properties of random networks [@newman01]. We apply here the same approximation and assume that the graph of forward connections extending from the pacemaker node represents a tree, so that any oscillator has only one incoming connection from the previous shell. Then the shell populations $N_h$ are given by $N_h=N_1 (pN)^{h-1}$ for $h=2,\ldots,H$, where $H$ is the total number of shells determined by $\sum_{h=1}^{H} {N}_{h}=N$. Because $pN$ is large, we have $N_{h} \ll N_H \simeq N$ for $h<H$, and thus $L \simeq
H$. Next, we estimate the numbers $m_{hk}$ of incoming connections leading from all elements in the $k$th shell to an oscillator in the $h$th shell. By definition of hierarchical shells, $m_{hk}=0$ if $k<h-1$. In the tree approximation, $m_{hk}=1$ for $k=h-1$. Because most of the population is concentrated in the last shell, reverse connections from other shells can be neglected. On the average, the number of reverse connections from the shell $H$ to an oscillator in the shell $h$ is $m_{hH}=p N_H$. Moreover, the relative statistical deviation from this average is of order $(pN)^{-1/2}$, and is thus negligible. Therefore, in this approximation all oscillators inside a particular shell have effectively the same number of connections from other shells, and a state with phase synchronization inside each shell is possible. In this state, all oscillators inside a shell have the same phase, i.e. $\phi_i=\theta _{h}$ for all oscillators $i$ in a shell $h$. Under entrainment, the phases of such a state can be found analytically as a solution of algebraic equations $$\begin{aligned}
&-&\frac{\kappa }{pN}\sum_{k=h-1}^{H}m_{hk} -\sin (\theta _{h}-\theta
_{k})=1\quad \mbox{for $h=2,\ldots,n$}, \nonumber \\
&-&\mu \sin (\theta _{1}-\theta _{0})-\frac{\kappa }{pN}
\sum_{k=2}^{H}m_{1k}\sin (\theta _{1}-\theta _{k})=1, \label{1}\end{aligned}$$ where $\theta_0 \equiv \phi_0$. For large $pN$, we can linearize $\sin(\theta_h-\theta_k)$ for $h,k \ge 2$ in the solution of Eqs. (\[1\]) \[it can be shown that $\theta_2-\theta_H$ is of order $O(1/pN)$\]. Furthermore, using that $N_H \simeq N \gg
N_{h}$ for $h<H$ and $L \simeq H$, we obtain for $h \ge 2$ that $$\sin(\theta_{h-1} - \theta_h) = \frac{pN}{\kappa} (1+pN)^{L-h}.
\label{phase}$$ Equations (4) determine the phases of oscillators in the considered synchronized state. Note that the explicit value of the phase $\theta_1$ in this state is not needed below.
The entrainment breakdown can, in principle, occur through destabilization of the synchronized state. Though the analytical proof of its stability is not yet available, our numerical simulations show that the synchronous entrained state with $0<\theta _{h-1}-\theta
_{h}<\pi /2$ is always stable when it exists. Thus, the breakdown of entrainment in the considered system takes place in a saddle-node bifurcation, through the disappearance of solutions of Eqs. (\[1\]). This occurs when $|\sin(\theta_{h-1}-\theta_h)|=1$ for certain $h$. For large enough $\mu$, we always have $0<\sin(\theta_0-\theta_1)<1$ (a sufficient condition is $\mu >1+\kappa$). Among the other terms, the term $\sin(\theta_1 - \theta_2)$ is always the largest one. Therefore, the solution disappears and breakdown occurs when $\sin (\theta
_{1}-\theta _{2})=1$. Substituting Eq. (\[phase\]) into this equation and solving it with respect to $\kappa$, we finally derive the dependence (\[exponential\]). Thus, we see that the entrainment breakdown occurs through the loss of frequency locking between the first shell and the rest of the network. The analytical estimate for the critical coupling strength, obtained using the tree approximation, agrees well with the numerical data, even for the networks which are not very large.
So far we have used the coupling strength which was rescaled as $\kappa \rightarrow \kappa /\Delta \omega $. Therefore, if the non-scaled coupling strength is fixed, Eq. (\[exponential\]) determines the maximum $\Delta \omega_{\rm c}$ at which the entrainment is still possible, $\Delta \omega_{\rm c} \propto \kappa
(1+pN)^{-L}$. The entrainment by a pacemaker can take place only if its frequency lies inside the interval $(\omega ,\omega +\Delta \omega_{\rm
c})$.
Thus, the entrainment window [*decreases exponentially*]{} with the depth of a network. This is the principal result of our study, which holds not only for standard random networks, where the above analytical estimate is available, but also for small-world graphs and other numerically investigated random topologies. In practice, it implies that only shallow random networks with small depths are susceptible to frequency entrainment.
Our results remain valid when, instead of a pacemaker, external periodic forcing acts on a subset of elements. We have checked that the reported strong dependence on the network depth remains valid for systems with larger network sizes, heterogeneity in frequencies of individual oscillators, and several other coupling functions. The study was performed for coupled phase oscillators which serve as an approximation for various real oscillator systems, including neural networks (see, e.g., [@kuramoto84; @kuramoto91; @kori]). Its conclusions should be applicable for a broad class of oscillator networks with random architectures.
Financial support by the Japan Society for Promotion of Science (JSPS) is gratefully acknowledged.
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[^1]: Note that the system (\[model\]) is invariant under transformation $\omega \to - \omega, \Delta \omega \to
-\Delta \omega, \phi \to -\phi$, and therefore the same entrainment behavior takes place when $\Delta \omega <0$.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present imaging and spectroscopic observations of the gravitationally lensed arcs in the field of RX J1347.5$-$1145, the most X-ray luminous galaxy cluster known. Based on the detection of the $\lambda$3727 emission line, we confirm that the redshift of one of the arcs is $z = 0.806$. Its color and line strength are consistent with those of distant, actively star forming galaxies. In a second arc, we tentatively identify a pair of absorption lines superposed on a red continuum; the lines are consistent with $\lambda$3933 (K) and $\lambda$3968 (H) at $z = 0.785$. We detected a faint blue continuum in two additional arcs, but no spectral line features could be measured. We establish lower limits to their redshifts based on the absence of emission, which we argue should be present and detectable in these objects. Redshifts are also given for a number of galaxies in the field of the cluster.'
author:
- 'Swara Ravindranath and Luis C. Ho\'
title: 'Magellan Spectroscopy of the Galaxy Cluster RX J1347.5$-$1145: Redshift Estimates for the Gravitationally Lensed Arcs'
---
Introduction
============
Gravitational lensing by galaxy clusters serves as a powerful probe of cosmological structure. The lensing phenomenon provides information on both the mass distribution of the lensing cluster and the nature of the background population of faint field galaxies (e.g., Smail et al. 1993; Fort & Mellier 1994). Measurements of cluster mass also place useful constraints on the nature of dark matter and the cosmological parameter $\Omega$ (e.g., Mellier, Fort, & Kneib 1993; Fort & Mellier 1994; Crone, Evrard, & Richstone 1994, 1996; White & Fabian 1995). The arcs that are often seen in deep images of galaxy clusters are the sheared and magnified images of faint, background galaxies (Paczyński 1987). The amplification provided by gravitational lensing enables spectroscopic studies of these faint galaxies, which would otherwise be extremely difficult, if not impossible. The spectra of the arcs yield information on the redshifts and stellar populations of the lensed galaxies (e.g., Smail et al. 1993; Bézecourt & Soucail 1997; Ebbels et al. 1998; Hall et al. 2000; Campusano et al. 2001). The redshifts of the arcs constrain models of the the cluster potential and provide robust estimates of the total cluster mass.
RX J1347.5$-$1145, at a redshift of 0.451, is the most luminous X-ray cluster known, with an X-ray luminosity in excess of 10$^{45}$ (Schindler et al. 1995, 1997; Ettori, Allen, & Fabian 2001). The mass estimates based on its X-ray properties (Schindler et al. 1995), the Sunyaev-Zel’dovich effect (Pointecouteau et al. 1999), weak-lensing models (Fischer & Tyson 1997), and strong-lensing models (Sahu et al. 1998; Cohen & Kneib 2002) have yielded discrepant results for the total mass of this cluster. Cohen & Kneib (2002) speculate that we may be witnessing the merging of two clusters along a direction perpendicular to our line of sight.
Strong constraints can be placed on the cluster mass based on the redshifts of lensed background galaxies. Schindler et al. (1995) discovered an arc system in RX J1347.5$-$1145; it comprises of two bright arcs located $\sim$35 from the central dominant galaxies, at diametrically opposite points along the North-South direction. This was confirmed by the observations of Fischer & Tyson (1997). For the bright northern arc (Arc 1), Sahu et al. (1998) reported the detection of an emission line plausibly identified with $\lambda$3727 at a redshift of 0.81. The bright southern arc (Arc 4) showed a faint blue continuum but no spectral line features. The high-resolution [*Hubble Space Telescope*]{} images of Sahu et al. (1998) also revealed three additional arcs in this cluster (Arcs 2, 3, and 5). Figure 1 shows an $R$-band image of the central region of the cluster with identification for the five arcs.
This paper presents new photometry and spectroscopy of the arc system in RX J1347.5$-$1145. We confirm the emission-line redshift previously reported for Arc 1, report the tentative detection of an absorption-line redshift for Arc 2, and give limits on the redshifts of Arcs 3 and 4. These measurements are in good agreement with the predictions from the lensing models of this cluster. We also give redshifts for a number of galaxies in the field of the cluster.
Observations and Data Reductions
================================
In preparation for the spectroscopic observations, on 2001 February 20 UT we acquired relatively deep $V$ (2$\times$1200 s) and $I$ (2$\times$900 s) images of RX J1347.5$-$1145 using the 2.5-m du Pont telescope at Las Campanas Observatory. The CCD was a thinned Tektronics $2048\times 2048$ chip with a pixel scale of 026, a gain of 3.0 $e^{-}$ per ADU, and a read noise of 7 $e^{-}$. The data were taken under photometric, sub-arcsecond ($\sim$08) conditions. The images were used for photometry of the arcs and to obtain accurate relative astrometry for the preparation of the slit mask for the spectroscopic observations. Photometric and astrometric calibrations were achieved by observing the open cluster M 67 (Montgomery, Marschall, & Janes 1993).
We obtained spectra of the brightest four arcs (Arcs 1–4 in the notation of Sahu et al. 1998) on 2001 May 15–16 UT, using the multislit LDSS-2 spectrograph (Allington-Smith et al. 1994) mounted on the 6.5-m Magellan I (Baade) telescope at Las Campanas. We did not include Arc 5 because it partly overlaps with Arc 1 along the dispersion direction. The 0.3cm 0.3cm 300 line mm$^{-1}$ grism blazed at 6000 Å covered the spectral range $\sim$4000 to 8000 Å with a dispersion of 5.3 Å pixel$^{-1}$. The two-dimensional spectra were recorded on a SITe $2048\times 2048$ CCD, which has a spatial resolution of 038 pixel$^{-1}$. In order to maximize the light input from the faint, low surface brightness arcs ($\mu_V \approx 25-26$ mag arcsec$^{-2}$), we chose the width of the slits to be 2. This gave a full width at half maximum (FWHM) spectral resolution of 25 Å. For simplicity, all the slits were oriented along the East-West direction, at the expense that for most of the arcs only a portion of each object intersected with the slit. The slits were required to have a minimum length of 10 in order to have enough area for sky subtraction. In addition to the arcs, the mask also contained slits for a number of galaxies that were likely to be associated with the cluster. The total integration time was 21,400 s ($\sim$6 hr), split into seven roughly equal exposures. The sky conditions were clear, the atmospheric seeing varied between 06 and 09, and the airmass ranged from 1.0 to 1.65.
The basic data reductions were carried out using the IRAF[^1] package. The du Pont images were bias-subtracted, flat-fielded with twilight skyflats, aligned, and then median combined after cosmic-ray rejection. Photometry of the arcs and galaxies was performed using the SExtractor software (Bertin & Arnouts 1996), which uses elliptical apertures to compute magnitudes within an isophotal radius that is twice the first-moment radius (Kron 1980). The maximum radius used to compute the first-moment radius corresponds to an isophotal value that is 1.5 $\sigma$ of the background. The formal errors on the magnitudes derived from SExtractor are $\sim$ 0.3 mag. The magnitudes were corrected for atmospheric extinction and transformed to the standard Johnson system using calibrations derived from photometry of the standard stars in M 67.
The spectroscopic frames were corrected for the overscan, flat-fielded with domeflats, aligned, and then median combined after cosmic-ray rejection. Accurate sky subtraction proved to be challenging especially for the lensed arcs. This is in part due to the extremely low surface brightness of the arcs and to the fact that the spectra are slightly tilted. After some experimentation, we found that two-dimensional background subtraction gave the most robust results. The background was estimated by fitting a third-order Chebyshev polynomial along the columns using all the rows excluding those that contain the object spectrum. We extracted one-dimensional spectra by summing the central 10 rows ($\sim$4); this extraction width was determined empirically to give the best signal-to-noise ratio (S/N). We established the wavelength scale by fitting a third-order polynomial to unblended emission lines of He and Ne in the comparison lamp spectra. Finally, the relative fluxes of the spectra were calibrated using longslit observations of the standard stars LTT 7987 and LTT 9491 (Stone & Baldwin 1983). Note that we did not attempt to perform absolute spectrophotometry because our primary goal was to obtain redshifts for the arcs and galaxies.
Properties of the Arcs
======================
Table 1 summarizes the integrated magnitudes and colors obtained from the broad-band images. Figure 2 shows the final spectra for the lensed arcs in RX J1347.5$-$1145. For each arc, we present both the two-dimensional background-subtracted spectrum (in greyscale) and the extracted, calibrated one-dimensional spectrum.
The spectrum of Arc 1 shows an unresolved emission line centered at $\sim$6728 Å. The emission-line knot is clearly visible in the two-dimensional sky-subtracted image, and it is unmistakable in each of the sub-exposures. This line was already reported by Sahu et al. (1998), who argued that it is likely to be redshifted $\lambda$3727. Adopting this interpretation gives a redshift of 0.806. The line has an equivalent width (EW) of 85$\pm$10 Å, where the error bars reflect uncertainties in the placement of the continuum level and in whether the line flux is measured by direct integration or profile fitting. We note that although our absolute fluxes are not reliable because of slit losses, the EW measurements are likely to be more robust, so long as there are not large spatial variations of the line-emitting regions within the galaxy. From the photometry, we measure a moderately red color of $V-I \approx 1.6$ mag, similar to the color $B_{J}-R$ = 1.1 mag given by Fischer & Tyson (1997).
Arc 2, for which there have been no previous spectroscopic observations, exhibits a pair of absorption lines at 7017 and 7084 Å superposed on a relatively red continuum. The color derived from the images is also rather red, with $V-I \approx 2.7$ mag. The lines fall on a fairly clean part of the spectrum and do not appear to be adversely affected by sky subtraction.
We clearly detected the continuum in Arcs 3 and 4, but the spectra do not reveal any distinct emission or absorption features that can provide a direct redshift measurement. The feature near 5200 Å in the spectrum of Arc 4 appears not to be real. It is not seen in the background-subtracted two-dimensional image and may have resulted from the addition of significant residuals within the extraction aperture . The spectrum of Arc 3 is substantially noisier than that of Arc 4, for two reasons. First, Arc 3 is $\sim$1.5 mag fainter. And second, the slit was oriented in an especially unfavorable position angle with respect to the object; the length of the arc runs North-South (see Fig. 1), exactly orthogonal to the slit. Arcs 3 and 4 are significantly bluer than Arcs 1 and 2.
We measure $V-I \approx 0.41$ and 0.58 mag for Arcs 3 and 4, respectively; for comparison, Fischer & Tyson (1997) give $B_{J}-R$ = 0.4 mag for Arc 4.
Redshifts for Galaxies in the Field of RX J1347.5$-$1145
========================================================
We obtained spectra for 22 galaxies in the field of the cluster; Table 2 gives redshifts and basic photometric data for 21 of these. The galaxy redshifts were determined using the cross-correlation technique of Tonry & Davis (1979) as implemented in the [*xcsao*]{} task in IRAF. Cross-correlation templates were taken from the spectrophotometric atlas of galaxies of Kennicutt (1992a). These templates cover the wavelength range 3650$-$7100 Å at a resolution of 5$-$8 Å. We correlated each galaxy spectrum with a set of seven template spectra chosen to represent a range of spectral types. The templates include NGC 3379 (E1), NGC 4472 (E2), NGC 4889 (E4), NGC 3921 (S0/a), NGC 3627 (Sb), NGC 6764 (Sbc), and NGC 6240 (I0). NGC 3921 has a post-starburst spectrum showing a mixed population with strong Balmer absorption lines typical of A stars along with K-giant features. Both absorption-line and emission-line templates were used for redshift determination, depending on the spectrum of the observed galaxy.
The two main contributions to the errors in the measured redshifts arise from template mismatch and errors in the wavelength calibration. The errors due to template mismatch can be estimated from the dispersion in the final redshifts resulting from different input templates. For redshifts based on absorption-line templates, these errors are typically $\sim$350 km s$^{-1}$. The errors are significantly smaller when the redshifts are determined using emission-line templates, with $\sim$100 km s$^{-1}$ being a typical value. The rms error in the wavelength calibration of our observed galaxy spectra was found to be $\sim$5 Å. Adding the two contributions in quadrature, we estimate a total error of $\sim$ 0.0014 for the derived redshifts. In addition, there can be errors in the velocity zeropoints due to wavelength calibration errors of the template galaxy spectra. However, these are of the order of few tens of km s$^{-1}$ (Yee, Ellingson, & Carlberg 1996), and we have not included them in our error estimation.
Our redshift estimates show excellent agreement with the results reported by Cohen & Kneib (2002) in the case of four galaxies common to both studies. We assign cluster membership based on the velocity distribution presented by Cohen & Kneib (2002) for 47 spectroscopically confirmed cluster members. Five galaxies in our study have redshifts close to the redshifts for the two central cD galaxies given by Cohen & Kneib (2002), and thus are likely to be cluster members. Among these, the galaxy with $z$ = 0.431 is at the tail end of the velocity distribution, but we include it as a cluster member because its red $V-I$ color is similar to that of the other cluster galaxies. Another probable cluster member is a galaxy at redshift $z$ = 0.426 that lies close to the range spanned by the velocity distribution. This galaxy shows an absorption-line spectrum, but its $V-I$ color is relatively blue compared to the other cluster members. Only one of the galaxies among the likely cluster members, G47195$-$4344, shows an emission-line spectrum; it is located at the outskirts of the cluster, $\sim 3^{\prime}$ from the central cD galaxies. A similar cluster member candidate was reported by Cohen & Kneib (2002); C47229\_4519 has a relatively blue color and is the only cluster galaxy in their sample other than the central cD galaxy (which hosts an active nucleus) to show the $\lambda$3727 emission line.
Discussion and Summary
======================
Our primary aim is to determine or place limits on the redshifts of the arcs in RX J1347.5$-$1145. The redshift for Arc 1 is relatively secure. Consistent with the study of Sahu et al. (1998), we detected a strong emission line at $\sim$6728 Å whose most likely identification is $\lambda$3727 at $z = 0.806$. As discussed by Sahu et al. (1998), Ly$\alpha$ $\lambda$1216 or $\lambda$1550 can be ruled out based on the photometric colors: the Lyman limit would render the $B_{J}-R$ color much redder than that reported by Fischer & Tyson (1997). More directly, our spectrum, which extends to $\sim$4500 Å, shows no sign of any continuum decrement. Strong optical lines such as H, 4959, 5007, or H can be ruled out trivially by the redshift (0.451) of the lensing cluster.
The detection of the line allows us to deduce a few basic properties concerning Arc 1. The measured EW of $\sim$75–95 Å compares well with values previously found in gravitationally lensed arcs seen toward other galaxy clusters (Ebbels et al. 1998). It is somewhat larger than in typical nearby late-type galaxies (EW $\approx$ 50 Å; Kennicutt 1992b), but lies in the upper end of the EW distribution seen in distant, faint galaxy samples (e.g., Colless et al. 1990; Hammer et al. 1997). The observed $V-I$ color is also consistent with that expected for the faint blue galaxy population at $z \approx 0.8$ (Forbes et al. 1996). Although slit losses prevent us from measuring accurate total fluxes, we can use the observed emission-line flux to set a lower limit to the formation rate of massive (ionizing) stars. Kennicutt (1998) gives the following empirical relation between luminosity ($L_{\rm [O~II]}$) and star formation rate (SFR): $${\rm SFR} (M_{\odot} \, {\rm yr^{-1}}) = (1.4 \pm 0.4) \times 10^{-41} \,
L_{\rm [O~II]} \,\, ({\rm erg \, s^{-1}}).$$ 0.3cm This derivation assumes a Salpeter (1955) stellar initial mass function and solar metallicity. For an observed $F_{\rm [O~II]} > 5.5\times10^{-16}$ and cosmological parameters $H_0$ = 75 Mpc$^{-1}$, $\Omega_{\rm m} = 0.3$, and $\Omega_{\lambda} = 0.7$, we obtain SFR $>$ 24 $M_{\odot}$ yr$^{-1}$. The above calculation accounts for a Galactic extinction of $A_B = 0.268$ mag (Schlegel, Finkbeiner, & Davis 1998) corrected using the extinction law of Cardelli, Clayton, & Mathis (1989). To obtain the [*intrinsic*]{} SFR, we need to know the flux magnification factor due to the lensing. According to the lensing model of Allen, Schmidt, & Fabian (2002), the magnification factor of Arc 1 is 7.7 (R. W. Schmidt, private communication). Thus, the true lower limit to the SFR for Arc 1 is $\sim$3 $M_{\odot}$ yr$^{-1}$. This is comparable to the level of star formation activity in nearby gas-rich spiral galaxies (e.g., Kennicutt 1983) but is significantly lower than those obtained for the arcs in Abell 2218 (Ebbels et al. 1996) and Abell 2390 (Bézecourt & Soucail 1997; Lémonon et al. 1998). It is unclear whether slit losses alone can make up for the difference.
For Arc 2, we tentatively identify the pair of absorption lines at 7017 and 7084 Å with $\lambda$3933 (K) and $\lambda$3968 (H) at a redshift of 0.785. This interpretation is plausible considering (1) the redness of the continuum, which is suggestive of an old stellar population, and (2) the absence of any strong emission lines blueward of the absorption lines. An old stellar population should show a more prominent 4000 Å break than observed, but the shape of the spectrum redward of $\sim$7200 Å is too uncertain (due to telluric molecular absorption bands and residuals from subtraction of sky lines) to be definitive on this point. For an elliptical galaxy at $z \approx 0.8$, the observed $V-I$ color of 2.7 mag corresponds to a present-day restframe $V-I \approx 1.2$ mag (Poggianti 1997), consistent with the colors of local elliptical galaxies (e.g., Fukugita, Shimasaku, & Ichikawa 1995). Allen et al. (2002) recently used [*Chandra*]{} data to refine the mass model for RX J1347.5$-$1145. Adjusting their model to reproduce the redshift measurement of Arc 1 by Sahu et al. (1998), Allen et al. (2002) predict the redshifts of the other arcs. For Arc 2, they give $z = 0.75 \pm 0.05$, in excellent agreement with our value.
Allen et al. (2002) predict $z = 0.97 \pm 0.05$ for Arcs 3 and 4. Since both of these objects have featureless continua very similar in shape to that of Arc 1 — indeed, they are [*bluer*]{}, suggesting an even younger stellar population — it is reasonable to expect that nebular emission should be present at a comparable, if not even greater, strength. However, no significant emission feature is discernible at $\sim$7340 Å, the expected location of at $z = 0.97$. Unfortunately, the quality of the spectrum in the region $\sim$7300–7400 Å is degraded by imperfect removal of the OH sky lines (see, e.g., Osterbrock & Martel 1992). Nevertheless, a narrow emission feature with a strength comparable to that of the line in Arc 1, and perhaps even a factor 2–3 weaker, would almost certainly have been detected in Arc 4, since the two objects have virtually identical continuum levels. These arguments suggest that the redshift of Arc 4 is greater than 1.04, where we have taken the upper limit of our bandpass to be 7600 Å. This redshift limit does not appear to be in serious conflict with the predicted value.
The faintness of Arc 3 makes its spectrum highly uncertain at the red end, and we limit our discussion to wavelengths 7200 Å, where the continuum is clearly detected and not severely affected by systematic effects. The absence of any emission features with equivalent widths greater than $\sim$10 Å suggests that the redshift is likely to be larger than 0.93. This is consistent with the value predicted by Allen et al. (2002).
This work is funded by NASA LTSA grant NAG 5-3556 and by NASA grants HST-AR-07527.03-A and HST-AR-08361.02-A from the Space Telescope Science Institute (operated by AURA, Inc., under NASA contract NAS5-26555). We are grateful to Paul Martini for obtaining the broad-band images of the cluster. We thank Andrew McWilliam, Patrick McCarthy, and Daniel Kelson for helpful discussions on the spectroscopic reductions. Robert Schmidt kindly communicated the magnification factors for the arcs. An anonymous referee gave helpful suggestions for improving the paper.
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[^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have made a multiwavelength study of the overlapping error boxes of the unidentified $\gamma$-ray sources TeV J2032+4130 and 3EG J2033+4118 in the direction of the Cygnus OB2 association ($d = 1.7$ kpc) in order to search for a point-source counterpart of the first unidentified TeV source. Optical identifications and spectroscopic classifications for the brighter X-ray sources in [*ROSAT*]{} PSPC and [*Chandra*]{} ACIS images are obtained, without finding a compelling counterpart. The classified X-ray sources are a mix of early and late-type stars, with one exception. The brightest source in the [*Chandra*]{} observation is a new, hard absorbed source that is both transient and rapidly variable. It lies $7^{\prime}$ from the centroid of the TeV emission, which places it outside of the claimed $2\sigma$ location ($r \approx 4.\!^{\prime}8$). A possible eclipse or “dip” transition is seen in its light curve. With a peak 1–10 keV luminosity of $\approx 7 \times 10^{32}\,(d/1.7\,{\rm kpc})^2$ ergs s$^{-1}$, this source could be a quiescent low-mass X-ray binary that lies beyond the Cyg OB2 association. A coincident, reddened optical object of $R = 20.4,\ J = 15.4,\ H = 14.2$, and $K = 13.4$ is observed, but not yet classified due to the lack of obvious emission or absorption features in its spectrum. Alternatively, this [*Chandra*]{} and optical source might be a considered a candidate for a “proton blazar,” a long hypothesized type of radio-weak $\gamma$-ray source. More detailed observations will be needed to determine the nature of this variable X-ray source, and to assess the possibility of its connection with TeV J2032+4130.'
author:
- 'R. Mukherjee, J. P. Halpern, E. V. Gotthelf, M. Eracleous, N. Mirabal'
title: 'Search for a Point-Source Counterpart of the Unidentified Gamma-Ray Source TeV J2032+4130 in Cygnus'
---
Introduction
============
The serendipitous detection of emission coming from the direction of the Cygnus OB2 stellar association by the HEGRA CT-System at La Palma (Daum et al. 1997) represents the first discovery of an unidentified TeV source, extending the mystery of the elusive $\gamma$-ray sources to the TeV regime. TeV J2032+4130 was discovered (Aharonian et al. 2002a) in observations originally devoted to Cygnus X–3 and the unidentified EGRET (Energetic Gamma-ray Experiment Telescope) source GeV J2035+4214 (Lamb & Macomb 1997). An analysis of these combined fields led to the detection of a source at (J2000) $20^{\rm h}32^{\rm m}07^{\rm s}\pm 9.\!^{\rm s}2_{\rm stat}\pm2.\!^{\rm s}2_{\rm sys},
\ +41^{\circ}30^{\prime}30^{\prime\prime}
\pm 2.\!^{\prime}0_{\rm stat} \pm 0.\!^{\prime}4_{\rm sys}$. Its error circle overlaps the edge of the 95% confidence error ellipse of another EGRET source, 3EG J2033+4118, and is $\approx 0.\!^{\circ}5$ north of Cyg X–3. It is not clear if TeV J2032+4130 is associated with 3EG J2033+4118, which is itself of uncertain origin. There is some evidence that the TeV emission is extended, with a Gaussian $1\sigma$ radius of $\sim 5.\!^{\prime}6 \pm 1.\!^{\prime}7$. Unlike the flaring blazars that have been detected so far, TeV J2032+4130 was found to be steady in repeated HEGRA observations from 1999 to 2001. The integrated flux measured above 1 TeV was found to be $(4.5\pm 1.3_{\rm stat})\times10^{-13}$ photons cm$^{-2}$ s$^{-1}$, which is $\approx 2.6\%$ of the flux of the Crab Nebula (Aharonian et al. 2002a).
The majority of the $\gamma$-ray sources detected above 100 MeV by the EGRET instrument on the [*Compton Gamma Ray Observatory*]{} (CGRO) are unidentified, some remaining so since the first surveys of the $\gamma$-ray sky with the COS–B satellite. Resolving the nature of these mysterious $\gamma$-ray sources is a challenge across all wavelengths. Their relatively large error boxes make counterpart searches difficult. Unidentified EGRET sources typically have positional uncertainties of $\sim 0.\!^\circ5-1^\circ$, making identification on the basis of position alone nearly impossible, especially in the Galactic plane. We have found that multiwavelength studies of EGRET fields on a case-by-case basis are often useful in finding a likely identification. Such an approach has been used to suggest counterparts for the EGRET unidentified sources 2EG J0635+0521 (Kaaret et al. 1999), 3EG J2227+6122 (Halpern et al. 2001a), 3EG J1835+5918 (Halpern et al. 2002; Mirabal et al. 2000,2001), 3EG J2006–2321 (Wallace et al. 2002), 3EG J2016+3657 (Mukherjee et al. 2000; Halpern et al. 2001b), 3EG J1621+8203 (Mukherjee et al. 2002), and 3EG 2021+3716 (Roberts et al. 2002), to name a few examples (see the review by Caraveo 2002).
Imaging atmospheric Cherenkov telescopes (IACTs) like HEGRA have the advantage of much better angular resolution, and therefore smaller error boxes for $\gamma$-ray source positions in comparison to EGRET. In this paper, we adopt for the moment the hypothesis that TeV J2032+4130 is [*not*]{} significantly extended, in which case one should search for and evaluate candidate point-source counterparts at other wavelengths. Accordingly, we studied the EGRET source 3EG J2033+4118, archival radio data from the NRAO-VLA Sky Survey (NVSS: Condon et al. 1998), and pointed [*ROSAT*]{} X-ray observations of the field of 3EG J2033+4118 and TeV J2032+4130. We also examined a recent [*Chandra*]{} observation that was centered on the TeV source, and obtained optical identifications of the brightest X-ray sources in this region using the MDM Observatory, Kitt Peak National Observatory (KPNO), and the [*Hobby-Eberly Telescope*]{} ([*HET*]{}).
Gamma-ray Observations of 3EG J2033+4118
========================================
The EGRET source 3EG J2033+4118 is located at $l=80.\!^{\circ}27$, $b=+0.\!^{\circ}73$ (Hartman et al. 1999), with a 95% error ellipse defined by semi-major and semi-minor axes of $0^\circ\!.31$ and $0^\circ\!.25$, respectively (Mattox, Hartman, & Reimer 2001). Figure 1 shows the 95% confidence EGRET ellipse superimposed on a [*ROSAT*]{} X-ray image that is described further in §3. 3EG J2033+4118 was detected by EGRET in several viewing periods, with the most significant detection in 1993 March (VP 212). Figure 2 shows the light curve measured by EGRET during 1991–1997. The points before 1996 come from the Third EGRET Catalog (3EG, Hartman et al. 1999), in which the summed exposure has a flux above 100 MeV of $(73.0\pm6.7) \times 10^{-8}$ photons cm$^{-2}$ s$^{-1}$. The corresponding background-subtracted spectrum of 3EG J2033+4118 is hard, with a photon spectral index of $1.96\pm0.10$ (Hartman et al. 1999).
This location was also in the EGRET field of view four times after the 3EG catalog. For two of these the source was almost $30^{\circ}$ from the axis, so we exclude them. The remaining two detections are $2.4\sigma$ and $4.2\sigma$. Their fluxes are shown as the last two points in Figure 2. A chi-square analysis was used to calculate the “variability index” defined by McLaughlin et al. (1996). This index is sometimes used to judge variability in EGRET sources, although it is somewhat arbitrary. The variability index of 3EG J2033+4118 was found to be $V=1.4$. For comparison, McLaughlin et al. (1996) interpret $V < 0.5$ as non-variable and $V \geq 1.0$ as variable.
X-ray and Radio Observations
============================
Figure 1 shows an X-ray image taken with the [*ROSAT*]{} Position Sensitive Proportional Counter (PSPC) in the energy range 0.2–2.0 keV, covering the field of 3EG J2033+4118/TeV J2032+4130. This image was created by co-adding exposure-corrected skymaps of 23.5 ks of data taken during 1991 and 1993. These fields were targeted originally to include the Cyg OB2 association and also Cyg X-3. The EGRET 95% error ellipse for 3EG J2033+4118 is superposed on the image, as is the $1\sigma$ error contour of the TeV J2032+4130 centroid position. The larger circle around the TeV position indicates the possible Gaussian $1\sigma$ extent of the TeV source (Aharonian et al. 2002a). Figure 3 shows a [*ROSAT*]{} HRI image, covering roughly the same field as that shown in Figure 1. The image was created by co-adding exposure corrected skymaps of $\approx 155$ ks of data taken during observations of Cyg OB2 and Cygnus X-3 in 1993–1995. The sources numbered in Figure 3 correspond to those in Figure 1 and Table 1.
A detailed description of [*ROSAT*]{} results on the sources in the Cyg OB2 association is given by Waldron et al. (1998). Several of the stars in the Cyg OB2 association are among the strongest stellar X-ray sources in the Galaxy. In Table 1 we list the coordinates of the brightest ROSAT sources as marked in Figures 1 and 3, along with optical identifications and positions. We have concentrated on obtaining optical identifications of several X-ray sources that are within the region of maximum likelihood for a point source of TeV emission. We find that all of these have ordinary stellar counterparts. Further details on their optical properties are presented in §4.2.
On 2002 August 11, [*Chandra*]{} made a 5 ks director’s discretionary observation (Butt et al. 2003) of the field of TeV J2032+4130 with the front-illuminated, imaging CCD array of the Advanced CCD Imaging Spectrometer (ACIS-I). Several X-ray sources near the centroid of the TeV source were detected. Figure 4 shows the [*Chandra*]{} image with the brightest point sources marked, which are those having at least 10 photons. Their positions and count rates are listed in Table 2. Cross-references to sources detected by both [*ROSAT*]{} and [*Chandra*]{} are indicated in Tables 1 and 2. The brightest [*Chandra*]{} source \#2 is notable in that it was [*not*]{} detected in any of the [*ROSAT*]{} or [*Einstein*]{} observations of this field. More details about this source are given in §5.1.
Table 3 is a list of eight 1.4 GHz sources from the NVSS within $10^{\prime}$ of the TeV centroid. The brightest is an extended source with flux density of only 18 mJy, which is at least an order of magnitude fainter than the weakest candidate radio sources for EGRET blazars considered by Mattox et al. (1999). Furthermore, none of the radio sources is coincident with an X-ray source. Thus, it is unlikely that TeV J2032+4130 has an ordinary blazar counterpart, although new types of blazars can be hypothesized (see §5.3).
Optical Observations
====================
Optical Imaging
---------------
In preparation for the identification of X-ray sources in the [*Chandra*]{} observation, on 2002 July 7 we obtained a deep $R$-band image of the field of TeV J2032+4130 using the MDM 1.3m telescope and a thinned, back-illuminated $2048 \times 2048$ pixel SITe CCD. This $17^\prime \times 17^\prime$ image covers all of the [*Chandra*]{} source positions marked in Figure 4. An astrometric grid was established for the image using 51 stars from the USNO–A2.0 catalog (Monet et al. 1996), with a resulting rms dispersion of $0.\!^{\prime\prime}49$. Differences between optical and X-ray positions for the most precisely located X-ray sources indicate that the X-ray aspect solution agrees with the USNO–A2.0 reference frame to within $0.\!^{\prime\prime}5$. Additional images in $B, V, R,$ and $I$ were obtained of the central $9^\prime \times 9^\prime$ of the same field using the MDM 2.4m telescope on 2002 August 23–28.
Likely optical identifications of [*Chandra*]{} sources are listed in Table 2, together with approximate $R$ magnitudes from our images or from the USNO–A2.0 where available. Only four of the [*Chandra*]{} sources have no optical counterpart to a limiting magnitude $> 23$. These happen to be the hardest sources in the image, with $>75\%$ of their photons above 2 keV. Thus they are likely to be AGNs that are highly absorbed by the Galactic ISM, and not, for example, nearby, old neutron stars. A higher-resolution image of the crowded region around [*Chandra*]{} source \#2 was obtained on the MDM 2.4m telescope on 2002 November 24. Figure 5 shows the possible identification of that source on the 2.4m image.
Optical Spectroscopy
--------------------
We obtained complete spectroscopic identifications for all [*ROSAT*]{} sources within $10^\prime$ of the centroid of TeV J2032+4130, using the Goldcam spectrograph on the KPNO 2.1m telescope on two runs, in 2002 June and October. The resulting spectra are shown in Figure 6. These are sources $a$, $b$, $c$, $e$, and $f$ in Table 1, and all are bright stars. They include one emission-line O star, one dMe star, and three foreground G stars, two of which are also listed in the Massey & Thompson (1991, hereafter MT91) compilation of stars in Cyg OB2. Magnitudes listed in Table 1 are from MT91, or from the USNO–A2.0 catalog (Monet et al. 1996).
A spectrum of the $R \approx 20.4$ counterpart of [*Chandra*]{} source \#2, also shown in Figure 6, was obtained with the [*HET*]{} and Marcario Low Resolution Spectrograph on 2002 December 8. However, its classification remains uncertain as it is of relatively poor signal-to-noise, having been observed through thin clouds. All of the apparent features in this spectrum can be attributed to imperfectly subtracted night-sky emission lines, leaving no definite intrinsic features in the 4600–9200 Å range.
A Transient X-ray Source in the Field of TeV J2032+4130
=======================================================
X-ray Properties
----------------
None of the X-ray sources in the immediate vicinity of TeV J2032+4130 are unusual in any way, except for [*Chandra*]{} source \#2, which lies $7^{\prime}$ from the TeV centroid. Although this is the brightest of the [*Chandra*]{} sources, with 195 photons detected, it is noticeably absent from any of the [*ROSAT*]{} or [*Einstein*]{} images. Thus, it may be described as a transient source. A light curve constructed from the 5 ks [*Chandra*]{} observation shows that the source was highly variable even during this brief period (Figure 7). After remaining faint for the first 3.5 ks, its count rate rose by about a factor of 10 for the final 1.5 ks.
Spectral analysis of source \#2, summarized in Table 4, is consistent with a power law of photon index $\Gamma \sim 2.0$, or with a hot plasma of $kT \sim 6$ keV. In addition to spectral fits for the summed observation, Table 4 presents fits for two intervals, corresponding to the first 3438 s (low state) and the final 1478 s (high state). Figure 8 shows the spectrum and best fitted power law for the summed observation. For either spectral model, a significant absorbing column of $N_{\rm H} \sim (1-2) \times 10^{22}$ cm$^{-2}$ is required, which is comparable to the largest X-ray and optical extinction values measured for stars in the Cyg OB2 association (MT91; Waldron et al. 1998). Thus, source \#2 is likely to be either embedded in the Cyg OB2 association at $d = 1740$ pc (MT91), or behind it. Its peak 1–10 keV flux of $\approx 2 \times 10^{-12}$ ergs cm$^{-2}$ s$^{-1}$ would correspond to a luminosity of $7 \times 10^{32}$ ergs s$^{-1}$ at $d = 1740$ pc. This value is somewhat larger than even the most luminous cataclysmic variables (Eracleous, Halpern, & Patterson 1991), and hints at a more compact source such as a neutron star or black hole in a quiescent low-mass X-ray binary (LMXB), or a very distant Be transient. If so, its location could be considerably beyond Cyg OB2.
The transition in the light curve of \#2 may be an egress from an eclipse, or, more likely for an LMXB, a “dip”. If so, the orbital period is probably longer than a few hours since the eclipse or dip is at least 1 hr long. A possible analog would be 4U 1624–49, the “big dipper,” which has period of 21 hr (Watson et al. 1985; Smale, Church, & Balucińska-Church 2001). However, assuming a distance of 8 kpc, the X-ray luminosity of \#2 would be only $1.5 \times 10^{34}$ ergs s$^{-1}$, much less than the $\sim 10^{38}$ ergs s$^{-1}$ luminosity of 4U 1624–49.
Optical and Infrared Properties
-------------------------------
As mentioned previously, the signal-to-noise ratio of the [*HET*]{} optical spectrum of the counterpart to source \#2 is too poor to classify it. However, it is likely to be the correct identification because the X-ray and optical positions differ by only $0.\!^{\prime\prime}37$. The absence of strong TiO absorption bands rules out a late K or M dwarf, which would otherwise be the expected spectral type for a star of its magnitude and color located between us and the Cyg OB2 association. Additional evidence that it is a more distant and luminous object comes from its detection in the 2MASS survey, with $J = 15.41 \pm 0.07,\ H = 14.17 \pm 0.06$, and $K = 13.45 \pm 0.06$. The absence of H$\alpha$ emission argues against a cataclysmic variable, as does the high-state X-ray luminosity of $\approx 7 \times 10^{32}\,(d/1.7\,{\rm kpc})^2$ ergs s$^{-1}$, although it cannot be ruled out that the optical spectrum is highly reddened emission from an accretion disk in a quiescent LMXB.
If we hypothesize a very large visual extinction of $A_V = 10$, which is compatible with the value of $N_{\rm H}$ fitted to the X-ray spectrum, then the dereddened magnitudes become $R = 12.9,\ J = 12.6,\ H = 12.3$, and $K = 12.3$ using the extinction curve of Cardelli, Clayton, & Mathis (1989). Such a flat color distribution would be compatible with accretion disk emission in an LMXB. Assuming a distance of 8 kpc, the corresponding absolute magnitude $M_R = -1.6$ is also in the range of LMXBs (van Paradijs & McClintock 1994). However, even at that large distance, the X-ray luminosity would be only $1.5 \times 10^{34}$ ergs s$^{-1}$, and then it is not clear that the accretion luminosity would dominate over the secondary star in the optical. Also, such a source would fall far from the relation between absolute magnitude, X-ray luminosity, and orbital period in LMXBs delineated by van Paradijs & McClintock (1994), having too small an X-ray luminosity.
Alternatively, if the distance and extinction are even larger, then it could be an early type star such as a B star in a transient high-mass X-ray binary system. However, the absence of H$\alpha$ emission argues against a Be star. Thus, there are no entirely satisfactory explanations of source \#2 in terms of any type of X-ray binary.
A Proton Blazar?
----------------
While an AGN classification for the optical spectrum of source \#2 cannot be immediately dismissed, the absence of any emission lines would favor a BL Lac identification, for which the lack of radio emission is highly unusual. Also, the known TeV blazars are highly episodic emitters, which is contrary to the steady nature of TeV J2032+4130. Thus, it would not be a simplification to hypothesize that \#2 is an AGN. Nevertheless, it has long been hypothesized that a class of radio-quiet blazars could exist (Mannheim 1993; Schlickeiser 1984) that are dominated by accelerated hadrons. In this “proton blazar” model, $\gamma$-ray emission arises from proton-induced cascades, and radio emission can be reduced if the ratio of accelerated electrons to protons is small. Alternatively, an extreme blazar whose synchrotron emission peaks at MeV energies, and inverse Compton at TeV energies, could be relevant (Ghisellini 1999). Because HEGRA performed a sensitive survey of the Galactic plane (Aharonian et al. 2002b), it may not be so surprising if it turns out that the first TeV selected blazar is discovered there.
Discussion and Conclusions
==========================
Spectroscopic optical identifications of most of the brighter X-ray sources in $\gamma$-ray error boxes of TeV J2032+4130 and 3EG J2033+4118 are O stars in the Cyg OB2 association at $d = 1.7$ kpc, or foreground late-type stars. Those [*Chandra*]{} sources that are identified with faint optical counterparts in the range $R \approx 17-20$, are probably M dwarfs, while the optically undetected sources with $R > 23$ are the most X-ray absorbed, thus are likely background AGNs. The only unusual X-ray source in this field is a transient one that is the brightest source in the recent [*Chandra*]{} observation. It has a peak 1–10 keV luminosity of $\approx 7 \times 10^{32}\,(d/1.7\,{\rm kpc})^2$ ergs s$^{-1}$. An optical spectrum of its $R = 20.4$ possible counterpart has, albeit with modest signal-to-noise, no strong emission or absorption features. The hard X-ray spectrum, rapid variability, and red optical/IR colors of this object suggest that it is a distant, quiescent X-ray binary system. On the other hand, it may also be the prototype of a new kind of AGN previously hypothesized to exist, the “proton blazar.” If so, its X-ray flaring behavior is a significant property, and related optical variability might be expected.
If we hypothesize that either TeV J2032+4130 or 3EG J2033+4118 is a point source, then we have a limited number of plausible point-source candidates at other wavelengths, perhaps only one. Without knowing the exact nature of [*Chandra*]{} source \#2, it is not a compelling identification for either TeV J2032+4130 or 3EG J2033+4118, especially since it lies outside both of their $2\sigma$ localization regions. It is $7^{\prime}$ from the centroid of the TeV source, while the $2\sigma$ position uncertainty of TeV J2032+4130 is given as $\approx 4.\!^{\prime}8$. However, since we are faced with the first unidentified TeV source, it is worthwhile to pursue whatever additional observations are needed to determine the nature of this variable X-ray candidate and to assess the possibility of its connection with TeV J2032+4130, no matter how remote.
If TeV J2032+4130 is truly an extended source, then it need not be centered on a point source counterpart at other wavelengths. Benaglia et al. (2001) suggested that colliding winds from the Cyg OB2 \#5 system and from other O stars in this association could be responsible for the EGRET source 3EG J2033+4118. Aharonian et al. (2002a) summarized those arguments, and hypothesized two possible origins for extended TeV emission that may be displaced from its originating source of energy. One is that TeV emission could arise from $\pi^0$ decay resulting from hadrons accelerated in shocked OB star winds and interacting with a local, dense gas cloud. The other is inverse Compton TeV emission in a jet-driven termination shock, either from an as-yet undetected microquasar, or from Cyg X-3. Another reason to investigate the nature of [*Chandra*]{} \#2 would be to find out if it could be such a jet source. We plan to pursue more detailed X-ray and optical studies of this source.
This publication makes use of data obtained from HEASARC at Goddard Space Flight Center and the SIMBAD astronomical database. It also makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This work was based in part on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, Pennsylvania State University, Stanford University, Ludwig-Maximillians-Universität München, and Georg-August-Universität Göttingen. The Marcario Low-Resolution Spectrograph is a joint project of the Hobby-Eberly Telescope Partnership and the Instituto de Astronomia de la Universidad Nacional Autónoma de Mexico. R. M. acknowledges support from NSF grant PHY-9983836. E.V.G. is supported by NASA LTSA grant NAG 5-7935. [*Chandra*]{} studies of unidentified $\gamma$-ray sources is supported by SAO grants GO2-3071X and GO2-3082X to J.P.H.
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[ccccllccc]{} $a$ & . . . & 20 32 18.1 +41 28 07 & 20 32 19.244 +41 27 57.37 & . . . & G8V & 15.4 & . . . & 15.0\
$b$ & . . . & 20 32 41.5 +41 27 44 & 20 32 41.454 +41 27 43.51 & . . . & M5Ve & 16.8 & . . . & 15.8\
$c$ & 18 & 20 32 16.1 +41 27 08 & 20 32 13.836 +41 27 12.33 & Cyg OB2 \#4 & O7III((f)) & 11.42 & 10.23 & 10.2\
$d$ & . . . & 20 33 03.9 +41 24 14 & 20 33 03.902 +41 24 08.48 & . . . & . . . & 16.4 & . . . & 16.9\
$e$ & 25 & 20 31 36.5 +41 23 38 & 20 31 37.267 +41 23 36.01 & MT91 \#115 & G6V & 13.90 & 13.10 & 13.1\
$f$ & 26 & 20 31 51.5 +41 23 21 & 20 31 51.319 +41 23 23.79 & MT91 \#152 & G3V & 13.40 & 12.74 & 13.1\
$g$ & . . . & 20 33 14.0 +41 20 14 & 20 33 14.144 +41 20 22.00 & Cyg OB2 \#7 & O3If & 11.94 & 10.50 & . . .\
$h$ & . . . & 20 33 15.3 +41 18 47 & 20 33 15.077 +41 18 50.50 & Cyg OB2 \#8A & O5.5I(f) & 10.08 & 8.99 & 9.1\
$i$ & . . . & 20 32 22.7 +41 18 17 & 20 32 22.432 +41 18 18.85 & Cyg OB2 \#5 & O7e & 10.64 & 9.21 & 8.1\
$j$ & . . . & 20 33 11.0 +41 15 07 & 20 33 10.733 +41 15 08.22 & Cyg OB2 \#9 & O5Iab:e & 12.61 & 10.78 & . . .\
$k$ & . . . & 20 32 41.4 +41 14 28 & 20 32 40.957 +41 14 29.30 & Cyg OB2 \#12 & B5Iab: & 14.41 & 11.40 & . . .\
$l$ & . . . & 20 32 32.1 +41 14 11 & 20 32 31.556 +41 14 08.48 & MT91 \#267 & . . . & 15.06 & 12.87 & 11.8\
$m$ & 27 & 20 31 38.0 +41 13 21 & 20 31 37.504 +41 13 21.05 & Cyg OB2 \#3 & O9: & 11.50 & 10.35 & 9.3\
$n$ & . . . & 20 33 09.4 +41 13 21 & 20 33 08.820 +41 13 18.00 & MT91 \#417 & O4III(f) & 13.59 & 11.5 & 9.3\
$o$ & . . . & 20 33 02.3 +41 11 19 & 20 33 01.793 +41 11 11.41 & MT91 \#384 & . . . & 17.28 & 16.73 & 16.1\
$p$ & . . . & 20 33 23.3 +41 09 07 & 20 33 23.477 +41 09 13.38 & MT91 \#516 & O5.5V((f)) & 14.04 & 11.84 & 11.0\
$q$ & . . . & 20 32 07.9 +41 08 37 & 20 32 07.330 +41 08 50.56 & . . . & . . . & 14.7 & 14.2 & 13.6\
$r$ & . . . & 20 33 41.6 +41 08 01 & 20 33 42.036 +41 07 53.70 & MT91 \#615 & . . . & 12.2 & 11.51 & 11.2\
[cccccclc]{} 1 & . . . & 20 31 56.426 +41 37 23.35 & 7 & 35 & . . . & . . . & $>23.2$\
2 & . . . & 20 31 43.755 +41 35 55.17 & 77 &118 & 20 31 43.739 +41 35 55.49 & . . . & 20.4\
3 & . . . & 20 32 10.248 +41 35 10.21 & 4 & 10 & 20 32 10.219 +41 35 11.31 & . . . & 18.3\
4 & . . . & 20 32 25.392 +41 34 01.92 & 16 & 4 & 20 32 25.346 +41 34 02.10 & MT91 \#249 & 12.2\
5 & . . . & 20 32 05.328 +41 33 12.38 & 3 & 10 & . . . & . . . & $>23.7$\
6 & . . . & 20 32 27.456 +41 32 50.78 & 3 & 10 & . . . & . . . & $>23.7$\
7 & . . . & 20 31 51.855 +41 31 18.91 & 3 & 22 & . . . & . . . & $>23.7$\
8 & . . . & 20 32 30.504 +41 31 01.74 & 6 & 5 & 20 32 30.412 +41 31 02.88 & . . . & 21.2\
9 & . . . & 20 31 39.192 +41 30 18.00 & 9 & 2 & 20 31 39.104 +41 30 18.49 & . . . & 17.1\
10 & . . . & 20 31 23.544 +41 29 48.77 & 13 & 9 & 20 31 23.573 +41 29 49.45 & . . . & 15.1\
11 & . . . & 20 32 18.845 +41 29 32.42 & 20 & 15 & 20 32 18.812 +41 29 32.83 & . . . & 19.3\
12 & . . . & 20 32 37.848 +41 28 52.39 & 10 & 9 & 20 32 37.831 +41 28 52.93 & . . . & 16.7:\
13 & . . . & 20 32 25.752 +41 28 42.53 & 8 & 7 & 20 32 25.731 +41 28 42.89 & . . . & 17.3\
14 & . . . & 20 32 28.080 +41 28 28.52 & 3 & 19 & 20 32 28.027 +41 28 29.02 & . . . & 18.8\
15 & . . . & 20 32 52.163 +41 28 15.92 & 10 & 14 & 20 32 52.206 +41 28 17.17 & . . . & 19.3:\
16 & . . . & 20 32 42.072 +41 27 48.13 & 17 & 1 & 20 32 42.014 +41 27 48.32 & . . . & 15.7\
17 & . . . & 20 32 14.688 +41 27 39.67 & 7 & 8 & 20 32 14.693 +41 27 40.09 & MT91 \#221 & 12.0\
18 & $c$ & 20 32 13.843 +41 27 12.13 & 27 & 8 & 20 32 13.837 +41 27 12.34 & Cyg OB2 \#4 & 10.2\
19 & . . . & 20 31 33.662 +41 26 51.46 & 15 & 22 & 20 31 33.650 +41 26 51.71 & . . . & 19.6\
20 & . . . & 20 32 27.598 +41 26 21.62 & 20 & 3 & 20 32 27.663 +41 26 22.44 & MT91 \#258 & 10.4\
21 & . . . & 20 32 45.432 +41 25 36.44 & 19 & 4 & 20 32 45.462 +41 25 37.51 & Cyg OB2 \#6 & 10.1\
22 & . . . & 20 32 55.344 +41 25 17.00 & 15 & 8 & 20 32 54.773 +41 25 16.09 & MT91 \#360 & 16.2\
23 & . . . & 20 32 06.877 +41 25 10.61 & 23 & 26 & 20 32 06.826 +41 25 10.68 & . . . & 20.3\
24 & . . . & 20 31 47.568 +41 24 48.02 & 12 & 15 & 20 31 47.535 +41 24 48.43 & . . . & 18.1\
25 & $e$ & 20 31 37.251 +41 23 35.44 & 29 & 3 & 20 31 37.267 +41 23 36.01 & MT91 \#115 & 13.1\
26 & $f$ & 20 31 51.321 +41 23 22.75 & 54 & 7 & 20 31 51.319 +41 23 23.79 & MT91 \#152 & 13.1\
27 & $m$ & 20 31 37.156 +41 13 17.42 &169 & 59 & 20 31 37.504 +41 13 21.05 & Cyg OB2 \#3 & 9.3\
[ccr]{} 20 31 31.18 & +41 25 12.6 & 2.1\
20 31 45.08 & +41 35 34.2 & 18.1\
20 31 48.17 & +41 33 58.3 & 6.1\
20 31 56.81 & +41 35 31.4 & 4.5\
20 32 01.34 & +41 37 22.7 & 14.0\
20 32 08.33 & +41 29 17.4 & 3.8\
20 32 17.04 & +41 26 21.4 & 2.3\
20 32 52.27 & +41 30 09.5 & 10.6\
[lccc]{} Counts (1–6 keV) & 169 & 35 & 134\
Exposure Time (s) & 4916 & 3438 & 1478\
Count Rate (s$^{-1}$) & 0.03 & 0.01 & 0.09\
Power Law: & & &\
$N_{\rm H}\ (10^{22}$ cm$^{-2}$) & $1.5 (0.6-2.3)$ & 1.3 & 1.7\
$\Gamma$ & $2.0 (1.2-2.7)$ & 2.1 & 2.0\
Flux ($10^{-13}$ ergs cm$^{-2}$ s$^{-1}$) & 6.6 & 1.8 & 18.0\
$\chi^2_{\nu}$(dof) & 0.6(13) & 1.0(2) & 0.4(10)\
Raymond-Smith: & & &\
$N_{\rm H}\ (10^{22}$ cm$^{-2}$) & $1.2 (0.6-1.8)$ & 0.9 & 1.6\
$kT$ (keV) & $6.0 (3.0-\infty)$ & 6.6 & 4.0\
Flux ($10^{-13}$ ergs cm$^{-2}$ s$^{-1}$) & 7.1 & 2.1 & 17.4\
$\chi^2_{\nu}$(dof) & 0.6(13) & 1.1(2) & 0.3(10)\
![[*ROSAT*]{} PSPC X-ray image. The bright source is Cyg X-3. The properties of the numbered sources are given in Table 1. The ellipse is the 95% uncertainty location of 3EG J2033+4118 from Mattox et al. (2001). The small circle is the $1\sigma$ uncertainty of the centroid of TeV J2032+4130, and the large circle is the estimated Gaussian $1\sigma$ extent of the TeV emission (Aharonian et al. 2002a). The squares are the fields of view of the CCDs in the subsequent [*Chandra*]{} observation (see Figure 4).[]{data-label="fig1"}](f1.eps){height="40pc"}
![EGRET $\gamma$-ray light curve for 3EG J2033+4118, 1991–1995 from Hartman et al. (1999), and 1996–1997 from this paper. Arrows are $2 \sigma$ upper limits. The horizontal error bars correspond to the extent of an individual observation.[]{data-label="fig2"}](f2.eps){height="20pc"}
![Optical spectra of five [*ROSAT*]{} sources in the field of TeV J2032+4130 taken with the KPNO 2.1m telescope, and one [*Chandra*]{} source with the [*HET*]{}.[]{data-label="fig6"}](f6.eps){height="40pc"}
![[*Upper panel*]{}: Light curve of [*Chandra *]{} source \#2 extracted from an aperture of radius $12^{\prime\prime}$, in 50 s time bins. The [*dots*]{} below the light curve are the arrival times of the individual photons. [*Lower panel*]{}: Local background extracted from an area 33 times larger than the source extraction region. Thus, background is demonstrated to be stable, and a negligible contaminant of the source light curve.[]{data-label="fig7"}](f7.eps){height="40pc"}
![Spectrum of [*Chandra*]{} source \#2 ([*crosses*]{}) and best fitted power-law model ([*solid line*]{}). This spectrum is for the entire time interval extracted from a 12$^{\prime\prime}$ radius aperture. The background is treated as negligible, as demonstrated in Figure 7.[]{data-label="fig8"}](f8.eps){height="40pc"}
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'A. Zsom$^{1}$, C.W. Ormel$^{1}$, C. P. Dullemond$^{1,2}$, Th. Henning$^{1}$'
subtitle: 'III. Sedimentation driven coagulation inside the snow-line'
title: |
The outcome of protoplanetary dust growth:\
pebbles, boulders, or planetesimals?
---
[The evolution of dust particles in protoplanetary disks determines many observable and structural properties of the disk such as the spectral energy distribution (SED), the appearance of disks, temperature profile, and chemistry. Dust coagulation is also the first step towards planet formation.]{} [We investigate dust growth due to settling in a 1D vertical column of a disk. We gradually build up the complexity of the models by considering the effects of porosity, different collision models, turbulence, and different gas models respectively. This way we can distinguish the effects of these physical processes on particle growth and motion. It is known from the 10 micron feature in disk SEDs, that small micron-sized grains are present at the disk atmosphere throughout the lifetime of the disk. We hope to explain such questions as what process can keep the disk atmospheres dusty for the lifetime of the disk and how does the particle properties change as a function of height above the midplane.]{} [We use a Monte Carlo code to follow the mass and porosity evolution of the particles in time. The used collision model is based on laboratory experiments performed on dust aggregates. As the experiments cannot cover all possible collision scenarios, the largest uncertainty of our model is the necessary extrapolations we had to perform. We simultaneously solve for the particle growth and motion. Particles can move vertically due to settling and turbulent mixing. We assume that the vertical profile of the gas density is fixed in time and only the solid component evolves.]{} [We find that the used collision model strongly influences the masses and sizes of the particles. The laboratory experiment based collision model greatly reduces the particle sizes compared to models that assume sticking at all collision velocities. We find that a turbulence parameter of $\alpha = 10^{-2}$ is needed to keep the dust atmospheres dusty, but such strong turbulence can produce only small particles at the midplane which is not favorable for planetesimal formation models. We also see that the particles are larger at the midplane and smaller at the upper layers of the disk. At 3-4 pressure scale heights micron-sized particles are produced. These particle sizes are needed to explain the 10 micron feature of disk SEDs. Turbulence may therefore help to keep small dust particles in the disk atmosphere.]{}
Introduction
============
The coagulation of dust particles is the first step towards planet formation. As dust particles are the main source of opacity [@Semenov2003], they also determine many observable quantities of disks, such as the spectral energy distribution and scattered light images (see below). Due to the vertical component of the stellar gravity, dust particles sediment towards the midplane of the disk. The relative settling velocity between particles of different aerodynamical properties drives coagulation and this process was investigated by several authors [@Safronov1969; @Nakagawa1986; @Schrapler2004; @Dullemond:2004p325; @Dullemond2005].
Observational evidence of the vertical sedimentation of grains exists for a large number of disks, although such evidence is usually indirect [@Henning2011]. Sub-micron grains are present at the disk surfaces as shown by scattered light images in the optical and near infrared (NIR) wavelengths. Such multi-wavelength scattered light images provide evidence for grain growth ([@Watson2007] and references therein). [@Pinte2007] showed by reproducing multi-band images of the binary system of GGTau that the dust scale height for 10 micron-sized particles is roughly half of that for micron-sized particles. The spectral energy distribution (SED) is also affected by settling. [@D'Alessio2006] showed that in order to explain the median SEDs of classical TTauri stars, the dust to gas ratio has to be reduced by a factor of 10 at the disk atmosphere compared to the standard value. There are also indications that the settling of grains is correlated with the age of the disk [@Sicilia-Aguilar2007]. However, the connection between the exact shape of the 10 micron feature of SEDs and sedimentation is not well understood [@Dullemond2008; @Juh'asz2010]. [@Apai2005] showed evidence for settling in disks around brown dwarf stars. They concluded that growth, crystallization and settling of dust happens around low mass stars in a similar manner as around intermediate and solar mass stars. This suggests that the first stage of planet formation is a robust process occurring in all kinds of disks.
Sedimentation also affects the vertical temperature structure of the disk. The simulations of [@Aikawa2006] showed that as the dust particles sediment towards the midplane, the opacity is reduced, and the temperature of the gas decreases. As the stellar radiation can now penetrate deeper in the disk, the temperature at intermediate heights increases. The change in the density and temperature structure as well as the radiation field naturally influences the chemistry of the disk atmospheres [@Bergin2007].
Recently [@Vasyunin2011] investigated the effects of dust growth on disk chemistry and found that the main effect of coagulation on the disk chemical composition comes mostly from sedimentation in their models.
Dust sedimentation not only affects the upper layers of the disk, but also the midplane regions. As long as the dust to gas ratio is much smaller than unity, the dust can be treated as a passive tracer in the gas flow. However, if the dust to gas ratio becomes comparable to unity, the dust will influence the gas. The dust can be accumulated by sedimentation around the midplane of the disk. In such a scenario, a shear is present between the dusty midplane layer and the less dusty upper layers. This shear triggers Kelvin-Helmholtz instability which develops into turbulence. This process was first recognized by [@Weidenschilling1980] and is still under active investigation (e.g. by [@Johansen2006], [@Chiang2008], [@Lee2010]).
A collaboration started between the lab-community and the modelers to better constrain the dust evolution in protoplanetary disks, using a realistic collision model that is based on the laboratory experiments. In [@Guttler2010] (henceforth Paper I), we introduced this collision model. In [@Zsom2010] (henceforth Paper II), we used this collision model for the first time in the Monte Carlo (MC) method of [@Zsom2008] (henceforth ZsD08). The models of Paper II were local box models, meaning that the dust evolution was only followed at one location of the disk. These models showed that bouncing plays an important role in dust evolution.
We further develop these models to simulate a 1D vertical column in the disk, thus investigating sedimentation-driven coagulation. We want to better understand the process of sedimentation and the role of particular physical phenomena like porosity of the aggregates, collision models and turbulence.
Previous work by [@Dullemond2005] (henceforth DD05) showed that without a mechanism that reduces the sticking probability of particles in the upper layers of the disk, or without a continuous source of small particles, the observed spectral energy distributions (SED) of TTauri stars should exhibit a very weak IR excess. In contrast, the observed SEDs of TTauri stars have strong IR excesses (e.g. [@Furlan2005; @Kessler-Silacci2006; @Bouwman2008]). Therefore some grain-retention mechanism is needed to explain the SEDs.
Previous models of grain evolution assumed a continuous cycle of growth and fragmentation, which provides the necessary amount of small particles (e.g. [@Brauer2008a], [@Birnstiel2009]). However, Paper I and II showed that particle evolution is halted by bouncing and no cycle of growth and fragmentation is present. In this paper we simulate dust evolution driven by Brownian motion, turbulence, and sedimentation in a 1D vertical column of the inner disk and the additional physics of Paper I and II are included. We investigate the time evolution of sedimentation-driven coagulation, and search for ways that can keep a sufficient amount of the small dust particles levitated at several pressure scale heights to explain the observed SEDs of young stars.
The paper is organized as follows. In Sect. \[sec:num\] we describe the numerical method used to follow the particle motion and coagulation. We validate the code and increase the complexity of the model step-by-step in Sect. \[sec:intro\]. We show the results in Sect. \[sec:res\], finally we discuss those results in Sect. \[sec:disc\] and provide a summary in Sect. \[sec:concl\].
Numerical method {#sec:num}
================
Basic considerations {#subsec:bas}
--------------------
The local box approach (or “particle-in-a-box” approach) in Paper II is based on two assumptions. 1.) The particles are homogeneously mixed inside the box. 2.) Particles do not enter or leave the box, i.e. it is closed. Due to these two assumptions, it was not necessary to follow the exact location of the particles.
In the models considered here, however, we place such boxes (or grid cells) on top of each other to simulate a 1D column in the disk and follow how particles settle towards the midplane. Inevitably, particles move from box to box during this process. Therefore the assumption that particles cannot enter or leave the boxes has to be relaxed. The first assumption of the method in Paper II is kept, we still assume that during the coagulation calculation, the particles inside a given box are homogeneously distributed (however, for particle motions, the individual positions of the particles are used). The second assumption is modified in the following way. 2.) The simulated column is closed, e.g. particles inside the column can move freely vertically. However neither do *new* particles enter from the “outside”, nor do particles from inside the column *leave*. As particles move through the boxes, it is necessary to follow the position of the particles (see Sect. \[subsec:pos\]) as we must find out in which box a particle is located.
The motion of particles imposes a limit on the time step of the simulation. We do not want the particles to move more than one box in a time step. A sedimenting particle should have the possibility to interact with all other particles along its way, it should not skip over boxes thus avoiding the particles in it. Therefore, we use an adaptive time stepping method. The maximum of all particle velocities is obtained ($v_{\mathrm{max}}$), and since we know the height of the boxes ($h_{\mathrm{box}}$), the maximal (safe) time step can be determined as $$\Delta t = C \frac{h_{\mathrm{box}}}{v_{\mathrm{max}}},$$ where $C$ is the Courant number which we typically set to be $0.1$.
The code schematically performs the following steps:
1. The velocities of the particles are calculated.
2. A safe time step is determined to avoid particle ‘jumps’.
3. The position of the particles is updated using their velocities, their previous positions, and the time step.
4. We determine the box in which each particle resides.
5. We call the coagulation subroutine described in Paper II to calculate the evolution of the particles separately in each box for the given time step. There can be multiple collisions per time step.
Initial conditions {#subsec:inicond}
------------------
We assume that the gas density profile is constant in time during the simulation. This assumption is valid if the simulated time is less or comparable to the viscous timescale of the gas. The viscous timescale can be calculated as $$t_{\mathrm{vis}}=r^2/ \nu_T,$$ where $r$ is the distance from the central star, $\nu_T$ is the turbulent viscosity. The value for $t_{\mathrm{vis}}$ at 1 AU varies between $10^3$ and $10^7$ yr depending on $\nu_T$. We assume that turbulence is parameterized by the [@Shakura1973] $\alpha$ parameter $$\nu_T=\alpha c_s H_g,
\label{eq:nuT}$$ where $c_s$ is the isothermal sound speed, and $H_g$ is the pressure scale height of the gas disk. The turbulence parameter $\alpha$ reflects the strength of the turbulence in the disk. Typical values range between $\alpha=10^{-6}$ and $10^{-2}$, where the former corresponds to the turbulent strength in dead zones, the latter describes turbulence in disk atmospheres. In this paper, we assume that $\alpha$ is constant as a function of height, which may be changed in future work (see Sect. \[sec:alpha\]).
The vertical structure of the disk is determined by the equilibrium between the vertical component of the gravitational force and the vertical pressure gradient in the gas. If the disk mass ($M_{\mathrm{disk}}$) is much smaller than the mass of the star ($M_*$), and the vertical thickness of the disk ($H_g$) is a small fraction of the radial distance (both conditions are safely met for the disk parameters described below), then the vertical density can be approximated as $$\rho_g (r,z) = \frac{\Sigma(r)}{\sqrt{2 \pi} H_g}\exp(-z^2/2H_g^2),$$ where $\Sigma(r)$ is the gas surface density at distance $r$, and $z$ is the height above the midplane. In this paper we choose $M_* = 0.5 M_\odot$, $r=1$ AU, $\Sigma(1$ AU$)=100$ g/cm$^2$ similarly to DD05. The pressure scale height can be calculated as $$H_g = c_s/\Omega,$$ where $\Omega$ is the orbital frequency at 1 AU. The isothermal sound speed is $$c_s = \sqrt{\frac{k_B T}{\mu m_p}},$$ where $k_B$ is the Boltzmann constant, $\mu$ is the molecular weight, which is 2.3 for molecular gas, $m_p$ is the mass of the proton, and $T$ is the temperature of the gas, which is 200 K for the stellar and disk parameters considered above (see DD05). We assume the temperature to be constant as a function of height. This is a reasonable assumption well below the photosphere if the temperature of the gas is solely determined by the stellar irradiation and the thermal coupling to the grains. The height of the photosphere may (and presumably will) change as the particles rain out, but we do not include this effect in this paper.
We need to quantify the proper number of particles and boxes to be used. As we know from ZsD08, a low number of particles is not desirable because the physics of the collision kernel will not be reproduced properly. However, many particles make the code computationally expensive and unfeasible to run. If too few boxes are used, the Gaussian profile of the gas disk will not be resolved properly. If too many boxes are used, sufficient number of particles per box is needed, which again renders the simulation computationally expensive. We found by numerical experiments that using $10^5$ particles and 40 evenly spaced boxes is a good compromise. We simulate 4 pressure scale heights (0.16 AU above the midplane), therefore there are 10 boxes per pressure scale height to resolve the Gaussain profile of the gas. $10^5$ particles are also enough to remove numerical artifacts from the simulation and reproduce the physics of the collision kernel properly. Using $10^5$ particles and 40 boxes also mean that there is at least 1 particle in the uppermost box in the beginning of the simulation.
Position update {#subsec:pos}
---------------
The vertical position of the particles are determined by vertical settling and turbulent diffusion. In principle, Brownian motion also contributes to the change of particle positions, but its effect is negligible compared to the other two effects.
The equation governing the diffusion and settling of the dust in a non-homogenous gas density field is [@Dubrulle1995; @Fromang:2006p324; @Ciesla2010] $$\frac{\partial \rho_d}{\partial t} = \frac{\partial}{\partial z} \left[D_d \rho_g \frac{\partial}{\partial z}
\left( \frac{\rho_d}{\rho_g} \right)\right] + \frac{\partial}{\partial z} (\Omega^2 t_s \rho_d z),$$ or equivalently: $$\frac{\partial \rho_d}{\partial t} = \frac{\partial}{\partial z}\left( D_d \frac{\partial \rho_d}{\partial z}\right) - \frac{\partial}{\partial z}\left( \rho_d \times D_d \frac{1}{\rho_g}\frac{\partial\rho_g}{\partial z}\right) + \frac{\partial}{\partial z} (\rho_d \times z \Omega^2 t_s)
\label{eq:diff}$$ where $\rho_d$ is the dust density, $D_d$ is the diffusion coefficient of the dust (see the next paragraph) and $t_s$ is the stopping time of the particle. The stopping time is the timescale a particle needs to react to the changes of the surrounding gas. We define the dimensionless Knudsen number as $$Kn = \frac{a}{\lambda_{\mathrm{mfp}}},$$ where $a$ is the size of the aggregate, and $\lambda_{\mathrm{mfp}}$ is the mean free path of the gas. A particle is in the Epstein regime if $Kn < 1$ (to be more precise, if $a < \frac{9}{4}\lambda_{\mathrm{mfp}}$), where the stopping time is ([@Epstein1924]): $$t_{s} = t_{\mathrm{Ep}} = \frac{3 m}{4 v_{\mathrm{th}} \rho_g A},
\label{eq:ts1}$$ where $m$ and $A$ are the mass and the aerodynamical cross-section of the particle, and $v_{\mathrm{th}}$ is the thermal velocity. If the Knudsen number is greater than 1 (at high gas densities where the mean free path is low or in the case of large particles), the Stokes regime applies and the stopping time becomes $$t_s = t_{\mathrm{St}} = \frac{3 m}{4 v_{\mathrm{th}} \rho_g A} \times \frac{4}{9} \frac{a}{\lambda_{\mathrm{mfp}}}.
\label{eq:ts2}$$
The first term on the right hand side of Eq. \[eq:diff\] is the diffusion term. Using only this term, particles with $t_s = 0$ (tracers) would be homogeneously distributed as a function of height over several diffusion timescales. The first and the second term together on the right hand side ensures that the tracer particles will be distributed according to the background gas density field. The third term describes the settling of the particles. Equation \[eq:diff\] is valid if the motion of the dust does not influence the motion of the gas (the back-reaction from the dust to the gas is negligible). This condition is met if the dust to gas ratio is $\ll 1$.
We note that we do not solve for the dust density directly. We follow the motion of dust *particles*, each of which represents a portion of the total dust mass inside the column. Thus we derive the corresponding velocities (or fluxes) for the first, second, and third terms of Eq. \[eq:diff\] to calculate the position update of the particles.
We calculate $D_d$, the diffusion coefficient of the dust, and define the diffusion velocity, $v_{D1}$. The diffusion coefficient of the gas can be defined as [@Shakura1973] $$D_g =\nu_T =\alpha c_s H_g.$$ Based on [@Youdin:2007p576], the diffusion coefficient of the dust can be calculated as $$D_d = D_g/(1+St^2),$$ where $St$ is the Stokes number $$St = t_s \Omega.$$ The average displacement of a particle in 1D during the time step of $\Delta t$ then is $$L = \sqrt{2 D_d \Delta t}.$$ The real displacement of the particle ($\Delta z$) is drawn from a Gaussian distribution which has zero mean and a half width of $L$. The “diffusion velocity” can then be calculated as $$v_{D1} = \Delta z/\Delta t.$$ This velocity component tries to smear out dust concentrations. It is important to note that the real, physical velocity of the particle during turbulent diffusion changes randomly every time the aggregate interacts with a turbulent eddy. The diffusion velocity defined above is a numerical construct to calculate the time-averaged velocity of the particle during a time-interval $\Delta t$.
The second term on the right side of Eq. \[eq:diff\] results in a systematic velocity term which pushes particles towards the density maxima of the gas. This velocity can be determined by $$v_{D2} = D_d\frac{1}{\rho_g}\frac{\partial\rho_g}{\partial z}.$$ Using these two velocity components ($v_{D1}$ and $v_{D2}$), the particles will be distributed according to the gas density profile in a diffusion timescale.
The fact that particles with $t_s>0$ settle towards the midplane and have a scale height less than $H_g$ is the result of the third term of Eq. \[eq:diff\], the settling velocity. The settling velocity of a particle is given by $$v_{\mathrm{set}}=-z \Omega^2 t_s.
\label{eq:set}$$
The new position of the particles can then be determined by using these three velocity terms: $$z = z_{\mathrm{old}} + (v_{D1}+v_{D2}+v_{\mathrm{set}})\Delta t.$$ This description os identical to the one used in [@Ciesla2010].
----- -------------- ---------------- ------------ ----------- ----------------------- ------------------------- ---------------------- --------------------- ----------- --
ID $\Sigma_{g}$ Coll. model Por. model $\alpha$ $\Psi_{\mathrm{max}}$ $<\Psi_{\mathrm{fin}}>$ $<m_{\mathrm{fin}}>$ $<St>$ $H_p$
\[g/cm$^2$\] \[g\] \[$H_g$\]
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
DD 100 hit&stick only compact 0 1 1 $2.5\times 10^{-3}$ $3.5\times 10^{-3}$ 0
DDa 100 hit&stick only Okuzumi 0 $10^7$ $1.1\times10^6$ $1.8\times 10^{8}$ $1.6$ 0
SB1 100 simpl. Br. Okuzumi 0 $10^4$ $2\times10^3$ $2.2\times 10^{-3}$ $8.0\times 10^{-5}$ 0
SB2 100 simpl. Br. Okuzumi $10^{-6}$ $10^4$ $1.6\times10^3$ $3.2\times 10^{-3}$ $7.0\times 10^{-5}$ 0.2
SB3 100 simpl. Br. Okuzumi $10^{-4}$ $5\times10^3$ 21 $2.2\times 10^{-4}$ $5.0\times 10^{-4}$ 0.6
CB1 100 compl. Br. Okuzumi $10^{-4}$ $10^3$ 3 $1.2\times10^{-2}$ $4.5\times 10^{-3}$ 0.2
CB2 100 compl. Br. Ormel $10^{-4}$ $10$ 2 $1.2\times10^{-2}$ $4.7\times 10^{-3}$ 0.2
CB3 100 compl. Br. Okuzumi $10^{-2}$ $3\times10^2$ 3 $3.6\times 10^{-7}$ $1.6\times 10^{-4}$ 0.95
CB4 100 compl. Br. Ormel $10^{-2}$ $10$ 2 $8.9\times 10^{-8}$ $1.2\times 10^{-4}$ 0.95
LD1 10 compl. Br. Okuzumi $10^{-4}$ $5\times10^2$ 2 $8.9\times 10^{-4}$ $9.7\times 10^{-3}$ 0.2
LD2 10 compl. Br. Okuzumi $10^{-2}$ $5\times10^1$ 23 $2.3\times10^{-9}$ $1.3\times10^{-4}$ 0.95
HD1 1000 compl. Br. Okuzumi $10^{-4}$ $5\times10^4$ 6 $8.3\times10^{-2}$ $6.7\times10^{-4}$ 0.5
HD2 1000 compl. Br. Okuzumi $10^{-2}$ $10^3$ 3 $1.2\times 10^{-5}$ $4.6\times10^{-5}$ 0.95
----- -------------- ---------------- ------------ ----------- ----------------------- ------------------------- ---------------------- --------------------- ----------- --
Col. 1 is the ID of the simulations (DD - Dullemond& Dominik models, SB - simplified Braunschweig model, CB - complete Braunschweig model, LD - low density model, HD - high density model), col. 2 is the gas surface density, col. 3 describes the used collision model (hit&stick, simplified Braunschweig model, or complete Braunschweig model), col. 4 indicates the used porosity model (based on [@Ormel:2007p93] or [@Okuzumi2009a]), col. 5 describes the value of the turbulence parameter $\alpha$, col. 6 is the maximum enlargement parameter of the aggregates during the simulation, col. 7, 8, and 8 are the final average enlargement parameter, mass and Stokes number of the particles respectively, and col. 10 is the scale height of the dust expressed in the scale height of the gas.
Coagulation {#subsec:coag}
-----------
The collision model used in this work is similar to the one used in Paper II. There are however two differences. The first difference is the additional source of relative velocity due to differential vertical settling (see Eq. \[eq:set\]): $$\Delta v_S = | v_{\mathrm{set1}}-v_{\mathrm{set2}} |.$$ The second difference concerns the calculation of the aerodynamical cross section which is used to calculate the stopping time. In Paper II we used the geometrical cross section of the particles [@Ormel:2007p93] $$A=r_c^2 \pi \Psi^{2/3},
\label{eq:cross1}$$ where $r_c$ is the compact radius of the aggregate (assuming that the mass of the particle is contained in a compact sphere of radius $r_c$), and $\Psi$ is the enlargement parameter. $\Psi=1$ for compact particles and the value is higher for fluffy particles [@Ormel:2007p93]. This formula works well for particles with fractal dimension above 2. However, we use the porosity model of [@Okuzumi2009a], and their model produces aggregates with fractal dimension below 2. If one calculates the stopping time of such an aggregate in the Epstein regime (Eq. \[eq:ts1\]) using the formula in Eq. \[eq:cross1\], one gets that the stopping time is less than the stopping time of a monomer. This is clearly unphysical. The reason for this low stopping time (large area) is that Eq. \[eq:cross1\] does not take into account the empty space between the ‘fractal branches’. To avoid such unphysical results for aggregates with low fractal dimensions, we use the aerodynamical cross section as defined in Eq. 47 in [@Okuzumi2009a].
![The collision rate between dust aggregates as a function of particle mass to particle mass measured one simulation of Paper II (the MMSN model with ID Mt1d-4m100). Initially particles grow due to Brownian motion and collisions between equal sized aggregates are frequent. When turbulent relative velocity dominates over Brownian motion (masses above $10^{8}$ g), collisions between equal sized aggregates are less frequent and collisions between different sized aggregates are dominant.[]{data-label="fig:coll_rate"}](coll_rate_MMSN.ps){width="49.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
{width="75.00000%"}
Porosity and collision models {#sec:poro}
-----------------------------
It is generally assumed that dust particles in the protoplanetary disk are aggregates, i.e. they consist of micron-sized spheres (monomers) which are connected via surface forces [@Henning1996]. The material of these monomers can be iron, silicate, organics (tar), different ices and a mixture of these (e.g., silicates with organic or icy mantels). The collision model described in Paper I is based on laboratory experiments performed on room-temperature dust aggregates made out of micron-sized silicate monomers. This naturally limits the region in the disk where our collision model is applicable. Therefore, our results are not applicable in regions where ices condensate (beyond the snow-line) or close to the central star where the effects of sintering (partial melting and crystallization) becomes important [@Poppe2003]. Another limitation that originates from the adopted collision model is that the laboratory experiments were performed using fractal dimension 3 aggregates. Therefore, we cannot reliably consider the initial fractal growth of the aggregates. This topic is further discussed in Sect. \[sec:collmod\].
The monomers are embedded in the gas and move relative to each other [@Beckwith2000]. The particles collide, stick, and grow due to their relative motions. There are two limiting cases for growth: cluster-cluster aggregation (CCA) and particle-cluster aggregation (PCA). CCA growth means that the aggregates tend to collide with other aggregates that consist of the same number of monomers. PCA growth occurs when an aggregate collides successively with much smaller aggregates (monomers) only. The resulting structure of the aggregates are different for the two growth types: CCA growth produces fluffy fractal aggregates with a fractal dimension of 2, while PCA growth produces compact structures with a fractal dimension of 3 [@Ossenkopf:1993p80].
Two monomers can perform different types of motions while they are connected if the available kinetic energy is high enough. These are rolling, sliding, twisting, and the connection can break due to pull-off [@Dominik:1997p89]. Most important of these is rolling as the aggregate can dissipate the initial collisional energy efficiently through rolling motions. The rolling energy ($E_{\mathrm{roll}}$) is defined as the energy that is needed to roll two monomers by 90 degrees. During the initial stages of growth, the collisional energy is lower than the rolling energy, therefore no significant restructuring occurs. The aggregates stick at the first contact and we refer to this as the hit&stick (S1) collision type.
We adopt the hit&stick porosity model of [@Okuzumi2009a] in this paper. In Paper II we used the porosity model of [@Ormel:2007p93] which only treats PCA and CCA collisions and constructs semi-analytical recipes if the size (or mass) ratio of the two colliding aggregates are intermediate. However, [@Okuzumi2009a] improved on this by modeling these intermediate regions, so called quasi-CCA collisions (QCCA), where two clusters with a given mass ratio can collide. The porosity model of Ormel et al. produces aggregates with a fractal dimension of 2.5. The Okuzumi et al. model results in fluffy, fractal dimension 2 aggregates. As the Ormel model analytically interpolates between PCA and CCA collisions, whereas the Okuzumi et al. model directly simulates these collisions, we prefer the Okuzumi model as the fiducial hit&stick porosity model.
As the particles grow, their collisional energy is increasing and we cannot neglect restructuring. This phase starts when the collisional energy is a few times larger than the rolling energy between the monomers. As compaction occurs, the fractal dimension of the aggregates increases. At the same time, the average number of connections a monomer has (coordination number) increases as well.
The further evolution of the dust aggregates is as of yet unclear and a matter of ongoing debate. Currently there are two competing collision models. The Braunschweig collision model (Paper I) is mostly based on laboratory experiments performed by using fractal dimension 3 aggregates (so called dust cakes and their pieces). As such, it postulates that the aggregates at one point during their evolution in the protoplanetary disk reach a rather compact state with a fractal dimension of 3. The validity of this assumption was not thoroughly checked yet (but see future work in Sect. \[sec:collmod\]). This collision model predicts that nine different collision types can occur between dust aggregates of different sizes and porosities (e.g., bouncing, mass transfer, cratering, etc.).
The other alternative comes from molecular dynamics models. [@Wada2008] performed a series of collisions using equal sized aggregates of fractal dimension 2 with an ever increasing relative velocity. They found that the aggregates end up with fractal dimension 2.5 due to restructuring before fragmentation sets in. This would imply that the collision model can be constructed from hit&stick collisions at low velocities, sticking with restructuring at higher velocities, and finally fragmentation. They did not observe any other collision type.
High relative velocities are required to initiate restructuring of equal sized collisions. Relative velocity due to Brownian motion is negligible for large aggregates, as $\Delta v_B$ decreases with mass. Relative velocity due to differential radial drift and settling is zero, as equal sized aggregates drift and settle with the same speed. Relative velocity due to turbulence is also zero or negligibly small, if both particles are well coupled to the gas [@Weidenschilling1993; @Ormel:2007p92]. If one particle is decoupled from the gas, the relative velocity between different and equal sized aggregates is roughly constant. However, one needs sufficiently large aggregates to enter this regime.
The relative velocity between different sized aggregates is larger than between equal sized aggregates during the initial stages of growth. As a result, collisions between different sized collisions are more frequent. To support this statement, we use one simulation result obtained in Paper II: one of the Minimum Mass Solar Nebula (MMSN) models with ID Mt1d-4m100. We produced a logarithmically spaced mass grid array and we calculated how many collisions happened between the aggregates within each grid cell during the simulation. The number of collisions per grid cell is indicated by the contour levels in Fig. \[fig:coll\_rate\]. We only used the first 300 yr of the simulation because particles at $t>300$ yr experience other collision types than sticking. Since the relative velocity of small particles is dominated by Brownian motion, similar sized aggregates collide (CCA-like growth) as long as the particle mass is less than $10^{-5}$ g. However, when the turbulent relative velocity dominates over Brownian motion (at masses above $10^{-5}$ g), the collision rate between equal sized aggregates is reduced. The large aggregates (mass of $10^{-4}$ g) preferentially collide with smaller ones (mass of $10^{-8}$ g). Therefore the results of [@Wada2008] (which is based on equal sized collisions) might not capture the full complexity of the aggregate restructuring phase.
Ideally, a restructuring collision model considering different sized aggregates is needed. As such a model – and its experimental verification – is not available yet, we for simplicity decide to stick with the Braunschweig lab-based model, as we did in previous works. We acknowledge the potential caveats of our adopted collision model, which we further discuss and quantify in Sect. \[sec:collmod\].
Prelude: the effects of porosity and collision models {#sec:intro}
=====================================================
We perform altogether 21 simulations to investigate the effects of different porosity models and turbulence, in which we gradually use more realistic collision models. The IDs and parameters of these simulations are shown in Tab. \[table:sedi\]. First we compare our model against the results of DD05 for compact particles (model DD in Tab. \[table:sedi\]). Then we use the hit&stick porosity model of [@Okuzumi2009a] to investigate the effects of porosity (model DDa). So far we assume that the aggregates stick at all relative velocities and that the turbulence parameter $\alpha$ is zero. In the next step we construct a more realistic collision model with sticking, bouncing and fragmentation (model SB1). We call this collision model the “simplified Braunschweig model” because it uses only three collision types out of 9, which is described in Paper I (the complete Braunschweig model). In the next step we turn on turbulence (models SB2-3) to examine the effects of turbulent stirring. Finally, we use the complete Braunschweig model with turbulence (models CB1-4) and use different disk models (models LD1-2 and HD1-2).
Test: comparison with the DD05 model {#sec:test}
------------------------------------
DD05 performed a simulation (S2 in their paper, DD in this paper), where the disk model is the same as the one described in Sect. \[subsec:inicond\]: the particles are compact, upon collision particles stick together at all collision energies, and the only source of relative velocity is Brownian motion and differential settling. They found that the ‘rain-out’ particles reach the midplane in 500 yr having attained sizes of a millimeter (10$^{-2}$ g in mass).
We performed the exact same simulation to validate our code. We find that the ‘rain-out’ particles in our simulation reach the midplane also in 500 yr and have masses of $10^{-2}$ g. Therefore we can conclude that our code works properly. We illustrate the mass evolution of particles as a function of their height at six different snapshots ($t=1$ yr, 10 yr, 100 yr, 316 yr, 10$^3$ yr, 10$^4$ yr) in Fig. \[fig:mass\_DD05\].
Brownian motion is essential in our simulations because growth by Brownian motion initializes the sedimentation-driven coagulation. The reason is that we have initially a *mono-disperse* particle size-distribution (meaning that all particles have the same size and mass). The aerodynamical properties of these particles are identical, thus there is no relative velocity due to settling between the monomers at a given height. If growth due to Brownian motion was not initiated (e.g. growth by Brownian motion did not introduce aggregates with different aerodynamical properties than that of the monomers), the monomers would simply sediment to the midplane without any growth. Although DD05 included Brownian motion, this effect would not have been present as that simulation started with a (narrow, but not infinitely narrow) size distribution.
As shown in Fig. \[fig:mass\_DD05\], growth by Brownian motion is faster at the midplane due to the higher gas and dust densities (at $t=1$ and 10 yr). Once particles in the upper layer also start to grow by Brownian motion, sedimentation-driven coagulation starts and particles at the upper layers grow much faster than the aggregates at the midplane (at $t=100$ yr) because the absolute value of the settling velocity increases with height (see Eq. \[eq:set\]). The heaviest particles sweep up the smaller particles while they sediment and further increase their settling velocity, resulting in a rain-out at $t=500$ yr. Once the first rain-out particles reach the midplane, they could only grow by Brownian motion, because at the midplane, the settling velocity of any particle is zero (see Eq. \[eq:set\]). But the relative velocity due to Brownian motion for such heavy aggregates is negligible. Therefore, the particles that have reached the midplane, do not increase in mass any longer.
The effects of the hit&stick porosity model
-------------------------------------------
In the previous section we used compact particles. In this section, we consider dust aggregates which are built from (sub)micron-sized solid spheres, called monomers. Such monomers collide with each other and form fluffy structures due to hit-and-stick collisions (collisions that result in sticking upon the first contact). In this section we assume that all collisions result in sticking. We include growth by Brownian motion and settling only.
We use here the hit&stick porosity model constructed by [@Okuzumi2009a]. As discussed in Sect. \[sec:poro\], this collision model defines a third type of aggregation next to the PCA and CCA collisions, that is QCCA (quasi cluster-cluster aggregation - collisions between clusters with a predefined mass ratio). This model is based on simulations of hit&stick collisions without restructuring.
The mass and the porosity evolution are shown in Figs. \[fig:mass\_DD05\_ok\] and \[fig:enpar\_DD05\_ok\]. The most striking property of this simulation is the maximum mass and porosity of the particles. We end up with particles of $10^{10}$ g in mass having an enlargement parameter of almost $10^8$ (while the compact radius of such an aggregate is some meters, the enlarged radius is several kilometers!). The stopping time in the Epstein regime (Eq. \[eq:ts1\]) is proportional to $m/A$. As the Okuzumi-model produces aggregates with a fractal dimension of $\sim 2$ (the mass scales with $a^2$, where $a$ is the particle radius), the stopping time only slightly increases with mass. Therefore, particles settle slowly and produce extremely fluffy structures. At one point, however, the aerodynamical cross section radius becomes larger than the mean free path of the gas, and the aggregate enters the Stokes regime (Eq. \[eq:ts2\]). As the stopping time is now proportional to $ma/A$, the stopping time more strongly increases with mass. This transition from Epstein to Stokes regimes happens at t=900 yr for the particles located at 1.7 $H_g$ above the midplane. Once the transition happens for a given particle, it settles to the midplane in a matter of years due to the heavy mass of the aggregate. Therefore, when the size of the aggregate becomes comparable to the mean free path of the gas ($a=\lambda_{\mathrm{mfp}}$), we reach a natural upper size limit where rain-out in any model is expected.
We emphasize that any collision model containing exclusively sticking is only valid, if no significant restructuring happens during the collisions (e.g. the collision energy is less than $5 E_{\mathrm{roll}}$, the rolling energy, which is the energy needed to roll two monomers by 90 deg). This condition is clearly not met at all times in our simulations, e.g. the rain-out particles can have collision velocities with the swept up particles as high as several 10 m/s in these simulations. Such collisions would result in catastrophic fragmentation. Therefore, the results presented in this section should be considered as toy models.
We conclude here that porosity alone is not sufficient to explain the observations. Therefore, the hit-and-stick assumption is insufficient to describe all settling stages, since we expect that the condition $E<5 E_{\mathrm{roll}}$ will be broken at some point. Clearly, a more realistic collision model that include compaction and fragmentation of dust aggregates, is then needed. Next, we will investigate the consequences of including these physical regimes, first by using a simplified prescription.
A simplified Braunschweig model
-------------------------------
In this section we use the simplified version of the collision model described in Paper I. We assume sticking and the increase of the porosity, if the collision energy is smaller than $5 E_{\mathrm{roll}}$. Bouncing with compaction is used if the collision energy is greater than $5 E_{\mathrm{roll}}$, but the relative velocity of the two aggregates is less than 1 m/s. Fragmentation occurs if the relative velocity of two aggregates is greater than 1 m/s. The recipe for mass and porosity evolution for bouncing and fragmentation is taken from Paper I (our hit & stick (S1), bouncing with compaction (B1), and fragmentation (F1) collision types). We still assume that the particles grow by Brownian motion and settling only (the effects of turbulence is discussed in the next section), and we use the porosity model of [@Okuzumi2009a] for the hit&stick phase.
The evolution of the mass is shown in Fig. \[fig:mass\_simpl\_not\_or\]. The mass distributions at $t=1$, 10, 100 yr are identical to Fig. \[fig:mass\_DD05\_ok\]. At $t=316$ and $1000$ yr we see the effects of bouncing at the intermediate energies. The rain-out particles cannot increase their mass further, when they suffer bouncing collisions. Therefore, their masses are $10^{-2}$ g when they arrive to the midplane and no larger aggregates are formed as opposed to the DDa model. As the rain-out particles are smaller, they settle slower, and reach the midplane only at $t=6000$ yr. The enlargement parameter at the end of the hit&stick phase is $\sim 10^{4}$ and it decreases significantly only in the midplane to values between 10-$10^4$ as the rain-out particles collide and bounce with the aggregates in the midplane. The particles in this simulation are always in the Epstein regime, as bouncing collisions prevent growth to sizes above the mean free path of the gas.
We conclude that realistic collision models reduce the particle sizes in sedimentation-driven coagulation and thus reduce the settling time. However these collision models with the combined effects of porosity are still insufficient to explain the long-term presence of the 10 micron feature in disk SEDs. We now include turbulence into the simulation.
Results {#sec:res}
=======
The effects of turbulence {#sec:turb}
-------------------------
So far all particles sooner or later ended up at the midplane because there was no effect that could counteract settling. In this section we examine the effects of a non-zero turbulence parameter, which can stir particles back up.
A small turbulence parameter ($\alpha = 10^{-6}$) does not significantly affect the masses of the rain-out particles compared to the SB1 model of the previous section (see model SB2 in Tab. \[table:sedi\]). As particles do not only settle but also diffuse downward (and upward) due to turbulence, the time the rain-out particles reach the midplane is somewhat shorter than for model SB1. In all previous simulations a dense dust layer formed at the midplane of the disk. However, even this low level of turbulence can prevent the formation of this layer and introduce a non-zero (although small) dust scale-height. The porosity of the aggregates are also similar to the results described for the SB1 simulation.
The influence of turbulence is more pronounced if $\alpha=10^{-4}$. The mass evolution of the aggregates is shown in Fig. \[fig:mass\_simpl\_1d-4\_or\]. The first rain-out particles reach the midplane already at $t=500$ yr due to downward diffusion, although these particles have lower masses than in model SB1 ($10^{-4}$ g – therefore, in the absence of turbulence, these particles would reach the midplane later than the particles in SB1). The porosity of the aggregates is strongly influenced by the higher turbulence value and bouncing. Due to the increased turbulent relative velocities, particles start bouncing before they reach the midplane. The average enlargement parameter is $2 \times 10^3$ at the end of the hit&stick phase. As the particles settle to the midplane and reach an equilibrium height, bouncing continues and gradually compactifies the aggregates. The enlargement parameter at $t=10^4$ yr is between 2 and 100. The dust distribution reaches a quasi-steady state at $t=10^{4}$ yr.
We see that the particle mass is constant as a function of height at $t=10^4$ yr. As turbulence effectively mixes the particles, and as bouncing prevents further growth or fragmentation (the dust growth is halted), both the masses and porosities of the aggregates are similar at all heights where dust is present. We will see in the next section that this is only true if the turbulence parameter is rather modest. A value of $\alpha=10^{-2}$ results in height-dependent particle mass.
We also see from these simulations that a higher turbulence value reduces the mass of particles and increases the dust scale height. If turbulence is strong enough, the dust scale height can be similar to the gas scale height and the disk atmosphere remains dusty at all times. However, such high turbulence value prevents any significant dust growth, which is not a fertile environment for planet formation (see next Section).
The complete Braunschweig collision model {#sec:comp_br}
-----------------------------------------
In this Section we use the complete Braunschweig model (see Paper I for details), the value of the turbulence parameter is $\alpha=10^{-4}$ and $10^{-2}$. The calculations are performed with both the Okuzumi porosity model (CB1, CB3) and the Ormel porosity model (CB2, CB4) because we want to investigate how sensitive the outcome of dust growth is to the used hit&stick porosity model within the context of our model.
In the simplified Braunschweig collision model, the growth is halted by bouncing immediately if the particles enter the bouncing regime. However, in the complete Braunschweig model, there channels for growth beyond the hit&stick border line (that is where $E_{\mathrm{coll}} > 5 E_{\mathrm{roll}}$). The most important area is in the regime where a small porous projectile collides with a heavy porous target (see Fig.11 of Paper I). Due to these “green” areas at intermediate collision energies, particles in the CB1 and CB2 simulations grow to higher masses than in the SB3 simulation. As a consequence, the scale height of the dust is lower in these simulations, as heavier particles are more difficult to stir up by turbulence. We illustrate the mass distribution at $t=1$, 10, 100, 316, $10^3$, $10^4$ yr in Fig. \[fig:mass\_br\_1d-4\_or\] for the CB1 model.
The particle evolution has two phases in these simulations. The first 1000 yr are identical for the SB3 and CB1/CB2 simulations, respectively (see also the first five snapshots of Figs. \[fig:mass\_simpl\_1d-4\_or\] and \[fig:mass\_br\_1d-4\_or\]). In this phase, particles start sedimenting, and the rain-out particles reach the midplane. The dust evolution in the SB3 model halts at this point as only bouncing collisions happen. However, during the second phase of the CB1 and CB2 simulations, particles can grow to higher masses because there are areas in the parameter space that is favorable for growth somewhat beyond the bouncing barrier (see Paper I and Paper II for a detailed explanation). Due to this growth, particles reach $10^{-2}$ g in mass for the CB1 and CB2 simulations.
The time evolution of the enlargement parameter is quite different in the two cases. Fluffy aggregates are produced with enlargement parameter $\Psi = 10^3$ when the Okuzumi porosity model is used. However, these aggregates are strongly compacted by bouncing and their final enlargement parameter is between 2 and 20. When the Ormel porosity model is considered, the enlargement parameter never exceeds 20 and by the end of the simulation the enlargement parameter is between 2 and 10. As we see, the enlargement parameter evolved through very different ways, but the final enlargement parameters are not so much different for the CB1 and CB2 simulations.
Figure \[fig:coll\_hist\_br\_1d-4\] illustrates the collision history of the CB1 simulation. If we compare this figure to Figs. 5, 7 and 10 of Paper II, we see that the features are more smeared out in Fig. \[fig:coll\_hist\_br\_1d-4\] than in the other figures. As we simulate here several boxes at different heights above the midplane, the physical conditions (e.g. gas density and sedimentation velocity) at the midplane and at the upper scale heights of the disk are different, which is responsible for the smeared out features of Fig. \[fig:coll\_hist\_br\_1d-4\].
We investigate the collision frequency of the nine collision types as a function of time (see Fig. \[fig:coll\_hist\_br\_1d-4\_or\]). If one compares this figure with Figs. 4c, 6c, and 9c of Paper II, we immediately see that the diversity of occurring collision types is much greater in the CB1 simulation, although the strength of the turbulence is the same in all cases ($\alpha=10^{-4}$). This can be explained by the presence of sedimentation. The particle population in a given box is not fixed as in Paper II. During the rain-out process, “heavy” particles coming from $z=1$-2 $H_g$ ‘hit’ the particle population at the midplane. Due to this process, the relative velocity between the small, midplane particle population and the generally larger rain-out population is increased. Thus collision types can occur that require larger collision energies.
We explore the effects of strong turbulence (CB3 and CB4 runs). In general we find that a turbulence parameter of $\alpha = 10^{-2}$ is able to keep the upper layers of the disk atmosphere dusty (see Fig. \[fig:mass\_br\_1d-2\_or\]). However the size of the particles at the midplane is several order of magnitude smaller than in the CB1 or CB2 simulations. This is not favorable for planetesimal formation via self-gravity assisted planetesimal formation [@Johansen:2007p65; @Cuzzi2008; @Youdin2011]. The high level of turbulence lowers the enlargement parameter in the CB3 simulation (see Fig. \[fig:enlargpar\_br\_1d-2\_or\]). The value of the enlargement parameter does not exceed $10^3$ during the simulation. The final enlargement parameter is between 2 and 10 for both the CB3 and 4 simulations.
Interestingly, the particle masses as function of height are constant in the simulations with $\alpha=10^{-4}$ or $10^{-6}$, but for higher $\alpha$ values, this is not the case. The particles in the upper layers are significantly smaller than particles at the midplane. The particles are mass-sorted (i.e. heavy particles are mostly located at the midplane and small particles at the upper layers). As 100 micron-sized particles from the midplane is mixed upwards by the strong turbulence, it looses mass along the way due to fragmenting collisions. Similarly, as a particle moves closer to the midplane, it is growing in mass due to sticking collisions. This can be explained by the decreasing gas density as a function of height. At low densities the stopping time (and therefore the relative velocity) of a particle is high. At high densities the particles are stronger coupled to the gas, their stopping time decreases. Therefore the relative velocity between these particles are low and they can coagulate. Figure \[fig:coll\_hist\_br\_1d-2\_or\] illustrates this effect as fragmentation (F1, F2 and F3) happens frequently at z=2 $H_g$ above the midplane (green shade for F1 collisions) but it is rare at the midplane (dark blue shade for F1 collisions).
We can also verify this behavior by comparing the viscous and collision timescales in this model. The viscous timescale is $$t_v = \frac{H_g^2}{\nu_T},$$ using the parameters of the disk model we get that $t_v=22$ yr. The collision timescale can be calculated as $$t_c = \frac{1}{\Delta v \sigma n},$$ where $\Delta v$ is the relative velocity of the two particles (in this case it is 1 m/s, the critical velocity for fragmentation), $\sigma$ is the cross section of the two particles (in this case two monomers with 1 micron size as such particles are present at 4 $H_g$), $n$ is the number density of the particles (calculated at the height of 4 $H_g$ assuming monomer-sized particles). Using these values, we get that $t_c=8$ yr. Therefore it is further verified that particles are fragmented while they are mixed upwards by turbulence as the timescale for collisions is shorter than the timescale for mixing.
It is also important to note that the particles at the upper layers of the disk have masses of $10^{-10}$ g and below (size of $\sim$ 1-10 micron). Although the upper layers of the disk are not well resolved, the obtained particle sizes are comparable to the sizes required to explain the 10 micron feature of SEDs [@boekel2003] and this vertical dust distribution has reached steady state after $t=2000$ yr. Therefore, we conclude that in the framework of our models, high values of turbulence (therefore energetic collisions with strong bouncing and fragmentation) are needed to explain why the disk atmospheres are dusty throughout the lifetime of the disk.
Different disk models
---------------------
We performed simulations in two additional disk models to explore the effects of the gas surface density. These are the low density (with $\Sigma_g = 10$ g/cm$^2$) and high density simulations (with $\Sigma_g = 1000$ g/cm$^2$). See also Tab. \[table:sedi\], simulations LD1-2 and HD1-2. The dust to gas ratio, the gas scale height and the gas temperature are the same as in all previous simulations, we change the turbulence parameter only.
We find that the final mass and Stokes number of the particles depend on the gas density (see Tab. \[table:sedi\]). The higher gas density increases the particle masses but decreases the Stokes numbers. Thus the low density model produces the smallest particles (10$^{-4}$ g if $\alpha=10^{-4}$) but these particles have the highest Stokes number amongst the three gas densities (almost $10^{-2}$ if $\alpha=10^{-4}$). This can be explained by considering the stopping time of these particles. The stopping time is inversely proportional to the gas density (see Eq. \[eq:ts1\]). Therefore particles in the low density model have greater stopping times than particles in the high density model. Particles with large stopping times are loosely coupled to the gas and have therefore high collision velocities. For this reason, the maximum particle mass proportional to the gas density.
The common feature of these simulations is that the dust scale height for $\alpha=10^{-2}$ is always similar to the gas scale height (see Table \[table:sedi\]). Therefore, we conclude that the disk atmospheres can be kept dusty in sparse and dense disks alike with sufficiently high turbulence values.
Discussion and future work {#sec:disc}
==========================
Validity of the porosity model {#sec:collmod}
------------------------------
The Braunschweig collision model is based on laboratory experiments that used fractal dimension 3 aggregates with an enlargement parameter typically between 7 and 2. As seen in Table \[table:sedi\], the maximum enlargement parameter of the aggregates can be several orders of magnitude larger than the dust aggregates used in the laboratory. More specifically, the hit&stick collision regime is followed by a collision type called sticking through surface effects (see Paper I) and bouncing at higher collisional energies. Thus we assume that the transition from fractal aggregates in the hit&stick regime to fractal dimension 3 aggregates in bouncing is an instantaneous one. Furthermore, our porosity model for bouncing is calibrated for dust cakes (enlargement parameter of 6), however the enlargement parameter at the end of the hit&stick phase is 2-3 orders of magnitude larger than the enlargement parameter of the dust cakes (see Table \[table:sedi\]). Therefore, it is questionable whether the porosity model for the restructuring phase is correct. Significant changes in the porosity of aggregates has the potential to significantly alter our collision model and thus the obtained results.
It is still debated how the porosity and the fractal dimension evolves during the evolution of dust aggregates. As this is a central issue in determining the dust properties, we need detailed information on the porosity evolution of dust aggregates. To resolve these issues, we plan to follow the hit&stick and restructuring phases of a particle distribution in a self-consistent way by combining the Monte Carlo model of ZsD08 with a molecular dynamics code in a follow-up paper.
Aggregates beyond the snow line {#sec:ice}
-------------------------------
The particle sizes at the midplane of the disk are rather small ($\sim$ 100 microns if $\alpha=10^{-2}$) as we see in the previous section. The question arises: under what conditions could planetesimals form via successive collisions of dust aggregates? Or how do large enough dust aggregates form that could, under favorable conditions, be concentrated in e.g. vortices or turbulent eddies, become self-gravitating, and form eventually planetesimals [@Johansen:2007p65; @Cuzzi2008; @Youdin2011; @Johansen2011]?
One answer to these questions could be icy aggregates. The molecular dynamics simulations of [@Wada2008; @Suyama2008] showed that icy aggregates are very resilient to restructuring. They observed sticking between icy aggregates at a relative velocity as high as 50 m/s. The uncertainty in these simulations is the microphysical parameters of the icy monomers (such as critical displacement, surface energy, Young modulus etc.). Recently laboratory experiments were performed using micron-sized ice monomers by [@Gundlach2011]. They measured the rolling energy between icy monomers and it turned out the previously assumed values by [@Wada2008; @Suyama2008] are in good agreement with the measured laboratory values. If experiments also confirm that icy aggregates can stick at relative velocities as high as 50 m/s, that would provide a way to form large enough dust aggregates beyond the snow-line in the solar nebula.
Is $\alpha$ constant as a function of height? {#sec:alpha}
---------------------------------------------
[@Gammie1996] proposed the concept of layered accretion disks. If the ionization fraction of the gas is not sufficient to support magneto-rotational instability (MRI - [@Balbus1991]), the turbulence parameter drops and a dead-zone forms at the midplane of the disk. The extent of the dead-zone is uncertain, as the ionization processes of the gas are not well-constrained. For typical TTauri disks it can extend between 0.1 - 4 AU [@D'Alessio1998]. Inside 0.1 AU, the thermal radiation from the star can keep the gas sufficiently ionized for MRI, and outside 4 AU, the gas surface density is typically below 100 g/cm$^2$, therefore cosmic rays can penetrate the disk and keep it MRI active at all heights.
It was proposed by [@Okuzumi2009] that negative charges on the surfaces of grains could prohibit growth. As we use a Monte Carlo code to follow the evolution of aggregates, it is possible to include a third particle property: the amount of charges present on the grain. In order to follow the charge evolution of the grains, we need to solve for the ionization state of the gas (i.e. amount of charges available in the gas phase), how efficiently these charges are collected by the dust grains and how the charges affect the relative velocity of the aggregates.
We plan to investigate how dust evolves in a layered disk model. Small dust particles can very efficiently sweep up charges in the gas. As shown by [@Turner2010], the dead-zone can extend to 2 $H_g$ for 1 micron-sized particles, but it shrinks below 0.5 $H_g$ for aggregates that are 100 micron in size. In a simulation like the one presented here, this would mean that as the particles grow, the dead-zone shrinks. When the dead-zone disappears, the whole disk becomes MRI active and the particles settled to the midplane might be fragmented and stirred back up. This could lead to initial oscillations before an equilibrium state is reached.
Summary {#sec:concl}
=======
We performed simulations in a 1D vertical column of a protoplanetary disk to better understand the process of sedimentation. We simultaneously solved for the particle motion and growth inside this column. The complexity of the models was gradually increased to examine the effects of different processes. The first simulation used a collision model that only contained sticking. We furthermore assumed that the particles were compact, and the turbulence parameter ($\alpha$) was set to zero. Later on we investigated the effects of different porosity models, more realistic collision models (with sticking, bouncing and fragmentation) and turbulence of different strengths. Below we summarize our results.
- Porosity helps to produce heavier particles by decreasing the stopping time of the aggregates, but porosity alone cannot prevent rain-out.
- If the size of the particle becomes greater than the mean free path of the gas, the drag law changes from Epstein to Stokes drag. At this point, rain-out becomes unavoidable due to the change in the drag law.
- Realistic collision models with bouncing and fragmentation limit the maximum particle sizes to be not more than 1 mm - 1 cm (particle masses between $10^{-3}$ - 1 g ). The exact value depends strongly on the strength of the turbulence, and the gas density.
- A higher value of turbulence decreases the particle masses but increases the dust scale height. Using the simplified Braunschweig model with $\alpha=10^{-6}$ results in a dust scale height of 0.2 $H_g$ and final particle mass of $10^{-3}$ g, however the dust scale height is $0.6 H_g$, and the final particle mass is $10^{-4}$ g when using $\alpha=10^{-4}$.
- The final particle size and Stokes number depends on the gas density. The mass of the particles is decreasing with decreasing gas density, however the Stokes number remains roughly constant.
- When using the most detailed collision model (the complete Braunschweig collision model), we obtain particle masses of $10^{-2}$ g (with an average radius of 1 mm, and an average Stokes number of $4 \times 10^{-3}$) and a dust scale height of 0.2 $H_g$ upon using $\alpha=10^{-4}$. However, the dust scale-height is almost 1 $H_g$ and the final particle mass at the midplane is $10^{-7}$ g (with an average radius of 100 micron, and an average Stokes number of $10^{-4}$) upon using $\alpha=10^{-2}$. Therefore, a sufficiently high turbulence value can keep the disk atmosphere dusty but the absence of significant dust growth is not favorable for planet formation.
- The maximum enlargement parameter of the aggregates during their evolution can be as high as $10^4$. However, our adopted porosity model for bouncing is based on laboratory experiments performed using enlargement parameter $\sim$ 6 dust cakes. Therefore the largest uncertainty of our model is the porosity evolution.
- The dust particle mass as a function of height is not constant if the turbulence parameter is $\alpha = 10^{-2}$. We see that the particle mass/size is a decreasing function of the height. Particles are 100 microns in size at the midplane and a few microns at 4 pressure scale heights.
- The micron-sized particles present in the upper layers are comparable to the sizes needed to explain the 10 micron feature of disk SEDs. Therefore in the framework of our model, high values of turbulence are needed to explain why disk atmospheres are dusty for $\sim 10^{6}$ yr.
A. Zsom acknowledges the support of the IMPRS for Astronomy & Cosmic Physics at the University of Heidelberg. C.W.O. acknowledges the financial support from the Alexander von Humboldt Foundation. We thank our referee (Hidekazu Tanaka) for his comments that greatly improved the quality and clarity of the paper. We thank Jürgen Blum and Carsten Güttler for the fruitful discussions during the project.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[neural networks, computational neuroanatomy, network science, spatial graph, robustness, recovery]{} Fifty years ago, John von Neumann compared the architecture of the brain with that of computers that he invented and which is still in use today. In those days, the organisation of computers was based on concepts of brain organisation. Here, we give an update on current results on the global organisation of neural systems. For neural systems, we outline how the spatial and topological architecture of neuronal and cortical networks facilitates robustness against failures, fast processing, and balanced network activation. Finally, we discuss mechanisms of self-organization for such architectures. After all, the organization of the brain might again inspire computer architecture.'
author:
- 'Marcus Kaiser[^1]'
title: 'Brain architecture: A design for natural computation'
---
\[firstpage\]
Introduction
============
The relation between the computer and the brain has always been of interest to scientists and the public alike. From the notion of ’thinking machines’ and ’artificial intelligence’ to applying concepts of neuroscience such as neural networks to solve problems in computer science. Also the earliest computers, using the von Neumann architecture still in use today, used memory and a central processing unit based on concepts of brain architecture [@vonNeumann1958]. Also, models of artificial neural networks were inspired by the function of individual neurons as integrators of incoming signals. Detailed models of neural processing, however, are often limited to single tasks (e.g., pattern recognition) and one modality (e.g., only visual information). In addition, artificial neural networks starting with Perceptrons [@Rosenblatt1959] are designed as a general purpose architecture whereas the architecture of natural neural systems shows a high specialization according to different tasks and functions. Global models, on the other hand, often deal with functional circuits (e.g. movement planning) without a direct link to the local structure of the neural network. Therefore, much of the complexity of neural processing in terms of combining local and global levels as well as integrating information from different domains is largely missing from current models.
About 50 years ago, John von Neumann—inventor of the current computer architecture—thought about where computers and the brain are the same or where they differ [@vonNeumann1958]. After 50 years of technological progress, how do the benchmark characteristics differ? The human brain consists of 10$^{10}$ neurons or processing units. The Internet, being the largest computer network, has only millions of processing units. However, the extension of the Internet to mobile services (pervasive computing) could lead to billions of processing nodes in the future. The human memory can be estimated from adjustable synaptic weights of connections between neurons. However, these 10$^{14}$ synapses/weights are only a first approximation of the hard-wired information storage as the position of synapses, both absolute on the target neuron and relative to other synapses influences signal integration. Computer memories have reached this level with some systems, such as the machines that store web information at Google, storing several petabytes (1 petabyte=10$^{15}$ bytes, see http://en.wikipedia.org/wiki/Petabyte). However, computer systems are still far-away from processing complex information like the human brain does. In spite of processing units or memory, the main difference between computers and brains is their hardware architecture—how they are wired up.
In this article, we present recent results on the topology (architecture) of complex brain networks. These results are not about standard (artificial) neural networks that deal with one single task, e.g. face recognition. Rather, we look at the high-level organization of the brain including modules for different tasks and different sensory modalities (e.g., sound, vision, touch). Nonetheless, similar organization [@Buzsaki2004] and processing [@Dyhrfjeld-Johnsen2007] has been found at the local level of connectivity within modules.
Cortical network organization
=============================
Cluster organization
--------------------
Cortical areas are brain modules which are defined by structural (microscopic) architecture. Observing the thickness and cell types of the cortical layers, several cortical areas can be distinguished [@Brodmann1909]. Furthermore, areas also show a functional specialization. Within one area further sub-units (cortical columns) exist, however, these units will not be covered in this review as there is not enough information about their connectivity. Using neuro-anatomical techniques, it can be tested which areas are connected, that means that projections in one or both directions between the areas do exist. If a fiber projection between two areas is found, the value ’1’ is entered in the adjacency matrix; the value ’0’ defines absent connections or cases where the existence of connections was not tested (figure \[fig1cortex\][*a*]{}).
Contrary to popular belief, cortical networks are not completely connected, i.e. [*not*]{} ’everything is connected to everything else’: Only about 30% of all possible connections (arcs) between areas do exist. Instead, highly connected sets of nodes ([*clusters*]{}) are found that correspond to functional differentiation of areas. For example, clusters corresponding to visual, auditory, somatosensory and fronto-limbic processing were found in the cat cortical connectivity network [@Hilgetag2004]. Furthermore, about 20% of the connections are unidirectional [@Felleman1991], i.e. a direct projection from area A to area B but not vice versa exists. Although some of these connections might be bidirectional as the reverse direction was not tested, there were several cases where it was confirmed that projections were unidirectional. Therefore, measures that worked for directed graphs were used.
![([*a*]{}) Adjacency Matrix of the cat connectivity network (55 nodes; 891 directed edges). Dots represent ’ones’ and white spaces the ’zero’ entries of the adjacency matrix. ([*b*]{}) Macaque cortex (95 nodes; 2,402 directed edges).[]{data-label="fig1cortex"}](Figure1.pdf){width="15cm"}
Until now, there is not enough information about connectivity in the human brain that would allow network analysis [@Crick1993]. However, several new non-invasive methods including diffusion tensor imaging [@Tuch2005] and resting state networks [@Achard2006] are under development and might help to define human connectivity in the future. At the moment, however, we are bound to analyze known connectivity in the cat and the macaque (rhesus monkey, Fig. \[fig1cortex\][*b*]{}) cortical networks [see also @Passingham2002; @Sporns2004]. Both networks exhibit clusters, i.e. areas belonging to a cluster have many existing connections between them but there are few connections to areas of different clusters [@Young1993; @Scannell1995]. These clusters are also functional and spatial units. Two connected areas tend to be spatially adjacent on the cortical surface and tend to have a similar function (e.g., both taking part in visual processing). Whereas there is a preference for short-length connections to spatially neighboring areas for the macaque, about 10% of the connections cover a long-distance ($\geq 40$ mm) – sometimes close to the maximum possible distance (69 mm) between two areas of one hemisphere [@Kaiser2004c].
Cortical networks show maximal structural and dynamic complexity which is thought to be necessary for encoding a maximum number of functional states and might arise as a response to rich sensory environments [@Sporns2000]. Using methods and concepts of network analysis [@Albert2002], we discuss small-world and scale-free properties as well as motifs and spatial network organization of cortical networks.
Small-world properties {#small-world-properties .unnumbered}
----------------------
Many complex networks exhibit properties of small-world networks [@Watts1998]. In these networks neighbors are better connected than in comparable Erdös-Rényi random networks [@Erdoes1960] (called random networks throughout the text) whereas the average path length remains as low as in random networks. Formally, the average shortest path (ASP, similar, though not identical, to characteristic path length $\ell$ [@Watts1999]) of a network with $N$ nodes is the average number of edges that has to be crossed on the shortest path from any one node to another: $$\label{asp}
ASP = \frac{1}{N (N-1)} \sum_{i, j} d(i,j) \ \ \ \ \ with \ i\ne j,$$ where $d(i,j)$ is the length of the shortest path between nodes $i$ and $j$.
The neighborhood connectivity is usually measured by the clustering coefficient. The clustering coefficient of one node $v$ with $k_v$ neighbors is $$\label{clustercoef}
C_v = \frac{|E(\Gamma_v)|}{{k_v \choose 2}},$$ where $|E(\Gamma_v)|$ is the number of edges in the neighborhood of $v$ and ${k_v \choose 2}$ is the number of possible edges [@Watts1999]. In the following analysis, we use the term clustering coefficient as the average clustering coefficient for all nodes of a network.
Small-world properties were found on different organizational levels of neural networks: from the tiny nematode [*C. elegans*]{} with about 300 neurons [@Watts1998] to cortical networks of the cat and the macaque [@Hilgetag2000b; @Hilgetag2004]. Whereas the clustering coefficient for the macaque is 49% (16% in random networks), the ASP is comparably low with 2.2 (2.0 in random networks). That is, on average only one or two intermediate areas are on the shortest path between two areas. Note that a high clustering coefficient does not necessarily correlate with the existence of multiple clusters. Indeed, the standard model for generating small-world networks by rewiring regular networks [@Watts1998] does not lead to multiple clusters.
Robustness and recovery
=======================
Compared to technical networks (power grids or communication networks), the brain is remarkably robust towards damage. On the local level, Parkinson’s disease in humans only becomes apparent after more than half of the cells in the responsible brain region are eliminated [@Damier1999]. On the global level, the loss of the whole primary visual cortex (areas 17, 18 and 19) in kittens can be compensated by another region, the postero-medial supra-sylvian area (PMLS) [@Spear1988]. On the other hand, the removal of a small number of nodes or edges of the network can lead to a breakdown of functional processing. As functional deficits are not related to the number or size of removed connections or brain tissue, it might be the role within the network that makes some elements more critical than others. Identifying these critical components has applications in neurosurgery where important parts of the brain should remain intact even after the removal of a brain tumour and its surrounding tissue.
Critical connections in neural systems {#critical-connections-in-neural-systems .unnumbered}
--------------------------------------
It was found that the robustness towards edge removal is linked to the high neighborhood connectivity and the existence of multiple clusters [@Kaiser2004d]. For connections within clusters, many alternative pathways of comparable length do exist once one edge is removed from the cluster (figure \[fig2sw\][*a*]{}). For edges between clusters, however, alternative pathways of comparable length are unavailable and removal of such edges should have a larger effect on the network. The damage to the macaque network was measured as the increase of the ASP after single edge removal. Among several measures, edge frequency (approximate measure of edge betweenness) of an edge was the best predictor of the damage after edge elimination (linear correlation r=0.8 for macaque). The edge frequency of an edge counts the number of shortest paths in which the edge is included.
Furthermore, examining comparable benchmark networks with three clusters, edges with high edge frequency are the ones between clusters. In addition, removal of these edges causes the largest damage as increase in ASP (figure \[fig2sw\][*b*]{}). Therefore, inter-cluster connections are critical for the network. Concerning random loss of fiber connections, however, in most cases one of the many connections within a cluster will be damaged with little effect on the network. The chances of eliminating the fewer inter-cluster connections are lower. Therefore, the network is robust to random removal of an edge [@Kaiser2004d].
![([*a*]{}) Schematic drawing of a network with three clusters showing examples for an intra- (gray dashed line) and inter-cluster (gray solid line) connection. ([*b*]{}) Edge frequency of the eliminated edge vs. ASP after edge removal (20 generated networks with three clusters, defined inter-cluster connections and random connectivity within clusters; inter-cluster connections: light-gray; connections within a cluster: black).[]{data-label="fig2sw"}](Figure2.pdf){width="\textwidth"}
Node removal behaviour similar to that of scale-free networks {#node-removal-behaviour-similar-to-that-of-scale-free-networks .unnumbered}
-------------------------------------------------------------
In addition to high neighborhood clustering, many real-world networks have properties of scale-free networks [@Barabasi1999]. In such networks, the probability for a node possessing $k$ edges is $P(k)\propto k^{-\gamma}$. Therefore, the degree distribution—where the degree of a node is the number of its connections—follows a power-law. This often results in highly connected nodes that would be unlikely to occur in random networks. Technical networks such as the world wide web of links between web pages [@Huberman1999] and the Internet [@Faloutsos1999] at the level of connections between domains/autonomous systems. Do cortical networks, as natural communication networks, share similar features?
In cortical networks, some structures (e.g. evolutionary older structures like the Amygdala) are highly connected. Unfortunately, the degree distribution can not be tested directly as less than 100 nodes are available in the cat and macaque cortical networks. However, using the node elimination pattern as an indirect measure, cortical networks were found to be similar to scale-free benchmark networks [@Kaiser2007EJN].
In that approach, we tested the effect on the ASP of the macaque cortical network after subsequently eliminating nodes from the network until all nodes were removed [@Barabasi2000a]. For random elimination, the increase in ASP was slow and reached a peak for a high fraction of deleted nodes before shrinking due to network fragmentation (figure \[fig3sf\][*a*]{}). When taking out nodes in a targeted way ranked by their connectivity (deleting the most highly connected nodes first), however, increase in ASP was steep and a peak was reached at a fraction of about 35%. The curves for random and targeted node removal were similar for the benchmark scale-free networks (figure \[fig3sf\][*b*]{}) but not for generated random or small-world [@Watts1998] networks [@Kaiser2007EJN]. Therefore, cortical as well as scale-free benchmark systems are robust to random node elimination but show a larger increase in ASP after removing highly connected nodes. Again, as for the edges, only few nodes are highly connected and therefore critical so that the probability to select them randomly is low.
![Average shortest path (ASP) after either random (dashed line) or targeted (gray solid line) subsequent node removal. ([*a*]{}) Macaque cortical network (73 nodes, 835 directed edges). ([*b*]{}) Scale-free benchmark network with the same number of nodes and edges (lines represent the average values over 50 generated networks and 50 runs each in the case of random node removal).[]{data-label="fig3sf"}](Figure3.pdf)
Processing
==========
Wiring constraints for processing {#wiring-constraints-for-processing .unnumbered}
---------------------------------
For microchips, increasing the length of electric wires increases the energy loss through heat dissipation. Inspired by these ideas, it was suggested that neural systems should be optimized to reduce wiring costs as well [@Cherniak1994]. In the brain, energy is consumed for establishing fibre tracts between areas and for propagating action potentials over these fibres. Thus, the total length of all wires should be kept as short as possible. This has led to the idea of optimal component placement in that modules are arranged in a way so that every rearrangement of modules would lead to an increase in total wiring length.
It has been proposed for several neural systems—including the [*C. elegans*]{} neural network and subsets of cortical networks—that components are indeed optimally placed [@Cherniak1994]. This means that all node position permutations of the network—while connections are unchanged—results in higher total connection length. Therefore, the placement of nodes is optimized to minimize the [*total*]{} wiring length. However, using larger data sets than used in the original study, we found that a reduction in wiring length by swapping the position of network nodes was possible.
For the macaque, we analyzed wiring length using the spatial three-dimensional positions of 95 areas and their connectivity. The total wiring length was between the case of only establishing the shortest-possible connections and establishing connections randomly regardless of distance (figure \[fig:wiring\][*a*]{}). A reduction of the wiring length was possible due to the number of long-distance connections in the original networks [@Kaiser2004c]; some of them even spanning almost the largest possible distance between areas. Why would these metabolically expensive connections exist in such large numbers? We tested the effect of removing all long-distance connections and replacing them by short-distance connections. Whereas several network measures improved, the value for the ASP increased when long-distance connections were unavailable (figure \[fig:wiring\][*b*]{}). Retaining a lower ASP has two benefits: First, there are fewer intermediate areas that might distort the signal. Second, as fewer areas are part of shortest paths, the transmission delay along a pathway is reduced. The propagation of signals over long distances, without any delay imposed by intermediate nodes, has an effect on synchronization as well: both nearby (directly connected) areas and faraway areas are able to get a signal at about the same time and could have synchronous processing [@Kaiser2006]. A low ASP might also be necessary because of the properties of neurons: John von Neumann, taking into account the low processing speed and accuracy of individual neurons, suggested that neural computation needed to be highly parallel with using a low number of subsequent processing steps [@vonNeumann1958]. But having a low ASP also brings a potential danger: How can it be prevented that information or activity flows uncontrolled through the entire network?
![([*a*]{}) Original placement of cortical areas. ([*b*]{}) Wiring length optimization leads to a reduction in total wiring length by 32% of the original length. ([*c*]{}) Placement after optimization for total wiring length.[]{data-label="fig:wiring"}](Figure4.pdf){width="\textwidth"}
Balanced network activation through hierarchical connectivity {#balanced-network-activation-through-hierarchical-connectivity .unnumbered}
-------------------------------------------------------------
Few processing steps enable the rapid transfer of activation patterns through cortical networks but this flow could potentially activate the whole brain. Such large-scale activations in the form of increased activity can be observed in the human brain during epileptic seizures: about 1% of the population is currently affected by epilepsy. In contrast to computer networks with a continuous flow of viruses and spam e-mails, the brain has some built-in mechanisms for preventing large-scale activation.
An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable network activations within a limited critical range. In this range, the activity of neural populations in the network persists between the extremes of quickly dying out, or activating a large part of the network as during epileptic seizures. The standard model would be to achieve such a balance by having interacting excitatory and inhibitory neurons. Whereas such models are of great value on the local level of neural systems, they are less meaningful when trying to understand the global level of connections between columns, areas, or area clusters.
Global corticocortical connectivity (connections between brain areas) in mammals possesses an intricate, nonrandom organization. Projections are arranged in clusters of cortical areas, which are closely linked among each other, but less frequently with areas in other clusters. Such structural clusters broadly agree with functional cortical subdivisions. This cluster organization is found at several levels: Neurons within a column, area or area cluster (e.g. visual cortex) are more frequently linked with each other than with neurons in the rest of the network [@Hilgetag2004].
![([*a*]{}) The hierarchical network organization ranges from cluster such as the visual cortex to sub-cluster such as V1 to individual nodes being cortical columns. ([*b*]{}) Schematic view of a hierarchical cluster network with five clusters containing five sub-clusters each. Examples for spread of activity in ([*c*]{}) random, ([*d*]{}) small-world and ([*e*]{}) hierarchical cluster networks ($i=100, i_0=150$), based on 20 simulations for each network. \[fig:spreading\]](Figure5.pdf){width="\textwidth"}
Using a basic spreading model without inhibition, we investigated how functional activations of nodes propagate through such a hierarchically clustered network [@Kaiser2007NJP]. The hierarchical network consisted of 1000 nodes made of 10 clusters with 100 nodes each. In addition, each cluster consisted of 10 sub-clusters with 10 nodes each (figure \[fig:spreading\][*a, b*]{}). Connections were arranged so that there were more links within (sub-)clusters than between (sub-)clusters. Starting with activating 10% of randomly chosen nodes, nodes became activated if at least six directly connected nodes were active. Furthermore, at each time step, activated nodes could become inactive with a probability of 30%.
The simulations demonstrated that persistent and scalable activation could be produced in clustered networks, but not in random or small-world networks of the same size (figure \[fig:spreading\][*c-e*]{}). Robust sustained activity also occurred when the number of consecutive activated states of a node was limited due to exhaustion. These findings were consistent for threshold models as well as integrate-and-fire models of nodes indicating that the topology rather than the activity model was responsible for balanced activity. In conclusion, hierarchical cluster architecture may provide the structural basis for the stable and diverse functional patterns observed in cortical networks. But how do networks with such properties arise?
Design vs. Self-organization
============================
Neural systems, rather than being designed, evolved over millions of years. Starting from diffuse homogeneous networks, network clusters evolved when different tasks had to be implemented. During individual brain development, the architecture is formed by a combination of genetic blueprint and self-organization [@Striedter2005].
What are the mechanisms of self-organization during network development? A possible algorithm for developing spatial networks with long-distance connections and small-world connectivity is spatial growth [@Kaiser2004b]. In this approach, the probability to establish a connection decays with the spatial (Euclidean) distance thereby establishing a preference for short-distance connections. This assumption is reasonable for neural networks as the concentration of growth factors decays with the distance to the source so that faraway neurons have a lower probability to detect the signal and sent a projection toward the source region of the growth factor. In addition, anatomical studies have shown that the probability of establishing a connections decreases with the distance between neurons.
In contrast to previous approaches that generated spatial graphs, the node positions were not determined before the start of connection establishment. Instead, starting with one node, a new node was added at each step at a randomly chosen spatial position. For all existing nodes, a connection between the new node $u$ and an existing node $v$ was established with probability $$\label{exponential}
P(u,v) = \beta \ e^{-\alpha \ d(u, v)},$$ where $d(u, v)$ was the spatial distance between the node positions, and $\alpha$ and $\beta$ were scaling coefficients shaping the connection probability. A new node that did not manage to establish connections was removed from the network. Node generation was repeated until the desired number of nodes was established. Parameter $\beta$ (“density”) served to adjust the general probability of edge formation. The nonnegative coefficient $\alpha$ (“spatial range”) regulated the dependence of edge formation on the distance to existing nodes. Depending on the parameters $\alpha$ and $\beta$, spatial growth could yield networks similar to small-world cortical, scale-free highway-transportation networks as well as networks in non-Euclidean spaces such as metabolic networks [@Kaiser2004b]. Specifically, it was possible to generate networks with similar wiring organization than the macaque cortical network [@Kaiser2004c]. Using different time domains for connection development, where several spatial regions of the network establish connections in partly overlapping time windows, allows the generation of multiple clusters or communities [@Kaiser2007NC].
Outlook
=======
Natural neural systems, such as cortical networks of connections between brain regions, have developed several properties that are desirable for computers as well. Cortical networks show an innate ability to compensate for and recover from damages to the network. Whereas removing the few highly-connected nodes has a large effect on network structure, a random removal of nodes or edges has a small effect in most of the cases. In addition, the spatial layout of cortical and neuronal networks exhibiting several long-distance connections ensures few processing steps and thus a faster response time. Speculating about the future, these mechanism for robust and rapid processing might provide new ideas for artificial neural network as well as for computer architecture. As the ’programme’ of the brain is implemented in its wiring organization, the topology of the brain might inspire theoretical work in the organization of parallel processing and integration.
Towards these topics, we currently work on three questions. First, to identify properties for robust processing in the brain. This includes understanding mechanisms for recovery in neural systems. These mechanisms will then be applied to computer networks to see if they can lead to faster recovery after failure. Second, to investigate epileptic spreading in cortical networks. We intend to determine how the network structure influences activity or, for the disease state, seizure spreading in cortical networks. The more general analysis of spreading in networks could give useful insights in how to prevent virus spreading in communication networks. Finally, to find principles that guide the development of neural networks over time. By looking at general constraints for network development such as space, resources for connection establishment and maintenance, or global performance of a network, the reasons for normal and disturbed network development can be assessed. Ideally, this knowledge might lead to artificial neural networks with brain-like topology as well as processing.
In conclusion, we hope that future advances in our understanding of neural systems might (again) inspire solutions to problems in computer systems.
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Watts, D. J. & Strogatz, S. H. (1998). Collective dynamics of ’small-world’ networks. , 393:440–442.
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Author Profile {#author-profile .unnumbered}
==============
[ ]{}
Born in Essen, Germany, Marcus Kaiser studied biology up to a Masters degree in 2002 at the Ruhr-University Bochum and continues studies of computer science at the distance university Hagen. He obtained his PhD in Neuroscience, funded by a fellowship from the German National Academic Foundation, from the Jacobs University Bremen in 2005. Directly after finishing his PhD, he started as RCUK Academic Fellow in Complex Neural Systems and Behaviour at Newcastle University. Having an appointment in Computing Science as well as at the Institute of Neuroscience, Marcus is interested in understanding the architecture and processing of the brain as a network. He works on development, spatial organisation, seizure spreading, and recovery in cortical networks. He gratefully acknowledges support from EPSRC (EP/E002331/1) and the Royal Society (RG/2006/R2). More information is available at http://www.biological-networks.org/
[^1]: Author for correspondence ([email protected]).
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---
author:
- |
[^1]\
Physikalisches Institut, University of Heidelberg\
E-mail:
title: Combination and QCD Analysis of the HERA Inclusive Cross Sections
---
Combined H1 and ZEUS Cross Section Measurements {#sec:data}
================================================
Deep inelastic scattering (DIS) at HERA has been crucial to the exploration of proton structure and quark-gluon interaction dynamics. The combination of the H1 and ZEUS data provides the most accurate measurements of DIS inclusive double differential cross-sections of neutral (NC) and charged current (CC) $e^\pm p$ scattering over an extended kinematic range of $0.045 < Q^2 < 30\,000$ GeV$^2$ and $6\times10^{-5}<x<0.65$. Therefore, these accurate measurements can be used as a sole input to the QCD analysis to determine the proton structure - parton distribution functions (PDFs) as described in the following sections, which can be used then for precise predictions for the LHC processes.
The combination of data uses the $\chi^2$ minimisation method and it takes into account the correlated systematic uncertainties for the H1 and ZEUS cross-section measurements [@h1pdf]. This combination procedure is applied for the following scenarios: to the published inclusive deep inelastic cross sections measured by the H1 and ZEUS collaborations in CC and NC unpolarised $ep$ scattering at HERA during the period 1994-2000, termed HERA I; to the preliminary inclusive deep inelastic cross sections measured by the H1 and ZEUS collaborations in NC unpolarised $ep$ scattering at HERA during its last months of operation in 2007 with reduced proton beam of $E_p=460 GeV$ and $E_p=575 GeV$ which provides additional PDF constraint in the low $Q^2$, low $x$ region; and to the preliminary inclusive deep inelastic NC and CC cross sections at high $Q^2$ measured by the H1 and ZEUS collaborations at HERA during the whole period which considerably improve PDF uncertainties at high-$x$ as described in these proceedings.
QCD Analysis settings {#sec:analysis}
=====================
The above combined data is used as a sole input into a QCD fit analysis to extract the proton’s PDFs. The HERA data have a minimum invariant mass of the hadronic system, $W$, of $15$ GeV and a maximum $x$ of $0.65$, such that they are in a kinematic region where there is no sensitivity to target mass and large-$x$ higher-twist contributions. In addition, to restrain to the region where perturbative QCD is valid, only data above $Q^2_{min}=3.5$ GeV$^2$ is used in the central fit. The fit procedure consists first in parametrising PDFs at a starting scale $Q^2_0=1.9~ \rm GeV^2$, chosen to be below the charm mass threshold. The parametrised PDFs are the valence distributions $xu_v$ and $xd_v$, the gluon distribution $xg$, and the $u$-type and $d$-type $x\bar{U}$, $x\bar{D}$, where $x\bar{U} = x\bar{u}$, $x\bar{D} = x\bar{d} +x\bar{s}$. The following standard functional form is used to parametrise them $$xf(x) = A x^{B} (1-x)^{C} (1 + D x + E x^2),
\label{eqn:pdf}$$ where the normalisation parameters, $A_{uv}, A_{dv}, A_g$, are constrained by the QCD sum-rules. The $B$ parameters $B_{\bar{U}}$ and $B_{\bar{D}}$ are set equal, $B_{\bar{U}}=B_{\bar{D}}$, such that there is a single $B$ parameter for the sea distributions. The strange quark distribution is already present at the starting scale and it is assumed here that $x\bar{s}= f_s x\bar{D}$ at $Q^2_0$. The strange fraction is chosen to be $f_s=0.31$ which is consistent with determinations of this fraction using neutrino induced di-muon production. In addition, to ensure that $x\bar{u} \to x\bar{d}$ as $x \to 0$, $A_{\bar{U}}=A_{\bar{D}} (1-f_s)$. The $D$ and $E$ are introduced one by one until further improvement in $\chi^2$ is found. The best fit results in a total of 10 free parameters.
The PDFs are then evolved using DGLAP evolution equations [@qcdnum]. at NLO and NNLO in the $\overline{MS}$ scheme with the renormalisation and factorisation scales set to $Q^2$. The QCD predictions for the structure functions are obtained by convoluting the PDFs with the calculable coefficient functions taking into account mass effect for the heavy quarks based on the general mass variable flavour scheme [@tr]. The uncertainties at HERA are classified in three categories: experimental, model, and parametrisation uncertainties. The consistency of the input data set and its small systematic uncertainties enable us to calculate the experimental uncertainties on the PDFs using the $\chi^2$ tolerance $\Delta\chi^2=1$. The model uncertainties are evaluated by varying the input assumptions, which are the variation of the starting scale and of the $Q^2_{min}$, the variations of the heavy quark masses which are set to the standard values of $m_c=1.4$ GeV and $m_b=4.75$ GeV for the central fit, and the variation of $f_s$. The parametrisation uncertainty is estimated as an envelope which is formed as a maximal deviation at each $x$ value from the central fit of 10 parameter fits with $D$ and $E$ non-zero from Equation \[eqn:pdf\].
Results and Comparisons {#sec:results}
=======================
The NLO QCD analysis has been performed first to the final HERA I data resulting into HERAPDF1.0 which has been published [@herapdf] and it will be used as a reference for the new studies. The NNLO fit results have been also performed using the same scheme as used for MSTW PDF sets, for different values of the strong coupling, $\alpha_s(M_Z) = 0.1176$ and $\alpha_s(M_Z) = 0.1145$ [@grid]. Fit results are shown in Figure \[Fig:1\]. However, the NNLO fits do not bring improvement in terms of the fit quality with respect to NLO fits (worse by about 65 and 50 units of $\chi^2$, respectively) for both $\alpha_S$ cases, with a preference for lower value of the strong coupling at NNLO.
The inclusion of the new preliminary data at low proton beam energy in the HERA QCD fits results in PDF distributions that agree well with the HERAPDF1.0, as shown in Figure \[Fig:1\]. However, a large sensitivity has been observed when the variation of the kinematic cut has been studied, i.e. $Q^2_{min}>5$ GeV$^2$, which yielded a different PDF shape for the gluon distribution with respect to the the central fit which uses $Q^2_{min}>3.5$ GeV$^2$. The QCD fit analysis of the combined HERA-I inclusive deep inelastic cross sections has been extended to include combined HERA II measurements at high $Q^2$ resulting into HERAPDF1.5 [@grid]. Figure \[Fig:2\] shows that the precision of the PDFs at high-$x$ is considerably improved, not only for the experimental uncertainties, but also for the parametrisation uncertainty - particularly in the valence sector, when compared to HERAPDF1.0. This leads to more precise predictions for the LHC process.
The predictions based on HERAPDFs from the DIS process agree well with the Tevatron jet production, $Z$ and $W$ cross sections from the $p\bar p$ process and provide a competitive prediction for the LHC $pp$ processes.
![Figure shows on the left hand side the summary plot at the $Q^2=10$ GeV$^2$ with gluon, sea (which are scaled by a factor of 0.05) and the valence distributions for HERAPDF1.0 at NLO (band) compared illustratively to NNLO fits using $\alpha_S(M_Z)=0.1145$ (dashed) and $\alpha_S(M_Z)=0.1176$ (dotted) lines. On the right hand side, it is shown the comparison between the HERAPDF1.0 (band) based on HERA I data and fits including the HERA II of lower energy proton beams with the kinematic cut variation $Q^2\ge 3.5$ (dotted) and $Q^2\ge 5.0$ (dashed).[]{data-label="Fig:1"}](radescu_figure_0.ps "fig:"){width="49.50000%"} ![Figure shows on the left hand side the summary plot at the $Q^2=10$ GeV$^2$ with gluon, sea (which are scaled by a factor of 0.05) and the valence distributions for HERAPDF1.0 at NLO (band) compared illustratively to NNLO fits using $\alpha_S(M_Z)=0.1145$ (dashed) and $\alpha_S(M_Z)=0.1176$ (dotted) lines. On the right hand side, it is shown the comparison between the HERAPDF1.0 (band) based on HERA I data and fits including the HERA II of lower energy proton beams with the kinematic cut variation $Q^2\ge 3.5$ (dotted) and $Q^2\ge 5.0$ (dashed).[]{data-label="Fig:1"}](radescu_figure_1.ps "fig:"){width="49.50000%"}
![Figure shows on the left hand side the summary plot for the HERAPDF1.5 at the $Q^2=10$ GeV$^2$ with gluon, sea (which are scaled by a factor of 0.05) and the valence distributions. The errors include the experimental (red), model (yellow) and the PDF parametrisation (green) uncertainties. On the right hand side, it is shown the comparison between the HERAPDF1.0 (light color) based on HERA I data and HERAPDF1.5 (dark color) based on HERA I and II data, using total uncertainty band at $Q^2=10$ GeV$^2$.[]{data-label="Fig:2"}](radescu_figure_2.ps "fig:"){width="49.50000%"} ![Figure shows on the left hand side the summary plot for the HERAPDF1.5 at the $Q^2=10$ GeV$^2$ with gluon, sea (which are scaled by a factor of 0.05) and the valence distributions. The errors include the experimental (red), model (yellow) and the PDF parametrisation (green) uncertainties. On the right hand side, it is shown the comparison between the HERAPDF1.0 (light color) based on HERA I data and HERAPDF1.5 (dark color) based on HERA I and II data, using total uncertainty band at $Q^2=10$ GeV$^2$.[]{data-label="Fig:2"}](radescu_figure_3.ps "fig:"){width="49.50000%"}
[99]{} H1 Collaboration, F. Aaron *et al. *Eur.Phys.J.C64 (2009) 561, arXiv 0904.0929. H1 Collaboration, ZEUS Collaboration, F. Aaron *et al. *, JHEP 1001, 109 (2010), arXiv:0911.0884. C. Amsler *et al. (Particle Data Group), Phys. Lett. **B667, (2008). R. S. Thorne code, revised in 2008. QCDNUM package, M. Botje, (2010), arXiv:1005.1481, `http://www.nikef.nl/h24/qcdnum/index.html` The LHAPDF grid files are located at `https://www.desy.de/h1zeus/combined`\_`results/index.php?do=proton`\_`structure`*******
[^1]: On behalf of the H1 and ZEUS collaborations
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Introduction {#sec:intro}
============
The ATLAS detector {#sec:detector}
==================
Data selection {#sec:selection}
==============
Simulated collisions {#sec:simulation}
====================
Selection for the mass distribution analysis {#sec:resonance}
============================================
Selection for the angular distributions analysis {#sec:angular}
================================================
Signal models {#sec:bsm}
=============
Limits {#sec:limits}
======
Conclusion {#sec:conclusion}
==========
Acknowledgements {#sec:acknowldge .unnumbered}
================
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\[pr\][Theorem]{}
\[pr\]
\[th\]
[**Kählerian Killing Spinors, Complex Contact\
Structures and Twistor Spaces**]{}
A. Moroianu –[*Centre de Mathématiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France\
*]{}
U. Semmelmann –[*Humboldt-Universität zu Berlin, Institut für reine Mathematik (SFB 288), Ziegelstr. 13A, D-10099 Berlin*]{}
[**Abstract**]{} - We collect our recent results (\[5\] and \[8\]) and we get the equivalence of the three notions of the title under some conditions. We then use this equivalence in order to prove some consequences about Sasakian manifolds, complex almost contact structures and complex k-contact structures.
[ - Contact structure, Sasakian manifold, spinor, twistor space.]{}
The notion of a [*complex contact structure*]{} was introduced in the late 50’s by S. Kobayashi (cf. \[6\]), in analogy to real contact structures.
In 1982 in \[9\], S. Salamon investigated quaternionic Kähler manifolds. In particular, he defined the [*twistor space*]{} over such a manifold as a generalization of the classical notion of twistor space over a self-dual 4-manifold.
In 1986, K.D. Kirchberg was led to define [*Kählerian Killing Spinors*]{}, in order to characterize Kähler spin manifolds of odd complex dimension admitting the smallest possible eigenvalue of the Dirac operator (cf. \[4\]). Some important contributions to this problem are also due to O. Hijazi (cf. \[1\]).
The aim of this paper is to collect our recent results (cf. \[5\] and \[8\]), in order to explain the close connection between these three notions and to derive some corollaries. .5cm
In this section we describe the three notions introduced above, and recall relevant results obtained in each of these directions.
Let $M$ be a compact spin Kähler manifold of odd complex dimension $m=n/2$ and positive scalar curvature $R$. Then, each eigenvalue $\lambda$ of the Dirac operator $D$ satisfies the inequality (cf. \[4\]) $$\lambda^2\ge{m+1\over 4m}\inf_M R.$$ In the limiting case of this inequality, $M$ is Einstein and any eigenspinor $\Psi$ of $D$ corresponding to the eigenvalues $\pm\sqrt{(m+1) R/4m}$ is a [*Kählerian Killing spinor*]{}, i.e., satisfies the following first-order differential equation (cf. \[1\], \[4\]): $$\nabla_X\Psi+{1\over n+2}X\.D\Psi+{1\over n+2}J(X)\.\tilde D
\Psi=0.$$ We call such $M$ a [*limiting manifold*]{}. Conversely, any compact Kähler manifold admitting Kählerian Killing spinors is a limiting manifold. The first known examples of such manifolds were the complex projective spaces $CP^{2k+1}$.
Using complex contact structures it is possible to construct other manifolds admitting Kählerian Killing spinors. We will shortly describe the construction of \[5\].
[*(cf.\[6\])*]{} Let $ M^{2m} $ be a complex manifold of complex dimension\
$ m = 2k+1 $. A [*complex contact structure* ]{} is a family ${\cal C } = \{ (U_i,\omega_i)\}$ satisfying the following conditions:
1. $\{ U_i\}$ is an open covering of $ M $.
2. $\omega_i$ is a holomorphic 1-form on $ U_i $.
3. $ \omega_i \wedge (\partial \omega_i)^k \in \Gamma (\Lambda
^{m,0}\,M ) $ is different from zero at every point of $U_i$.
4. $\omega_i = f_{ij} \omega_j $ in $ U_i \cap U_j \quad $, where $ f_{ij} $ is a holomorphic function on $ U_i \cap U_j . $
Let ${\cal C } = \{(U_i,\omega_i)\}$ be a complex contact structure. Then there exists an associated holomorphic line subbundle $ L_{{\cal C }} \subset \Lambda^{1,0} (M) $ with transition functions $ \{ f_{ij}^{-1} \} $ and local sections $\omega_i$. From condition (iii) immediately follows the isomorphism $ L_{{\cal C
}}^{k+1} \; \cong \; K , $ where $K = \Lambda^{m,0} (M)$ denotes the canonical bundle of $M$. If we assume $k$ to be an odd integer then $M$ admits a canonical spin structure. It is given by the isomorphism $$\label{ident}
L _{{\cal C } }^{\frac{k+1}{2}}
\quad \cong \quad K^{\frac{1}{2}} \quad \cong \quad S_0 .$$ Here $S_0$ is the subbundle of the spinor bundle $S$ which is defined as the eigenspace of $\Omega$ for the eigenvalue $- i \,m $, where the Kähler form $\Omega$ is considered as endomorphism of $S$. We construct now a section $\Psi_{{\cal C } }$ of the spinor bundle which is associated to the contact structure $\cal C$. For doing so we fix $\; (U, \omega) \in {\cal C } \; $ and define $\; \Psi_{{\cal
C } } \;$ over the open set $U$ by $$\label{spinor}
\Psi_{{\cal C } } \; \big|_{U }
:= | \Psi_{\omega} |^{-2} \; \bar{\eta}_{\omega} \; \cdot
\;\Psi_{\omega} \;,$$ where $ \Psi_{\omega} \in \Gamma ( S_0 \; \big|_{U }) $ is the local section in $S_0$ corresponding to $\omega^{\otimes \,
\frac{k+1}{2}}$ under the identification (\[ident\]) and $\eta_{\omega} := \omega \wedge (\partial\omega)^{\frac{k-1}{2}} $. From the condition (iv) it follows that the spinor $ \Psi_{{\cal
C
} }$ is globally defined. We have the following
[*(cf. \[5\])*]{} \[kks\] Let $ ( M,g,J) $ be a compact Kähler-Einstein manifold of complex dimension $ m = 2k+1 $ with k odd, and let ${\cal C } $ be a complex contact structure on $ M $. Then the spinor $\Psi_{{\cal C } }$ associated with ${\cal C } $ satisfies the equation $$D^2 \, \Psi_{{\cal C } } \; = \; \frac{m+1}{4m} \; R
\;\Psi_{{\cal C } } ,$$ where $R$ is the scalar curvature of $(M,g)$. In particular, the spinors\
$ \Psi_{{\cal C } }^{\pm} := \lambda_1 \Psi_{{\cal C } }
\pm D \Psi_{{\cal C } } $ are Kählerian Killing spinors, where $
\lambda_1 = \sqrt{\frac{m+1}{4m} R}$.
A class of manifolds satisfying the assumptions of Proposition \[kks\] are the twistor spaces of quaternionic Kähler manifolds introduced by S. Salamon (cf. \[9\]).
A [*quaternionic Kähler manifold* ]{} is defined to be an oriented 4n-dimensional Riemannian manifold whose restricted holonomy group is contained in the subgroup $Sp(n) Sp(1) \subset SO(4n) \; (n \ge 2)$. Salamon’s idea is to construct over each such manifold $M$ a natural $ C P^1$– bundle Z, admitting a Kähler metric such that the bundle projection is a Riemannian submersion. He called this bundle the [*twistor space* ]{} of $M$.
[*(cf.\[9\])*]{} \[salamon\] Let $ M^{4k} $ be a quaternionic Kähler manifold with positive scalar curvature. Then its twistor space $Z$ admits a Kähler Einstein metric of positive scalar curvature and a complex contact structure. Moreover, $Z$ is spin for odd k and $Z$ is spin for even k iff $Z = CP^{2k+1}$.
From Propositions \[kks\] and \[salamon\] we obtain that all the twistor spaces of quaternionic Kähler manifolds $M^{4k} \; (k
\equiv 1(2))$ with positive scalar curvature admits Kählerian Killing spinors, i.e. they are limiting manifolds.
The only explicitly known manifolds of this kind are the following three families:
- $ \quad Sp(k+1) / Sp(k) \times U(1) \quad \cong \quad
CP^{2k+1} $,
- $ \quad SU(k+2) / S(U(k) \times U(1) \times U(1)) $,
- $ \quad SO(k+4) / S( O(k) \times O(3) \times O(2) ) $.
and the 15–dimensional exceptional space $ \quad F_4 / Sp(3) U(1)$.
It is now interesting to see that each such limiting manifold (i.e. each spin Kähler manifold of odd complex dimension and positive scalar curvature admitting Kählerian Killing spinors) has to be a twistor space. This is due to the following classification result:
[*(cf. \[8\])*]{} \[andrei\] The limiting manifolds of complex dimension $4l+3$ are exactly the twistor spaces associated to quaternionic Kähler manifolds of positive scalar curvature. The only limiting manifold of complex dimension $4l+1$ is $CP^{4l+1}$.
The idea of the proof is the following. Take a limiting manifold $M$ and consider a maximal root of the canonical line bundle with some hermitian metric. The associated principal $U(1)$-bundle over $M$, say $P_M$, with a carefully chosen metric, is spin, and any spinor on $M$ induces a [*projectable*]{} spinor on $P_M$. Moreover, a Kählerian Killing spinor induces a projectable real Killing spinor on $P_M$. This forces $P_M$ to admits a regular Sasakian 3-structure and $M$ to be the twistor space over the quotient of $P_M$ by the Sasakian 3-structure.
The last part of the proposition follows from the fact that the only spin twistor space of complex dimension 4l+1 is $CP^{4l+1}.$
Combining the above propositions we have
\[gesamt\] Let $M$ be a compact spin Kähler manifold of positive scalar curvature and complex dimension $4l+3$. Then the following statements are equivalent:
[*(i)*]{} $\ \ M$ admits Kählerian Killing spinors;
[*(ii)*]{} $\ M$ is Kähler-Einstein and admits a complex contact structure;
[*(iii)*]{} $\ M$ is the twistor space of some quaternionic Kähler manifold of positive scalar curvature.
As an immediate corollary we have the following result:
\[c1\] If $M$ is a Kähler-Einstein manifold of complex dimension $4l+3$ which admits a complex contact structure, then $M$ is the twistor space of some quaternionic Kähler manifold of positive scalar curvature.
This corollary is in fact part of a very recent theorem of C. LeBrun (cf. \[7\]). He proves the same statement but without the restriction on the dimension and also using a different method. The interest of our proof lies in the unexpected appearance of the Dirac operator. As a less obvious corollary we have the following
\[reg\] Let $M$ be a Riemannian manifold of real dimension $n=8l+7$, admitting a Sasakian 3-structure which is regular in one direction. Then it is regular in all directions.
Let $V$ be the Killing vector field in the regular direction. We denote by $N$ the quotient of $M$ by the $S^1$-action in the direction of $V$. Regularity just means that $N$ is a manifold. Now a simple calculation (cf. \[2\]) shows that $N$ is a Kähler–Einstein manifold admitting a complex contact structure.
Corollary \[c1\] yields that $N$ is the twistor space of some quaternionic Kähler manifold $Q$, of positive scalar curvature. Using \[2\] once again, we see that the 2-distribution given by the two other Killing vector fields of the Sasakian 3-structure, projects on the 2-distribution $\Theta$ which gives the complex contact structure on $N$. So the quotient of $M$ by the Sasakian 3-structure is diffeomorphic to the space of leaves of $\Theta$, which is exactly the manifold $Q$. Thus our Sasakian 3-structure is regular.
Corollary [*\[reg\]*]{} is also true for $n=8l+3$. We just have to use the result of C. LeBrun [*(\[7\])*]{} instead of Corollary [*\[c1\]*]{} in the above proof.
In \[3\] S. Ishihara and M. Konishi introduced the concept of [*complex almost contact structures*]{}. These are the hermitian manifolds of odd complex dimension $2n+1$ whose structure group can be reduced to $U(1) \, \times \, (Sp(n) \, \otimes \, U(1)) $. They proved that each such manifold under an additional normality condition admits a Kähler–Einstein metric and also a complex contact structure. In \[2\] they also showed the existence of a normal complex almost contact structure on the $S^1$–quotient of a 3–Sasakian space which is regular in one direction. From Theorem \[gesamt\] we then have
Let $M$ be a complete Hermitian manifold with a complex almost contact structure. Then the structure is normal iff $M$ is the twistor space of some quaternionic Kähler manifold of positive scalar curvature.
To give a last application of Theorem \[gesamt\] we consider a generalization of complex contact structures. For this let ${\cal C}
= \{ U_i, \omega_i \}$ be a family of (local) r–forms which again satisfies conditions (i) – (iv) of Definition 1, where (iii) has to be changed into:
[*(iii)*]{} ’ $$\quad \omega_i \wedge
(\partial \omega_i)^s \in \Gamma (\Lambda ^{m,0}M \, \big|_{U_i } )
\quad \mbox{is different from zero at each point of}
\quad U_i.$$ Here $s = \frac{m-r}{r+1}$ must be an integer. Such a family was called a complex r–contact structure in \[5\]. If $s$ is an odd integer then $M$ again admits a canonical spin structure. In this situation it is once more possible to construct a Kählerian Killing spinor $\psi_{\cal C}$ (similar to (\[spinor\])). Theorem \[gesamt\] then implies
Let $(M^{2m}, g, J)$ be a compact Kähler–Einstein manifold with positive scalar curvature which admits a complex r–contact structure such that $s = (m-r)/(r+1)$ is an odd integer. Then $M$ is a complex contact manifold.
As it was recently pointed out to us by K. Galicki, Corollary \[reg\] is an old result of S. Tanno \[10\].
[ This note was finished during our stay at the E. Schrödinger Institut in Vienna. We would like to thank the institute for support and hospitality. ]{}
[12]{}
O. Hijazi, Eigenvalues of the Dirac Operator on Compact Kähler Manifolds, Commun. Math. Phys. [**160**]{} (1994), 563-579.ß
S. Ishihara and M. Konishi, Real Contact 3-Structure and Complex Contact Structure, Sea Bull. Math [**3**]{} (1979), 151-161.ß
S. Ishihara and M. Konishi, Complex almost Contact Manifolds, Kodai Math. J. [**3**]{} (1980), 385-396.ß
K.-D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Global Anal. Geom. [**3**]{} (1986), 291-325. ß
K.-D. Kirchberg and U. Semmelmann, Complex Contact Structures and the first Eigenvalue of the Dirac Operator on Kähler Manifolds, to appear. ß
S. Kobayashi, Remarks on Complex Contact Manifolds, Proc. Amer. Math. Soc. [**10**]{} (1959), 164-167.ß
C. LeBrun, Fano Manifols, Contact Structures, and Quaternionic Geometry, to appear.ß
A. Moroianu, La première valeur propre de l’opérateur de Dirac sur les variétés kählériennes compactes, C.R. Acad. Sci. Paris, t.[**319**]{}, Série [**I**]{} (1994), 1057-1062.ß
S. Salamon, Quaternionic Kähler Manifolds, Invent. Math. [**67**]{} (1982), 143-171. ß
S. Tanno, Killing vectors on contact Riemannian manifolds and fiberings related to the Hopf fibrations, Tohoku Math. J. [**23**]{}, 1971, 314-333.
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---
abstract: 'Suppose you have one unit of stock, currently worth 1, which you must sell before time $T$. The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see $a$ units of time into the future, and base our stopping rule on that; we should be able to do better than expected value 1. But how much better can we do? And how would we exploit the additional information? The optimal solution to this problem will never be found, but in this paper we establish remarkably close bounds on the value of the problem, and we derive a fairly simple exercise rule that manages to extract most of the value of foresight.'
author:
- 'Philip Ernst, L.C.G. Rogers, and Quan Zhou'
bibliography:
- 'VF.bib'
title: The value of foresight
---
*We dedicate this work to our colleague, mentor, and friend, Professor Larry Shepp (1936-2013)*
Introduction. {#intro}
=============
What is the value of foresight in a financial market? This is a question that intrigued Larry Shepp (see page 2 of [@SheppSem]) and seems an interesting question in the context of insider trading; if we could know one minute in advance what the price of a stock was going to do, what would we be prepared to pay for that information? Of course, it is rather fanciful to imagine that we could possibly be told the price of the stock at some time in the future, but we might imagine a situation where some market participants received information only after a delay, which would confer the same kind of advantage on those who got the information earlier. In a modern financial market, any such differences would be measured in microseconds, a timescale on which conventional models of stock prices could not be trusted, but one of the first observations of this paper is that the value of foresight can be equivalently interpreted in terms of the value of a fixed-window lookback option; at a time of your choosing, you may sell the stock for the best price which the stock achieved in the previous $a$ units of time. This transforms the question into an American option pricing problem, but not one that is possible to solve in closed form, since the state variable at time $t$ is the entire history of the stock from time $t-a$ to time $t$ -a path-valued state. Moreover, even if we were to discretize time, the state vector will be high dimensional, so existing numerical methods will struggle to cope. Nevertheless, recent developments allow good progress to be made on the question, as we shall see.\
To begin with, we set some notation. We shall take the sample space to be the path space $C({ {\mathbb R} }^+,{ {\mathbb R} })$ with canonical process $W$, Wiener measure $P$ and the usual $P$-augmentation $({{\mathcal F}}_t)$ of the canonical filtration of $W$. We denote by ${{\mathcal T}}$ the class of all $({{\mathcal F}}_t)$-stopping times. We fix some $a>0$ which represents the foresight available to the insider. The insider may choose to stop at any stopping time of the larger filtration $({{\mathcal F}}'_t) \equiv ({{\mathcal F}}_{t+a})$; we denote the set of all $({{\mathcal F}}'_t)$-stopping times by ${{\mathcal T}}'$.
At an abstract level, this set-up can be considered an example of [*grossissement*]{} — the enlargement of filtrations. This theory was developed from the late 1970’s on, starting with the works of Barlow [@Barlow], [@Bayraktar], Jeulin & Yor [@Yor1], [@Yor2], [@Yor3], and further developed by others including Yoeurp [@Yoeurp], and by Itô’s extension of the stochastic integral (see [@Ito]). The topic has since continued to flourish, primarily because of its natural connection with insider trading. Varied formulations of the insider trader’s advantage over other agents in a financial market are addressed in [@Amendinger], [@Back], [@Elliott], [@Pontier], [@Imkeller3], [@Imkeller2], [@Imkeller1], [@Jacod], [@Karatzas], [@Mansuy], and [@Protter], among others. It has to be understood that the theory of enlargement of filtrations is not a [*universal*]{} theory; results are only established for particular classes of enlargement, such as filtrations enlarged with an honest time, or filtrations with an initial enlargement. The results proved say that if the enlargement has one of these particular structural forms, then any $({{\mathcal F}}_t)$-local martingale is a $({{\mathcal F}}'_t)$-semimartingale, and the semimartingale decomposition is then identified. The following proposition shows that none of these general results can be applied to the problem we consider here.
\[prop0\] The process $W$ is not a semimartingale in the filtration $({{\mathcal F}}'_t)$.
Consider the simple integrands $$H^n_t \equiv n^{-1/2} \sum_{j=1}^n { \hbox{\rm sgn} }(\Delta^n_j)\; I_{ \{ (j-1)a < nt \leq ja \}},
\label{Hn}$$ where $$\Delta^n_j \equiv W( ja/n ) - W( (j-1)a/n).
\label{Deltadef}$$ The processes $H^n$ are left-continuous, bounded, and $({{\mathcal F}}'_t)$-previsible; indeed, $H^n_t$ is measurable on ${{\mathcal F}}'_0 \equiv {{\mathcal F}}_a$. Now consider the (elementary) stochastic integral $$H^n \cdot W = n^{-1/2} \sum_{j=1}^n |\Delta^n_j| = n^{-1} \sum_{j=1}^n \sqrt{n}|\Delta^n_j| \;.$$ The random variables $\sqrt{n} \Delta^n_j$ are independent zero-mean gaussians with common variance $a$. By the Weak Law of Large Numbers, $H^n \cdot W$ converges in probability to $E|W_a|$ as $n \rightarrow \infty$. But the Dellacherie-Bichteler theorem (see, for example, Theorem IV.16.4 in [@RW2]) says that $W$ is a semimartingale if and only if whenever a sequence $H^n$ of bounded previsible simple processes tends uniformly to zero, then the simple stochastic integrals $H^n\cdot W$ tend to zero in probability. We conclude that $W$ is not an $({{\mathcal F}}'_t)$-semimartingale.
The message from Proposition \[prop0\] is that none of the results from enlargement of filtrations will help us here — the problem addressed is [*concrete, challenging, and not amenable to general theory*]{} — which is why it appealed to Larry Shepp.
So what [*are*]{} we able to do? To begin with, applying the methods of [@bobs], we obtain remarkably tight upper and lower bounds on the value of foresight. This methodology is based on simulations, so there is no simple interpretation of the exercise rule which arises. However, in Section \[S3\] we develop simple and transparent rules based on heuristic arguments which we are then able to compare with the bounds from Section \[S2\]; the resulting (explicit) rules essentially achieve the lower bound which comes from the simulation approach of [@bobs] applied in Section \[S2\].
Preliminaries. {#preliminaries. .unnumbered}
--------------
To start with, we set out some notation and make various standardizations of the question. There is a fixed time horizon $T>a>0$ by which time the investor must have sold the stock. The stock price process $S$ will be the solution to an SDE $$dS_t = S_t( \sigma dW_t + \mu_t dt)I_{\{t \leq T\} }, \qquad S_0 = 1,
\label{dS}$$ where $\mu$ is a bounded $({{\mathcal F}}_t)$-previsible process, and $\sigma>0$ is a constant. Notice that $S_t = S_T$ for all $t \geq T$, and where necessary we make the convention that $S_t = 1$ for $t<0$. It is immediate from the definitions that $\tau' \in {{\mathcal T}}'$ if and only if $\tau'+a \in {{\mathcal T}}$. We let $Q$ denote the pricing measure, given in terms of the Cameron-Martin-Girsanov martingale $$d\Lambda_t = \Lambda_t (r-\mu_t) \; dW_t, \qquad \Lambda_0 = 1,$$ by $$\frac{dQ}{dP}\biggr\vert_{{{\mathcal F}}_t} = \Lambda_t.$$
What is the time-0 value of stopping at $\tau' \in {{\mathcal T}}'$? The obvious immediate answer to this question is just $E^Q[ \exp(-r\tau') S_{\tau'}]$, but is it clear that there is no issue arising from the fact that $S_{\tau'}$ is not ${{\mathcal F}}_{\tau'}$-measurable? We can see that this is in fact correct by the following argument. At time $\tau'$, the investor receives $S_{\tau'}$ which he can then place in the bank account for $a$ units of time, so that at $({{\mathcal F}}_t)$-stopping time $\tau = \tau'+a$ he has $e^{ra} S_{\tau'} = e^{ra} S_{\tau-a}$. This random variable is ${{\mathcal F}}_\tau$-measurable, so the time-0 value of it is just given by the usual expression, $$E^Q[ e^{-r\tau} e^{ra} \,S_{\tau-a}] = E^Q[ e^{-r\tau'}\, S_{\tau'}],$$ as expected. So what we find is that the time-0 value of stopping at $\tau' \in {{\mathcal T}}'$ is the $Q$-expectation of the discounted value of the stock at the time of exercise. We may as well therefore work with the discounted stock, and work in the pricing measure $Q$. Equivalently, we may (and shall) assume for simplicity that $$r = \mu = 0.
\label{simple1}$$ Since the choice of $\sigma$ amounts to the choice of a time unit, we may and shall assume that $$\sigma = 1.
\label{simple2}$$ Our model for the asset dynamics is therefore $$S_t = \exp( W_t - { {\scriptstyle \frac{1}{2} }\, }t) \equiv \exp( X_t),
\label{eq2}$$ where $X$ is the drifting Brownian motion $$X_t = W_t + ct,
\label{Xdef}$$ with the special value[^1] $c = -{ {\scriptstyle \frac{1}{2} }\, }$. Notice that we have by convention fixed $S_0 = 1$.\
What we want to understand then is $$v(a) \equiv \sup_{\tau' \in {{\mathcal T}}',\; 0 \leq \tau' \leq T} E[\; S_{\tau'}\; ] \equiv \sup_{\tau \in {{\mathcal T}}, \; 0 \leq \tau \leq T+a}
E[\; S_{\tau-a} \; ].
\label{eq3}$$ It is clear that $v$ will be increasing, and $v(0) = 1$, but our aim is to determine as accurately as possible what $v(a)$ is, and to identify a good approximation to the optimal stopping time. The first item of business, dealt with in Section \[S2\], is to show that the value $v$ can be alternatively expressed as $$v(a) = \sup_{\tau \in {{\mathcal T}}, \; 0 \leq \tau \leq T} E[ \; Z_\tau \; ],
\label{eq7}$$ where[^2] $$Z_t \equiv \sup\{ S_u: t-a \leq u \leq t \}.
\label{Zdef}$$ In other words, the value $v(a)$ is the value of an [*American fixed-window lookback option.*]{}
Foresight as lookback. {#S2}
======================
Recalling the convention that $S_u = 1$ for $u<0$, and $S_u = S_T$ for $u \geq T$, we have a simple proposition.
\[prop1\] With $\tau$ denoting a generic $({{\mathcal F}}_t)$-stopping time, $$v(a) \equiv \sup_{a \leq \tau \leq T+a} E[\; S_{\tau-a} \; ] = \sup_{0 \leq \tau \leq T} E[ \; Z_\tau \; ],
\label{eq9}$$ where $Z_t \equiv \sup\{ S_u: t-a \leq u \leq t \}$.
0.1 in [Proof.]{} Because $S_u = S_T$ for all $u \geq T$, it is clear that $Z_t \leq Z_{t \wedge T}$. Therefore, for any stopping time $\tau$ such that $ a \leq \tau \leq T+a$, we have $$S_{\tau-a} \leq Z_\tau \leq Z_{\tau \wedge T}.$$ Therefore $$v(a) \equiv \sup_{a \leq \tau \leq T+a} E[\; S_{\tau-a} \; ] \leq \sup_{0 \leq \tau \leq T} E[ \; Z_\tau \; ].
\label{eq10}$$ For the reverse inequality, suppose that $\tau$ is a stopping time, $0 \leq \tau \leq T$, and define a new random time $\tilde \tau$ by $$\tilde\tau = \inf\{ u \geq \tau\vee a: S_{u-a} = Z_\tau\}.
\label{tildetau}$$ Clearly $\tau \vee a \leq \tilde\tau \leq \tau + a$. We claim that $\tilde\tau$ is a stopping time, as follows: $$\begin{aligned}
\{ \tilde\tau \leq v \} &=& \{ \hbox{\rm for some $u \in [\tau \vee a,v], \;
S_{u-a} = Z_\tau$ } \}
\\
&=& \{ \tau\vee a \leq v\} \cap \{
\hbox{\rm for some $u \in [(\tau \vee a)\wedge v ,v], \;
S_{u-a} = Z_{\tau\wedge v}$ } \}
\\
& \in & {{\mathcal F}}_v \; ,\end{aligned}$$ since the event $ \{\exists u \in [(\tau\vee a)\wedge v ,v], \;
S_{u-a} = Z_{\tau\wedge v} \}$ is ${{\mathcal F}}_v$-measurable, as is $(\tau\vee a) \wedge v$. Now we see that $$Z_\tau = S_{\tilde\tau - a},$$ and therefore $$E[ \; Z_\tau \; ] = E[ \; S_{\tilde\tau - a} \; ]
\leq \sup_{a \leq \tau \leq T+a} E[\; S_{\tau-a} \; ],
\label{prop1_conc}$$ since $ a \leq \tilde\tau \leq T+a$. Since $0 \leq \tau \leq T$ was any stopping time, we deduce that $$\sup_{0 \leq \tau \leq T} E[ \; Z_\tau \; ] \leq \sup_{a \leq \tau \leq T+a} E[\; S_{\tau-a} \; ],$$ and the proof is complete. 0.1 in $\square$
The importance of Proposition \[prop1\] is that it turns the problem of calculating the value of foresight into the calculation of an American fixed-window lookback option. By discretizing the time, we shall instead calculate numerically the value of a Bermudan fixed-window lookback option. Of course, we need to account for the difference between American and Bermudan prices, but this is in essence a solved problem; see Broadie, Glasserman & Kou [@BGK].
So we shall take the standardized asset dynamics , fix some time horizon $T>0$ which is subdivided into $N_T$ steps of length $h = T/N_T$, and consider the problem of bounding $$v_h(a) \equiv \sup_{0 \leq \tau_h \leq T} E[ \; Z^{(h)}_{\tau_h} \; ],
\label{vh}$$ where $\tau_h$ is a stopping time taking values in the set $h { {\mathbb Z} }^+$, and $$Z^{(h)}_t = \max\{ S_{kh}: t-a \leq kh \leq t \}.
\label{Zh}$$ Since this discretization is now fixed for the rest of the section, we shall drop the appearance of $h$ in the notation and refer to $Z$ for $Z^{(h)}$, $v$ for $v_h$, $\tau$ for $\tau_h$. We are now exclusively considering the optimal stopping of a functional of a discrete-time Markov process. A moment’s thought shows that the process $Z$ is not Markov, but the process $$x_t = (S_{t-mh},\; \ldots,\; S_t) \qquad (t \in h { {\mathbb Z} }^+)
\label{xdef}$$ is Markovian (where $mh = a$), and the payoff process $Z$ is simply a function of the Markov process $x$: $$Z_t = g(x_t),$$ where $g(x) \equiv \max\{x_0, \ldots, x_m \}$ is the largest component of the $(m+1)$-vector $x$. We use the approach of [@bobs], which is a combination of several techniques developed over the last twenty or so years, and in summary consists of the four steps:
- pretend that the stopping reward process $Z$ is itself Markovian, and by discretizing $Z$ onto a suitably-chosen finite set of values estimate the transition probabilities of this finite state Markov chain by simulation (this is the approach of Barraquand & Martineau [@BM]);
- solve the optimal stopping problem for this finite state Markov chain by dynamic programming;
- use the solution to generate a stopping rule whose performance is evaluated by simulation;
- use the dual characterization of the value of the problem (see [@R1], [@HK], [@AB]) to find a hedging martingale.
The method is fully explained in [@bobs], and illustrated with examples, so there is no need to discuss it further here, except to highlight one point, which is used in various places in [@bobs] and is needed here. Suppose that $(M_t)_{t \geq 0}$ is any strictly positive martingale, $M_0=1$. Then we may equivalently express the value $$\begin{aligned}
v(a) &\equiv& \sup_{0 \leq \tau \leq T} E[ \; Z_\tau \; ]
\nonumber
\\
&=& \sup_{0 \leq \tau \leq T} E[ \; M_\tau \, (Z_\tau/M_\tau) \; ]
\nonumber
\\
&=& \sup_{0 \leq \tau \leq T} \tilde E[ \;Z_\tau/M_\tau \; ],
\label{numeraire}\end{aligned}$$ where the probability $\tilde P$ equivalent to $P$ is defined by using the likelihood-ratio martingale $M$: $$\frac{d\tilde P}{dP}\biggr\vert_{{{\mathcal F}}_t} = M_t.
\label{dPtildedP}$$ In the present application, it is natural to use $M_t = S_t$ as the change-of-measure martingale, and the effect of this is to change the Brownian motion $W$ into a Brownian motion with drift 1. Thus when we do simulations, we simulate a Brownian motion with drift 1 in place of $W$, and the stopping reward process is changed to $Z_t/S_t$. This is a good thing to do, because the value of $Z_t$ will be close to $S_t$, so the binning procedure of the Barraquand-Martineau approach should achieve a lot better accuracy than we would get without this measure transformation. [Moreover, after the change of measure the stopping rule depends on $(Z_t/S_t, t)$ instead of $(S_t, t)$, which is more sensible and resembles the stopping rule we will propose below.]{} We do indeed find that the accuracy is substantially improved by doing this change of measure; the results are reported in Table 1.
$a/h$ Lower SE(low) Upper SE (up) Gap(%)
------- ------- --------- ------- --------- --------
1 1.054 0.43 1.055 0.53 0.09
2 1.074 0.62 1.076 0.77 0.18
3 1.088 0.73 1.092 0.90 0.37
4 1.100 0.83 1.104 1.0 0.37
5 1.109 0.95 1.114 1.1 0.45
6 1.117 1.0 1.123 1.2 0.53
7 1.123 1.1 1.131 1.2 0.71
8 1.129 1.2 1.137 1.3 0.70
9 1.135 1.3 1.144 1.4 0.79
10 1.140 1.4 1.149 1.4 0.78
11 1.144 1.4 1.154 1.5 0.87
12 1.149 1.5 1.159 1.6 0.86
13 1.152 1.6 1.164 1.7 1.03
14 1.156 1.6 1.168 1.7 1.03
15 1.159 1.7 1.172 1.8 1.11
16 1.163 1.8 1.175 1.8 1.02
17 1.166 1.8 1.179 1.9 1.10
18 1.168 1.9 1.182 1.8 1.18
19 1.171 1.9 1.185 1.9 1.18
20 1.174 2.0 1.188 1.9 1.18
: Upper and lower bounds of $\sup_{\tau< T} E[Z_\tau]$ from simulation using method of [@bobs] with $h = 1/2500$, $N_T = 250$. ‘SE’ is the standard error, in basis points (1e-4). Simulation parameters: number of bins = 200, number of samples per bin = 200, lower-bound simulation = 50,000, upper-bound simulation = 10,000, sub-simulation per step = 50.
\[table:bermuda\]
Explicit stopping rules. {#S3}
========================
As was stated earlier, the stopping rules which are derived in Section \[S2\] are the output of a simulation; they have no particular structure or interpretation, and a different simulation run will generate a different stopping rule. The methodology of [@bobs] is generic, and works just the same for essentially any Markov process, and any stopping function of the process, but we may hope to improve in individual applications by exploiting the specific structure of that application, which is what we shall do here.
We work in terms of the log price $X_t = W_t + ct$, and use the notations $$\bar X_t \equiv \sup_{0 \leq u \leq t} X_u, \qquad
\bar X_{[t-a,t]} \equiv \sup_{t-a \leq u \leq t} X_u \equiv \log Z_t\; .
\label{notation}$$ The first time we may need to consider stopping is the stopping time $$\tau_0 \equiv \inf \{\; t: \bar X_{[t-a,t]} = X_{t-a} \; \},
\label{tau0def}$$ because up until that time we have $X_{t-a} < \bar X_{[t-a,t]}$, so it will be suboptimal to exercise - waiting a little longer may improve, and will not make the reward less. But at time $\tau_0$, continuation means we have to let go of the good value $X_{\tau_0-a}$ in the hope of doing better in the future, and we may in fact do worse. Whether we should optimally continue will depend on the entire path of $X$ from $\tau_0-a$ to $\tau_0$, but we will simplify to consider only rules where continuation is decided by the value of $X_{\tau_0} - X_{\tau_0 -a}\;$; if this is higher than some threshold $q <0$ we shall continue, otherwise we stop.
An important observation is the fact that $$\bar X_{[\tau_0-a,\tau_0]} = \bar X_{\tau_0},
\label{simplify}$$ as a moment’s thought will reveal. Therefore we have $$\tau_0 = \inf \{\; t: \bar X_t = X_{t-a} \; \}.
\label{tau0def_bis}$$
Now it is clear that the choice of threshold value $q$ will depend on the time to go when we have to make the stop/continue decision, and this makes complete solution of the problem much more complicated. So what we shall do is to propose a [*modified*]{} problem, where the time by which we must sell is not a fixed time $T$ but an independent random time $\alpha \sim \exp(\eta)$ for some $\eta>0$. This gives us a renewal property that allows us to make progress, and obtain explicit expressions. When it comes to converting the solution to the modified problem into an exercise rule for the original problem, what we do is set the threshold according to the value of $\eta$ which makes the expectation of $\alpha$ equal to the time to go.
For this modified problem, we define a stopping rule $R(q)$ depending on the chosen threshold as follows:
- Wait until $\tau_0 \wedge \alpha$;
- If $\alpha < \tau_0$, stop and receive $Z_\alpha = \exp(\bar X_\alpha)$;
- If $\tau_0 < \alpha$ and $X_{\tau_0} - \bar X_{\tau_0} < q$, stop and receive $Z_{\tau_0}= \exp(\bar X_{\tau_0} )$;
- If $\tau_0 < \alpha$ and $X_{\tau_0} - \bar X_{\tau_0} > q$, forget and continue.
In the final eventuality (iii.b), ‘forget and continue’ means that we wipe away the whole path of $X$ in the time interval $[0, \tau_0)$, keeping only the value $X_{\tau_0}$, and restart the rule from that point. If the value of this strategy is $K$, then we have the identity $$\begin{aligned}
K &=& E[ \;\exp(\bar X_\alpha) : \alpha < \tau_0]
+ E[\; \exp(\bar X_{\tau_0}) : \tau_0 < \alpha, X_{\tau_0} - \bar X_{\tau_0} < q \; ]
\nonumber
\\
&& \qquad\qquad\qquad + K E[ \; \exp(X_{\tau_0}): \tau_0 < \alpha, X_{\tau_0} - \bar X_{\tau_0} > q \; ].
\label{Keq}\end{aligned}$$ It is therefore apparent that we can evaluate this particular stopping rule provided we can find explicit expressions for $$\begin{aligned}
A_0 &=& E[ \;\exp(\bar X_\alpha) : \alpha < \tau_0\;], \label{A0}
\\
A_-(q) &=& E[\; \exp(\bar X_{\tau_0}) : \tau_0 < \alpha, X_{\tau_0} - \bar X_{\tau_0} < q \; ], \label{A-}
\\
A_+(q) &=& E[ \; \exp(X_{\tau_0}): \tau_0 < \alpha, X_{\tau_0} - \bar X_{\tau_0} > q \; ]. \label{A+}\end{aligned}$$ We can. If $\varphi$ denotes the density of the standard $N(0,1)$ distribution, and $\Phi$ denotes its distribution function, we have the following result.
\[prop2\]For $q<0$, denote $$\begin{aligned}
{\beta}&=& \sqrt{c^2+2\eta},
\label{bdef}
\\
\nu &=& \frac{2}{\sqrt{a}}\; \varphi({\beta}\sqrt a) + 2{\beta}\Phi({\beta}\sqrt a) - c - {\beta},
\label{nudef} \\
\nu_a &=& \frac{2}{\sqrt{a}} \; \varphi(c\sqrt{a}) - 2c\, \Phi(-c\sqrt a),
\label{nuadef} \\
\nu_\alpha &=& 2{\beta}(\Phi({\beta}\sqrt a) - { {\scriptstyle \frac{1}{2} }\, }) -2c(\Phi(c\sqrt a) - { {\scriptstyle \frac{1}{2} }\, })
\nonumber\\
&&\qquad\qquad \qquad\quad
+ \frac{2}{\sqrt a} \bigl\lbrace\; \varphi({\beta}\sqrt a) - \varphi(c\sqrt a)
\;\bigr\rbrace ,
\label{nualphadef}
\nonumber \\
\psi_0(q)&=& e^{-\eta a} \biggl\lbrace\;
\frac{2}{\sqrt a} \; \varphi( (q-ca)/\sqrt a) - 2c\, \Phi( (q-ca)/\sqrt a)
\;\biggr\rbrace,
\label{psi0def}
\\
\psi_1(q)
&=& e^{(c+{ {\scriptstyle \frac{1}{2} }\, }-\eta)a} \biggl[ \; \frac{2}{\sqrt a}\;\bigl\lbrace
\; \varphi(\bar c \sqrt a)
-\varphi( (q-\bar c a)/\sqrt a) \;\bigr\rbrace
\nonumber \\
&& \qquad\qquad\qquad + 2\bar c\bigl\lbrace \; \Phi(\bar c \sqrt a) -
\Phi( -(q-\bar c a)/\sqrt a) \; \bigr\rbrace \; \biggr],
\label{psi1def}\end{aligned}$$ where $\bar c \equiv 1 + c$. Then: $$\begin{aligned}
A_0 &=& \frac{\nu_\alpha+ (1-e^{-\eta a}) \nu_a}{\nu-1},
\label{A0_ans}
\\
A_-(q) &=& \frac{\psi_0(q)}{\nu -1},
\label{A-ans}
\\
A_+(q) &=& \frac{\psi_1(q)}{\nu -1}.
\label{A+ans}\end{aligned}$$
[Proof.]{} The process $Y_t \equiv X_t - \bar X_t$ is a diffusion taking values in the negative half-line, and reflecting from zero. The process $\bar X_t$ is its local time at zero, and the path of $Y$ can be decomposed into a Poisson process of excursions, as Itô [@itopp] explained. For a more extensive discussion of excursion theory, in particular, the notion of marked excursion processes, see [@GTTE] or Chapter VI.8 of [@RW2].
The process $X$ runs until $\tau_0 \wedge \alpha$, which happens in the first excursion of $Y$ which [*either*]{} lasts at least $a$, [*or*]{} contains an $\eta$-mark. Let $n$ denote the excursion measure, $\zeta$ denote the lifetime of an excursion, $P^c$ denote the law of $X_t = W_t +ct$, and $H_b$ denote the first time $X$ hits $b$.[ The excursion law can be characterized as the limit as $\varepsilon \downarrow 0$ of the (rescaled) law of $X$ started at $ -\varepsilon <0$ run until it hits zero; see VI.50.20 in [@RW2]. The rescaling required is to multiply by $n (f: \inf f < -\epsilon )$, and it is known that $ n (f: \inf f < -\epsilon )\sim \epsilon^{-1}$ (see VI.51.2 in [@RW2]). We therefore have]{} $$\begin{aligned}
n(\zeta>a) &=& \lim_{\varepsilon\downarrow 0}\varepsilon^{-1}P^c[\; H_0 >a \;|\; X_0 = -\varepsilon] \nonumber
\\
&=&\lim_{\varepsilon\downarrow 0}\varepsilon^{-1}P^c[\; \bar X_a < \varepsilon \;|\; X_0 = 0] \nonumber
\\
&=& \lim_{\varepsilon\downarrow 0}\varepsilon^{-1}
\biggl\lbrace\;
\Phi\biggl( \frac{\varepsilon-ca}{\sqrt{a}} \biggr)
- e^{2c\varepsilon} \Phi\biggl( \frac{-\varepsilon-ca}{\sqrt{a}} \biggr)
\;\biggr\rbrace
\label{eq29}
\\
&=& \frac{2}{\sqrt{a}} \; \varphi(c\sqrt{a}) - 2c\, \Phi(-c\sqrt a),
\label{eq30}
\\
&\equiv & \nu_a, \nonumber\end{aligned}$$where is a standard result on Brownian motion; see, for example, [@BS]. By similar reasoning, $$\begin{aligned}
n(\alpha < \zeta < a) &=&\lim_{\varepsilon\downarrow 0}\varepsilon^{-1}
\int_0^a P^c[ H_0\in ds | X_0 = -\varepsilon] \; (1-e^{-\eta s})
\nonumber\\
&=& \lim_{\varepsilon\downarrow 0}\varepsilon^{-1}
\int_0^a \frac{\varepsilon e^{-\varepsilon^2/2s}}{\sqrt{2\pi s^3}}\;
e^{c\varepsilon- c^2 s/2} (1-e^{-\eta s}) \; ds
\nonumber\\
&=& \int_0^a \frac{ e^{-c^2 s/2}}{\sqrt{2\pi s^3}}\;
(1-e^{-\eta s}) \; ds
\nonumber\\
&=& 2{\beta}(\Phi({\beta}\sqrt a) - { {\scriptstyle \frac{1}{2} }\, }) -2c(\Phi(c\sqrt a) - { {\scriptstyle \frac{1}{2} }\, })
\nonumber\\
&&\qquad\qquad \qquad\quad + \frac{2}{\sqrt a} \bigl\lbrace\; \varphi({\beta}\sqrt a) - \varphi(c\sqrt a)
\;\bigr\rbrace \label{eq31}
\\
&\equiv & \nu_\alpha \nonumber\end{aligned}$$ where we recall that ${\beta}\equiv \sqrt{c^2 + 2 \eta}$. The rate of excursions which [*either*]{} last at least $a$, [*or*]{} contain an $\eta$-mark is $$n(\zeta >a, \; \hbox{\rm or} \; \alpha < \zeta)
= n(\alpha < \zeta<a) + n(\zeta >a)
\label{eq33}$$ and this is simply the sum of the two expressions and , which is therefore known explicitly. In fact, a few calculations confirm that it is the expression $\nu$ defined at . Immediately from Itô excursion theory:
- $\bar X_{\tau_0 \wedge \alpha} \sim \exp( \;
\nu
\;)$;
- $
P[ \; \alpha < \tau_0 \;]
= \{ \;\nu_\alpha+ (1-e^{-\eta a}) \nu_a\; \}/\nu
$;
- $P[\; \tau_0 < \alpha \;] = \nu_a\, e^{-\eta a} /\nu$;
and from this the expression for $A_0$ follows.
To deal with $A_\pm$, we need to find the measure of excursions which get to time $a$ without killing, and which are in $dy$ at time $a$. We use the reflection principle and the Cameron-Martin-Girsanov theorem to derive $$\begin{aligned}
g(y)dy &\equiv& n(a < \zeta\wedge\alpha, Y_a \in dy)
\label{gdef}
\\
&=& \lim_{\varepsilon\downarrow 0} \varepsilon^{-1} P^c[ \;
H_0\wedge \alpha >a, Y_a \in dy \; \vert \; Y_0 = -\varepsilon
\;] \nonumber
\\
&=&\lim_{\varepsilon\downarrow 0} \varepsilon^{-1} e^{c(y+\varepsilon)-c^2a/2
-\eta a}
\{ \; \varphi( (y + \varepsilon)/\sqrt{a}) - \varphi( (y -\varepsilon)/\sqrt{a})
\;\} dy /\sqrt{a}
\nonumber
\\
&=& \frac{-2y}{a^{3/2}} \; \varphi((y-ca)/\sqrt a)\; e^{-\eta a} \; dy.
\label{g_ans}\end{aligned}$$ Straightforward calculations lead us to $$\begin{aligned}
\int_{-\infty}^q g(y) \; dy
&=& e^{-\eta a} \biggl\lbrace\;
\frac{2}{\sqrt a} \; \varphi( (q-ca)/\sqrt a) - 2c\, \Phi( (q-ca)/\sqrt a)
\;\biggr\rbrace,
\\
&\equiv & \psi_0(q), \\
\int_q^0 e^y g(y) \; dy
&=& e^{(c+{ {\scriptstyle \frac{1}{2} }\, }-\eta)a} \biggl[ \; \frac{2}{\sqrt a}\;\bigl\lbrace
\; \varphi(\bar c \sqrt a)
-\varphi( (q-\bar c a)/\sqrt a) \;\bigr\rbrace
\nonumber \\
&& \qquad\qquad\qquad 2\bar c\bigl\lbrace \; \Phi(\bar c \sqrt a) -
\Phi( -(q-\bar c a)/\sqrt a) \; \bigr\rbrace \; \biggr],
\\
& \equiv & \psi_1(q).\end{aligned}$$ Hence $P( Y_{\tau_0} < q \;\vert \;\tau_0 < \alpha ) = \psi_0(q)/\psi_0(0)$. We therefore have $$\begin{aligned}
A_-(q) &=& E[\; \exp(\bar X_{\tau_0}) : \tau_0 < \alpha, X_{\tau_0} - \bar X_{\tau_0} < q \; ]
\\
&=& \frac{\nu}{\nu-1} \; P[\;\tau_0 < \alpha, X_{\tau_0} - \bar X_{\tau_0} < q \; ]
\\
&=& \frac{\nu}{\nu-1}\cdot \frac{\nu_ae^{-\eta a}}{\nu}\cdot \frac{\psi_0(q)}{\psi_0(0)}
\\
&=& \frac{\psi_0(q)}{\nu -1}\end{aligned}$$ after some simplifications. This is the form of $A_-$ claimed at .
Finally, we deal with $A_+(q)$. Observe that $$E[\; \exp( \, Y_{\tau_0} \, )I_{\{ Y_{\tau_0}>q\}}\; \vert \; \tau_0 < \alpha\;]
=\frac{\psi_1(q)}{n(a < \zeta\wedge\alpha)} \; .$$ Hence we have $$\begin{aligned}
A_+(q) &=& E[ \; \exp(X_{\tau_0}): \tau_0 < \alpha, X_{\tau_0} - \bar X_{\tau_0} > q \; ]
\\
&=& E[\; \exp(\bar X_{\tau_0}) \;] \cdot
E[ \; \exp(Y_{\tau_0}): \tau_0 < \alpha, Y_{\tau_0} > q \; ]
\\
&=& \frac{\nu}{\nu-1}\cdot P(\tau_0 < \alpha ) \cdot \frac{\psi_1(q)}{n(a < \zeta\wedge\alpha)}
\\
&=& \frac{\psi_1(q)}{\nu -1}\end{aligned}$$ after some simplifications. This is the form claimed at .
0.1 in $\square$
Now we can draw all the pieces together. Proposition \[prop2\] was a step on the way to evaluating $K$, the value of the proposed strategy; we found $K$ expressed in terms of $A_0$, $A_-(q)$ and $A_+(q)$ at , and it remains just to write the answer cleanly.
\[thm1\]The value of the rule $R(q)$ is given by $$K = \frac{A_0 + A_-(q)}{1-A_+(q)} = \frac{\nu_\alpha +(1-e^{-\eta a}) \nu_a + \psi_0(q)}{\nu-1-\psi_1(q)} \; .
\label{K_ans}$$ The denominator $\nu - 1 - \psi_1(q)$ is always positive.
[Proof.]{} The expression is a trivial arrangement of , so all that remains is to deal with the final assertion. Using the fact that $c = -{ {\scriptstyle \frac{1}{2} }\, }$, we see that $$\psi_1(-\infty) = \;
{\uparrow\lim}_{q \downarrow -\infty} \psi_1(q)
= e^{-\eta a}\biggl\lbrace \; \frac{2}{\sqrt a}\; \varphi(\sqrt{a}/2) -\Phi(-\sqrt{a}/2)\;
\biggr\rbrace.$$ Therefore $$\begin{aligned}
\nu - 1- \psi_1(q) &>& \nu - 1 - \psi_1(-\infty)
\\
&=& \nu_a + \nu_\alpha - 1 - \psi_1(-\infty)
\\
&>& \nu_a - 1 - \psi_1(-\infty)
\\
&=& \frac{2}{\sqrt a}\; \varphi(\sqrt{a}/2) + \Phi(\sqrt{a}/2)- 1 - \psi_1(-\infty)
\\
&=& (1-e^{-\eta a}) \biggl\lbrace \; \frac{2}{\sqrt a}\; \varphi(\sqrt{a}/2) -\Phi(-\sqrt{a}/2)\;
\biggr\rbrace
\\
&>& 0.\end{aligned}$$
0.1 in $\square$
Using the stopping rule $R(q)$. {#using-the-stopping-rule-rq. .unnumbered}
-------------------------------
[**Rule 1.**]{} How is the stopping rule $R(q)$ analysed in the preceding section relevant to the original problem? Holding $\eta$ fixed, there will be an optimal $q^* = q^*(\eta)$ which maximizes the value $K$ given by . Now recall how the stopping rule works; we let the process run until the stopping time $$\tau_0 = \inf \{\; t: \bar X_{[t-a,t]} = X_{t-a} \; \} =
\inf \{\; t: \bar X_t = X_{t-a} \; \},$$ and at that moment we stop if and only if $Y_{\tau_0} \equiv X_{\tau_0}
-\bar X_{\tau_0} < q$, else we forget and continue. What value of $q$ do we use? A natural choice is to take $$q = q^*(\; (T- \tau_0)^{-1}\;).
\label{q_choice}$$ This is because at time $\tau_0$ there is time $(T-\tau_0)$ still to go, and an exponential random variable with rate $(T-\tau_0)^{-1}$ has this as its mean. The result of using this rule is shown in the left-hand panel of Figure \[fig:sim\]. The dots (evaluated by simulating 50,000 runs of the rule) are visibly close to (but below) the lower bound we obtained by the Barraquand-Martineau technique of [@bobs].\
[**Rule 2.**]{} Can we do better than this? Indeed we can. Firstly, we observe that $$K(q^*(\eta)) = \exp( -q^*(\eta)).
\label{Kq*}$$ This is because when we arrive at time $\tau_0$, we have to choose between stopping and receiving $S_{\tau_0-a}$, or continuing and receiving $K(q^*(\eta))\cdot S_{\tau_0}$ in expectation. Optimal behaviour requires us to continue if and only if $$K(q^*(\eta)) \cdot S_{\tau_0} > S_{\tau_0-a}.$$ On the other hand, the rule $R(q^*(\eta))$ requires us to continue if and only if $$S_{\tau_0} > S_{\tau_0-a} \cdot \exp( q^*(\eta)).$$ Since $q^*(\eta)$ was chosen optimally, must therefore hold.\
Secondly, if $a = T$, then the value will just be the expectation of the overall maximum, $E \exp( \bar X_T )$. Routine but tedious calculations give $$E[\; \exp( \bar X_a) \;] = \lambda(a) \equiv {\left(2 + \dfrac{a}{2}\right)}\Phi{\left(\dfrac{\sqrt{a}}{2}\right)} + \sqrt{a} \varphi {\left(\dfrac{a}{4}\right)}. \label{smax.distr}$$ If we had arrived at time $\tau_0$ and it turns out that $\tau_0 = T-a$, then by forgetting and continuing we will actually receive expected reward $\lambda(a) S_{\tau_0}$, whereas if we used Rule 1 we would think we were going to receive $K(q^*(a^{-1}))\cdot S_{\tau_0} = e^{-q^*(a^{-1})} \, S_{\tau_0}$. This suggests that we modify Rule 1, replacing by $$q = q^*(\;(T-\tau_0)^{-1}\;) - q^*(a^{-1}) - \log\lambda(a).
\label{q_choice_2}$$ By making this modification, the stop/continue decision we make in the event that $T- \tau_0 = a$ will be exactly correct.
Of course, the argument just presented is only a rough heuristic, but if we look at the right-hand panel in Figure \[fig:sim\] we see the results of using Rule 2. The dots are now essentially coincident with the lower bounds, which is very encouraging, and argues for the use of Rule 2 rather than Rule 1. This rule is something which can be clearly motivated and precisely specified, in contrast to the randomly-generated rules which come from the Barraquand-Martineau technique in Section \[S2\]. One could continue to search for other explicit rules which do even better, but such a study is beyond the scope of this paper.
![The simulation results with $h = 1/2500$ and $N_T = 250$.[]{data-label="fig:sim"}](rule1.png "fig:"){width="0.49\linewidth"} ![The simulation results with $h = 1/2500$ and $N_T = 250$.[]{data-label="fig:sim"}](rule2.png "fig:"){width="0.49\linewidth"}
$a/h$ Rule 1 Rule 2 Lower bound
------- -------- -------- -------------
1 1.055 1.054 1.054
2 1.073 1.073 1.074
3 1.086 1.087 1.088
4 1.096 1.098 1.100
5 1.104 1.107 1.109
6 1.112 1.116 1.117
7 1.119 1.121 1.123
8 1.126 1.128 1.129
9 1.131 1.134 1.135
10 1.136 1.139 1.140
11 1.139 1.143 1.144
12 1.145 1.148 1.149
13 1.148 1.153 1.152
14 1.152 1.155 1.156
15 1.155 1.160 1.159
16 1.158 1.163 1.163
17 1.160 1.165 1.166
18 1.164 1.168 1.168
19 1.168 1.172 1.171
20 1.170 1.174 1.174
: $E[Z_\tau]$ estimates from simulation of Rules 1 and 2 with $h = 1/2500$ and $N_T = 250$, averaged over $50,000$ sample paths. Standard errors are 0.001 (to one significant figure) in all cases. []{data-label="table:rq"}
Supplemental Materials
======================
The code for all simulations can be found on <http://stat.wharton.upenn.edu/~ernstp/blb.cpp>. The pseudocode is available on <http://stat.wharton.upenn.edu/~ernstp/Bermuda_code.pdf>
[^1]: Later on, we shall derive expressions for various probabilities and expectations associated with $X$, and it turns out to be notationally cleaner to work with a general drift, which is why we write $X$ as .
[^2]: ... using the convention that $S_u = 1$ for $u<0$, and $S_u = S_T$ for $u \geq T$ ...
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Learning a set of tasks over time, also known as continual learning (CL), is one of the most challenging problems in artificial intelligence. While recent approaches achieve some degree of CL in deep neural networks, they either (1) grow the network parameters linearly with the number of tasks, (2) require storing training data from previous tasks, or (3) restrict the network’s ability to learn new tasks. To address these issues, we propose a novel framework, Self-Net, that uses an autoencoder to learn a set of low-dimensional representations of the weights learned for different tasks. We demonstrate that these low-dimensional vectors can then be used to generate high-fidelity recollections of the original weights. Self-Net can incorporate new tasks over time with little retraining and with minimal loss in performance for older tasks. Our system does not require storing prior training data and its parameters grow only logarithmically with the number of tasks. We show that our technique outperforms current state-of-the-art approaches on numerous datasets—including continual versions of MNIST, CIFAR10, CIFAR100, and Atari—and we demonstrate that our method can achieve over 10X storage compression in a continual fashion. To the best of our knowledge, we are the first to use autoencoders to sequentially encode sets of network weights to enable continual learning.'
author:
- 'Blake Camp[^1]'
- 'Jaya Krishna Mandivarapu$^{*}$'
- Rolando Estrada
bibliography:
- 'bibliography.bib'
title: 'Self-Net: Lifelong Learning via Continual Self-Modeling'
---
Introduction {#sec:introduction}
============
Lifelong or continual learning (CL) is one of the most challenging problems in machine learning, and it remains a significant hurdle in the quest for artificial general intelligence (AGI) [@investigation_of_catastrophic_forgetting; @measuring_catastrophic_forgetting]. In this paradigm, a single system must learn to solve new tasks without forgetting previously learned information. Different tasks might require different data (e.g., images vs. text) or they might process the same data in different ways (e.g., classifying an object in an image vs. segmenting it). Crucially, in CL there is no point at which a system stops learning; it must always be able to update its representation of its problem domain(s).
![**Framework overview:** Our proposed system has a set of reusable *task-specific networks* (TN), a *Buffer* for storing the latest $m$ tasks, and a lifelong, *auto-encoder* (AE) for long-term storage. Given new tasks $\{t_{k+1}, ... ,t_{k+m}\}$, where $k$ is the number of tasks previously encountered, we first train $m$ task-networks to learn optimal parameters $\{\theta_{k+1},..., \theta_{k+m}\}$ for these tasks. These networks are temporarily stored in the Buffer. When the Buffer fills up, we incorporate the new networks into our long-term representation by retraining the AE on both its approximations of previously learned networks and the new batch of networks. When an old network is needed (e.g. when a task is revisited), we reconstruct its weights and load them onto the corresponding TN (solid arrow). Even when the latent representation $e_i$ is asymptotically smaller than $\theta_i$, the reconstructed network closely approximates the performance of the original.[]{data-label="fig:overview"}](Pictures/flat_Self-Net_w_buffer_v4.png){width="90.00000%"}
CL is particularly challenging for deep neural networks because they are trained end-to-end. In standard deep learning we tune all of the network’s parameters based on training data, usually via backpropagation [@Rumelhart1986]. While this paradigm has proven highly successful for individual tasks, it is not suitable for continual learning because it overwrites existing weights (a phenomenon evocatively dubbed *catastrophic forgetting* [@robins1995catastrophic]). For example, if we first train a network on task A and then on task B, the latter training will modify the weights learned for A, thus likely reducing the network’s performance on this task.
There are several approaches that can achieve some degree of continual learning in deep networks. However, existing methods suffer from at least one of three limitations: they either **(1)** restrict the network’s ability to learn new tasks by penalizing changes to existing weights [@overcoming_catastrophic_forgetting; @synaptic_intelligence; @progress_and_compress; @variational_continual_learning]; **(2)** expand the model size linearly as the number of tasks grows [@progressive_networks; @LwF] (or dynamically define task-specific sub-networks [@dynamically_expandable_networks; @context_dependent_gating], which is asymptotically equivalent); or **(3)** retrain on old tasks. In the latter, we either **(a)** store some of the old training data directly [@playing_atari_deep_reinforcement; @iCarl; @variational_continual_learning], thus increasing storage requirements linearly (and at a faster rate than increasing the network, since data tends to be higher dimensional), or **(b)** they use compressed data [@lifelong_generative_modeling; @fearNet; @deep_generative_replay; @encoder_based_lifelong_learning], which complicates training.
In this paper, we propose a novel approach, Self-Net, that overcomes the aforementioned limitations by decoupling how it *learns* a new task from how it *stores* it. Figure \[fig:overview\] provides an overview of our proposed framework. Our system grows only *logarithmically* with the number of tasks, while retaining excellent performance across all learned tasks. Our approach is loosely inspired by the role that the hippocampus is purported to play in memory consolation [@hipp_memory_index_theory]. As noted in [@interplay_hippo_prefrontal_cortex], during learning the brain forms an initial neural representation in cortical regions; the hippocampus then consolidates this representation into a form that is optimized for storage and retrieval. These complementary biological mechanisms enable continual learning by efficiently consolidating knowledge and prior experiences. In this spirit, we propose a system that consists of three components: **(1)** a set of reusable *task-networks* (TNs), **(2)** a *Buffer* in which we store the latest $m$ learned weights exactly, and **(3)** a lifelong *autoencoder* (AE) with which we can encode an arbitrary number of older tasks. The AE learns a low-dimensional representation for each of the high-dimensional parameter vectors that define weights of the TNs. Thus, our system *self-models* its own behavior, allowing it to approximate previously learned parameters instead of storing them directly. In short, when our system learns a new task, it firsts trains an appropriate TN using standard deep learning and then stores a copy of the weights in the Buffer. When the Buffer fill up, the AE learns a set of compact, latent vectors for the weights in the Buffer. The Self-Net then discards the original weights, freeing up the Buffer to store new tasks. If our system needs to solve a previously learned task, it generates an approximation of the original weights by feeding the corresponding latent vector through the AE and then loading the reconstructed weights onto a TN.
Our approach leverages the flexibility of conventional neural networks while avoiding their inability to remember old tasks. More specifically, a TN is free to modify its parameters as needed to learn a new task, since previously learned weights are encoded by the AE. Our AE doesn’t simply memorize old weights; our experiments show that an AE can encode a very large number of networks while retaining excellent performance on all tasks (Section \[sec:cl\_large\_tasks\]). Our framework can even incorporate fine-tuning by initializing a TN with the weights from a previous, related task. Below, we first overview existing CL methods for deep network and then detail our approach in Section \[sec:methodology\].
Prior work {#sec:priorWork}
==========
Several methods have recently emerged for continual learning in deep networks, although, as noted above, existing approaches either **(1)** restrict new learning, **(2)** grow the number of parameters linearly, or **(3)** require old training data. Notable examples of the first type include Elastic Weight Consolidation (EWC) [@overcoming_catastrophic_forgetting], Synaptic Intelligence [@synaptic_intelligence], Variational Continual Learning [@variational_continual_learning] (which also reuses old data), and Progress & Compress [@progress_and_compress]. These approaches reuse the same network for each new task, but they apply a regularization method to restrict changes in weights over time. Hence, they typically use constant space[^2]. EWC, in particular, uses the diagonal of the Fisher information matrix between the weights learned for the new task vs. the old tasks. Like our proposed approach, Progress & Compress also uses both a task-network and a long-term storage network; however, it uses EWC to update the weights of the latter, so it has very similar performance to this first method.
The second category includes Progressive Networks [@progressive_networks], Dynamically Expandable Networks [@dynamically_expandable_networks], and Context-Dependent Gating [@context_dependent_gating]. These methods achieve excellent performance, but they grow the network linearly with the number of tasks, which is asymptotically the same as using independent networks. Thus, they cannot scale to large numbers of tasks. Their advantage is in facilitating *transfer learning*, i.e., using previous learning to speed up new learning.
Finally, some methods store a fraction of the old training data and use it to retrain the network on previously learned tasks. Key approaches include Experience Replay [@playing_atari_deep_reinforcement] iCarl [@iCarl], Variational Continual Learning [@variational_continual_learning], and Learning without Forgetting [@LwF]. Unfortunately, this paradigm combines the drawbacks of the previous two. First, most of these methods use a single network, so they cannot continually learn a large number of tasks well. Second, their storage requirements grow linearly in the number of tasks because they have to store old training data. Moreover, data usually takes up orders of magnitude more space than the network itself because a trained network is effectively a compressed representation of the training set [@2016arXiv160605908D]. A few methods reduce this storage requirement by storing a compressed representation of the data. Methods of this type include Lifelong Generative Modeling [@lifelong_generative_modeling], FearNet [@fearNet], and Deep Generative Replay [@deep_generative_replay]. Our proposed approach uses a similar idea but instead stores the *networks themselves*, rather than the data. Our scheme has two advantages over compressing the data. First, networks are much smaller, so we can encode them more quickly, using less space. Second, by reconstructing the networks directly, we do not need to retrain task-networks on data from previous tasks.
Methodology {#sec:methodology}
===========
Figure \[fig:overview\] provides a high-level overview of our proposed approach. Our Self-Net system uses a set of reusable task-networks (TNs), a Buffer for storing newly learned tasks, and a lifelong autoencoder (AE) for storing older tasks. In addition, we store an $O(\log{(n)})$ latent vector for each task. Each TN is just a standard neural network, which can learn regression, classification, or reinforcement learning tasks (or some combination of the three). For ease of discussion, we will focus on the case where there is a single TN and the Buffer can hold only one network; the extension to multiple networks and larger Buffers is trivial. The AE is made up of an *encoder* that compresses an input vector into a lower-dimensional, latent vector $e$ and a *decoder* that maps $e$ back to the higher-dimensional space. Our system can produce high-fidelity recollections of the learned weights, despite this intermediate compression. In our experiments, we used a contractive autoencoder (CAE) [@CAE] due to its ability to quickly incorporate new values into its latent space.
In CL, we must learn $k$ different tasks sequentially. To learn these tasks independently, one would need to train and save $k$ networks, with $O(n)$ parameters each, for a total of $O(kn)$ space. In contrast, we propose using our AE to encode each of these $k$ networks as an $O(\log(n))$ latent vector. Thus, our method uses only $O(n + k\log{(n)})$ space, where the $O(n)$ term accounts for the TNs and the fixed-size Buffer. Despite this compression, our experiments show that we can obtain a high-quality approximation of previously learned weights, even when the number of tasks exceeds the number of parameters in the AE (Sec. \[sec:cl\_large\_tasks\]). Below, we first describe how to encode a single task-network before discussing how to encode multiple tasks in a continual fashion.
Single-network encoding
-----------------------
Let $t$ be a task (e.g., recognizing faces) and let $\theta$ be the $O(n)$-dimensional vector of parameters of a network trained to solve $t$. That is, using a task-network with parameters $\theta$, we can achieve performance $p$ on $t$ (e.g., a classification accuracy of 95%). Now, let $\hat{\theta}$ be the approximate reconstruction of $\theta$ by our autoencoder and let $\hat{p}$ be the performance that we obtain by using these reconstructed weights for task $t$. Our goal is to minimize any performance loss w.r.t. the original weights. If the performance of the reconstructed weights is acceptable, then we can simply store the $O(\log{(n)})$ latent vector $e$, instead of the $O(n)$ original vector $\theta$.
If we had access to the test data for $t$, we could assess this difference in performance directly and train our AE until we achieve an acceptable margin $\epsilon$: $$\begin{aligned}
p - \hat{p} \leq \epsilon.\end{aligned}$$ For example, for a classification task we could stop training our AE if the drop in accuracy is less than $1\%$.
In a continual learning setting, though, the above scheme requires storing validation data for each old task. Instead, we measure a distance between the original and reconstructed weights and stop training when we achieve a suitably close approximation. Empirically, we determined that the cosine similarity, $$\begin{aligned}
\label{cos_sim}
cos(\theta, \hat{\theta}) = \dfrac{\theta \cdot \hat{\theta}}{\|\theta\| \|\hat{\theta}\|} = \dfrac{\sum_{i=1}^{n}\theta_i \hat{\theta}_i}{\sqrt{\sum_{i=1}^{n}\vphantom{\hat{\theta}_i^2} \theta_i^2 } \sqrt{\sum_{i=1}^{n}\hat{\theta}_i^2}},\end{aligned}$$ is an excellent proxy for a network’s performance. Unlike the mean-squared error, this distance metric is scale-invariant, so it is equally suitable for weights of different scales. As detailed in Section \[sec:experiments\], a cosine similarity of 0.997 or higher yielded excellent performance for a wide variety of tasks and architectures.
In addition, one can improve the efficacy with which the AE learns a new task by encouraging the parameters of all task-networks to remain in the same general neighborhood. This can be accomplished by fine-tuning all networks from a common source and penalizing large deviations from this initial configuration with a regularization term. Formally, let $\theta^{*}$ be the source parameters, ideally optimized for some highly-related task. Without loss of generality, we can define the loss function of task-network $\theta_{i}$ for task $t_{i}$ as: $$\begin{aligned}
\label{eqn:regularized_task_loss}
TaskNetLoss_{i} = TaskLoss + \lambda MSE(\theta^{*}, \theta_{i})\end{aligned}$$ where $\lambda$ is the regularization coefficient determining the importance of remaining close to the source parameters vs. optimizing for the current task. By encouraging the parameters for all task-networks to remain close to one another, we make it easier for the AE to learn a low-dimensional representation of the original space.
Continual encoding {#sec:continual-encoding}
------------------
We will now detail now to use our Self-Net to encode a sequence of trained networks in a continual fashion. Let $m$ be the size of the Buffer, and let $k$ be the number of tasks which have been previously encountered. As noted above, we train each of these $m$ task-networks using conventional backpropagation, one per task. Now, assume that our AE has already learned to encode the first $k$ task-networks. We will now show how to encode the most recent batch of $m$ task-networks corresponding to tasks $\{t_{k+1}, ... ,t_{k+m}\}$ into compressed representations $\{e_{k+1}, ... ,e_{k+m}\}$ while still remembering all previously trained networks.
Let ***T*** be the set of all Tasks encountered during the lifetime of the system Let $m$ be the size of the **Buffer** ***E*** = \[\] initialize **AE** Set cosine\_threshold
- Intitialize **TN**
- Train the **TN** for curr\_task until optimized - **Buffer**.append(**TN**)
***R*** = \[\]
r = **AE**.Decoder(encoded-network) ***R***.append(r)
flat\_network = extract and flatten parameters from network ***R***.append(flat\_network)
average\_cosine\_similarity = $0.0$ ***E*** = \[\] calculate AE\_loss using Equation . back-propagate **AE** w.r.t **r** update average\_cosine\_similarity using cos(**r**,**AE**(**r**)) ***E***\[r\_idx\] = **AE**.Encoder(**r**) empty **Buffer**
\[alg:csm\_algo\]
Let $E$ be the set of latent vectors for the first $k$ networks. In order to integrate $m$ new networks $\{\theta_{k+1}, ..., \theta_{k+m}\}$ into the latent space, we first recollect all previously trained networks by feeding each $e \in E$ as input to the decoder of the AE. We thus generate a set $R$ of recollections, or approximations, of the original networks (see Fig. \[fig:overview\]). We then append each network $\theta_{i}$ in the Buffer to $R$ and retrain the AE on all $k+m$ networks until it can reconstruct them, i.e., until the average of their respective cosine similarities is above the predefined threshold. Algorithm \[alg:csm\_algo\] summarizes our continual learning strategy.
As we show in our experiments, our compressed network representations still achieve excellent performance compared to the original parameters. Since each $\hat{\theta} \in R$ is simply a vector of network parameters, it can easily be loaded back onto a task-network with the correct architecture. This allows us to discard the original networks and store $k$ networks using only $O(k\log{(n)})$ space. In addition, our framework can efficiently encode many different types and sizes of networks in a continual fashion. In particular, we can encode a network of arbitrary size $q$ using a constant-size AE (that takes inputs of size $n$) by splitting the input network into $r$ subvectors[^3], such that ($n = q/r$). As we verify in Section \[sec:experiments\], we can effectively reconstruct a large network from its subvectors and still achieve a suitable performance threshold.
As Fig. \[fig:robustness\_combined\] illustrates, we empirically found a strong correlation between a reconstructed network’s performance and its cosine similarity w.r.t. to the original network. Intuitively, this implies that vectors of network parameters that have a cosine similarity approaching 1 will exhibit near-identical performance on the underlying task. Thus, the cosine similarity can be used as a terminating condition during retraining of the AE. That is, there exists a cosine similarity threshold above which the performance of the reconstructed network can be expected to be sufficiently similar to that of the original. In practice, we found a threshold of .997 to be sufficient for most experiments. Below, we offer empirical results which demonstrate the efficacy and flexibility of our approach.
Experimental Results {#sec:experiments}
====================
In order to evaluate the continual-learning performance of Self-Net, we carried out a range of experiments on a variety of datasets, in both supervised and reinforcement-learning (RL) settings. We first performed a robustness analysis to establish the degree to which an approximation of a network can deviate from the original and still retain comparable performance on the underlying task (Section \[sec:robust\]). Then, we evaluated the performance of our approach on the following continual-learning datasets: Permuted MNIST [@overcoming_catastrophic_forgetting], Split MNIST [@variational_continual_learning], Split CIFAR-10 [@synaptic_intelligence], Split CIFAR-100 [@synaptic_intelligence], and successive Atari games [@playing_atari_deep_reinforcement] (we describe each dataset below). As our experiments show, Self-Net can effectively encode each of these different types of networks in sequential fashion, effectively achieving continual learning and outperforming several competing techniques. Finally, we also analyzed our system’s performance under three additional scenarios: **(1)** very large numbers of tasks, **(2)** different sizes of AEs, and **(3)** different task-network architectures. We detail each experiment below.
Robustness analysis {#sec:robust}
-------------------
Our approach relies upon *approximations* of previously learned networks, and we assume no access to validation data for previously learned tasks. Thus, we require a method for estimating the performance of a reconstructed network which does not rely upon explicit testing on a validation set.
Figure \[fig:robustness\_combined\] shows the relationship between performance and deviations from the original parameters as measured by cosine similarity, for three datasets. There is a clear correlation between the amount of parameter dissimilarity and the probability of a decrease in performance. That is, given an approximate network that deviates from the original by some amount, the potential still exists that such a network will retain comparable performance. However, as the degree of deviation increases, the probability that the performance remains high falls steadily. Thus, in order to assume, with reasonable confidence, that the performance of a reconstructed network will be sufficiently high, the AE must minimize the degree of deviation as much as possible.
Empirically, we established a cosine similarity threshold above which the probability of high task-performance stabilizes, as seen in Figure \[fig:robustness\_combined\]. This threshold can be used as a terminating condition during retraining of the AE, and it allows the performance of a reconstructed network to be approximated *without* access to any validation data. In our experiments, a common threshold yields good performance across a variety of different types and sizes of networks.
Experiments on CL datasets {#sec:cl_experiments}
--------------------------
**Permuted MNIST**: As an initial evaluation of Self-Net’s CL performance, we trained convolutional feed-forward neural networks with 21,840 parameters on successive tasks, each defined by distinct permutations of the MNIST dataset [@726791], for 10-digit classification. We used networks with 2 convolution layers (kernels of size 5x5, and stride 1x1), 1 hidden layer (320x50), and 1 output layer (50x10). Our CAE had three, fully connected layers with 21,840, 2000, and 20 parameters, resp. Thus, our latent vectors were of size 20. For this experiment, we used a Buffer of size 1. Each task network was encoded by our lifelong AE in sequential fashion, and the accuracies of all reconstructed networks were examined at the end of each learning stage (i.e., after learning a new task). Figure \[fig:combined\_comparisons\] (top) shows the mean performance after each stage. Our technique almost perfectly matched the performances achieved by independently trained networks, and it dramatically outperformed other state-of-the-art approaches including EWC [@overcoming_catastrophic_forgetting], Online EWC (the correction to EWC proposed in [@HuszarE2496]), and Progress & Compress [@progress_and_compress]. As a baseline, we also show the results for SGD (no regularization), L2-based regularization in which we compare new weights to all the previous weights, and Online L2, which only measures deviations from the weights learned in the previous iteration. Not only does our technique allow for superior knowledge retention, but it does not inhibit knowledge acquisition necessary for new tasks. The result is minimal degradation in performance as the number of tasks grow.
**Split MNIST:** We performed a similar continual learning task but with different binary classification objectives on subsets of the MNIST dataset (Split MNIST) [@variational_continual_learning]. Our task-networks, CAE, and Buffer size were the same as for Permuted MNIST (except that the outputs of the task-networks were binary, instead of 10 classes). Tasks were defined by tuples comprised of the positive and negative digit class(es), e.g., (\[pos={1}, neg={6,7,8,9}\], \[pos={6}, neg={1,2,3,4}\], etc.). Here, the training and test sets consisted of approximately 40% positive examples and 60% negative examples. In this domain, too, our technique dramatically outperformed competing approaches, as seen in Figure \[fig:combined\_comparisons\] (middle).
![CL performance comparisons with average test set accuracy on all observed tasks at each stage for **(top)** Permuted MNIST, **(middle)** Split MNIST, and **(bottom)** Split CIFAR-10.[]{data-label="fig:combined_comparisons"}](Pictures/combined_comparisons_compressed_v4.png){width=".9\linewidth"}
**Split CIFAR-10:** We then verified that our proposed approach could reconstruct larger, more sophisticated networks. Similar to the Split MNIST experiments above, we divided the CIFAR-10 dataset [@krizhevsky2009learning] into multiple training and test sets, yielding 10 binary classification tasks (one per class). We then trained a task-specific network on each class. Here, we used TNs having an architecture which consisted of 2 convolutional layers, followed by 3 fully connected hidden layers, and a final layer having 2 output units. In all, these task networks consisted of more than 60K parameters. Again, for this experiment we used a Buffer of size 1. Our CAE had three, fully connected layers with 20442, 1000, and 50 parameters, resp. As noted below, we split the 60K networks into three subvectors to encode them with our autoencoder. The individual task-networks achieved accuracies ranging from 78% to 84%, and a mean accuracy of approximate 81%. Importantly, we encoded these larger networks using almost the same CAE architecture as the one used in the MNIST experiments. This was achieved by *splitting* the 60K parameter vectors into three subvectors. As noted in Section \[sec:methodology\], by splitting a larger input vector into smaller subvectors, we can encode networks of arbitrary sizes. As seen in Figure \[fig:combined\_comparisons\] (bottom), the accuracies of the reconstructed CIFAR networks also nearly matched the performances of their original counterparts, while also outperforming all other techniques.
![CL performance comparisons with average test set accuracy on all observed tasks at each stage for CIFAR-100.[]{data-label="fig:mean_accuracy_CIFAR100"}](Pictures/mean_accuracy_CIFAR100_compressed_v3.png){width="0.9\linewidth"}
**Split CIFAR-100:** We applied the same learning approach for the CIFAR-100 dataset [@krizhevsky2009learning]. We split the dataset into 10 distinct batches comprised of 10 classes of images each. This resulted in 10 separate datasets, each designed for 10-class classification tasks. We used the same task-network architecture and Buffer size as in our CIFAR-10 experiments, modified slightly to accommodate a 10-class classification objective. The trained networks achieved accuracies ranging from 46% to 59%. We then encoded these networks using the same CAE architecture described in the previous experiments, again accounting for the input size discrepancy by splitting the task-networks into smaller subvectors. As seen in Figure \[fig:mean\_accuracy\_CIFAR100\], our technique almost perfectly matched the performances achieved by independently trained networks.
**Incremental Atari:** To evaluate the CL performance of Self-Net in the challenging context of reinforcement learning, we used the code available at [@a3c_repo] to implement a modified Async Advantage Actor-Critic (A3C) framework, originally introduced in [@a3c_paper], to attempt to learn successive Atari games while retaining good performance across all games. A3c simultaneously learns a policy and a value function for estimating expected future rewards. Specifically, the model we used was comprised of 4 convolutional layers (kernals of size 3x3, and strides of size 2x2), a GRU layer (800x256), and two ouput layers: an Actor (256xNum\_Actions), and Critic (256x1), resulting in a complex model architecture and over 800K parameters. Critically, this entire model can be flattened and encoded by the single AE in our Self-Net framework having three, fully connected layers with 76863, 2000, and 200 parameters, resp. For these experiments we also used a Buffer of size 1.
Similar to previous experiments, we trained our system on successive tasks, specifically the following Atari games: Boxing, Star Gunner, Kangaroo, Pong, and Space Invaders. Figure \[fig:cl\_atari\] shows the near-perfect retention of performance on each of the 5 games over the lifetime of the system. This was accomplished by training on each game only once, never revisiting the game for training purposes. The dashed, vertical lines demarcate the different stages of continual learning. That is, each stage indicates that a new network was trained for a new game, over 40M frames. Afterwards, the mean (dashed, horizontal black lines) and standard-deviation (solid, horizontal black lines) of the network’s performance were computed by allowing it to play the game, unrestricted, for 80 episodes. After each stage, the performances of all reconstructed networks were examined by re-playing each game with the appropriate reconstructed network. As Figure \[fig:cl\_atari\] shows, the cumulative means and SD’s of the reconstructed networks closely mimic those achieved by their original counterparts.
![**CL on five Atari games with Self-Net:** To evaluate the reconstruction score at each stage, we ran the reconstructed networks for 80 full game episodes. The cumulative mean score is nearly identical to the original TN at each stage.[]{data-label="fig:cl_atari"}](new_atari_compressed.png){width="0.9\linewidth"}
Performance and storage scalability {#sec:cl_large_tasks}
-----------------------------------
In CL, there is a trade-off between storage and performance. Using different networks for $k$ tasks yields optimal performance but uses $O(kn)$ space, while regularized methods such as Online EWC only require $O(n)$ space but suffer a steep drop in performance as the number of tasks grows. Our experiments on CL datasets show that our approach achieves much better performance retention than existing approaches by using slightly more space, $O(n + k\log{(n)})$. More precisely, we can quantify performance with respect to implicit storage compression. For example, by the tenth task, Online EWC [@HuszarE2496] has essentially performed 10x compression because it uses 1/10th of the overall storage required by ten different networks; however, its performance by this point is very poor. In contrast, our system achieves 10X compression when the size of the stored latent vectors grows to $10n$. In the following experiments, we verified that our method retains excellent performance even when reaching 10X compression, thus confirming that our AE is not simply memorizing previously learned weights.
![**10X Compression for Split-MNIST:** Orange lines denote the average accuracy achieved by individual networks, one per task. Green lines denote the average accuracy when training the AE to encode all networks as a single batch. Blue lines indicate the average accuracy obtained by Self-Net at each CL Stage. **Top:** 50 tasks with latent Vectors of size 5 and a Buffer of size 5. **Middle:** 100 tasks with latent vectors of size 10 and Buffer of size 10. The x-axis (top and middle) denotes the compression factor achieved at each learning stage. **Bottom:** the training epochs required by the 5-dimensional AE to incorporate new networks decreases rapidly over time.[]{data-label="fig:selfNet_cl_combined"}](Pictures/selfNet_cl_combined_v2.png){width=".78\linewidth"}
The top two plots of Fig. \[fig:selfNet\_cl\_combined\] show the mean performance for 50 and 100 Split-MNIST tasks, with latent vectors of size 5 and 10, resp. As before, the AE had 21432 input parameters. For comparison, we also plotted the original networks’ performance and the performance of the reconstructions when the AE learns all the tasks in a single batch. The line with dots represents the CL system, where each dot indicates the point where the AE had to encode a new set of $m$ networks because the Buffer had filled up. For these experiments, we used a Buffer size $m$ of 5 and 10, resp.; these values were chosen so that each new batch of networks yielded an integer compression ratio, e.g., after encoding 15 networks with a latent vector of size 5, the Self-Net achieved 3X compression. Here, we fine-tuned all networks from the mean of the initial set of $m$ networks and penalized deviations from this source vector (using $\lambda = 0.001$), as described in Section \[sec:methodology\]. This regularization allowed the AE to incorporate subsequent networks with very little additional training, as seen in stages 4-10 (bottom image of Fig. \[fig:selfNet\_cl\_combined\]).
For 10X compression, the Self-Net with a latent vector of size 5 retained $\sim$95.7% average performance across 50 Split-MNIST tasks, while the Self-Net with 10-dimensional latent vectors retained $\sim$95.2% across 100 tasks. This represents a relative change of only $\sim$3.3% compared to the original performance of $\sim$99%. In contrast, existing methods dropped to $\sim$50% performance for 10X compression on this dataset (Fig. \[fig:combined\_comparisons\]).
Splitting networks and using multiple architectures
---------------------------------------------------
Splitting larger networks into smaller sub-vectors allows us to use a smaller AE. As an additional analysis, we verified that the smaller AE can be trained in substantially less time than a larger one. Figure \[fig:split\_multi\_arch\_combined\] (left) shows the respective training rates of an AE with 20,000 input units (blue line)—trained to reconstruct 3 sub-vectors of length 20,000—compared to that of a larger one, with 61,000 input units (yellow line), trained on a 60K CIFAR-10 network. Clearly, using more inputs for a smaller AE enables us to more quickly encode larger networks. Finally, we also validated that the same AE can be used to encode trained networks of different sizes and architectures. Figure \[fig:split\_multi\_arch\_combined\] (right) shows that the same AE can simultaneously reconstruct 5 MNIST networks and 1 CIFAR network so that all approach their original baseline accuracies.
![**Additional analyses:** **Left:** the AE training efficiency is improved when large networks are split into smaller subvectors. **Right:** a single AE can encode networks of different architectures and sizes.[]{data-label="fig:split_multi_arch_combined"}](cifar10splitting_v3 "fig:"){width="0.49\linewidth"} ![**Additional analyses:** **Left:** the AE training efficiency is improved when large networks are split into smaller subvectors. **Right:** a single AE can encode networks of different architectures and sizes.[]{data-label="fig:split_multi_arch_combined"}](multi-Arch_v2 "fig:"){width="0.49\linewidth"}\
Conclusions and future work
===========================
In this paper, we introduced a scalable approach for multi-context continual learning that decouples learning a set of parameters from storing them for future use. Our proposed framework makes use of state-of-the-art autoencoders to facilitate lifelong learning via continual self-modeling. Our empirical results confirm that our method can efficiently acquire and retain knowledge in continual fashion, even for very large numbers of tasks. In future work, we plan to improve the efficiency with which the autoencoder can continually model vast numbers of task networks. Furthermore, we will explore how to use the latent space to extrapolate to new tasks based on existing learned tasks with little or no training data. We also intend to compress the latent space even further (e.g., using only $\log{(k)}$ latent vectors for $k$ tasks). Promising approaches include clustering the latent vectors into sets of closely related tasks and using sparse latent representations. Finally, we will also investigate how to infer the current task automatically, without a task label.
[^1]: Both authors contributed equally.
[^2]: Although, as noted in [@HuszarE2496], standard EWC stores an $O(n)$ set of Fisher weights for each task, so it actually grows linearly. The modified version proposed in [@HuszarE2496] does use constant space.
[^3]: We pad with zeros whenever $q$ and $n$ are not multiples of each other.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider D-term hybrid inflation in the framework of superconformal supergravity. In part of the parameter space, inflation continues for subcritical inflaton field value. Consequently, a new type of inflation emerges, which gives predictions for the scalar spectral index and the tensor-to-scalar ratio that are consistent with the Planck 2015 results. The potential in the subcritical regime is found to have a similar structure to one in the simplest class of superconformal $\alpha$ attractors.'
author:
- Koji Ishiwata
title: Superconformal Subcritical Hybrid Inflation
---
I. introduction
===============
The observations of the cosmic microwave background (CMB) strongly support inflation as the paradigm of early universe. To discover the nature of inflation, intensive analysis of the CMB has been performed. The latest results by the Planck collaboration [@Ade:2015lrj; @Ade:2015xua] provide the bounds on the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ of the primordial density fluctuations, $$\begin{aligned}
&n_s=0.9655\pm0.0062\,\,
(68\%\,{\rm CL}) \,,
\nonumber \\
&r<0.10\,\, (95\%\,{\rm CL})\,.
\label{eq:nsr_obs}\end{aligned}$$ In fact, some inflation models, such as canonical chaotic inflation [@Linde:1983gd] and hybrid inflation [@Linde:1993cn], are already disfavored due to the bounds. Although they are not supported by the current observations, the models are simple and still attractive in theoretical point of view.
Recently Refs.[@Buchmuller:2012ex; @Buchmuller:2013zfa] studied the hybrid inflation in the framework of superconformal supergravity [@Einhorn:2009bh; @Kallosh:2010ug; @Ferrara:2010yw; @Ferrara:2010in]. It was found that the Starobinsky model [@Starobinsky:1980te] emerges in the supersymmetric D-term hybrid inflation [@Binetruy:1996xj; @Halyo:1996pp; @Kallosh:2003ux], to give a good accordance with the Planck observations. On the other hand, the D-term hybrid inflation was considered in a different context. In a shift symmetric Kähler potential [@Kawasaki:2000yn], a ‘chaotic regime’ was found in the subcritical value of the inflaton field [@Buchmuller:2014rfa]. In the framework, inflation lasts even after the critical point of the hybrid inflation to give rise to different predictions from chaotic inflation. The following study [@Buchmuller:2014dda] showed that the energy scale of inflation coincides with the Grand Unification (GUT) scale using the Planck 2013 data [@Planck:2013jfk]. However, there is a tension between the predictions and the observations, especially the Planck 2015 data [@Ade:2015lrj; @Ade:2015xua].
In this letter we revisit D-term hybrid inflation in superconformal framework. It will be shown that there exists a single slow-rolling field in the subcritical value of the inflaton field. Since inflation continues for sufficiently long period, cosmic strings are unobservable as in Refs.[@Buchmuller:2014rfa; @Buchmuller:2014dda]. The potential in the subcritical region turns out to be in a general class of superconformal $\alpha$ attractors [@Kallosh:2013yoa; @Kallosh:2013xya], especially similar to the simplest version of the model. Consequently, non-trivial behavior and different predictions from the simplest ones are discovered.
II. subcritical regime in superconformal D-term inflation
=========================================================
We consider D-term hybrid inflation in supergravity with superconformal matter [@Buchmuller:2012ex; @Buchmuller:2013zfa]. In the model three chiral superfields $S_\pm$ and $\Phi$, which have local U(1) charge $\pm q$ ($q>0$) and $0$, respectively, are introduced. The superpotential and Kähler potential after fixing a gauge for the local conformal symmetry are respectively given by, $$\begin{aligned}
& W=\lambda S_+ S_- \Phi \,, \\
& K=-3\log \Omega^{-2}\,,
\label{eq:Kahler}\end{aligned}$$ with, $$\begin{aligned}
\Omega^{-2}=1-\frac{1}{3}\left(|S_+|^2+|S_-|^2+|\Phi|^2\right) -
\frac{\chi}{6}\left(\Phi^2+\bar{\Phi}^2\right)\,,
\nonumber \\\end{aligned}$$ where $\lambda$ and $\chi$ are constants.[^1] The term proportional to $\chi$ in the Kähler potential breaks superconformal symmetry explicitly. In the model the Fayet-Iliopoulos (FI) term can be accommodated. Then, the D-term potential in the Einstein frame is [@Buchmuller:2012ex], $$\begin{aligned}
V_D=\frac{1}{2}g^2\left(q\Omega^2(|S_+|^2-|S_-|^2)-\xi\right)^2\,,\end{aligned}$$ where $g$ is the gauge coupling and $\xi$ is the FI term, which is taken as a constant. (See Refs.[@Binetruy:2004hh; @Komargodski:2010rb; @Dienes:2009td; @Catino:2011mu; @Wieck:2014xxa; @Domcke:2014zqa] for the subtleties of this issue in supergravity.) The F-term potential in the Einstein frame, on the other hand, is given in a simple form without exponentially growing terms [@Buchmuller:2012ex; @Einhorn:2012ih], $$\begin{aligned}
V_F &=\Omega^{4} \lambda^2\biggl[
|\Phi|^2\left(|S_+|^2+|S_-|^2\right)+|S_+S_-|^2 \nonumber \\
&
-\frac{\chi^2|S_+S_-\Phi|^2}
{3+\frac{\chi}{2}\left(\Phi^2+\bar{\Phi}^2\right)+\chi^2|\Phi|^2} \biggr]\,.\end{aligned}$$ As in the canonical hybrid inflation, $S_-$ is stabilized to its origin meanwhile $S_+$ suffers from the tachyonic instability depending on the field value of $\Phi$. The nature of $\Phi$ depends on the value of $\chi$. In the Kähler potential there is a shift symmetry under ${\rm Re}\,\Phi \,({\rm Im}\,\Phi) \to {\rm Re}\,\Phi\,
({\rm Im}\,\Phi)\,+$const. for $\chi=-1\,(+1)$, and ${\rm Re}\,\Phi
\,({\rm Im}\,\Phi)$ can play a role of inflaton, as mentioned in Ref.[@Buchmuller:2012ex]. We consider $\chi \le -1$ in the later discussion without loss of generality. Then, the total potential is given by the waterfall field $s\equiv \sqrt{2}|S_+|$ and the inflaton field $\phi\equiv \sqrt{2}{\rm Re}\,\Phi$, $$\begin{aligned}
V_{\rm tot}(\phi,s)&=V_F+V_D \nonumber \\
&=\frac{\Omega^4(\phi,s) \lambda^2}{4} s^2 \phi^2
+ \frac{g^2}{8}\left(q \Omega^2(\phi,s)s^2-2\xi\right)^2\,, \\
\Omega^{-2}(\phi,s)&=1-\frac{1}{6}\left(s^2+(1+\chi)\phi^2\right)\,.\end{aligned}$$ The waterfall field becomes tachyonic below the critical value $\phi_c$ of the inflaton field, $$\begin{aligned}
\phi_c^2=\frac{6qg^2\xi}{3\lambda^2+(1+\chi)qg^2\xi}\,.\end{aligned}$$ After the tachyonic growth, the waterfall field is expected to reach its local minimum, which is obtained by $\partial V_{\rm
tot}(\phi,s)/\partial s =0$, $$\begin{aligned}
s_{\rm min}^2
&=\frac{2\xi\Omega^{-2}(\phi,0)}{q(1+\tilde{\xi})}
\frac{1-\Psi^2}{1+\frac{\tilde{\xi}}{1+\tilde{\xi}}\Psi^2}\,,
\label{eq:smin}\end{aligned}$$ where $\tilde{\xi}\equiv \xi/3q$ and, $$\begin{aligned}
\Psi\equiv \frac{\Omega(\phi,0)\phi}{\Omega(\phi_c,0)\phi_c}
=\frac{\Omega(\phi,0)\phi}{\sqrt{2qg^2\xi/\lambda^2}}\,.\end{aligned}$$ The expression for the local minimum given in Refs.[@Buchmuller:2014rfa; @Buchmuller:2014dda] corresponds to the case for $\chi=-1$ (and $q=1$) from the facts that $\Omega(\phi,0)|_{\chi=-1}=1$ and $\tilde{\xi}\sim {\cal O}(10^{-4})$ in our targeted parameter space. Following Refs.[@Buchmuller:2014rfa; @Asaka:2001ez] (see also Appendix), we have confirmed numerically that the waterfall field reaches to the local minimum after ${\cal O}(1/H_c)$ where $H_c(=g\xi/\sqrt{6})$ is the Hubble parameter at the critical point, and then it becomes a single field inflation. Since the inflation lasts well over ${\cal
O}(10^{2}/H_c)$, cosmic strings, which are produced during the tachyonic growth, are unobservable. After the waterfall field relaxed to the local minimum, the dynamics of the inflaton is described by the potential, $$\begin{aligned}
V&\equiv V_{\rm tot}(\phi,s_{\rm min}) \nonumber \\
&= g^2\xi^2(1+\tilde{\xi})\Psi^2
\frac{1-\frac{\Psi^2}{2(1+\tilde{\xi})}}
{1+2\tilde{\xi}\Psi^2} \,.
\label{eq:V}\end{aligned}$$ As in Eq., it is easily to see that the potential $V$ with $\chi=-1$ agrees with one given in Refs.[@Buchmuller:2014rfa; @Buchmuller:2014dda] up to ${\cal
O}(\xi)$.[^2]
We note that non-zero $\lambda$ explicitly breaks the shift symmetry for ${\rm Re}\,\Phi$ as well as $\chi$ that deviates from $-1$ does. Thus, a parameter $\lambda \ll 1$ and $\chi\simeq-1$ is consistent with each other under the approximate shift symmetry. In addition, $\chi\simeq -1$ is required for $\lambda \ll 1$ otherwise $\phi^2_c$ gets negative. As it will be seen, the observational data indeed implies such a parameter space.
III. cosmological consequences
==============================
![Scalar spectral index and tensor-to-scalar ratio for various values of $\delta\chi$ ($0<\delta\chi<1$) and $N_{*}=50$ and $60$ as ‘superconf.’. Result in the shift symmetric Kähler potential in Ref.[@Buchmuller:2014dda] is also given as ‘shift sym.’ (updated using the Planck 2015 results). Here $q=g=1$ and appropriate values of $\lambda$ and $\xi$ to satisfy the observed scalar amplitude is taken. []{data-label="fig:nsr"}](nsr.pdf)
The slow roll parameters for the inflaton dynamics are given as, $$\begin{aligned}
\epsilon(\phi) = \frac{1}{2}\left(\frac{V'}{V}\right)^2\,,
\quad \quad
\eta(\phi) =\frac{V''}{V}\,,\end{aligned}$$ where $V'=dV/d\hat{\phi}$ and $V''=d^2V/d\hat{\phi}^2$. Here $\hat{\phi}$ is canonically-normalized inflaton field that is related to $\phi$ as, $$\begin{aligned}
\frac{d \phi}{d\hat{\phi}}=K_{\Phi \bar{\Phi}}^{-1/2}\,,
\label{eq:dphidphihat}\end{aligned}$$ where $K_{\alpha\bar{\alpha}}\equiv\partial^2
K/\partial\alpha\partial{\bar{\alpha}}$. $|S_-|=0$, $\Phi=\bar{\Phi}=\phi/\sqrt{2}$, and $|S_+|=s_{\rm min}/\sqrt{2}$ are implicit here. (Parametrically $s_{\rm min}\simeq 0$ is a good approximation as discussed later.) Inflation ends at $\phi=\phi_f\equiv{\rm
Max}\{\phi_{\epsilon}$,$\phi_{\eta}$} where $\epsilon(\phi_\epsilon)=1$ and $|\eta(\phi_\eta)|=1$, and the last $e$-folds $N_*$ before the end of inflation is obtained by, $$\begin{aligned}
N_*=\int_{\phi_f}^{\phi_*}d\phi \frac{V}{dV/d\phi}K_{\Phi\bar{\Phi}}\,.\end{aligned}$$ The cosmological observables, [*i.e.,*]{} the scalar amplitude $A_s$, the spectral index, and the tensor-to-scalar ratio, are then determined by, $$\begin{aligned}
A_s=&\frac{V(\phi_*)}{24\pi^2 \epsilon(\phi_*)}\,,
\\
n_s=1+2\eta(\phi_*)-6\epsilon&(\phi_*)\,, \quad \quad
r=16 \epsilon(\phi_*)\,.\end{aligned}$$ We normalize the scalar amplitude by using the Planck 2015 data[@Ade:2015xua] $A_s=2.198^{+0.076}_{-0.085}\times 10^{-9}$ and compute $n_s$ and $r$ for a given $N_*$.
As we have stated before, our target is the parameter space $\lambda
\ll 1$. To search such a region, it is convenient to parametrize $\chi$ as, $$\begin{aligned}
\chi=-1-\frac{3\lambda^2}{qg^2\xi}\delta \chi \quad \quad
(0<\delta\chi<1)\,,\end{aligned}$$ to satisfy $\phi_c^2=2qg^2\xi/\lambda^2(1-\delta\chi)>0$.
![Allowed region for $N_*=55$ to 60 from the bounds on $n_s$ (68% CL) and $r$ (95% CL). Line contents are the same as Fig.\[fig:nsr\]. Here we have updated the result for the shift symmetric Kähler case by using the Planck 2015 data.[]{data-label="fig:lamxi"}](Alwdxi.pdf)
Now we are ready to discuss the cosmological consequences. Fig.\[fig:nsr\] shows the predictions of $n_s$ and $r$ in our model. Here $q=g=1$ is taken (see footnote 2), and $\lambda$ and $\xi$ are determined for a $\delta \chi$ and $N_*$ by using the scalar amplitude observed by the Planck collaboration. In Fig.\[fig:lamxi\], the allowed regions due to the bounds on $n_s$ and $r$ are shown for $N_*=55$–60.[^3] The upper and lower bounds on $\xi$ corresponds to the upper limit on $r$ and lower limit on $n_s$, respectively. In the $n_s$-$r$ plane, smaller values of $n_s$ and $r$ are obtained for larger $\lambda$ (and smaller $\xi$). In Fig.\[fig:nsr\] the result in the previous work [@Buchmuller:2014dda], [*i.e.*]{}, the shift symmetric Kähler potential case, is also given as ‘shift sym.’.[^4] We have checked that the result for $\delta \chi=0$ agrees with it numerically and the similar behavior is seen around $\delta \chi\simeq 0$. When $\delta \chi$ gets close to unity, on the contrary, a different behavior is observed. It is seen that $r$ gets smaller meanwhile $n_s$ tends to stay in the same value, which is within the Planck bounds. As a result, a wider allowed parameter space is obtained, which is seen in Fig.\[fig:lamxi\].
It is seen $\lambda\sim 10^{-4}$–$10^{-3}$ and $\sqrt{\xi}\sim
10^{16}$GeV are consistent region with the Planck observation. Although the allowed region becomes larger, $\sqrt{\xi}$ tends to sit around the GUT scale even for $\delta \chi=0.9$. As a consequence, the predicted $r$ is not extremely small. For example, $r>0.0020$ (0.075) for $N_*=60$ (50) for $\delta \chi=0.9$. The value of $\chi$ in the allowed region, on the other hand, is found as $-1.41$ $(-1.016)<\chi<-1.0046$ ($-1.0092$) for $N_*=60$ (50). Therefore, the parameter space $\lambda \ll 1$ and $\chi\sim -1$ is indeed favored by the observations.
In order to interpret the results, it is instructive to consider a canonically-normalized inflaton field $\hat{\phi}$. Although the r.h.s of Eq. is complicated, it can be approximated in the parameter space we are considering as, $$\begin{aligned}
\frac{d \phi}{d\hat{\phi}}
\simeq \sqrt{1-\frac{1}{6}(1+\chi)\phi^2}\,.
\label{eq:dphidphihat_app}\end{aligned}$$ Then it is solved analytically, $$\begin{aligned}
\phi=\frac{1}{\sqrt{\beta}}\sinh\sqrt{\beta}(\hat{\phi}+C)\,,
\label{eq:phiinphihat}\end{aligned}$$ where $C$ is a constant and, $$\begin{aligned}
\beta\equiv -\frac{1+\chi}{6}=\frac{\lambda^2}{2qg^2\xi}\delta\chi
=\frac{\delta \chi}{\phi^2_c(1-\delta\chi)}\,.
\label{eq:beta}\end{aligned}$$ We have found that $C=0$ is appropriate choice. Then $\Psi$ is simply given as, $$\begin{aligned}
\Psi\simeq \delta \chi^{-1/2} \tanh\sqrt{\beta} \hat{\phi}\,, \end{aligned}$$ to express the potential in terms of $\hat{\phi}$, $$\begin{aligned}
V\simeq g^2\xi^2 \delta \chi^{-1} \tanh^2\sqrt{\beta} \hat{\phi}
\biggr[1-\frac{\delta \chi^{-1}}{2} \tanh^2\sqrt{\beta}
\hat{\phi}\biggl]\,.
\label{eq:Vcano}\end{aligned}$$ This potential is valid in $\hat{\phi}\le \hat{\phi}_c=
\frac{1}{\sqrt{\beta}}\sinh^{-1}\sqrt{\beta}\phi_c$. It is straightforward to check that the r.h.s is equal to $g^2 \xi^2/2$ for $\hat{\phi}=\hat{\phi}_c$, and $\hat{\phi}_c\to \infty$ for $\delta
\chi\to 1$. We note that the potential coincides with a general class of superconformal $\alpha$ attractors [@Kallosh:2013yoa]. It especially resembles to the simplest class of the model, $$\begin{aligned}
V_{\alpha \mathchar`-attr}=
\Lambda^4 \tanh^{2m}\frac{\hat{\phi}}{\sqrt{6\alpha}}\,.
\label{eq:Valpha}\end{aligned}$$ Due to the additional term, however, it has a different asymptotic behavior as we will see below.
In the small $\lambda$ (and large $\xi$) region, $\beta$ gets small, then the potential reduces to, $$\begin{aligned}
V\simeq
g^2\xi^2(1-\delta\chi) \frac{\hat{\phi}^2}{\phi_c^2}
\left[1-\frac{1+(4/3)\delta\chi}{2(1-\delta\chi)}
\frac{\hat{\phi}^2}{\phi^2_c}\right]\,.\end{aligned}$$ This is nothing but the potential for the shift symmetric Kähler case given in Refs.[@Buchmuller:2014rfa; @Buchmuller:2014dda] in the limit $\delta \chi \to 0$, which leads to $\hat{\phi}\to \phi$. This feature is clearly seen in Fig.\[fig:nsr\]. We note that the quadratic term is rewritten as $(\lambda^2\xi/2q)\hat{\phi}^2$, which is independent of $\delta\chi$. Therefore, $n_s$ and $r$ approach to those in quadratic chaotic inflation in small $\lambda$ limit (while $\lambda^2 \xi\simeq$ constant), independent of $\delta \chi$. (Such a region is excluded, thus it is not shown in Fig.\[fig:lamxi\].) The potential $V_{\alpha \mathchar`-attr}$, on the other hand, has a similar structure, $$\begin{aligned}
V_{\alpha \mathchar`-attr}\simeq \frac{\Lambda^4}{6\alpha}\hat{\phi}^{2m}
\left[1-\frac{m\hat{\phi}^2}{9\alpha}\right]\,.\end{aligned}$$ Although it coincides with $V$ in the limit $\hat{\phi}\to 0$ for $m=1$ and $\Lambda^4/6\alpha=\lambda^2\xi/2q$, it is not possible to get the same factor for the quartic term.
![Potential as function of canonically-normalized inflaton field $\hat{\phi}$. $\delta\chi=0.9$ and $1$ cases (‘superconf.’) are shown, which are compared with superconformal $\alpha$ attractors (‘$\alpha$ attr.’), $R^2$ inflation (‘$R^2$’), and the shift symmetric Kähler case (‘shift sym.’). The field values $\hat{\phi}_f$ and $\hat{\phi}_*$ at the end of inflation and the last 60 $e$-folds, respectively, are also indicated for $\delta\chi=0.9$ and $1$ cases. []{data-label="fig:V"}](potentials.pdf)
In large $\lambda$ (and small $\xi$) region, on the contrary, $\beta$ increases, which leads us to expand $\Psi$ in large $\sqrt{\beta}\hat{\phi}$ limit to obtain, $$\begin{aligned}
V\simeq \frac{1}{2}g^2\xi^2(2-\delta\chi^{-1})
\left[1+a_1e^{-2\sqrt{\beta}\hat{\phi}}
-a_2e^{-4\sqrt{\beta}\hat{\phi}}\right]\,,
\label{eq:Vlargefield}\end{aligned}$$ with $a_1=8(1-\delta\chi)/(2\delta\chi-1)$ and $a_2=16(2-\delta\chi)/(2\delta\chi-1)$. This expression should be compared with Eq. in the $\alpha\ll 1$ limit. As shown in Ref.[@Kallosh:2013yoa], it reduces to the potential in $R^2$ inflation [@Whitt:1984pd] at large field value[^5], $$\begin{aligned}
V_{\alpha \mathchar`-attr}\simeq \Lambda^4\left[
1-4me^{-\frac{2\hat{\phi}}{\sqrt{6\alpha}}}\right]\,.\end{aligned}$$ Now it is clear that the form of the potential with $\delta\chi=1$ (in large $\lambda$ region) reduces to $R^2$ inflation, or the simplest class of superconformal $\alpha$ attractors in $\alpha\ll 1$ limit. To be specific, a choice of $\Lambda^4=g^2\xi^2/2$ and $\alpha=1/24\beta$ leads to the same asymptotic form. Then, we get $n_s\simeq
1-2/(N_*+1)-3qg^2\xi/8\lambda^2(N_*+1)^2$, $r\simeq
qg^2\xi/\lambda^2(N_*+1)^2$, while satisfying $\lambda^2 \xi\simeq$ constant. Namely, when $\lambda$ increases $n_s$ approaches to $1-2/(N_*+1)$ and $r$ gets smaller and smaller. We have confirmed this behavior using Eq. with $\delta\chi=1$. Recall that, however, the critical value becomes infinity, which is unphysical.
Such a behavior, on the contrary, can not be seen for $\delta \chi\neq
1$ case shown in Fig.\[fig:nsr\]. This arises from non-zero $a_1$ in Eq.. This is why we have seen the different cosmological consequences.
In Fig.\[fig:V\], the potential as function of canonically-normalized inflaton field is plotted for $\delta \chi=0.9$ and $1$. Here $\lambda=9.4 \times 10^{-4}$ ($1.9\times 10^{-3}$), $\sqrt{\xi}=1.3\times 10^{16}$ ($5.7 \times 10^{15}$)GeV for $\delta
\chi=0.9$ (1) to give $n_s=0.966$ and $r=0.051$ (0.00052) for $N_*=60$. In the plot the potentials in superconformal $\alpha$ attractors $V_{\alpha\mathchar`-attr}$, $R^2$ inflation, and the shift symmetric Kähler case, are also shown for comparison. Each potential is normalized to unity when $\hat{\phi}$ reaches to the critical point ($\delta \chi=0.9$ case and the shift symmetric Kähler case) or infinity ($\delta \chi=1$ case, $\alpha$ attractors, and $R^2$ inflation). For the shift symmetric Kähler case, the critical point is taken to the same value as $\delta\chi=0.9$ case. We take the parameters for $\alpha$ attractors and $R^2$ inflation to have the same asymptotic form as $\delta
\chi=1$ case in large field limit. It is seen that $\delta \chi=1$ case is similar to $\alpha$ attractors, but not exactly the same. As a result, the predictions for the slow-roll quantities are different, [*i.e.,*]{} $n_s=0.967$ and $r=0.00044$. It is clear, on the other hand, that $\delta \chi=0.9$ case shows a different behavior from the others. To summarize, the model has a nature of both the shift symmetric Kähler case and the simplest superconformal $\alpha$ attractors, and the slow-roll predictions change accordingly.
IV. conclusion
==============
We have revisited superconformal D-term hybrid inflation. After reaching its critical value, the inflaton field is slowly rolling thus inflation continues for a small coupling $\lambda$ of inflaton to the other fields. Because of a sufficiently long period of slow-roll regime, cosmic strings, which are formed during the tachyonic growth of the waterfall field, are unobservable. The potential which determines the dynamics of the canonically-normalized inflaton in the subcritical regime has been found to resemble to the simplest version of superconformal $\alpha$ attractors but with an additional term. Consequently, different predictions for the slow-roll parameters are obtained. For $\lambda\sim 10^{-4}$–$10^{-3}$ and $\sqrt{\xi}\sim
10^{16}\,{\rm GeV}$, $n_s$ and $r$ are consistent with the Planck data.
The predictions depend on a parameter $\chi$ that explicitly breaks superconformal symmetry in the Kähler potential. In addition, the Kähler potential with $|\chi|=1$ has a shift symmetry for the inflaton field, which is explicitly broken by non-zero $\lambda$ in the superpotential. On the other hand, $|\chi|\simeq 1$ is required from the consistency of the model setting, thus $\lambda\ll 1$ is parametrically natural. It has been found that the observational bounds indeed prefer such a parameter space.
[*Acknowledgments*]{}\
We are grateful to Wilfried Buchmüller for valuable discussions and helpful comments on the manuscript. This work was supported by JSPS KAKENHI Grant Numbers JP17H05402, JP17K14278 and JP17H02875.
Appendix: Tachyonic growth of waterfall field
=============================================
Around the critical value of the inflaton field, the dynamics of the waterfall field is governed by its tachyonic growth. For the evaluation, we define canonically-normalized waterfall field $\hat{s}$ around the critical point, $$\begin{aligned}
\frac{ds}{d\hat{s}}\simeq K_{S_+\bar{S}_+}^{-1/2}(\phi_c,0)\,.\end{aligned}$$ The rest parts are parallel to Refs.[@Buchmuller:2014rfa; @Buchmuller:2014dda]. We expand the potential $V_{\rm tot}(\phi,s)$ near the critical point, [*i.e.,*]{} $\phi\simeq
\phi_c+\dot{\phi}_c t$ as $$\begin{aligned}
V_{\rm tot}(\phi,s)=
\frac{g^2\xi^2}{2}-\frac{d^3}{2}t s^2 +{\cal O}(t^2, s^4)\,,\end{aligned}$$ to leads the interaction term in the equation of motion of $\hat{s}$, $$\begin{aligned}
\frac{\partial V_{\rm tot}(\phi,s)}{\partial \hat{s}}&=
K_{S_+\bar{S}_+}^{-1/2}(\phi,s) \frac{\partial V(\phi,s)}{\partial s}
\nonumber \\
&=-\hat{d}^3\hat{s} +{\cal O}(t^2, s^3)\,,\end{aligned}$$ where $\hat{d}^3=(2qg^2\xi)^2 |\dot{\phi}_c|/\lambda^2\phi_c^3$ and $|\dot{\phi}_c|=-(\partial V_{1l}/\partial \phi)/(3H_c)=
\sqrt{6}\log 2\,
q^2g^3\lambda^2\xi/4\pi^2\phi_c(3\lambda^2+qg^2\xi(1+\chi)^2)$. The equation of motion for $\hat{s}$ gives rise to that of momentum mode $\hat{s}_k$ of the quantum fluctuation[@Buchmuller:2014rfa; @Buchmuller:2014dda; @Asaka:2001ez], $$\begin{aligned}
\ddot{\hat{s}}_k +
\Bigl[k^2e^{-2H_ct}-\frac{9}{4}H_c^2-\hat{d}^3t\Bigr]\hat{s}_k=0\ .\end{aligned}$$ Here we have used $V_{1l}$ given in Ref.[@Buchmuller:2012ex]. Solving the equation numerically, we obtain the variance $\langle \hat{s}^2 (t)\rangle$.
After the decoherence time $t_{\rm dec}$ we match the variance with the classical motion of the waterfall field as $s(t_{\rm
dec})=K_{S_+\bar{S}_+}^{-1/2}(\phi_c,0)\sqrt{\langle \hat{s}^2
(t_{\rm dec})\rangle}$, and solve the equations of motion for $\phi$ and $s$ $$\begin{aligned}
&3H\dot{\phi}=-K_{\Phi\bar{\Phi}}^{-1}(\phi,s)
\frac{\partial V(\phi,s)}{\partial \phi}\,,
\\
&3H\dot{s}=-K_{S_+\bar{S}_+}^{-1}(\phi,s)
\frac{\partial V(\phi,s)}{\partial s}\,.\end{aligned}$$ We have confirmed numerically that the obtained solutions coincide with the dynamics described by $V$ in Eq..
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[^1]: Throughout this letter we use the same notation for chiral superfields and scalar fields and take the reduced Planck mass $M_{\rm pl}=1$ unit.
[^2]: $q$ and $g$ can be absorbed by the redefinition of $\lambda$ and $\xi$, $\bar{\lambda}\equiv \lambda /\sqrt{qg}$ and $\bar{\xi}\equiv g \xi$ if we ignore terms proportional to $\tilde{\xi}$ that are irrelevant numerically. Although we will use $\lambda$ and $\xi$ in the following discussion, the results in terms of $\bar{\lambda}$ and $\bar{\xi}$ can be obtained by $q\to 1$, $g\to 1$, $\lambda \to \bar{\lambda}$, and $\xi \to \bar{\xi}$.
[^3]: There is no allowed region for $N_*=50$ except for $\delta \chi=0.9$.
[^4]: Do not confuse with the shift symmetric Kähler case with the present superconformal case where the shift symmetry is (weakly) broken in the Kähler potential.
[^5]: To be precise, $\alpha=1$ gives the original $R^2$ inflation. The factor $4m (>0)$ is quantitatively irrelevant for the slow-roll predictions.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau’s explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m \geq 4.971 \times 10^9$.'
author:
- |
<span style="font-variant:small-caps;">Adrian W. Dudek</span>\
Mathematical Sciences Institute\
The Australian National University\
`[email protected]`
bibliography:
- 'biblio.bib'
nocite: '[@*]'
title: |
**An Explicit Result for\
Primes Between Cubes**
---
Introduction
============
Legendre’s conjecture is the assertion that there is at least one prime between any two consecutive squares. Confirmation of this seems to be out of reach, for applying modern techniques on the assumption of even the Riemann hypothesis does not suffice in forming a proof (see Titchmarsh’s [@titchmarsh1986theory] classic text for a discussion). It is thus the aim of this paper to study the weaker problem of primes between cubes, where some progress has already been made.
Consider first the more general problem of showing the existence of at least one prime in the interval $(x,x+x^{\theta})$ for some $\theta \in (0,1)$ and for all sufficiently large $x$. These are *short* intervals, so called for the relative (compared to the size of $x$) length of such an interval tends to zero as $x$ goes to infinity.
In 1930, Hoheisel [@hoheisel] was able to solve the problem for $\theta = 1-1/33000$, that is, there will be a prime in the interval
$$(x,x+x^{32999/33000})$$ for all sufficiently large $x$. His idea relied on having asymptotic estimates on the distribution of zeroes of the Riemann zeta-function $\zeta(s)$, namely a zero-free region and a zero-density estimate. Landau’s explicit formula for the Riemann zeta-function then allows a connection to be made between the zeroes and the primes. Using Hoheisel’s ideas, Ingham [@ingham] was able to prove a more general theorem, specifically that if one has a bound of the form
$$\zeta(1/2+it) = O(t^c )$$ for some $c>0$, then one can take
$$\theta = \frac{1+4c}{2+4c}+\epsilon.$$
This result arises through using the bound for $\zeta(1/2+it)$ to construct a zero-density estimate, and this in turn furnishes a value for $\theta$ through the explicit formula. Notably, Hardy and Littlewood were able to give a value of $c=1/6+\epsilon$, which corresponds to $\theta = 5/8+\epsilon$. From this, one sets $x = n^3$ and it follows that for all sufficiently large $n$ there exists a prime in the interval
$$(x,x+x^{5/8}) = (n^3,n^3+n^{15/8+ \epsilon}) \subset (n^3, (n+1)^3).$$ That is, there is a prime between any two consecutive cubes so long as these are sufficiently large. The reader should, however, note that consideration of the interval
$$(x,x+3x^{2/3})$$ is sufficient for primes between cubes and as such is the interval we use throughout this paper.
We wish to make the above result explicit, in that we might determine numerically a lower bound for which this result would hold onwards. By working through that of Ingham [@ingham], we can do this thanks to Ford’s [@ford] explicit zero-free region, Ramaré’s [@ramare] explicit zero-density theorem and an effective version of the explicit formula (see Theorem \[explicitformula\]). We employ these estimates to prove our main theorem.
\[maintheorem\] There is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$.
We should note that a result has been given by Cheng [@cheng], in which he purports to show the above theorem for the range $n \geq \exp(\exp(15))$. We should, however, note that he incorrectly goes from
$$n^3 \geq \exp(\exp(45))$$ to
$$n \geq \exp(\exp(15))$$ in establishing his result. There are some other errors also, notably in his proof of Theorem 3 in his paper [@cheng], the first inequality sign is backwards and he has used Chebyshev’s $\psi$-function instead of the $\theta$-function.
On the other hand, we have the result of Ramaré and Saouter [@ramaresaouter] that every interval
$$(x(1-\Delta^{-1}),x]$$ contains a prime for $x > x'$ and for various values of $\Delta(x')$ (given in Table 1 of their paper). It is a straightforward task to use this table to verify that there is a prime in $(x,x+3x^{2/3})$ for all $x \leq e^{60}$. One simply works through the table whilst checking that the Ramaré–Saouter interval is contained in the short interval.
We should also mention the striking result of Baker, Harman and Pintz [@bakerharmanpintz], that the interval $(x,x+x^{0.525})$ contains a prime for all sufficiently large $x$. This is tantalisingly close to $\theta = 1/2$, which would furnish a proof of Legendre’s conjecture with at most finitely many exceptions.
An effective version of the explicit formula
============================================
The result
----------
Our considerations begin with the von Mangoldt function:
$$\Lambda(n) = \left\{
\begin{array}{ll}
\log p & : \hspace{0.1in} n=p^m, \text{ $p$ is prime, $m \in \mathbb{N}$}\\
0 & : \hspace{0.1in} \text{otherwise.}
\end{array}
\right.$$
As usual, we introduce the sum $\psi(x) = \sum_{n \leq x} \Lambda(n)$ to study the distribution of the prime numbers. The usefulness of this function presents itself in the *explicit formula*
$$\psi(x) = x - \sum_{\rho} \frac{x^\rho}{\rho}-\log 2\pi - \frac{1}{2} \log(1-x^{-2})$$
where $x$ is any positive non-integer and the sum is over all nontrivial zeroes $\rho$ of $\zeta(s)$ (see Titchmarsh [@titchmarsh1986theory] for details). One can see that feeding information regarding the zeroes of $\zeta(s)$ into the above formula yields estimates for the prime powers, allowing us then to obtain estimates regarding the primes. However, the explicit formula as seen above relies on estimates over *all* of the nontrivial zeroes of $\zeta(s)$ and so is impractical for certain applications.
The Riemann Hypothesis asserts the fixedness of these zeroes to the line $\Re(s) = 1/2$. As this grand statement is yet to be confirmed, we often find more use in a *truncated* version of the explicit formula in which one inputs information regarding the zeroes $\rho$ up to some height $T$, that is, with $| \Im \rho | < T$. It is our first intention to provide such a formula but with an explicit error term. Our result is as follows.
\[explicitformula\] Let $x>e^{60}$ be half an odd integer and suppose that $T \in (50,x)$ is not the ordinate of any zero of $\zeta(s)$. Then
$$\label{explicitequation}
\psi(x) = x - \sum_{|\Im \rho| < T} \frac{x^{\rho}}{\rho} + O^*\Big( \frac{ 2 x \log^2 x}{T} \Big)$$
where the notation $f = g + O^* (h)$ means $|f-g| < h$.
In the remainder of this paper, unless otherwise mentioned, we shall assume that $x$ and $T$ are in the ranges specified in Theorem \[explicitformula\], and that $x$ is half an odd integer (this assumption minimises the error term in the above theorem, as will become evident in the proof of Lemma $\ref{boundbigsum}$).
The author attempted to locate an effective explicit formula in the literature and found that which was given by Skewes [@skewes1955difference], though the error term here is of better order. Liu and Wang [@liuwang] give a version of Theorem \[explicitformula\] with an improved constant, but holding only for $T$ as a certain function of $x$, which is useful for estimates on Goldbach’s weak conjecture but not for our application.
The method of proof for the explicit formula is well-known: we employ Perron’s formula to express $\psi(x)$ as a contour integral over a vertical line. We then truncate this integral to one over a finite line segment. This is where we will pick up the bulk of the error term, and so more precision is needed here than anywhere else in this paper. Our line of integration is then shifted so as to acknowledge the residues and introduce the sum which accompanies $\psi(x)$ in Theorem \[explicitformula\]. The crux of our working involves keeping track of the errors.
We proceed as laid out in Davenport’s [@davenport1980multiplicative] well known expository text, though by working carefully we will obtain an explicit result. We should also note that the frame of this derivation can be applied to other arithmetic functions attached to an appropriate Dirichlet series.
Truncating the Line Integral
----------------------------
For $c>0$, we define the contour integral
$$\delta(x) = \frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} \frac{x^s}{s} ds.$$ A good exercise (see Murty’s [@murtyproblems] problem book) for budding students of analytic number theory (or complex analysis) is to show that
$$\delta(x) =
\left\{
\begin{array}{ll}
0 & \mbox{if } 0 < x < 1 \\
1/2 & \mbox{if } x=1 \\
1 & \mbox{if } x > 1.
\end{array}
\right.$$
The importance of this integral becomes apparent when one wishes to study the sum of an arithmetic function up to some value $x$, particularly when that function is generated by a Dirichlet series. In our case, we consider the following for a positive non-integer $x$ and $c>1$:
$$\begin{aligned}
\psi(x) = \sum_{n \leq x} \Lambda(n) & = & \sum_{n = 1}^{\infty} \Lambda(n) \delta\Big(\frac{x}{n} \Big) \\
& = & \sum_{n=1}^{\infty} \Lambda(n) \bigg[ \frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} \Big( \frac{x}{n} \Big)^s \frac{ds}{s} \bigg] \\
& = & \frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} \Big( \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s} \Big) \frac{x^s}{s} ds.\end{aligned}$$
Notice that keeping $c>1$ gives absolute convergence to the series in the above equation, and thus justifies the interchange of integration and summation. The Dirichlet series in the above equation is known to be equal to $-\zeta'(s)/\zeta(s)$, and so we have that
$$\sum_{n \leq x} \Lambda(n) = \frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} \Big( - \frac{\zeta'(s)}{\zeta(s)} \Big) \frac{x^s}{s} ds.$$ In a more general form this is known as Perron’s formula. We may thus estimate the sum of the von Mangoldt function through some knowledge of certain analytic properties of $\zeta'(s)/\zeta(s)$. Our first step is to truncate the path of the integral to a finite segment, namely $(c-iT, c+iT)$. We define for $T > 0$ the truncated integral
$$I(x,T) = \frac{1}{2 \pi i} \int_{c-i T}^{c+i T} \frac{x^s}{s} ds.$$
The next lemma is a variant of the first lemma in Davenport [@davenport1980multiplicative Ch.17], and will bound the induced error term upon estimating $\delta(x)$ by $I(x,T)$. The proof is omitted here, for it is almost identical except in that we maintain the multiplicative constants $1/2 \pi$ throughout which arise from the contour integrals.
For $x>0$ with $x \neq 1$, $c>0, T>0$ we have
$$\delta(x) = I(x,T) + O^*\Big( \frac{x^c}{\pi T | \log x |} \Big).$$
We can now employ this bound to estimate $\psi(x)$ in the following way.
$$\begin{aligned}
\psi(x) & = & \sum_{n = 1}^{\infty} \Lambda(n) \delta\Big(\frac{x}{n} \Big) \\ \\
& = & \sum_{n = 1}^{\infty} \Lambda(n) \Big[I(\frac{x}{n},T) + O^*\Big(\frac{1}{\pi T} \Big(\frac{x}{n}\Big)^c \Big| \log \frac{x}{n} \Big|^{-1} \Big)\Big] \\ \\
& = & \frac{1}{2 \pi i} \int_{c-i T}^{c+i T} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds + \frac{1}{\pi T} O^* \Big( \sum_{n=1}^{\infty} \Lambda(n) \Big(\frac{x}{n}\Big)^c \Big| \log \frac{x}{n} \Big|^{-1} \Big).\end{aligned}$$
We proceed to bound the sum in the above formula, by splitting it up and estimating it in parts.
\[boundbigsum\] Let $x > e^{60}$ be half an odd integer and set $c = 1+1/\log x$. Then
$$\sum_{n=1}^{\infty} \Lambda(n) \Big( \frac{x}{n} \Big)^c \Big| \log \frac{x}{n} \Big|^{-1} < 2.8 x \log^2 x.$$
Some care needs to be taken here. When $x$ and $n$ are quite close, the reciprocal $\log$ will become large. Thus, we introduce the parameter $\alpha \in (1,2)$ and break up the infinite sum:
$$\begin{aligned}
\sum_{n=1}^{\infty} & = & \sum_{n=1}^{[x/\alpha]} + \sum_{n=[x/\alpha] + 1}^{[x]-1} + \sum_{n = [x]}^{[x]+1} + \sum_{n = [x]+2}^{[\alpha x]}+\sum_{n=[\alpha x]+1}^{\infty} \end{aligned}$$
On the right side of the above formula, denote the $i$th sum by $S_i$. The reader should be convinced by this division; $S_3$ deals with the most inflated terms, namely when $n$ is either side of $x$. Then $S_2$ and $S_4$ measure the remainder of the region which is *close* to $x$. We also note that $S_1$ and $S_5$ contribute little and can be estimated almost trivially.
Considering the range of $n$ in $S_1$ and $S_5$, we have
$$\Big| \log \frac{x}{n} \Big| > \log \alpha.$$ Inserting this into these sums, pulling out terms which are independent on $n$, and extending the range of summation to $\mathbb{N}$ we arrive at
$$S_1 + S_5 < \frac{x^c}{\log \alpha} \sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^c} = \frac{x^c}{\log \alpha} \Big( -\frac{\zeta'(c)}{\zeta(c)} \Big).$$ We then use Lemma 3.1 from [@ford] to obtain
$$\label{s1s5}
S_1+S_5 < \frac{ex \log x}{\log \alpha}.$$
We now turn our attention to $S_3$, which is the sum of only two things. It follows, using the fact that $[x]=x-1/2$ and the trivial bound $\Lambda(n) \leq \log n$, that
$$\begin{aligned}
S_3 & = & \Big( \frac{x}{x-\frac{1}{2}} \Big)^c \Lambda(x-1/2) \Big| \log \frac{x}{x-\frac{1}{2}} \Big| + \Big( \frac{x}{x+\frac{1}{2}} \Big)^c \Lambda(x+1/2) \Big| \log \frac{x}{x+\frac{1}{2}} \Big| \\
& < & 4x \log x \Big( \frac{x}{x-\frac{1}{2}} \Big)^c.\end{aligned}$$
This will actually be of little consequence to the final sum (as we will soon see), and so we feel no remorse in collecting here the weaker bound
$$\label{s3}
S_3 < 5 x \log x.$$
In consideration of $S_2$, we estimate $x/n < \alpha$ and $\Lambda(n) \leq \log n$ to get
$$S_2 < \alpha^c \log x \sum_{n=[x/\alpha]+1}^{[x]-1} \Big| \log \frac{x}{n} \Big|^{-1}.$$ If we let $n = [x] - v$, then the problem becomes that of summing over $v=1,2,\ldots, [x]-[x/\alpha]-1$. We have
$$\Big| \log \frac{x}{n} \Big| = \log \frac{x}{n} > \log \frac{[x]}{n} = - \log \Big(1- \frac{v}{[x]} \Big) > \frac{v}{[x]}$$ and thus
$$S_2 < \alpha^c x \log x \sum_{v=1}^{[x]-[x/\alpha]-1} \frac{1}{v}.$$ One can estimate this by the known bound $\sum_{n \leq x} 1/n \leq \log x + \gamma + 1/x$ where $\gamma \approx 0.5772\ldots$ is Euler’s constant to get:
$$\label{s2}
S_2 < \alpha^c x \log x \Big( \log(x - x/\alpha)) + \gamma + \frac{1}{x-x/\alpha} \Big).$$
The sum $S_4$ is dealt with similarly to get
$$\label{s4}
S_4 < 2 \frac{\log( \alpha x)}{3 - \alpha} \Big( \log( \alpha x - x) + \gamma + \frac{1}{\alpha x - x} \Big).$$
Finally, one may combine $(\ref{s1s5}), (\ref{s3}), (\ref{s2}), (\ref{s4})$ to get an inequality of the form
$$\sum_{n=1}^{\infty} \Lambda(n) \Big( \frac{x}{n} \Big)^c \Big| \log \frac{x}{n} \Big|^{-1} < f(\alpha, x).$$
The result follows now from choosing $\alpha = 1.194$ and letting $x>e^{60}$.
Error Bounds
------------
The immediate result of Lemma $\ref{boundbigsum}$ is that
$$\begin{aligned}
\psi(x) & = & \frac{1}{2 \pi i} \int_{c-i T}^{c+i T} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds + O^* \Big( \frac{2.8 x \log^2 x}{\pi T} \Big)\end{aligned}$$
for $x>e^{60}$, $c=1+1/\log x$ and $T>0$. We now look to shifting the line of integration so that we might involve the residues of the integrand. In doing so, we incur errors which only slightly increase the above error term. That is, the bulk of the error has already been obtained, and so we can be excused for not pursuing the best possible bound in the remainder of this section. Let $U>0$ and define the line segments
$$C_1 = [c-iT,c+iT] \hspace{1in} C_2 = [c+iT,-U+iT]$$
$$C_{3} = [-U+iT,-U-iT] \hspace{0.8in} C_{4} = [-U-iT,c-iT]$$ and their union $C$ along with the corresponding integrals
$$I_i = \frac{1}{2 \pi i} \int_{C_i} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds.$$ We also denote by $I$ the integral around the rectangle $C$. Note that we need to account for the fact that while $T$ is stipulated not to be the ordinate of a zero of $\zeta(s)$, it might be undesirably close to such. We show in Lemma $\ref{choicet}$ that there is always some good choice of $T$ nearby, and so some work will be required later to shift our horizontal paths. Also note that any work we do in bounding $I_2$ will also hold for $I_{4}$ and so it follows that
$$\label{psi}
| \psi(x) - I | < 2 |I_2|+|I_{3}| + 2.8 \frac{x \log^2 x}{\pi T}.$$
One can use Cauchy’s theorem (see Davenport [@davenport1980multiplicative] for full details) to show that
$$I = x - \sum_{|\gamma| < T} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)}+\sum_{0 < 2m < U} \frac{x^{-2m}}{2m}.$$
Noting that the rightmost summation is a partial sum of the series for $\log(1-x^{-2})/2$, we can write that
$$\psi(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} + O^*\big(E(x,T,U)\big)$$ where
$$\label{bigerror}
E(x,T,U) < \frac{\zeta'(0)}{\zeta(0)} + \frac{1}{2} \log(1-x^{-2}) + 2 |I_2|+|I_{3}| + 2.8 \frac{x \log^2 x}{\pi T}.$$
It so remains to bound $|I_2|$ and $|I_{3}|$ by deriving and making use of explicit estimates for $| \zeta'(s)/\zeta(s)|$ in appropriate regions.
Explicit Bounds for $|\zeta'(s)/\zeta(s)|$
------------------------------------------
We first establish a bound on the lengths of the rectangle $C$ that intersect with the half-plane $\sigma \leq -1$. Our contour, or rather $U$, is chosen so that we might avoid the poles of $\tan \pi s/2$ which occur at the odd integers.
\[bound1\] Let $U$ be an even integer. Then
$$\Big| \frac{\zeta'(s)}{\zeta(s)} \Big| < 9 + \log | s|$$
on the intersection of $C$ with $\sigma \leq -1$.
Consider the logarithmic derivative of the functional equation
$$\frac{\zeta'(1-s)}{\zeta(1-s)} = -\log 2\pi - \frac{1}{2} \pi \tan \frac{\pi s}{2} + \frac{\Gamma'(s)}{\Gamma(s)} + \frac{\zeta'(s)}{\zeta(s)}.$$ Let $\sigma \geq 2$ (so that $1-\sigma \leq -1$) and notice that $| \frac{1}{2} \pi \tan \frac{\pi s}{2}| < 2$ so long as $s$ is distanced by at least $1$ from odd integers on the real axis (this justifies our choice of $U$). We can then use
$$\label{gamma}
\frac{\Gamma'(s)}{\Gamma(s)} = \log s - \frac{1}{2s} - \int_0^{\infty} \frac{[u]-u+1/2}{(u+s)^2}du$$
to bound $|\Gamma'(s)/\Gamma(s)|$ trivially. The result then follows by observing that
$$\label{trivialzetabound}
\Big| \frac{\zeta'(s)}{\zeta(s)} \Big| \leq - \frac{\zeta'(2)}{\zeta(2)} < \frac{3}{5}$$
and putting it all together.
We now look to the harder task of establishing a bound over the region that includes the critical strip, as is essential for the estimation of $I_2$.
\[zetaaszeroes\] Let $s = \sigma +it$ where $\sigma > -1$ and $t >50$. Then
$$\frac{\zeta'(s)}{\zeta(s)} = \sum_{\rho} \Big(\frac{1}{s-\rho}-\frac{1}{2+it-\rho} \Big) + O^*(2 \log t),$$
We start with the equation (see 12.8 of Davenport [@davenport1980multiplicative])
$$\label{zetazeroes}
-\frac{\zeta'(s)}{\zeta(s)} = \frac{1}{s-1} - B -\frac{1}{2} \log \pi + \frac{\Gamma'(\frac{s}{2}+1)}{2 \Gamma(\frac{s}{2}+1)} - \sum_{\rho} \Big(\frac{1}{s-\rho}+\frac{1}{\rho} \Big)$$
where $B = -\gamma/2 - 1 +\frac{1}{2} \log 4 \pi$. Successively, we set $s_0 = 2+it$ and $s = \sigma +it$ and then find the difference between the two expressions. The terms involving the $\Gamma$-function are dealt with using (\[gamma\]), whereas the rest are estimated either trivially or with (\[trivialzetabound\]) to arrive at the result.
We can estimate the sum in Lemma \[zetaaszeroes\] by breaking it into two smaller sums $S_1$ and $S_2$, where $S_1$ ranges over the zeroes $\rho = \beta+i \gamma$ with $|\gamma-t| \geq 1$ and $S_2$ is over the remaining zeroes.
Let $s = \sigma +it$, where $\sigma > -1$ and $t >50$. Then
$$S_1 = \sum_{|t-\gamma| \geq 1} \Big(\frac{1}{s-\rho}-\frac{1}{2+it-\rho} \Big) = O^*(16 \log t).$$
We can estimate the summand as follows:
$$\Big| \frac{1}{s-\rho} - \frac{1}{2+it-\rho} \Big| < \frac{3}{(t - \gamma )^2}.$$
We then have that
$$S_1 < \sum_{|t - \gamma | \geq 1} \frac{3}{(t-\gamma)^2} \leq \sum_{\rho} \frac{6}{1+(t-\gamma)^2}.$$ By letting $\sigma' = 2$, taking real parts in (\[zetazeroes\]) and estimating as before one has
$$\sum_{\rho} \Re \Big( \frac{1}{s-\rho} + \frac{1}{\rho} \Big) < \frac{2}{3} \log t$$ for $t > 50$. We then use the two simple facts
$$\Re \Big( \frac{1}{s-\rho} \Big) = \frac{2 - \beta}{(2-\beta)^2+(t-\gamma)^2} > \frac{1}{4+(t-\gamma)^2}$$ and
$$\Re \Big( \frac{1}{\rho} \Big) = \frac{\beta}{|\rho|^2} > 0$$ to get
$$\sum_{\rho} \frac{1}{4+(t-\gamma)^2} < \frac{2}{3} \log t.$$ Putting it all together we have
$$\begin{aligned}
S_1 & < & \sum_{\rho} \frac{6}{1+(t-\gamma)^2}\\
&<& 24 \sum_{\rho} \frac{1}{4+(t-\gamma)^2} \\
& < & 16 \log t.\end{aligned}$$
We now wish to estimate the remaining sum
$$S_2 = \sum_{|\gamma-t|<1} \Big( \frac{1}{s-\rho} - \frac{1}{2+it - \rho} \Big).$$ To do this, we first note that as $|2+it-\rho| > 1$ the contribution of the second term to the sum can be estimated trivially by
$$N(t+1)-N(t-1)$$ where $N(T)$ denotes the number of zeroes of $\zeta(s)$ in the critical strip up to height $T$. We can use Corollary 1 of Trudgian [@trudgianargument] with $T_0 = 50$ to get
$$\label{boundzerocount}
N(t+1) - N(t-1) < \log t$$
so long as $t > 250000$. Using <span style="font-variant:small-caps;">Mathematica</span>, we can additionally verify this for $50<t \leq 250000$. At this point we have that
$$\frac{\zeta'(s)}{\zeta(s)} = \sum_{| t-\gamma|<1} \frac{1}{s-\rho} + O^*(19 \log t).$$
Finally, we are concerned with bounding the magnitude of
$$S_2' = \sum_{|\gamma-t|<1} \frac{1}{s-\rho}.$$ Of course, the problem here is that $s$ might be close to a zero $\rho$, and this will give us trouble when we attempt to bound the line integrals. We search instead for a better value of $t$, say $t_0 \in (t-1,t+1)$, which will give a better bound. We will use this in the next section to shift our horizontal line of integration to a better height.
\[choicet\] Let $t> 50$. There exists $t_0 \in (t-1,t+1)$ such that
$$\label{bound2}
\bigg| \sum_{|\gamma-t|<1} \frac{1}{(\sigma+it_0)-\rho} \bigg| < \log^2 t + \log t$$
By $(\ref{boundzerocount})$, there are at most $ \log t$ terms in $S_2'$. As such, we can partition the region into no more than $\log t +1$ zero-free regions. We can see that there will always be such a region of height
$$\frac{2}{ \log t + 1}$$ and choosing the midpoint, say $t_0$, of this region will guarantee a distance of
$$\frac{1}{ \log t +1}$$ from any zero. As such, we have, letting $s = \sigma + i t_0$, that
$$\sum_{|\gamma-t|<1} \frac{1}{|s- \rho |} \leq \sum_{|\gamma-t|<1} \frac{1}{|\gamma - t|} \leq \sum_{|\gamma-t|<1} ( \log t +1) \leq \log^2 t + \log t.$$
Finally, we can put the previous three lemmas together to get the following.
Let $\sigma > - 1, t > 50$. There exists $t_0 \in (t-1,t+1)$ such that
$$\Big| \frac{\zeta'(\sigma+it_0)}{\zeta(\sigma+it_0)} \Big| < \log^2 t+ 20\log t.$$
That is, if our contour is somewhat close to a zero, we can shift it slightly to a region where we have good bounds.
Integral Estimates
------------------
We now bound the error term $E(x,T,U)$ in $(\ref{bigerror})$, by estimating each integral trivially. Using Lemma \[bound1\], we have
$$\begin{aligned}
|I_3| & = &\frac{1}{2\pi} \Big| \int_{-U-iT}^{-U+iT} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds \Big|\\ \\
& < & \int_{-T}^{T} \frac{9 + \log \sqrt{U^2+T^2}}{2 \pi x^U T}dt \\ \\
&=& \frac{18+2\log \sqrt{U^2+T^2}}{2 \pi x^U}.\end{aligned}$$
We save this, for soon we will combine our estimates and bound them in unison upon an appropriate choice for $U$. Consider now the problem of estimating $I_2$, and the issue that $T$ might be close to the ordinate of a zero. From Lemma \[choicet\], there exists some $T_0 \in (T-1,T+1)$ that we should integrate over instead. We thus aim to shift the line of integration from $C_2$ to
$$C_2 ' = [-U+iT_0, c +iT_0].$$
It follows from Cauchy’s theorem that
$$|I_2| < \sum_{T-1 < \rho < T+1} \Big| \frac{x^{\rho}}{\rho} \Big| + |I_5| + |I_6| +|I_7|+|I_8|$$ where
$$\begin{aligned}
I_5 & = & \frac{1}{2 \pi i} \int_{-U+iT}^{-U+iT_0} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds \hspace{0.7in} I_6 = \frac{1}{2 \pi i} \int_{-U+iT_0}^{-1+iT_0} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds\\ \\
I_7 & = & \frac{1}{2 \pi i} \int_{-1+iT_0}^{c+iT_0} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds \hspace{0.8in} I_8 = \frac{1}{2 \pi i} \int_{c+iT}^{c+iT_0} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds.\end{aligned}$$
From $(\ref{boundzerocount})$, we can estimate the sum by
$$\sum_{T-1 < \Im \rho < T+1} \Big| \frac{x^{\rho}}{\rho} \Big| < \sum_{T-1 < \Im \rho < T+1} \frac{x}{T-1} < \frac{2 x \log T}{T-1}.$$
We can bound $I_5$ in the same way as $I_3$ to obtain
$$|I_5| <\frac{18+2\log \sqrt{U^2+(T+1)^2}}{2 \pi x^U T}.$$
Bounding $I_6$ is done using Lemma \[bound1\]:
$$|I_6| < \frac{9+\log \sqrt{U^2+(T+1)^2}}{2 \pi x (T-1)}$$
We also use Lemma \[choicet\] to get
$$|I_7| < \frac{e}{2 \pi (T-1)} (\log^2 (T+1) + \log (T+1)).$$
To get an upper bound for $I_8$, we notice that $\Re s = 1+ 1/\log x$ and so following the line of working which led to $(\ref{s1s5})$ gives
$$\Big| \frac{\zeta'(s)}{\zeta(s)} \Big| < \log x.$$ This is by the working involved in $(\ref{s1s5})$. Following through we get the bound
$$|I_8| < \frac{ex \log x}{\pi(T-1)}.$$
Now, throwing all of our estimates for the terms in $(\ref{bigerror})$ together, implanting the information that $T \leq x$, $x > e^{60}$ and letting $U$ be equal to the even integer closest to $x$ we obtain Theorem \[explicitformula\]. It now remains to apply this result to the problem of primes between cubes.
Primes between cubes
====================
An indicator function for intervals
-----------------------------------
We define the Chebyshev $\theta$-function as
$$\theta(x) = \sum_{p \leq x} \log p$$ and consider that
$$\theta_{x,h} = \theta(x+h)-\theta(x) = \sum_{x < p \leq x+h} \log p$$ is positive if and only if there is at least one prime in the interval $(x,x+h]$. Many questions involving the primes can be phrased in terms of $\theta_{x,h}$. For example, the twin prime conjecture is equivalent to $\theta_{p,2}$ taking on a positive value infinitely often where $p$ is a prime.
We are interested in setting $h = 3 x^{2/3}$ to tackle the problem of primes between cubes. For if
$$\theta_{x,3x^{2/3}} > 0$$ for all $x > x_0$, then there is a prime in the interval
$$(x,x+3x^{2/3}]$$ for all $x > x_0$. If we then set $x = y^3$ then there is a prime in the interval
$$(y^3, y^3 + 3 y^{2}] \subset (y^3, (y+1)^3)$$ for all $y > y_0 = x_0^{1/3}$. It is our intention to determine explicitly a value for $x_0$ and thus $y_0$.
We introduce Chebyshev’s $\psi$-function, given by
$$\psi(x) = \sum_{p^r \leq x} \log p.$$ Let $h$ be some positive real number. Substituting $x+h$ and then $x$ into Theorem \[explicitformula\] and taking the difference, we find that:
$$\label{psiinterval}
\psi(x+h) - \psi(x) > h - \bigg| \sum_{|\gamma| < T} \frac{(x+h)^{\rho} - x^{\rho}}{\rho}\bigg| - \frac{4 (x+h) \log^2 (x+h)}{T}.$$
Whilst the above will tell us information about prime powers, we are actually interested in primes. We thus require the following lemma; a combination of Proposition 3.1 and 3.2 of Dusart [@dusart].
Let $x \geq 121$. Then
$$0.9999 x^{1/2} < \psi(x) - \theta(x) < 1.00007 x^{1/2} + 1.78 x^{1/3}.$$
It follows from (\[psi\]) and the above lemma that
$$\begin{aligned}
\label{big}
\theta_{x,h} & > & h - \bigg| \sum_{|\gamma| < T} \frac{(x+h)^{\rho} - x^{\rho}}{\rho}\bigg| - \frac{4 (x+h) \log^2 (x+h)}{T} \nonumber \\ \nonumber \\
&-& 1.00007 (x+h)^{1/2}-1.78 (x+h)^{1/3} + 0.9999 x^{1/2}.\end{aligned}$$
Given that we are interested in $h = 3 x^{2/3}$, it remains to choose $T=T(x)$ and find $x_0$ such that $\theta_{x,h}$ is positive for all $x>x_0$.
Estimating the sum over the zeroes
----------------------------------
In consideration of (\[big\]), we let
$$S = \bigg| \sum_{|\gamma| < T} \frac{(x+h)^{\rho} - x^{\rho}}{\rho}\bigg|.$$ We then have that
$$\begin{aligned}
S & = & \bigg| \sum_{|\gamma| < T} \int_{x}^{x+h} t^{\rho-1} \bigg| \leq \sum_{|\gamma| < T} \int_{x}^{x+h} t^{\beta-1} \leq h \sum_{| \gamma| < T} x^{\beta-1}.\end{aligned}$$
From the identity
$$\begin{aligned}
\sum_{|\gamma| < T} ( x^{\beta-1} - x^{-1}) & = & \sum_{|\gamma| < T} \int_0^{\beta} x^{\sigma - 1} \log x d \sigma \\
& = & \int_0^1 \sum_{ \beta> \sigma, |\gamma| < T } x^{\sigma-1} \log x d \sigma,\end{aligned}$$
it follows that
$$\label{summypoos}
\sum_{|\gamma| < T} x^{\beta -1} = 2 x^{-1} N(T) + 2 \int_0^{1} N(\sigma,T) x^{\sigma-1} \log x d \sigma,$$
where $N(\sigma, T)$ denotes the number of zeroes $\rho$ of the Riemann zeta-function with $0<\Im \rho < T$ and $\Re(\rho) > \sigma$.
We can estimate the above sum, and thus $S$, with the assistance of some explicit bounds. Firstly note, that by Corollary 1 of Trudgian [@trudgianargument] we have that
$$N(T) < \frac{T \log T}{2 \pi}$$ for all $T>15$, say. Explicit estimates for $N(\sigma,T)$ are rare, though have come to light recently through the likes of Kadiri [@kadiri] and Ramaré [@ramare], who have produced zero-density estimates of rather different shape to each other. Ramaré’s estimate, which is an explicit and asymptotically better version of Ingham’s [@ingham] original density estimate, is required for the problem of primes between cubes. We give the result here, which is a corollary of Theorem 1.1 of [@ramare].
\[ramaredensity\] Let $T \geq 2000$ and $\sigma \geq 0.52$. Then
$$N(\sigma,T) \leq 9.7 (3T)^{8(1-\sigma)/3} \log^{5-2\sigma} T + 103 \log^2 T.$$
The following zero-free region, given by Ford, will also be required.
\[fordregion\] Let $T \geq 3$. Then there are no zeroes of $\zeta(s)$ in the region given by $\sigma \geq 1 - \nu(T)$ where
$$\nu(T) = \frac{1}{57.54 \log^{2/3} T (\log \log T)^{1/3}}.$$
It is useful to carry out the bulk of the calculations with $A$ in place of the constant 9.7 in Lemma \[ramaredensity\] and $c$ in place of the $57.54$ in Lemma \[fordregion\]. Doing so allows us later on to see the importance of improvements of these constants, and thus gives direction to future efforts on this problem.
We split the integral in (\[summypoos\]) into two parts, one over the interval $0 \leq \sigma \leq 5/8$, where $N(\sigma,T)$ may as well be bounded by $N(T)$, and another over $5/8 \leq \sigma \leq 1-\nu(T)$. By inserting the relevant estimates, we get
$$\begin{aligned}
\label{crackers}
\sum_{|\gamma| < T} x^{\beta-1} & < & 2 x^{-1} N(T) + 2 x^{-1} N(T) \log x \int_0^{5/8} x^{\sigma} dx \nonumber \\
& + & 2 A x^{-1} (3T)^{8/3} \log x \log^5 T \int_{5/8}^{1-\nu(T)} \bigg( \frac{x}{(3T)^{8/3} \log^2 T} \bigg)^{\sigma} d \sigma \nonumber \\
& + & 103 x^{-1} \log x \log^2 T \int_{5/8}^{1-\nu(T)} x^{\sigma} d \sigma.\end{aligned}$$
The working out is routine, yet tedious. We give the qualitative details to the extent that the reader can follow the process. We introduce the parameter $k \in (\frac{2}{3},1)$, which will play a part in the relationship between $T$ and $x$. The reasons for the range of values of $k$ will become clear soon. Now let $T=T(x)$ be the solution to the equation
$$\frac{x}{(3T)^{8/3} \log^2 T} = \exp( \log^k x).$$
Upon performing the integration in (\[crackers\]), we directly substitute in the above relationship, along with the bound for $N(T)$ and the fact that $\log T < (3/8) \log x$, to get
$$\begin{aligned}
\label{dong}
\sum_{|\gamma| < T} x^{\beta-1} & < & \frac{ e^{-\frac{3}{8} \log^k x} \log^{1/4} x}{3^{3/4} 8^{1/4} \pi}+ \frac{27A}{256} \log^{4-k} x ( e^{-\nu(T) \log^k x}- e^{-(3/8) \log^k x}) \nonumber \\
& + & \frac{927 A}{32} \log^2 x (e^{- \nu(T) \log x} - x^{-3/8}).\end{aligned}$$
There is some cancellation in the above. First, we need to estimate one of the exponential terms involving $\nu(T)$. We have that
$$\begin{aligned}
e^{-v(T) \log x} & = & \exp\Big( - \frac{\log x}{c \log^{2/3} T (\log \log T)^{1/3}} \Big) \\
& < & \exp\Big( - \frac{4}{3^{2/3} c} \Big( \frac{\log x}{ \log \log x} \Big)^{1/3} \Big).\end{aligned}$$
Now, upon expansion of (\[dong\]) and using the above we can notice that
$$- \frac{27 A}{256} (\log x)^{4-k} e^{- (3/8) \log^k x}+ \frac{927 A}{32} \log^2 x (e^{- \nu(T) \log x} - x^{-3/8}) < 0.$$ This is clear if one looks at the dominant terms. It follows that
$$\begin{aligned}
\sum_{|\gamma| < T} x^{\beta-1} & < & \frac{ e^{-\frac{3}{8} \log^k x} \log^{1/4} x}{3^{3/4} 8^{1/4} \pi}+ \frac{27A}{256} (\log x)^{4-k} e^{-\nu(T) \log^k x}.\end{aligned}$$
The remaining exponential term involving $\nu(T)$ is dealt with as before to get
$$\begin{aligned}
S \leq h \sum_{|\gamma| < T} x^{\beta-1} & < & \frac{ h e^{-\frac{3}{8} \log^k x} \log^{1/4} x}{3^{3/4} 8^{1/4} \pi}+ \frac{27Ah}{256} (\log x)^{4-k} \exp\Big(- \frac{4}{3^{2/3}c} \frac{\log^{k-2/3} x}{(\log \log x)^{1/3}} \Big).\end{aligned}$$
Estimates for inequalities
--------------------------
It is now clear that we may write $(\ref{big})$ as
$$\theta_{x,h} > h - f(x,h,k,A,c) - g(x,h,k)-E(x,h,k)$$ where
$$\begin{aligned}
f(x,h,k,A,c) & = & \frac{27 A h}{256} (\log x)^{4-k} \exp\Big( - \frac{4}{3^{2/3} c} \frac{\log^{k-2/3} x}{(\log \log x)^{1/3}}\Big),\\ \\
g(x,h,k) & = & 12 \Big(\frac{3}{8} \Big)^{3/4} \frac{(x+h) \log^{11/4} (x+h)}{x^{3/8}} \exp(\frac{3}{8} \log^k x),\\ \\
E(x,h,k) & = & - \frac{ h (\log x)^{1/4} \exp(-\frac{3}{8} \log^k x)}{6^{3/4} \pi}-1.00007(x+h)^{1/2} \\\
&-&1.78 (x+h)^{1/3}+0.9999 x^{1/2}.\end{aligned}$$
First, we look to bound the error. Noting that $x > e^{60}$, we set $h = 3x^{2/3}$ and use the fact that $k=2/3$ will give us the worst possible error to get
$$\frac{E(x,3 x^{2/3},2/3)}{3x^{2/3}} < 10^{-3}.$$ Thus, one can show that positivity holds if the following two inequalities are simultaneously satisfied:
1. $f(x,h,k,A,c) < \frac{1}{2} (1-10^{-3}) h$,
2. $g(x,h,k) < \frac{1}{2} (1-10^{-3}) h.$
This splitting simplifies our working greatly whilst perturbing the solution negligibly. To be convinced of this, one could consider the right hand side of each of the above inequalities as being equal to $h$, in some better-than-possible scenario. It turns out that the improvements would hardly be noticeable. We will, however, mention at the end of this paper some direction for future attempts at improving the work on primes between cubes.
Now, in the first inequality, we take the logarithm of both sides and set $x = e^y$ to get
$$\label{1}
\log\Big(\frac{27 A}{256}\Big) + (4-k) \log y - \frac{4}{3^{2/3} c} \frac{y^{k-2/3}}{\log^{1/3} y} < \log (\frac{1}{2} (1-10^{-3}))$$
This is easy to solve using <span style="font-variant:small-caps;">Mathematica</span>, given knowledge of $A$, $k$ and $c$. There are some notes to make here first. We can see that $A$, the constant in front of Ramaré’s zero-density estimate has little contribution, for being in the argument of the logarithm. However, $c$ plays a much larger part from where it is positioned. We can also see the reason for $k > 2/3$, in that it guarantees a solution.
We deal with the second inequality in the same way, but first we notice that
$$\frac{g(x,h,k)}{h} < \frac{2 \log^{11/4} x}{x^{1/24}} \exp( \frac{3}{8} \log^k x).$$ This is obtained using the main result of Ramaré and Saouter [@ramaresaouter] to bound
$$x+h < \frac{x}{1-\Delta^{-1}}$$ where $\Delta = 28 314 000$ as given in their paper. Thus, using the same approach as before we get
$$\label{2}
\frac{11}{4} \log y + \frac{3}{8} y^k-\frac{1}{24} y < \log( \frac{1}{4} (1-10^{-3})).$$
We notice here our reason for having $k<1$. One can also see the reason for leaving $k$ free to vary in $(2/3,1)$. There should be an optimal value of $k$, where the solution range of the above two inequalities are equal and their intersection is minimised.
No rigorous analysis needs to be conducted; we set $A=9.7$, $c=57.54$ and use the `Manipulate` function of <span style="font-variant:small-caps;">Mathematica</span> to “hunt” for a good value of $k$. It turns out that upon choosing $k=0.9359$, we have that both inequalities are satisfied for $y>8 \times 10^{14}$, or $x^{1/3} > \exp(\exp(33.217))$, which proves our main result.
Notes for future improvements
-----------------------------
Using the explicit methods of this paper, better estimates for zero-densities, zero-free regions and the error term of Landau’s explicit formula could effectively be implemented to furnish a new estimate. The following preemptive discussion might be useful for one looking to do such a thing.
Let’s consider first improving the zero-density estimate given by Ramaré. Say, for the sake of discussion, one could obtain a value of $A=10^{-4}$. Then we would obtain our result instead with $n \geq \exp(\exp(32.7))$, an improvement which would probably not be worth the efforts required to obtain such a value of $A$.
Ramaré has communicated that one could use the Brun-Titchmarsh theorem to remove a power from the logarithm in the error term of (\[explicitformula\]). This does not seem to improve the overall result; a shortcoming, perhaps, of the numerical methods used by the author.
There are other parameters where one might wish to direct future efforts. In Ramaré’s zero density estimate, one might consider the power $5-2\sigma$ of the logarithm to be $L-2\sigma$. The main difference in our working would be $(L-1-k)$ in place of $(4-k)$ in the reduced form of our second inequality. The following table summarises the improvements which would follow; a prime between $n^3$ and $(n+1)^3$ for all $n \geq n_0$.
$L$ $\log \log n_0$
----- -----------------
5 33.217
4 31.8
3 29.8
2 22.19
Turning now to the error term of Theorem \[explicitformula\] one could also consider a smaller constant in place of $2$. This constant, however, would appear in the logarithm of the right hand side of (\[2\]), and thus make little difference.
Wolke has derived the explicit formula with an error term which is
$$O\Big(\frac{x \log x}{T \log (x/T)}\Big) = O\Big( \frac{x}{T} \Big)$$ for the choice of $T(x)$ used in this paper. One may propose all sorts of “good” explicit constants for the above error term and try them via the methods of this paper, but there will be no major improvements.
Changes in the constant $c$ are more effective, though seemingly much more difficult to obtain. A value of $c=40$ would yield only $n \geq \exp(\exp(31.88))$, and $c=20$ would give $n \geq \exp(\exp(29.6))$. The removal of the $(\log \log T)^{1/3}$ would give a similar result.
Thus one expects a major result, or perhaps many minor ones, to make significant progress on this problem.
Higher powers
-------------
In lieu of a complete result on the problem of primes between cubes, we consider instead primes between $m$th powers, where $m$ is some positive integer. Appropriately, we choose $h = m x^{1-1/m}$, and we are able to prove the following result.
\[mpowers\] Let $m \geq 4.971 \times 10^9$. Then there is a prime between $n^m$ and $(n+1)^m$ for all $n \geq 1$.
The result seems absurd on a first glance as the value of $m$ is quite large. We shall leave it to others to attempt to bring the value down.
We now prove the above theorem as follows; for our choice of $h$, it follows that inequality (\[2\]) becomes
$$\label{new2}
\frac{11}{4} \log y - \Big( \frac{3}{8} - \frac{1}{m} \Big) y+\frac{3}{8} y^k < \log\Big( \frac{m}{12} (1-10^{-3})\Big)$$
whereas inequality (\[1\]) remains the same. As before, we can, for some given $m$, choose $k$ and find $n_0$ such that there is a prime between $n^m$ and $(n+1)^m$ for all $n \geq n_0$ by solving both inequalities. Some results are given in the following table.
$m$ $k$ $\log \log n_0$
------ -------- -----------------
4 0.9635 29.240
5 0.9741 27.820
6 0.9796 27.230
7 0.983 26.427
1000 0.9998 19.807
One can see that this method has its limitations, even in the case of higher powers. Nonetheless, we have that there is a prime in $(n^{1000}, (n+1)^{1000})$ for all $n \geq \exp(\exp(19.807))$. It follows that, for $m \geq 1000$, there is a prime between $n^m$ and $(n+1)^m$ for all
$$\label{lower}
n \geq \exp\bigg( \frac{1000 \exp(19.807)}{m}\bigg).$$
We could choose $m = 1000 \exp(19.807) \approx 4 \times 10^{11}$ to get primes between $n^m$ and $(n+1)^m$ for all $n \geq e$. Betrand’s postulate improves this to all $n \geq 1$.
However, we can use Corollary 2 of Trudgian [@trudgianpomerance] to improve on this value of $m$. This states that for all $x \geq 2898239$ there exists a prime in the interval
$$\bigg[ x,x\bigg(1+\frac{1}{111\log^2 x}\bigg) \bigg].$$ If we set $x = n^m$, we might ask when the above interval falls into $[n^m,n^m+m n^{m-1}]$. One can rearrange the inequality
$$n^m \bigg(1+\frac{1}{111\log^2 (n^m)}\bigg) < n^m+m n^{m-1}$$ to get
$$\label{upper}
\frac{n}{\log^2 n} < 111 m^3.$$
We wish to choose the lowest value of $m$ for which the solution sets of (\[lower\]) and (\[upper\]) first coincide. It is not to hard to see that this equates to solving simultaneously the equations
$$n = \exp\bigg( \frac{1000 \exp(19.807)}{m}\bigg)$$
and
$$\frac{n}{\log^2 n} = 111 m^3.$$
We do this by substituting the first equation directly into the second to get
$$\exp\bigg( \frac{1000 \exp(19.807)}{m}\bigg) = 111 (1000 \exp(19.807))^2 m$$ which can easily be solved with <span style="font-variant:small-caps;">Mathematica</span> to prove Theorem \[mpowers\].
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'It is known that, when an excited atom spontaneously emits one photon, two effects are produced. First, the atom’s internal and external states are entangled with the states of the emitted photon. Second, the atom receives a momentum transferred from the photon. In this work, the dynamics of such an atom in vacuum is studied. Through a specific calculation, it is demonstrated that these effects cause the atom to experience, on average, a friction force opposite to its initial velocity. Properties of the force are also discussed.'
author:
- 'Wei Guo[^1]'
title: Radiative friction on an excited atom moving in vacuum
---
Because of atom-vacuum interaction, an excited atom is able to emit a photon spontaneously into any direction [@man95], and consequently has its internal and external states entangled with photonic states [@rz92; @guo10]. Thus, even inside the vacuum, the atom’s motion can never be a free motion. Already pointed out was that, in the presence of such photon-atom entanglement, the atom’s position-momentum uncertainty can be temporarily lower than the lower bound specified by the Heisenberg uncertainty relation [@guo10]. It is certainly desirable to know what other effects the atom-vacuum interaction can have on the atom’s motion.
Note, the frequency of the emitted photon depends on the direction in which the photon is emitted according to the Doppler effect. In particular, when the photon is in the same direction as the atom’s velocity, it must have a frequency larger than the frequency it has when emitted in the opposite direction. On the other hand, the photon emitted in any two directions perpendicular to the velocity must have the same frequency. Since a higher frequency means that a larger momentum is carried away by the photon, and that a larger momentum is transferred to the atom from the photon, the excited atom, when moving, must receive, on average, a nonzero momentum transferred from the photon, and should experience, in addition to the photon-atom entanglement, a radiative force, on average, opposite to its velocity. The present paper is devoted to the derivation of the radiative force $\vec{F}_{r}$. Since the radiative force is opposite to the atom’s velocity, it is addressed as radiative friction in the paper. In the literature, radiative forces on an atom are often discussed when the atom is exposed to laser beams [@da85; @man95], and are seldom analyzed when the atom is simply in the vacuum. Unlike the laser beams, the vacuum has far more electromagnetic modes to be considered. It will become evident in the following discussion that $\vec{F}_{r}$ also depends critically on the photon-atom entanglement.
Denote the momentum operator, position operator, and mass of the excited atom as $\vec{P}$, $\vec{R}$, and $m$ respectively. For simplicity, the atom is assumed to have two energy states: an excited state $\vert E \rangle$ with energy $\hbar \omega _{E}$ and the ground state $\vert G \rangle$ with energy $\hbar \omega _{G}$. The difference between $\omega _{E}$ and $\omega _{G}$ is known as the atomic transition frequency $\omega _{0}\equiv \omega _{E}-
\omega _{G}$. The interaction between the vacuum modes and atom is assumed, as usual, to be through the atom’s electric dipole moment $\vec{\mu}$ and is described by an operator $V_{I}$ defined as $$\label{e1}
V_{I}=\sum _{\alpha}\Big (\vec{\mu}_{GE}\cdot \vec{g} _{\alpha}e^{-i
\vec{k}_{\alpha}\cdot \vec{R}}a^{\dagger}_{\alpha}\vert G \rangle
\langle E \vert +\vec{\mu}_{EG}\cdot \vec{g}^{\ast}_{\alpha}
e^{i\vec{k}_{\alpha}\cdot \vec{R}} a_{\alpha} \vert E \rangle
\langle G \vert \Big ),$$ where $\vec{\mu}_{GE}$ is the matrix element of $\vec{\mu}$ between $\vert G \rangle$ and $\vert E \rangle$, and $\vec{\mu}_{EG}$ the complex conjugate of $\vec{\mu}_{GE}$. The frequency and amplitude (containing a polarization unit vector $\vec{\epsilon}_{\alpha}$) of mode $\alpha$ are represented by $\omega _{\alpha}$ and $\vec{g}_{\alpha}=i\sqrt{2\pi \hbar \omega ^{2}_{0}/(L^{3}
\omega _{\alpha})} \vec{\epsilon}_{\alpha}$ respectively. The quantization volume is $L^{3}$. Also used in $V_{I}$ are $\vec{k}_{\alpha}$, the wave vector of mode $\alpha$ ($\mid \vec{k}_{\alpha}\mid =\omega _{\alpha}/c$), and $a^{\dagger}_{\alpha}$ ($a_{\alpha}$), the creation (annihilation) operator for the same mode. Throughout the paper, $c$ is the speed of light in the vacuum, and $\ast$ denotes the complex conjugate of a quantity. Still for simplicity, $V_{I}$ is treated under the rotating-wave approximation [@R1]. The Hamiltonian $H$ of the atom-vacuum system is constructed by adding to $V_{I}$ the unperturbed Hamiltonian $H_{0}=P^{2}/(2m)+\hbar
\omega _{E} \vert E \rangle \langle E \vert +\hbar \omega _{G}
\vert G \rangle \langle G \vert +\sum _{\alpha}\hbar \omega _{\alpha}
a^{\dagger}_{\alpha}a_{\alpha}$: $$\label{e2}
H=H_{0}+V_{I}.$$ Since it is constant and unimportant to the atom-vacuum system’s evolution, the zero point energy of the vacuum is ignored in $H$. Also ignored is the Röntgen interaction, because this interaction [@wi94; @bo02] is roughly of the order of $vc^{-1}
V_{I}$ (where $v$ is the atom’s speed), much weak compared with $V_{I}$ in the present nonrelativistic analysis. For example, in a typical experiment on spontaneous emission [@ku97], $v$ is merely of the order of $10^{3}m/s$.
The atom’s initial external state $\vert \psi (0) \rangle =\int
d\vec{p}_{0}f(\vec{p}_{0}) \vert \vec{p}_{0}\rangle$ is expressed in terms of the eigenstates $\vert \vec{p}_{0} \rangle$ of the momentum operator. To mimic the initial condition, assumed without loss of generality, that the atom moves along the positive direction $\hat{x}$ of the $x$-axis of a coordinate system stationary in the vacuum, the eigenvalues $\vec{p}_{0}$ of these states are assumed to be all along $\hat{x}$. Since the atom is in the excited state $\vert E \rangle$, and no photons are present $\vert 0 \rangle$, evolution of the atom-vacuum system must start from such an initial state $\vert \phi (0) \rangle = \vert
\psi (0) \rangle \otimes \vert E \rangle \otimes \vert 0
\rangle$. At time $t$, the state of the system $\vert \phi (t)
\rangle$ consequently becomes $$\label{e3}
\vert \phi (t) \rangle =e^{-\frac{iHt}{\hbar}}\vert \phi (0)
\rangle =-\frac{1}{2\pi i}\int ^{\infty}_{-\infty}
dq \frac{e^{-iqt/\hbar }}{q-H} \vert \phi (0) \rangle .$$ As, for example, in the discussion of the Ehrenfest theorem [@sa94] and the atomic motion in the laser beams [@da85; @man95], the radiative friction $\vec{F}_{r}$ is defined to be proportional to the second-order time derivative of the expectation value of the position operator $\vec{R}$ taken with respect to state $\vert \phi (t) \rangle$: $$\label{e4}
\vec{F}_{r}=m\frac{d^{2}}{dt^{2}}\langle \phi (t) \vert \vec{R} \vert
\phi (t) \rangle .$$
Consider the $x$-component of the force $\vec{F}_{r}$. From Eq. (\[e3\]), the time-dependent expectation value of $R_{x}$ (the $x$-component of $\vec{R}$), $$\label{e5}
\langle R_{x} \rangle (t)=\langle \phi (t) \vert R_{x} \vert \phi (t)
\rangle= \langle \phi (0) \vert e^{\frac{iHt}{\hbar}} R_{x}
e^{-\frac{iHt}{\hbar}} \vert \phi (0) \rangle ,$$ and its first-order derivative, $$\begin{aligned}
\label{e6}
\frac{d}{dt} \langle R_{x} \rangle (t)&=&\frac{i}{\hbar} \langle
\phi (0) \vert e^{\frac{iHt}{\hbar}}[H,R_{x}]
e^{-\frac{iHt}{\hbar}} \vert \phi (0) \rangle \nonumber\\
&=& \frac{1}{m} \langle \phi (0) \vert e^{\frac{iHt}{\hbar}}
P_{x} e^{-\frac{iHt}{\hbar}} \vert \phi (0) \rangle ,
\end{aligned}$$ are first obtained. In Eq. (\[e6\]), $P_{x}$ is the $x$-component of the momentum operator $\vec{P}$. From the preceding equation, it is then a straightforward matter to find the second-order time derivative of $\langle R_{x} \rangle (t)$: $$\begin{aligned}
\label{e7}
\frac{d^{2}}{dt^{2}} \langle R_{x} \rangle (t)&=& \frac{i}{m}
\langle \phi (t) \vert \sum _{\alpha} (\vec{\mu}_{GE}\cdot
\vec{g}_{\alpha})a^{\dagger}_{\alpha} \vert G \rangle
\langle E \vert k_{\alpha x} e^{-i\vec{k}_{\alpha}\cdot \vec{R}}
\vert \phi (t) \rangle \nonumber\\
& &- \frac{i}{m} \langle \phi (t) \vert \sum _{\alpha}
(\vec{\mu}_{EG}\cdot \vec{g}^{\ast}_{\alpha})a_{\alpha} \vert E
\rangle \langle G \vert k_{\alpha x}
e^{i\vec{k}_{\alpha}\cdot \vec{R}} \vert \phi (t) \rangle ,
\end{aligned}$$ where $k_{\alpha x}$, as the $x$-component of the mode vector $\vec{k}_{\alpha}$, illustrates the momentum exchange between the photon and atom when the photon is created or annihilated.
The evolution of the atom-vacuum system is accompanied by two simultaneous processes: spontaneous emission and associated atomic recoil from the emitted photon. The net result of these processes is that, as noted before, the atomic and photonic states are entangled [@guo10]: $$\begin{aligned}
\label{e8}
\vert \phi (t) \rangle &=&
-\frac{1}{2\pi i} \int dq e^{-iqt/\hbar}
\int d\vec{p}_{0} \frac{f(\vec{p}_{0})}{q-p^{2}_{0}/(2m)-\hbar
\omega _{E} -B} \vert \vec{p}_{0} \rangle \otimes \vert E \rangle
\otimes \vert 0 \rangle \nonumber\\
& & -\frac{1}{2\pi i} \int dq e^{-iqt/\hbar} \int
d\vec{p}_{0} \frac{f(\vec{p}_{0})}{q-p^{2}_{0}/(2m)-\hbar \omega _{E}
-B} \nonumber\\
& &\times \sum _{\alpha} \frac{ (\vec{\mu} _{GE} \cdot
\vec{g}_{\alpha})
e^{-i\vec{k}_{\alpha}\cdot \vec{R}} \vert \vec{p}_{0} \rangle
\otimes \vert G \rangle
\otimes \vert 1_{\alpha} \rangle }{q-(\vec{p}_{0}-\hbar
\vec{k}_{\alpha })^{2}
/(2m)-\hbar \omega _{G}-\hbar \omega _{\alpha}},
\end{aligned}$$ where $p_{0}=\vert \vec{p}_{0} \vert$, and $$\begin{aligned}
\label{9}
B &\simeq &-\frac{\Gamma _{0}\hbar }{2\omega _{0} \pi}
\Big [ \Omega +\omega _{0} \ln \big (\frac{\Omega -\omega _{0}}{
\omega _{0}}\big )\Big ]-i \frac{\Gamma _{0}\hbar }{2}\nonumber\\
&\equiv&B_{r}+iB_{i}.
\end{aligned}$$ In the expression of $B$, the quantity $\Gamma _{0}=4\mid
\vec{\mu}_{GE}\mid ^{2}\omega ^{3}_{0}/(3\hbar c^{3})$ is the spontaneous emission rate of a stationary atom in the vacuum, and $\Omega $ a cut-off frequency needed to make the nonrelativistic Hamiltonian $H$ applicable in the present discussion [@co89]. Note, in Eq. (\[e8\]), $(\vec{p}_{0}-\hbar
\vec{k}_{\alpha})^{2}/(2m)$ is the atom’s kinetic energy after the atom recoils from the photon emitted into mode $\vert 1_{\alpha}
\rangle$.
Substitute Eq. (\[e8\]) into Eq. (\[e7\]) to get $$\label{e10}
\frac{d^{2}}{dt^{2}}\langle R_{x} \rangle (t) = -
\frac{\omega ^{2}_{0}\vert \vec{\mu}_{GE} \vert ^{2}}{3m^{2}c^{5}}
(\omega _{0}+B_{r}/\hbar )^{2} e^{-\Gamma _{0}t}
\int d\vec{p}_{0} \vert f(\vec{p}_{0})\vert ^{2} p_{0}.$$ In the derivation of Eq. (\[e10\]), $\vec{\mu}_{GE}$ is averaged over its orientation to conform to the fact that the orientation is usually unknown. Since the dependence of the atom’s spontaneous emission on the atom’s speed is weak [@guo08], the contribution from the Doppler effect is ignored in the frequency of the emitted photon $(\omega _{0}+B_{r}/\hbar)$. Also used in Eq. (\[e10\]) is the mode-continuum approximation; see, for example, Ref. [@guo05]. A comparison of Eqs. (\[e7\]) and (\[e8\]) shows that the derivative in Eq. (\[e10\]) can never survive unless the photonic and atomic states are entangled as in Eq. (\[e8\]).
Similarly, it is found that the second-order time derivatives of $\langle R_{y} \rangle (t)$ and $\langle R_{z} \rangle (t)$ both vanish, where $R_{y}$ and $R_{z}$ are the $y$- and $z$-components of $\vec{R}$ respectively. Thus, the radiation friction $\vec{F}_{r}$ on the atom must be $$\begin{aligned}
\label{e11}
\vec{F}_{r} &=& - \frac{\omega ^{2}_{0}\vert \vec{\mu}_{GE}
\vert ^{2}}{3mc^{5}} (\omega _{0}+B_{r}/\hbar )^{2}
e^{-\Gamma _{0}t} \int d\vec{p}_{0} \vert f(\vec{p}_{0})\vert
^{2} \vec{p}_{0} \nonumber\\
&=&-\frac{(\omega _{0}+B_{r}/\hbar)^{2}\hbar \Gamma _{0}}{4
m\omega _{0}c^{2}}
e^{-\Gamma _{0}t} \int d\vec{p}_{0} \vert f(\vec{p}_{0})\vert
^{2} \vec{p}_{0} .
\end{aligned}$$
Three properties of $\vec{F}_{r}$ are recognized. First, the magnitude of $\vec{F}_{r}$ decays with time exponentially. This observation is understandable, because the force is only present when the atom and photon interact during spontaneous emission, and spontaneous emission is largely an exponential process. For a discussion of spontaneous emission beyond the rotating-wave approximation, see, for example, Ref. [@guo05]. Second, the force is proportional to the average (initial) momentum of the atom, a character of the average Langevin force if the atom’s motion is viewed as a Brownian motion [@pat96]. Physically, the average momentum determines the difference between the photonic frequencies when the photon is emitted in or opposite to the direction of the atomic velocity, and, thus, should also determine the strength of the friction force. A stationary atom, whose momentum is zero, certainly does not experience the friction force $\vec{F}_{r}$. Finally, since it is proportional to the initial momentum of the atom, the force $\vec{F}_{r}$ must depend on the coordinate system in which it is observed. As Eq. (\[e7\]) shows, such dependence comes from the fact that $\frac{d^{2}}{d
t^{2}}\langle R_{x} \rangle (t)$ is related to the wave vector $\vec{k}_{\alpha}$ of the photon, which, when combined with $i\omega _{\alpha}/c$, is a 4-vector, and is different in different systems. The friction force is not an invariant under the Lorentz transformation.
In conclusion, it is demonstrated that the motion of an excited atom in the vacuum is subject to a friction force. The force comes not only from the photon-atom entanglement but also from the momentum transferred from the emitted photon to the atom.
[99]{} L. Mandel and E. Wolfe, [*Optical Coherence and Quantum Optics*]{} (Cambridge University Press, New York, 1995). K. Rzazewski and W. Zakowicz, J. Phys. B [**25**]{}, L319 (1992). W. Guo and Y. Aktas, Opt. Commun. [**283**]{}, 4324 (2010). J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B [**2**]{}, 1707 (1985). For a discussion on the validity to apply the rotating-wave approximation to a two-level system, see, for example, W. Guo, Phys. Rev. A [**80**]{}, 033828 (2009). M. Wilkens, Phys. Rev. A [**49**]{}, 570 (1994). L. G. Boussiakou, C. R. Bennett, and M. Babiker, Phys. Rev. Lett. [**89**]{}, 123001 (2002). C. Kurtsiefer, et al., Phys. Rev. A [**55**]{}, R2539 (1997). J. J. Sakurai, [*Modern Quantum Mechanics*]{} (Addison-Wesley, New York, 1994). C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, [*Photons and atoms: Introduction to quantum electrodynamics*]{} (Wiley, New York, 1989). W. Guo, Phys. Rev. A [**77**]{}, 062111 (2008). W. Guo, Opt. Commun. [**250**]{}, 137 (2005). R. K. Pathria, [*Statistical Mechanics*]{}, second edition (Butterworth-Heinemann, Oxford, 1996).
[^1]: Electronic address: [email protected]
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'Jörg Bünemann^,^, Florian Gebhard^^, Tobias Schickling^^, and Werner Weber^^'
title: Numerical Minimisation of Gutzwiller Energy Functionals
---
Introduction
============
In solid-state theory, multi-band Hubbard models are used to study transition metals and their compounds. In these models only the local (atomic) part of the Coulomb interaction is explicitly taken into account. All non-local terms are included on the level of a ‘Density-Functional Theory’ calculation, which is used to set up a proper tight-binding Hamiltonian, see Sect. \[sec1\].
Despite the relative simplicity of Hubbard models, as compared to the full electronic Hamiltonian, calculating their properties still constitutes a very difficult many-particle problem. In recent years, significant progress has been made in this direction by the systematic study of models in the limit of infinite spatial dimensions ($D\to \infty$). The exact solution of Hubbard models in this limit leads to the Dynamical Mean Field Theory (DMFT), in which the original lattice model is mapped onto an effective single-impurity system that has to be solved numerically [@metzner1989; @vollhardt1989; @vollhardt1993; @georges1996; @gebhard1997]. Although significant progress has been made in recent years in developing numerical techniques for the solution of the DMFT equations, it is still quite challenging and can be carried out only with limited accuracy.
An alternative method, that also relies on infinite-$D$ techniques, is the Gutzwiller variational approach. It allows for the approximate study of ground-state properties and single-particle excitations with much less numerical effort than within DMFT and has been applied in a number of works in recent years [@buenemann1997c; @buenemann1998; @buenemann2003; @attaccalite2003; @buenemann2003b; @ferrero2005; @julien2005; @buenemann2005; @buenemann2007b; @buenemann2007d; @buenemann2008; @lanata2008; @ho2008; @deng2009; @zhouang2009; @borghi2009; @hofmann2009; @wang2010; @zhou2010; @buenemann2011; @yao2011; @buenemann2011e]. A related approach that leads to the same energy functional for multi-band models is the slave-boson mean field theory [@lechermann2007; @buenemann2007c; @ferrero2009; @isidori2009; @lechermann2009; @buenemann2010; @piefke2011]. Starting from the approximate ground-state description, it is also possible to study two-particle excitations within the ‘time-dependent Gutzwiller theory’ [@seibold1998b; @seibold2003; @lorenzana2003; @seibold2004; @seibold2004b; @lorenzana2005; @seibold2005; @seibold2006; @seibold2007; @seibold2008; @seibold2008b; @guenther2010; @buenemann2011b; @buenemann2011c].
The main numerical problem in the Gutzwiller theory is the minimisation of the energy functional with respect to the variational parameters since their number can be quite large in investigations of multi-band models. We have developed an efficient numerical scheme for this minimisation which has already been applied successfully in our studies on nickel [@buenemann2003; @buenemann2008] and iron-pnictides [@buenemann2011; @buenemann2011e]. In particular, the studies on the spin-orbit coupling effects in nickel were numerically demanding since they required a rather fine energy resolution and the handling of up to 8000 variational parameters [@buenemann2008]. To the best of our knowledge, no Gutzwiller minimisation of similar complexity has been reported in other works. We are therefore convinced that our minimisation algorithm will be of significant interest for all researchers who intend to apply the Gutzwiller theory to real materials. It is the purpose of this work to give detailed account of our method. Note that an alternative method for the minimisation of a restricted class of Gutzwiller energy functionals has been proposed in a recent work [@lanata2011].
Our presentation is organised as follows. In Sections \[sec1\] and \[chap2b\] we summarise the main results on multi-band Gutzwiller wave functions and their energy functionals in infinite spatial dimensions. Our minimisation algorithm is described in detail in Section \[app7\]. Some technical parts of the presentation are referred to four appendices.
Multi-Band Hubbard models {#sec1}
=========================
We aim to study the physics of multi-band Hubbard models $$\label{h2}
\hat{H}=\sum_{i\neq j} \sum_{\sigma,\sigma'}
t^{\sigma,\sigma'}_{i,j} {\hat{c}^{\dagger}}_{i,\sigma}{\hat{c}^{\phantom{\dagger}}}_{j,\sigma'}+
\sum_i \hat{H}_{i,{\rm loc}}\;.$$ Here, we introduced the ‘hopping parameters’ $t^{\sigma,\sigma'}_{i,j}$ and the operators $\hat{c}^{(\dagger)}_{i,\sigma}$, which annihilate (create) an electron with spin-orbital index $\sigma$ on a lattice site $i$. The local Hamiltonian $$\begin{aligned}
\label{4.10a}
\hat{H}_{i;{\rm loc}}&=&\sum_{\sigma_1,\sigma_2}\varepsilon_{i;\sigma_1,\sigma_2}
{\hat{c}^{\dagger}}_{i,\sigma_1} {\hat{c}^{\phantom{\dagger}}}_{i,\sigma_2}\\ \nonumber
&&+\sum_{\sigma_1,\sigma_2,\sigma_3,\sigma_4}
U_i^{\sigma_1,\sigma_2,\sigma_3,\sigma_4}
{\hat{c}^{\dagger}}_{i,\sigma_1} {\hat{c}^{\dagger}}_{i,\sigma_2}{\hat{c}^{\phantom{\dagger}}}_{i,\sigma_3} {\hat{c}^{\phantom{\dagger}}}_{i,\sigma_4}\;\end{aligned}$$ is determined by the orbital-dependent on-site energies $\varepsilon_{i;\sigma_1,\sigma_2}$ and by the two-particle Coulomb interaction matrix elements $U_i^{\sigma_1,\sigma_2,\sigma_3,\sigma_4}$. We assume that the $2N$ spin-orbital states $\sigma$ are ordered in some arbitrary way, $\sigma= 1,\ldots,2N$ where $N$ is the number of orbitals per lattice site. In order to set up a proper basis of the local Hilbert space, we introduce the following notations for the $2^{2N}$ possible configurations.
- An atomic configuration $I$ is characterised by the electron occupation of the orbitals, $$\begin{aligned}
\label{4.20a}
I &\in& \{\emptyset;
(1),\ldots,(2N);
(1,2),\ldots,(2,3),\\\nonumber
&&\ldots (2N-1,2N);
\ldots;
(1,\ldots,2N)
\}\;,\end{aligned}$$ where the elements in each set $I=(\sigma_1,\sigma_2,\ldots)$ are ordered, i.e., it is $\sigma_1<\sigma_2<\ldots$. The symbol $\emptyset$ in (\[4.20a\]) means that the site is empty. In general, we interpret the indices $I$ as sets in the usual mathematical sense. For example, in the atomic configuration $I\backslash I'$ only those orbitals in $I$ that are not in $I'$ are occupied. The complement of $I$ is $\overline{I}\equiv(1,2,\ldots,2N)\backslash I$, i.e., in the atomic configuration $\overline{I}$ all orbitals but those in $I$ are occupied.
- The absolute value $|I|$ of a configuration is the number of elements in it, i.e., $$\begin{aligned}
\label{4.25a}
&&|\emptyset|=0;|(\sigma_1)|=1;|(\sigma_1,\sigma_2)|=2;\\\nonumber
&&\ldots;|(1,\ldots,2N)|=2N
\;. \end{aligned}$$
- A state with a specific configuration $I$ is given as $$\label{4.30a}
{\left| I \right \rangle }=\hat{C}_{I}^{\dagger}{\left| 0 \right \rangle }\equiv\prod_{\sigma \in I}{\hat{c}^{\dagger}}_{\sigma}{\left| 0 \right \rangle }=
{\hat{c}^{\dagger}}_{\sigma_1}\dots{\hat{c}^{\dagger}}_{\sigma_{|I|}}{\left| 0 \right \rangle }\;,$$ where the operators ${\hat{c}^{\dagger}}_{\sigma}$ are in ascending order, i.e., it is $\sigma_1<\sigma_2\ldots<\sigma_{|I|}$. Products of annihilation operators, such as $$\label{4.35a}
\hat{C}_{I}^{}\equiv\prod_{\sigma\in I}{\hat{c}^{\phantom{\dagger}}}_{\sigma}={\hat{c}^{\phantom{\dagger}}}_{\sigma_1}\dots{\hat{c}^{\phantom{\dagger}}}_{\sigma_{|I|}},$$ will be placed in descending order, i.e., with $\sigma_1>\sigma_2\ldots>\sigma_{|I|}$. Note that we have introduced the operators $\hat{C}_{I}^{\dagger}$ and $\hat{C}_{I}^{}$ just as convenient abbreviations. They must not be misinterpreted as fermionic creation or annihilation operators.
- The operator $\hat{m}_{I,I'}\equiv {\left| I \right \rangle }{\left \langle I' \right |}$ describes the transfer between configurations $I'$ and $I$. It can be written as $$\label{4.50a}
\hat{m}_{I,I'}=
\hat{C}_{I}^{\dagger}
\hat{C}_{I'}^{}
\prod_{\sigma''\in J}(1-\hat{n}_{\sigma''})$$ where $J\equiv \overline{I\cup I'}$. A special case, which derives from (\[4.50a\]), is the occupation operator $$\label{4.52a}
\hat{m}_{I}\equiv {\left| I \right \rangle }{\left \langle I \right |}=\prod_{\sigma\in I}\hat{n}_{\sigma}
\prod_{\sigma'\in \bar{I}}(1-\hat{n}_{\sigma'})\;.$$
The states ${\left| I \right \rangle }$ form a basis of the atomic Hilbert space. Therefore, we can write the eigenstates of the local Hamiltonian (\[4.10a\]) as $$\label{4.60a}
|\Gamma\rangle =\sum_{I}T_{I,\Gamma}{\left| I \right \rangle }$$ with coefficients $T_{I,\Gamma}$. With these eigenstates, the atomic Hamiltonian has the form $$\begin{aligned}
\label{eft}
\hat{H}_{i,{\rm loc}}&=&\sum_{\Gamma}
E_{i;\Gamma}\hat{m}_{i;\Gamma,\Gamma}\;,\\
\hat{m}_{i;\Gamma,\Gamma'}&\equiv&
| \Gamma{\rangle\hspace{-0.2cm}{\phantom{\rangle}}_{i}
\hspace{0.05cm}}
{\hspace{-0.1cm}{\phantom{\rangle}}_{i}
\hspace{-0.05cm} \langle\hspace{0.03cm} } \Gamma'| =
\sum_{I,I'}T_{I,\Gamma}T^*_{I',\Gamma'}|
I{\rangle\hspace{-0.2cm}{\phantom{\rangle}}_{i}
\hspace{0.05cm}}
{\hspace{-0.1cm}{\phantom{\rangle}}_{i}
\hspace{-0.05cm} \langle\hspace{0.03cm} } I'|
\;.\end{aligned}$$
Gutzwiller Energy Functional {#chap2b}
============================
Multi-band Gutzwiller wave-functions have the form $$\label{1.3}
|\Psi_{\rm G}\rangle=\hat{P}_{\rm G}|\Psi_0\rangle=\prod_{i}\hat{P}_{i}|\Psi_0\rangle\;,$$ where $|\Psi_0\rangle$ is a normalised single-particle product state and the local Gutzwiller correlator is defined as $$\label{1.4b}
\hat{P}_{i}=\sum_{\Gamma,\Gamma^{\prime}}\lambda_{i;\Gamma,\Gamma^{\prime}}
|\Gamma \rangle_{i} {}_{i}\langle \Gamma^{\prime} |\equiv
\sum_{\tilde{\Gamma}}\lambda_{i;\tilde{\Gamma}}
| \tilde{\Gamma} \rangle_{i} {}_{i}\langle \tilde{\Gamma} | \;,$$ where we introduced the matrix of variational parameters $\lambda_{i;\Gamma,\Gamma^{\prime}}$ which allows us to optimise the occupation and the form of the eigenstates $|\tilde{\Gamma} \rangle_{i}$ of $\hat{P}_{i}$.
The evaluation of expectations values with respect to the wave function (\[1.3\]) is a difficult many-particle problem, which cannot be solved in general. As shown in Refs. [@buenemann1998; @buenemann2005], one can derive analytical expressions for the variational ground-state energy in the limit of infinite spatial dimensions ($D\to \infty$). Using this energy functional for the study of finite-dimensional systems is usually denoted as the ‘Gutzwiller approximation’. This approach is the basis of most applications of Gutzwiller wave functions in studies of real materials and it will also be addressed in this work. One should keep in mind, however, that the Gutzwiller approximation has its limitations and the study of some phenomena requires an evaluation of expectation values in finite dimensions [@buenemann2011d].
Local basis
-----------
In general, the local density matrix for non-interacting electrons $$\label{xc}
C_{i;\sigma,\sigma'}=\langle {\hat{c}^{\dagger}}_{i,\sigma}{\hat{c}^{\phantom{\dagger}}}_{i,\sigma'} \rangle_{\Psi_0}$$ is non-diagonal with respect to $\sigma,\sigma'$. For a fixed state ${\left| \Psi_0 \right \rangle }$, one can always find a local basis with a diagonal density matrix. This will turn out to be quite useful in the minimisation with respect to the variational parameters $\lambda_{i;\Gamma,\Gamma^{\prime}}$ because, with such a basis, the energy functional has a much simpler form. We introduce the explicit expression of this simplified functional in the following Sects. \[con1\] and \[exp\]. If one minimises the energy with respect to ${\left| \Psi_0 \right \rangle }$, however, the diagonality of (\[xc\]) is only ensured in systems with high symmetries. Therefore, we also need the general expression for the variational ground-state energy with an arbitrary local basis. This is given in Appendix \[ka\].
Note that, in general, the [*correlated*]{} density matrix $$\label{xc1}
C^{\rm c}_{i;\sigma,\sigma'}=\langle {\hat{c}^{\dagger}}_{i,\sigma}{\hat{c}^{\phantom{\dagger}}}_{i,\sigma'}
\rangle_{\Psi_{\rm G}}$$ is different from the [*non-interacting*]{} density matrix (\[xc\]). In the following, however, we will frequently use the short term ‘density matrix’ for (\[xc\]) since the correlated density matrix (\[xc1\]) is not considered in this work. Moreover, we only study systems and wave functions which are translationally invariant. Therefore we drop lattice site indices whenever this does not create ambiguities.
Constraints {#con1}
-----------
As shown in Refs. [@buenemann1998; @buenemann2005], it is most convenient for the evaluation of Gutzwiller wave functions in infinite dimensions to impose the following (local) constraints $$\begin{aligned}
\label{1.10a}
\langle\hat{P}^{\dagger}\hat{P}^{}\rangle_{\Psi_0}&=&1\;,\\
\label{1.10b}
\langle \hat{c}^{\dagger}_{\sigma} \hat{P}^{\dagger}\hat{P}^{} \
\hat{c}^{}_{\sigma'}\rangle_{\Psi_0}&=&\langle
\hat{c}^{\dagger}_{\sigma}\hat{c}^{}_{\sigma'} \rangle_{\Psi_0}\;. \end{aligned}$$ Note that moving the operator $\hat{P}^{\dagger}\hat{P}^{}$ relative to $\hat{c}^{\dagger}_{\sigma}$ or $\hat{c}^{}_{\sigma'}$ in (\[1.10b\]) does not alter the whole set of constraints. With the explicit form of the correlation operator (\[1.3\]) and an orbital basis with a diagonal local density matrix, $$\label{xc2}
C_{\sigma,\sigma'}=\delta_{\sigma,\sigma'}n_{\sigma}\;,$$ the constraints read as $$\begin{aligned}
\label{5.5}
\sum_{\Gamma,\Gamma_1,\Gamma_2}
\lambda_{\Gamma,\Gamma_1}^{*}\lambda_{\Gamma,\Gamma_2}^{}
m^{0}_{\Gamma_1,\Gamma_2}&=&1\;,\\\label{5.5b}
\sum_{\Gamma,\Gamma_1,\Gamma_2}
\lambda_{\Gamma,\Gamma_1}^{*}\lambda_{\Gamma,\Gamma_2}^{}
m^{0}_{\Gamma_1\cup \sigma,\Gamma_2\cup \sigma'}
&=&\delta_{\sigma,\sigma'}n_{\sigma}\;,\end{aligned}$$ where $$\begin{aligned}
\label{wet}
|\Gamma\cup \sigma \rangle
&\equiv& \hat{c}^{\dagger}_{\sigma}|\Gamma \rangle
=\sum_{I(\sigma \notin I)}T_{I,\Gamma}|I \cup \sigma\rangle\;,\\\label{wet2}
m^{0}_{\Gamma,\Gamma'}&=&\langle \hat{m}_{\Gamma,\Gamma'} \rangle_{\Psi_0}=
\sum_{I}T_{I,\Gamma}T^*_{I,\Gamma'}m^{0}_{I}\;,\\
m^{0}_{I}&=&\prod_{\sigma \in I}n_{\sigma}\prod_{\sigma \notin I}(1-n_{\sigma}) \;.\end{aligned}$$ For a general orbital basis the explicit form of the constraints is given in Appendix \[ka\].
Expectation values {#exp}
------------------
Each local operator $\hat{O}_i$, e.g., the local Hamiltonian (\[4.10a\]), can be written as $$\hat{O}_i=\sum_{\Gamma,\Gamma'}O_{\Gamma,\Gamma'}\hat{m}_{i;\Gamma,\Gamma'}\;.$$ In infinite dimensions, its expectation value with respect to (\[1.3\]) is given as $$\langle
\hat{O}\rangle_{\Psi_{\rm G}}
=\sum_{\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4}
O_{\Gamma_2,\Gamma_3}
\lambda_{\Gamma_2,\Gamma_1}^{*}\lambda_{\Gamma_3,\Gamma_4}^{}
m^0_{\Gamma_1,\Gamma_4}\;,$$ where the expectation values $m^0_{\Gamma,\Gamma'}$ have been introduced in (\[wet2\]). Hence, the expectation value of the local Hamiltonian (\[eft\]) becomes $$\label{kdr}
\langle
\hat{H}_{i,{\rm loc}}\rangle_{\Psi_{\rm G}}
=\sum_{\Gamma,\Gamma_1,\Gamma_2}E_{\Gamma}
\lambda_{\Gamma,\Gamma_1}^{*}\lambda_{\Gamma,\Gamma_2}^{}
m^0_{\Gamma_1,\Gamma_2}\;.$$
The expectation value for a hopping operator in infinite dimensions has the form $$\label{8.410}
\big \langle \hat{c}_{i,\sigma_1}^{\dagger}\hat{c}_{j,\sigma_2}^{\phantom{+}} \big \rangle_{\Psi_{\rm G}}
=\sum_{\sigma'_1,\sigma'_2}q_{\sigma_1}^{\sigma'_1}\left( q_{\sigma_2}^{\sigma'_2}\right)^{*}\big \langle
\hat{c}_{i,\sigma'_1}^{\dagger}\hat{c}_{j,\sigma'_2}^{\phantom{+}} \big \rangle_{\Psi_{0}}\;,$$ where, for an orbital basis with diagonal local density matrix, the (local) renormalisation matrix reads $$\begin{aligned}
\nonumber
q_{\sigma}^{\sigma'}&=&\frac{1}{n_{\sigma'}}
\sum_{\Gamma_1\ldots\Gamma_4}\lambda^{*}_{\Gamma_2,\Gamma_1}
\lambda^{}_{\Gamma_3,\Gamma_4}
\langle \Gamma_2|
{\hat{c}^{\dagger}}_{\sigma}
|\Gamma_3\rangle\\\label{8.460}
&&\times \Big \langle
\big (|\Gamma_1 \rangle
\langle \Gamma_4 | {\hat{c}^{\phantom{\dagger}}}_{\sigma'}\big )
\Big \rangle_{\Psi_0}\;.\end{aligned}$$ The expressions for the on-site energy and the renormalisation matrix with a general orbital basis are given in Appendix \[ka\].
Energy functional
-----------------
In a translationally invariant system, the expectation values, which we introduced in the previous section, lead to the following variational energy functional (per lattice site) $$\begin{aligned}
\label{ap7.1}
E_{\rm G}\big (\lambda_{\Gamma,\Gamma'},{\left| \Psi_0 \right \rangle }\big)&=&
\sum_{\substack{\sigma_1,\sigma_2 \\ \sigma'_1,\sigma'_2}}
q^{\sigma'_1}_{\sigma_1}\left(q^{\sigma'_2}_{\sigma_2}\right)^*
E_{\sigma_1,\sigma_2,\sigma'_1,\sigma'_2}\\\nonumber
&&+
\sum_{\Gamma,\Gamma_1,\Gamma_2}E_{\Gamma}
\lambda_{\Gamma,\Gamma_1}^{*}\lambda_{\Gamma,\Gamma_2}^{}
m^0_{\Gamma_1,\Gamma_2}\;.\end{aligned}$$ Here, we introduced the tensor $$\begin{aligned}
\label{ap7.2}
E_{\sigma_1,\sigma_2,\sigma'_1,\sigma'_2}&\equiv&
\frac{1}{L}\sum_{i\neq j} t^{\sigma_1,\sigma_2}_{i,j}
\langle
{\hat{c}^{\dagger}}_{i,\sigma'_1}{\hat{c}^{\phantom{\dagger}}}_{j,\sigma'_2}
\big \rangle_{\Psi_0}\\
&=& \frac{1}{L}\sum_{{{\bm k }}}
\varepsilon_{{{\bm k }};\sigma_1,\sigma_2}
\big \langle
{\hat{c}^{\dagger}}_{{{\bm k }},\sigma'_1}{\hat{c}^{\phantom{\dagger}}}_{{{\bm k }},\sigma'_2}
\big \rangle_{\Psi_0}\end{aligned}$$ with the bare dispersion $$\varepsilon_{{{\bm k }};\sigma,\sigma'}\equiv\frac{1}{L}\sum_{i\neq j}
t^{\sigma,\sigma'}_{i,j}
e^{{{\rm i}}{{\bm k }}({{\bm R }}_{i}-{{\bm R }}_{j}) }\;.$$ The energy (\[ap7.1\]) is a function of $\lambda_{\Gamma,\Gamma'}$ and ${\left| \Psi_0 \right \rangle }$ where ${\left| \Psi_0 \right \rangle }$ enters (\[ap7.1\]), (\[ap7.2\]) solely through the (non-interacting) density matrix $\tilde{\rho}$ with the elements $$\rho_{(i\sigma),(j\sigma')}\equiv
\langle \hat{c}_{j,\sigma'}^{\dagger}
\hat{c}_{i,\sigma}^{\phantom{+}}\rangle_{\Psi_0}\;.$$ Therefore, the energy $$E_{\rm G}=E_{\rm G}(\lambda_{\Gamma,\Gamma'},\tilde{\rho})$$ has to be minimised with respect to the variational parameters $\lambda_{\Gamma,\Gamma'}$ and the density matrix $\tilde{\rho}$ obeying the constraints (\[5.5\]), (\[5.5b\]), (or (\[qwa\]), (\[qwa2\])) and $$\label{16}
\tilde{\rho}^2=\tilde{\rho}\;.$$ This additional constraint ensures that $\tilde{\rho}$ corresponds to a single-particle wave function.
Numerical Minimisation of the Gutzwiller Energy Functional {#app7}
==========================================================
In principle, it is conceivable to minimise the energy with respect to the variational parameters $\lambda_{\Gamma,\Gamma'}$ and the density matrix $\tilde{\rho}$ simultaneously. However, we found it more efficient to use consecutive cycles of ‘inner minimisations’ (with respect to $\lambda_{\Gamma,\Gamma'}$ and with fixed $\tilde{\rho}$) and ‘outer minimisations’ (with respect to $\tilde{\rho}$ and with fixed $\lambda_{\Gamma,\Gamma'}$) until a self-consistent minimum is reached.
In the following we assume that all quantities in the energy functional and in the constraints are real. This is allowed since, in case of complex variational parameter or constraints (\[5.5\]), (\[5.5b\]), we may introduce the (independent) real and imaginary parts of these quantities.
‘Inner’ Minimisation {#app7.1}
--------------------
Before we explain our minimisation algorithm in Sect. \[inmin\], it is essential to resolve the fundamental structure of our energy function.
### Structure of the energy function
For a fixed density matrix $\tilde{\rho}$, the energy function is given as $$\begin{aligned}
\nonumber
E_{\rm G}({{\bm v }})&=&
\sum_{\sigma_1,\sigma_2,\sigma'_1,\sigma'_2}
q^{\sigma'_1}_{\sigma_1}({{\bm v }})q^{\sigma'_2}_{\sigma_2}({{\bm v }})
E_{\sigma_1,\sigma_2,\sigma'_1,\sigma'_2}\\\label{ap7.1dd}
&&+\sum_{Z,Z'}U_{Z,Z'}
v_Zv_{Z'}\;,\end{aligned}$$ where we used the abbreviation $v_Z$ for the $n_{\rm v}$ variational parameters $$\label{ap7.1d}
v_Z=\frac{\lambda_{\Gamma,\Gamma'}}{\sqrt{m_{\Gamma}^0 m_{\Gamma'}^0}}\;,$$ which are considered as the elements of a vector ${{\bm v }}$. In our numerical calculations we found that the inner minimisation, as it will be described in Sect. \[inmin\], is much faster if we use the variational parameters (\[ap7.1d\]) instead of $\lambda_{\Gamma,\Gamma'}$.
The renormalisation matrix $$\label{ap7.3}
q^{\sigma'}_{\sigma}({{\bm v }})=\sum_{Z,Z'}S^{\sigma'}_{\sigma}(Z,Z')
v_Zv_{Z'}$$ and the $n_{\rm c}$ (independent) constraints (\[5.5\]), (\[5.5b\]), which we denote as $$\label{ap7.7}
g_{l}({{\bm v }})=\sum_{Z,Z'}f_{l}(Z,Z')v_Zv_{Z'}-g^0_{l}=0\;\;\;\;\;(l=1,\ldots,n_{\rm c})\;,$$ are quadratic functions of the variational parameters $v_Z$. The numbers $g^0_{l}$ in (\[ap7.7\]) correspond to the r.h.s. of Eqs. (\[5.5\]), (\[5.5b\]). Note that, for a fixed density matrix $\tilde{\rho}$, the coefficients $C_{Z,Z'}=\{S^{\sigma'}_{\sigma}(Z,Z'),f_{l}(Z,Z'),U_{Z,Z'}\}$ need to be calculated only once. Moreover, we are free to work with an orbital basis with a diagonal local density matrix, which allows us to calculate these coefficients with the simplified energy expressions introduced in Sect. \[chap2b\]. It is important in our algorithm that the coefficients $C_{Z,Z'}$ are stored in the main memory of the computer because, in this way, derivatives of all quadratic functions can be calculated very fast, see below. Even for large numbers $n_{\rm v}$ of variational parameters this can be achieved, since only a small fraction of the coefficients $C_{Z,Z'}$ is, in fact, finite and needs to be stored. In case that the main-storage capacity is exceeded, there are several strategies to reduce the number of variational parameters, which we have tested. They are discussed in Appendix \[redpar\].
The energy functional can be further simplified if we introduce the matrix $$\label{ap7.5}
r^{\sigma'}_{\sigma}({{\bm v }})
\equiv\sum_{Z,Z'}R^{\sigma'}_{\sigma}(Z,Z')
v_Zv_{Z'}$$ with the coefficients $$\label{ap7.6}
R^{\sigma'_1}_{\sigma_1}\equiv
\sum_{\sigma_2,\sigma'_2}E_{\sigma_1,\sigma_2,\sigma'_1,\sigma'_2}
S^{\sigma'_2}_{\sigma_2}(Z,Z')\;.$$ It allows us to write the energy as $$\label{ap7.4}
E_{\rm G}({{\bm v }})=
\sum_{\sigma_1,\sigma'_1}
q^{\sigma'_1}_{\sigma_1}({{\bm v }})r^{\sigma'_1}_{\sigma_1}({{\bm v }})+\sum_{Z,Z'}U_{Z,Z'}
v_Zv_{Z'}\;.$$ Note that the coefficients in (\[ap7.5\]) also need to be calculated only once in an inner minimisation and should be stored in the main memory. In this way, the energy (\[ap7.4\]) and its gradient ${{\bm E }}({{\bm v }})$ with the elements $$\begin{aligned}
\nonumber
E^{}_Z({{\bm v }})&\equiv &\frac{\partial }{\partial v_{Z}}
E_{\rm G}({{\bm v }})= 2\sum_{Z'}\Big[
\sum_{\sigma_1,\sigma'_1}
\big(q^{\sigma'_1}_{\sigma_1}({{\bm v }})R^{\sigma'_1}_{\sigma_1}(Z,Z')\\\label{ap7.9}
&&+
r^{\sigma'_1}_{\sigma_1}({{\bm v }})S^{\sigma'_1}_{\sigma_1}(Z,Z')\big)
+U_{Z,Z'}\Big]v_{Z'}
\;\end{aligned}$$ can be calculated very fast. The same holds for the gradients ${{\bm F }}^{l}({{\bm v }})$ of the constraints which have the elements $$\label{ap7.8}
F^l_Z({{\bm v }})\equiv \frac{\partial }{\partial v_{Z}}
g_{l}({{\bm v }})= 2\sum_{Z'}f_l(Z,Z')
v_{Z'}\;.$$ Note that in (\[ap7.9\]) and (\[ap7.8\]) we have used the symmetry $C_{Z,Z'}=C_{Z',Z}$, which we are free to impose.
### Algorithm for the inner minimisation {#inmin}
We aim at a minimisation of the energy (\[ap7.4\]) in the manifold $\mathcal{M}_{\rm c}$ defined by the constraints (\[ap7.7\]). To this end, we can always start our minimisation in the uncorrelated limit, i.e., at the point ${{\bm v }}_0$ (with $\lambda_{\Gamma,\Gamma'}=\delta_{\Gamma,\Gamma'}$) for which ${{\bm v }}_0 \in \mathcal{M}_{\rm c}$ is automatically fulfilled. We found numerical strategies that try to move [*exactly*]{} along $\mathcal{M}_{\rm c}$ to be quite cumbersome. Therefore, starting from a certain point ${{\bm v }}_0\in \mathcal{M}_{\rm c}$, we allow the minimisation algorithm to violate the constraints by making ‘short’ steps to points ${{\bm v }}_1\notin \mathcal{M}_{\rm c}$. To keep the violation of the constraints minimal, these steps have to take place in the subspace $\mathcal{M}_{\parallel}({{\bm v }}_0)$ that is tangential to $\mathcal{M}_{\rm c}$ at the point ${{\bm v }}_0$. The optimal direction of a step in $\mathcal{M}_{\parallel}({{\bm v }}_0)$ is determined by the tangential component of the gradient ${{\bm E }}({{\bm v }}_0)$ since it leads to a decrease of the energy. In summary, and more precisely, these ideas lead to the following algorithm for the inner minimisation:
- Find a point ${{\bm v }}_0$ in the variational parameter space $\mathcal{V}$ that obeys the constraints (\[ap7.7\]) (i.e, ${{\bm v }}_0\in \mathcal{M}_{\rm c}$).
- Determine the gradients ${{\bm F }}^{l}({{\bm v }}_0)$ and ${{\bm E }}({{\bm v }}_0)$.
- Calculate the component ${{\bm E }}_{\parallel}({{\bm v }}_0)$ of ${{\bm E }}({{\bm v }}_0)$ in $\mathcal{M}_{\parallel}({{\bm v }}_0)$ by the following procedure. The gradient ${{\bm E }}({{\bm v }}_0)$ is written as $$\label{ap7.10}
{{\bm E }}({{\bm v }}_0)={{\bm E }}_{\parallel}({{\bm v }}_0)+
{{\bm E }}_{\perp}({{\bm v }}_0)\;,$$ where the tangential component ${{\bm E }}_{\parallel}({{\bm v }}_0)$ is defined by $$\label{ap7.11}
{{\bm E }}_{\parallel}({{\bm v }}_0)\cdot {{\bm F }}^{l}({{\bm v }}_0)=0
\;\;\;\forall l\;.$$ The perpendicular component can be expressed as a linear combination $$\label{ap7.12}
{{\bm E }}_{\perp}({{\bm v }}_0)= \sum_{l=1}^{n_{\rm c}}\alpha_l
{{\bm F }}^{l}({{\bm v }}_0)$$ of the vectors ${{\bm F }}^{i}({{\bm v }}_0)$. In order to determine the coefficients $\alpha_i$, we multiply equation (\[ap7.10\]) with a vector ${{\bm F }}^{m}({{\bm v }}_0)$ and use the expansion (\[ap7.12\]). This leads to $$\begin{aligned}
\label{ap7.13}
{{\bm E }}({{\bm v }}_0)\cdot {{\bm F }}^{m}({{\bm v }}_0)
&=&\sum_{l}
{{\bm F }}^{l}({{\bm v }}_0)\cdot{{\bm F }}^{m}({{\bm v }}_0)\alpha_l\\\nonumber
&=&
\sum_{l}W_{m,l}({{\bm v }}_0)\alpha_l\;,
\end{aligned}$$ where we used equation (\[ap7.11\]) and introduced the (symmetric) matrix $\tilde{W}({{\bm v }})$ with the elements $$\label{ap7.14}
W_{m,l}({{\bm v }})\equiv {{\bm F }}^{l}({{\bm v }})\cdot{{\bm F }}^{m}({{\bm v }})\;.$$ The linear equations (\[ap7.13\]) for $\alpha_l$ have a unique solution, as long as the vectors ${{\bm F }}^{l}({{\bm v }}_0)$ are linearly independent. A linear dependency of these vectors can only arise if certain constraints (\[ap7.7\]) are redundant. In that case, the redundant constraints have to be eliminated right from the start. With the coefficients $\alpha_l$, we calculate the tangential component $$\label{ap7.16}
{{\bm E }}_{\parallel}({{\bm v }}_0)={{\bm E }}({{\bm v }}_0)-
\sum_{l}\alpha_l{{\bm F }}^{l}({{\bm v }}_0)\;.$$ of ${{\bm E }}({{\bm v }}_0)$.
- Make a ‘proper’ step in the direction of $-{{\bm E }}_{\parallel}({{\bm v }}_0)$ to a new vector $$\label{ap7.17}
\bar{{{\bm v }}}_1={{\bm v }}_0-\beta {{\bm E }}_{\parallel}({{\bm v }}_0)\;.$$ For the choice of the parameter $\beta$, various strategies are conceivable. Since the point $\bar{{{\bm v }}}_1$ is not in $\mathcal{M}_{\rm c}$, the energy gain is not necessarily a useful criterion and it is also rather time consuming to be determined. Instead, we calculate $$\label{ap7.18}
\Delta g(\bar{{{\bm v }}}_1) \equiv \sum_{l}[g_l(\bar{{{\bm v }}}_1)]^2\geq 0$$ as a measure for the violation of the constraints and choose the parameter $\beta$ such that $\Delta g$ does not exceed a certain critical value $\Delta g_{\rm c}$. This critical value should be automatically adjusted by the algorithm to ensure that, after returning to the hyper-surface $\mathcal{M}_{\rm c}$, there is a sufficient energy gain.
- In order to return to $\mathcal{M}_{\rm c}$ from the point $\bar{{{\bm v }}}_1\notin \mathcal{M}_{\rm c}$, the following algorithm turned out to be very useful. We seek a vector ${{\bm v }}_1$ that solves the constraint equations $g_l({{\bm v }}_1)=0$ and is as close as possible to $\bar{{{\bm v }}}_1$. To this end, we could calculate the gradients ${{\bm F }}^{l}(\bar{{{\bm v }}}_1)$ and try to solve the set of equations $$\label{ap7.19}
g_l\bigg(\bar{{{\bm v }}}_1+\sum_m \gamma_m{{\bm F }}^{m}(\bar{{{\bm v }}}_1)\bigg)=0$$ by a proper choice of the coefficients $\gamma_m$. Such an exact solution of equations (\[ap7.19\]), however, is quite time consuming. Therefore, we consider the linear set of equations $$\label{ap7.20}
g_l(\bar{{{\bm v }}}_1)+\sum_{m}W_{l,m}(\bar{{{\bm v }}}_1)\gamma_m=0\;,$$ which results from an expansion of (\[ap7.19\]) to leading order in $\gamma_m$. Equations (\[ap7.20\]) can be readily solved with respect to $\gamma_m$. This yields a new vector $$\label{ap7.22}
\bar{{{\bm v }}}_1 \to\bar{{{\bm v }}}'_1=\bar{{{\bm v }}}_1+
\sum_m\gamma_m{{\bm F }}^{m}(\bar{{{\bm v }}}_1)\;.$$ which, in general, is not yet a solution of $g_l(\bar{{{\bm v }}}'_1)=0$. However, this vector is closer to $\mathcal{M}_{\rm c}$ than $\bar{{{\bm v }}}_1$ because $\Delta g(\bar{{{\bm v }}}'_1)<\Delta g(\bar{{{\bm v }}}_1)$. By an iteration of equations (\[ap7.20\])-(\[ap7.22\]) we eventually approach a vector ${{\bm v }}_1\in\mathcal{M}_{\rm c}$. Note that the fast convergence of this procedure is crucial for our algorithm. We have tried several other ways to return to $\mathcal{M}_{\rm c}$ that all turned out to be much slower.
- If $E_{\rm G}({{\bm v }}_1)<E_{\rm G}({{\bm v }}_0)$ we restart the procedure at point ii) with ${{\bm v }}_0$ replaced by ${{\bm v }}_1$. In case that $E_{\rm G}({{\bm v }}_1)>E_{\rm G}({{\bm v }}_0)$, the critical value $\Delta g_{\rm c}$ has to be lowered and the algorithm continues with point iv). A useful measure for the convergence of the whole iteration is the norm of ${{\bm E }}_{\parallel}$. This number goes to zero near a minimum ${{\bm v }}_{\rm min}$ of the energy functional $E_{\rm G}({{\bm v }})$ for vectors ${{\bm v }}\in \mathcal{M}_{\rm c}$.
‘Outer’ Minimisation {#app7.2}
--------------------
With the optimum variational parameters ${{\bm v }}^{\rm min}$ from the inner minimisation, described in Sect. \[app7.1\], we have to minimise the energy $$\begin{aligned}
\label{4.7}
E_{\rm G}(\tilde{\rho})&=&\sum_{i\ne j}\sum_{\sigma,\sigma'}
\bar{t}^{\sigma,\sigma'}_{i,j}(\tilde{\rho})\rho_{(j\sigma'),(i\sigma)}\\\nonumber
&&+L\sum_{Z,Z'}U_{Z,Z'}(\tilde{\rho})
v^{\rm min}_Zv^{\rm min}_{Z'}\end{aligned}$$ with respect to $\tilde{\rho}$. Here we introduced the renormalised hopping parameters $$\label{4.7b}
\bar{t}^{\sigma_1,\sigma_2}_{i,j}(\tilde{\rho})=\sum_{\sigma'_1,\sigma'_2}
q^{\sigma_1}_{\sigma'_1}(\tilde{\rho})q^{\sigma_2}_{\sigma'_2}(\tilde{\rho})
t^{\sigma'_1,\sigma'_2}_{i,j}$$ and the renormalisation factors $$\label{ap7.3sss}
q^{\sigma'}_{\sigma}(\tilde{\rho})=\sum_{Z,Z'}
S^{\sigma'}_{\sigma}(Z,Z';\tilde{\rho})
v^{\rm min}_Zv^{\rm min}_{Z'}\;.$$ In addition, the (independent) constraints (\[qwa\]), (\[qwa2\]), $$\begin{aligned}
\label{ap7.7b}
&&g_{l}(\tilde{\rho})=\sum_{Z,Z'}f_{l}(Z,Z',\tilde{\rho})
v^{\rm min}_Zv^{\rm min}_{Z'}-g^0_{l}=0\\\nonumber
&&(l=1,\ldots,n_{\rm c})\;,\end{aligned}$$ and (\[16\]) need to be obeyed.
The local elements of the density matrix $$\label{782}
C_{\sigma,\sigma'} =\rho_{(i\sigma'),(i\sigma)}$$ play a special role in the energy function because only they enter the coefficients in (\[4.7\]), (\[ap7.3sss\]), (\[ap7.7b\]), $$\begin{aligned}
U_{Z,Z'}(\tilde{\rho})&=&U_{Z,Z'}(\tilde{C})\;\;, \\
S^{\sigma'}_{\sigma}(Z,Z';\tilde{\rho})&=&S^{\sigma'}_{\sigma}(Z,Z';\tilde{C})\;\; , \;\;\\
f_{l}(Z,Z',\tilde{\rho})&=&f_{l}(Z,Z',\tilde{C})\;.\end{aligned}$$ If they are kept fixed, only the hopping term in (\[4.7\]) and the constraint (\[16\]) need to be taken into account in the minimisation with respect to $\tilde{\rho}$. This leads to a minimisation strategy which we discuss in Sect. \[sw2\]. An alternative way of minimising (\[4.7\]) with respect to [*all*]{} elements of $\tilde{\rho}$ will be introduced in Sect. \[sw1\].
The Hermiticity of the density matrix, $\tilde{\rho}^{\dagger}=\tilde{\rho}^{}$, is a constraint which is obeyed automatically in our outer minimisation algorithm in Sect. \[sw1\]. To this end, however, the functional dependence of the energy with respect to $\tilde{\rho}^{}$, which is not unique, must be chosen such that $$\label{axd}
\frac{\partial E_{\rm G}}{\partial \rho_{(i\sigma),(j\sigma')}}=
\left(\frac{\partial E_{\rm G}}{\partial \rho_{(j\sigma'),(i\sigma)}}\right)^*\;.$$ This can always be achieved by employing the Hermiticity of $\tilde{\rho}$. We further assume that equation (\[axd\]) is also satisfied by the constraints (\[ap7.7b\]).
### Fixed local density matrix {#sw2}
If the local density matrix is fixed, we have to minimise $$\label{4.7sss}
E_{{\rm G},0}(\tilde{\rho})\equiv\sum_{i\ne j}\sum_{\sigma,\sigma'}
\bar{t}^{\sigma,\sigma'}_{i,j}\rho_{(j\sigma'),(i\sigma)}$$ with respect to $\tilde{\rho}$ obeying the constraints (\[16\]) and (\[782\]). We impose these constraints by means of Lagrange parameters $\eta_{\sigma,\sigma'}$ and $\Omega_{(i\sigma),(j\sigma')}$, which leads to the ‘Lagrange functional’ $$\begin{aligned}
\nonumber
L_{\rm G}&\equiv& E_{{\rm G},0}(\tilde{\rho})
-\sum_{\sigma,\sigma'}\eta_{\sigma,\sigma'}
\sum_i(C_{\sigma,\sigma'}-\rho_{(i\sigma'),(i\sigma)})\\\label{4.7bss}
&&-\sum_{i,j}\sum_{\sigma,\sigma'}\Omega_{(i\sigma),(j\sigma')}
[\tilde{\rho}^2-\tilde{\rho}]_{(j\sigma'),(i\sigma)}\;.
\end{aligned}$$ As recalled in Appendix \[ap3\], the minimisation of (\[4.7bss\]) with respect to $\tilde{\rho}$ leads to the effective single-particle Hamiltonian $$\label{tzs}
\hat{H}^{\rm eff}_0=\sum_{i\ne j}\sum_{\sigma,\sigma'}(\bar{t}^{\sigma,\sigma'}_{i,j}
+\delta_{i,j}\eta_{\sigma,\sigma'}){\hat{c}^{\dagger}}_{i,\sigma}{\hat{c}^{\phantom{\dagger}}}_{j,\sigma'}\;.$$ The optimum single-particle state ${\left| \Psi_0 \right \rangle }$ is the ground state of $\hat{H}^{\rm eff}_0$ where the parameters $\eta_{\sigma,\sigma'}$ have to be chosen such that $C_{\sigma,\sigma'}=\langle {\hat{c}^{\dagger}}_{i,\sigma}{\hat{c}^{\phantom{\dagger}}}_{i,\sigma'} \rangle_{\Psi_0}$ is satisfied.
With the state ${\left| \Psi \right \rangle }_0$, we may determine a new tensor (\[ap7.2\]) and start another run of the inner minimisation until self-consistency with respect to ${\left| \Psi \right \rangle }_0$ is reached. In this way, we find the ground-state energy $E=E_0(\tilde{C} )$ for a fixed local density matrix $C_{\sigma,\sigma'}$. To obtain the total variational ground-state energy, $E_0(\tilde{C} )$ still needs to be minimised with respect to $\tilde{C}$ with the constraint of total particle number conservation, $\sum_{\sigma}C_{\sigma,\sigma}=N/L$. Alternatively, one may start a self-consistency cycle of inner and outer minimisation for a fixed set of ‘effective crystal fields’ $\eta_{\sigma,\sigma'} $ (and a fixed particle number). This defines an energy function $E_0(\tilde{\eta})$ which has to be minimised with respect to $\eta_{\sigma,\sigma'}$.
Obviously, these two ways of minimising the energy are feasible only when the number $n_{\rm i}$ of independent elements in $\tilde{C}$ (or fields $\tilde{\eta}$) is small. It can also be useful, when there are physical reasons to minimise $E_0(\tilde{C} )$ (or $E_0(\tilde{\eta})$) only in some subspace of possible density matrices $\tilde{C}$ (or fields $\tilde{\eta}$)). Such a strategy has been used, e.g., in our calculations on the spin-orbit coupling effects in nickel. There, we could clearly identify the relevant fields $\eta_{\sigma}$: the dominant term in nickel is the effective exchange splitting accompanied by a smaller orbital-energy splitting and an effective spin-orbit coupling. In this way, the energy $E_0(\tilde{\eta})$ had to be minimised only in a $3$-dimensional subspace of fields $\tilde{\eta}$. However, such a procedure is bound to fail when the number $n_{\rm i}$ of parameters $\eta_{\sigma,\sigma'}$ is too large and cannot be reduced by any physical arguments. In that case, one may use the algorithm which we introduce in the following section.
### Unrestricted outer minimisation {#sw1}
In order to minimise the energy with respect to [*all*]{} elements of the density matrix we impose the constraints (\[ap7.7b\]) by means of Lagrange parameters $\Lambda_l$. This leads us to the functional $$\begin{aligned}
\label{sgh}
L_{\rm G}&\equiv&
E_{\rm G}(\tilde{\rho})-\sum_l\Lambda_lg_{l}(\tilde{\rho})\\\nonumber
&&-\sum_{i,j}\sum_{\sigma,\sigma'}\Omega_{(i\sigma),(j\sigma')}
[\tilde{\rho}^2-\tilde{\rho}]_{(j\sigma'),(i\sigma)}\end{aligned}$$ where $E_{\rm G}(\tilde{\rho})$ has been defined in (\[4.7\]). The minimisation with respect to $\rho$ yields again an effective single-particle Hamiltonian of the form (\[tzs\]) where the fields $\eta_{\sigma,\sigma'}$ are now given as $$\label{sdfj}
\eta_{\sigma,\sigma'}=\frac{\partial}{\partial C_{\sigma,\sigma'}}E_{\rm G}(\tilde{\rho})-
\sum_l\Lambda_l\frac{\partial}{\partial C_{\sigma,\sigma'}}g_{l}(\tilde{\rho})\;.$$ To determine these fields we need to calculate the Lagrange parameters $\Lambda_l$. This can by achieved if we use the fact that, in the variational ground state, the Lagrange functional (\[sgh\]) is also minimal with respect to the variational parameters $v_Z$. This leads to the equations $$\frac{\partial}{\partial v_{Z}}E_{\rm G}(\tilde{\rho},{{\bm v }})\Big|_{{{\bm v }}
={{\bm v }}^{\rm min}}-\sum_l\Lambda_l\frac{\partial}{\partial v_{Z}}
g_{l}(\tilde{\rho},{{\bm v }})\Big|_{{{\bm v }}={{\bm v }}^{\rm min}}=0$$ which can be written in matrix-vector form as $$\label{dfg}
\tilde{G}{{\bm \Lambda }}={{\bm E }}\;,$$ where $\tilde{G}$ and ${{\bm E }}$ have the elements $$\begin{aligned}
\label{dfg7}
\tilde{G}_{l,Z}&\equiv& \frac{\partial}{\partial v_{Z}}
g_{l}(\tilde{\rho},{{\bm v }})\Big|_{{{\bm v }}={{\bm v }}^{\rm min}}\;,\\
E_{Z}&\equiv&\frac{\partial}{\partial v_{Z}}
E_{\rm G}(\tilde{\rho},{{\bm v }})\Big|_{{{\bm v }}
={{\bm v }}^{\rm min}}\;.\end{aligned}$$ The number of equations in (\[dfg\]) is usually much larger then the number of parameters $\Lambda_l$. For physical reasons, however, Eq. (\[dfg\]) must have a unique solution. Therefore we can alternatively solve the equation $$\label{dfgd}
\tilde{G}^{\rm T}\tilde{G}{{\bm \Lambda }}=\tilde{G}^{\rm T}{{\bm E }}\;,$$ since it gives us the same solution for ${{\bm \Lambda }}$ as (\[dfg\]).
Note that the calculation of the derivatives in (\[sdfj\]) is much easier if we work with an orbital basis with a diagonal density matrix, see Appendix \[ka4\]. This leads us to the following algorithm for the outer minimisation.
- Set $q^{\sigma'}_{\sigma}=\delta_{\sigma,\sigma'}$ and choose a reasonable set of fields $\eta^{({\rm i})}_{\sigma,\sigma'}$, e.g., $\eta^{({\rm i})}_{\sigma,\sigma'}=\varepsilon_{\sigma,\sigma'}$ with the bare on-site energies $\varepsilon_{\sigma,\sigma'}$ in the local Hamiltonian (\[4.10a\]).
- Find the ground state ${\left| \Psi_0 \right \rangle }$ of the effective Hamiltonian (\[tzs\]) with $\eta_{\sigma,\sigma'}=\eta^{({\rm i})}_{\sigma,\sigma}$ and determine $C_{\sigma,\sigma'}$. If $C_{\sigma,\sigma'}$ is not diagonal, find an orbital basis with a diagonal local density matrix. Continue the algorithm with this new basis and its values for $C_{\sigma,\sigma'}=\delta_{\sigma,\sigma'}n_{\sigma}$ and $E_{\sigma_1,\sigma_2,\sigma'_1,\sigma'_2}$.
- Carry out an inner minimisation, as described in section \[app7.1\], and determine the Lagrange parameters $\Lambda_l$ by solving Eq. (\[dfgd\]).
- Use Eq. (\[sdfj\]) to determine a new set of parameters $\eta^{({\rm o})}_{\sigma,\sigma'}$. Set $\eta^{({\rm i})}_{\sigma,\sigma'}\equiv \eta^{({\rm o})}_{\sigma,\sigma'}$ and go back to ii) until self-consistency, $\eta^{({\rm o})}_{\sigma,\sigma'}\approx
\eta^{({\rm i})}_{\sigma,\sigma'} $ is reached.
This algorithm obviously relies on a certain ‘proximity’ to the true variational ground-state, in particular, when there is more than one (local) minimum. In the latter case, the algorithm may have to be supported by a preliminary manual scan of the variational space as described in Sect. \[sw2\]. Moreover, it can be necessary to introduce some kind of ’damping‘ by setting $$\eta^{({\rm i})}_{\sigma,\sigma'}\equiv
\eta^{({\rm i})}_{\sigma,\sigma'}
+\beta
(\eta^{({\rm o})}_{\sigma,\sigma'}-\eta^{({\rm i})}_{\sigma,\sigma'})$$ with $0<\beta<1$ instead of $\eta^{({\rm i})}_{\sigma,\sigma'}\equiv \eta^{({\rm o})}_{\sigma,\sigma'}$ in step iv). The value of $\beta$ must be small enough to ensure that the energy decreases in each step of the cycle. In our numerical tests, we found that $\beta$ may sometimes have to be smaller than $1$ even in the immediate vicinity of the variational ground state.
Note that the calculation of the derivatives in (\[sdfj\]) and (\[dfg7\]) in steps iii) and iv) of the algorithm is very much simplified by the fact that the local density matrix is diagonal with respect to ${\left| \Psi_0 \right \rangle }$. This does [*not*]{} mean, however, that the derivatives with respect to non-diagonal elements $C_{\sigma,\sigma'}$ necessarily vanish, see Appendix\[ka4\]. Therefore, the orbital basis will, in general, be changing in each cycle of the algorithm until a self-consistent minimum is reached.
Summary
=======
In summary, we have given a detailed account of a numerical scheme for the minimisation of Gutzwiller energy functionals, which we found to be quite efficient in previous studies on transition metals and transition metal compounds. We are confident that our algorithm is of significant interest for other researchers who intend to apply the multi-band Gutzwiller theory to other materials.
Energy functional for an arbitrary local density matrix {#ka}
=======================================================
The constraints (\[5.5\]), (\[5.5b\]) for a general orbital basis read $$\begin{aligned}
\label{qwa}
\sum_{\Gamma,\Gamma_1,\Gamma_2}
\lambda_{\Gamma,\Gamma_1}^{*}\lambda_{\Gamma,\Gamma_2}^{}
m^{0}_{\Gamma_1,\Gamma_2}&=&1\;,\\\label{qwa2}
\sum_{\Gamma,\Gamma_1,\Gamma_2}
\lambda_{\Gamma,\Gamma_1}^{*}\lambda_{\Gamma,\Gamma_2}^{}
m^{0}_{\Gamma_1\cup \sigma,\Gamma_2\cup \sigma'}
&=&C_{\sigma,\sigma'}\;,\end{aligned}$$ where $$\begin{aligned}
|\Gamma\cup \sigma \rangle
&\equiv& \hat{c}^{\dagger}_{\sigma}|\Gamma \rangle
=\sum_{I(\sigma \notin I)}T_{I,\Gamma}|I \cup \sigma\rangle\;,\\\label{utr}
m^{0}_{\Gamma,\Gamma'}&=&\langle \hat{m}_{\Gamma,\Gamma'} \rangle_{\Psi_0}=
\sum_{I,I'}T_{I,\Gamma}T^*_{I',\Gamma'}m^{0}_{I,I'}\;,\\
m^{0}_{I,I'}&=&\langle \hat{m}_{I,I'} \rangle_{\Psi_0}\;.\end{aligned}$$ The result for the local energy is the same as in Eq. (\[kdr\]) only with $m^{0}_{\Gamma_1,\Gamma_2}$ given by Eq. (\[utr\]).
With Wick’s theorem, the expectation values $m^{0}_{I,I'}$ in (\[utr\]) can be written as the determinant $$\label{miiprime}
m^{0}_{I,I'}=\left|
\begin{array}{cc}
\Omega^{I,I'}&-\Omega^{I,J}\\
\Omega^{J,I'}&\bar{\Omega}^{J,J}
\end{array}
\right|\;.$$ Here, $\Omega_{I,I'}$ are the matrices $$\Omega_{I,I'}=\left(
\begin{array}{cccc}
C_{\sigma_1,\sigma'_1}&C_{\sigma_1,\sigma'_2}&\ldots&C_{\sigma_1,\sigma'_{|I'|}}\\
C_{\sigma_2,\sigma'_1}&C_{\sigma_2,\sigma'_2}&\ldots&C_{\sigma_2,\sigma'_{|I'|}}\\
\ldots&\ldots&\ldots&\ldots\\
C_{\sigma_{|I|},\sigma'_1}&C_{\sigma_{|I|},\sigma'_2}&\ldots&C_{\sigma_{|I|},\sigma'_{|I'|}}
\end{array}
\right)\;,$$ in which the entries are the elements of the uncorrelated local density matrix (\[xc\]), that belong to the configurations $I=(\sigma_1,\ldots,\sigma_{|I|})$ and $I'=(\sigma'_1,\ldots,\sigma'_{|I'|})$. The matrix $\bar{\Omega}^{J,J}$ in (\[miiprime\]) is defined as $$\bar{\Omega}_{J,J}=\left(
\begin{array}{cccc}
1-C_{\sigma_1,\sigma_1}&-C_{\sigma_1,\sigma_2}&\ldots&-C_{\sigma_1,\sigma_{|J|}}\\
-C_{\sigma_2,\sigma_1}&1-C_{\sigma_2,\sigma_2}&\ldots&-C_{\sigma_2,\sigma_{|J|}}\\
\ldots&\ldots&\ldots&\ldots\\
-C_{\sigma_{|J|},\sigma_1}&-C_{\sigma_{|J|},\sigma_2}&\ldots&1-C_{\sigma_{|J|},\sigma_{|J|}}
\end{array}
\right)\;,$$ with $\sigma_i\in J\equiv (1,\ldots,N)\backslash (I\cup I') $.
The renormalisation matrix in (\[8.410\]) has the form $$\begin{aligned}
\label{qmat}
q_{\sigma}^{\sigma'}&=&\sum_{\Gamma_1,\ldots,\Gamma_4}\lambda^{*}_{\Gamma_2,\Gamma_1}
\lambda^{}_{\Gamma_3,\Gamma_4}\langle \Gamma_2|\hat{c}^{\dagger}_{\sigma}
|\Gamma_3 \rangle\\\nonumber
&&\times \sum_{I_1,I_4}T_{I_1,\Gamma_1}T^{*}_{I_4,\Gamma_4}
H^{\sigma'}_{I_1,I_4}\;,\end{aligned}$$ where the matrix $H^{\sigma'}_{I_1,I_4}$ contains three different contributions depending on whether the index $\sigma'$ is an element of $I_1\cap I_4$, $I_4\backslash (I_1\cap I_4)$, or $J=(1,\ldots,N)\backslash(I_1\cup I_4)$. With the abbreviation $f_{\sigma,I}\equiv\langle I |\hat{c}^{\dagger}_{\sigma}\hat{c}^{}_{\sigma} |I \rangle$ we can write $H^{\sigma'}_{I_1,I_4}$ as $$\begin{aligned}
\label{8sgdd}
H^{\sigma'}_{I_1,I_4}&\equiv&(1-f_{\sigma',I_1})\langle I_4 |\hat{c}^{}_{\sigma'} |I_4\cup \sigma' \rangle
m^{0}_{I_1,I_4\cup \sigma'}\\\nonumber
&&+\left(
f_{\sigma',I_4}m^{0}_{I_1\backslash \sigma',I_4}+
(1-f_{\sigma',I_4})m^{0;\sigma'}_{I_1\backslash \sigma',I_4}
\right)\\\nonumber
&&\times \langle I_1 \backslash \sigma' |\hat{c}^{}_{\sigma'} |I_1 \rangle
\;.\end{aligned}$$ The expectation value $m^{0;\sigma'}_{I_1\backslash \sigma',I_4}$ in (\[8sgdd\]) has the same form as the one in (\[miiprime\]), except that the index $J$ has to be replaced by $J \backslash \sigma'$.
Strategies to treat large numbers of ‘inner’ variational parameters {#redpar}
===================================================================
Our algorithm is particularly fast for the inner minimisation if we can store all the second-order coefficients $C_{Z,Z'}$ in the main memory of our computer, see Sect. \[app7.1\]. Unfortunately, this cannot always be achieved in multi-band studies, in particular, when we include non-diagonal variational parameters $\lambda_{\Gamma,\Gamma'}$. In this case we may try to reduce the number of variational parameters, e.g., by symmetry considerations, see Appendix \[redparb\]. Alternatively, one can employ additional numerical schemes that complement our inner minimisation algorithm, see Appendix \[redparc\].
Reduction of the variational space {#redparb}
-----------------------------------
It is obvious that, due to symmetries, many parameters $\lambda_{\Gamma,\Gamma'}$ vanish automatically in the variational ground state and can be discarded from the outset. In order to identify these parameters one may use, e.g., the expectation values (\[wet2\]) which vanish for such parameters.
A further reduction can be achieved if we take only those variational parameters into account which couple states $|\Gamma\rangle,|\Gamma'\rangle$ that belong to the same (degenerate) multiplet of the atomic Hamiltonian in (\[4.10a\]). Such a strategy has been used in our calculations on the spin-orbit coupling effects in nickel [@buenemann2008]. Although clearly an approximation, this scheme is justified since one is usually bound to make similar approximations already on the level of the operators in the local Hamiltonian (\[4.10a\]). For example, in studies on transition metals and their compounds a spherical approximation is often used which allows one to express all Coulomb-interaction parameters by the three Racah or the three Slater–Condon parameters. To go beyond this spherical approximation is actually simple within the Gutzwiller theory, however, it increases the number of independent Coulomb-interaction parameters significantly. Since there exists no established way to calculate these parameters from first principles, they have to be determined by some fitting procedure, which only makes sense if their number is not too large.
For sufficiently large Coulomb interactions, atomic charge fluctuations are significantly suppressed. For example, in elementary nickel with its approximately nine $3d$ electrons per atom the occupation of states with less than six $3d$-electrons is negligibly small. Hence, the variational parameters $\lambda_{\Gamma,\Gamma'}$ of such shells may be assumed to be diagonal or even to vanish.
Additional numerical schemes {#redparc}
----------------------------
In case that, even after all symmetry considerations, the number of variational parameters $\lambda_{\Gamma,\Gamma'}$ is still too large for our inner minimisation algorithm, one may employ one of the following numerical schemes.
The simplest scheme is to split up the whole set of variational parameters into sub-sets, for which the main storage of our computer is adequate and the minimisation algorithm in Sect. \[inmin\] can be applied. The minimisation with respect to each of these sub-sets of parameters has then to be repeated until a total minimum is reached.
Another scheme is based on the observation that the multiplet states $|\Gamma\rangle$ do not necessarily have to be the eigenstates of our local Hamiltonian (\[4.10a\]). Instead, the states $|\Gamma\rangle$ themselves are considered as variational objects in the following algorithm.
- Choose a certain basis of multiplets states $|\Gamma\rangle^{\rm (i)}$
- Set $|\Gamma\rangle=|\Gamma\rangle^{\rm (i)}$ and determine the most ‘relevant’ non-diagonal variational parameters $\lambda_{\Gamma,\Gamma'}$ such that their number still allows for the use of the minimisation algorithm in Sect. \[inmin\]. A criterion for the ‘relevance’ of the parameters $\lambda_{\Gamma,\Gamma'}$ may be the size of the non-interacting expectation value (\[wet2\]). Alternatively one could use the corresponding correlated expectation value which can be calculated in a preceding calculation with a diagonal variational parameter matrix $\lambda_{\Gamma,\Gamma}$.
- Determine the optimum values $\lambda^{\rm opt}_{\Gamma,\Gamma'}$ of the parameters chosen in (ii). Calculate the eigenstates $|\Gamma\rangle^{\rm (o)}$ of the optimal correlation operator $$\hat{P}^{\rm opt}=\sum_{\Gamma,\Gamma'}\lambda^{\rm opt}_{\Gamma,\Gamma'}
\hat{m}_{\Gamma,\Gamma'}\;.$$
- Set $|\Gamma\rangle^{\rm (i)}=|\Gamma\rangle^{\rm (o)}$ and go back to (ii) until self-consistency $|\Gamma\rangle^{\rm (i)}\approx|\Gamma\rangle^{\rm (o)}$ is reached.
We have tested both numerical schemes, discussed in this Appendix. From these preliminary calculations, however, we are not yet able to draw any final conclusions on the efficiency of both approaches.
Minimisation of functions with respect to non-interacting density matrices {#ap3}
==========================================================================
We consider a general function $E(\tilde{\rho})$ of a non-interacting density matrix $\tilde{\rho}$ with the elements $$\rho_{\gamma,\gamma'}=\langle {\hat{c}^{\dagger}}_{\gamma'} {\hat{c}^{\phantom{\dagger}}}_{\gamma}\rangle_{\Phi_0}\;.$$ The fact that $\tilde{\rho}$ is derived from a single-particle product wave function ${\left| \Phi_0 \right \rangle }$ is equivalent to the matrix equation $\tilde{\rho}^2=\tilde{\rho}$. Hence, the minimum of $E(\tilde{\rho})$ in the ‘space’ of all [*non-interacting*]{} density matrices is determined by the condition $$\frac{\partial}{\partial \rho_{\gamma',\gamma}}L(\tilde{\rho})=0\;,$$ where we introduced the ‘Lagrange functional’ $$\begin{aligned}
\label{sfg}
L(\tilde{\rho})&\equiv& E(\tilde{\rho})-\sum_{l,m}\Omega_{l,m}
\big[\tilde{\rho}^2-\tilde{\rho}\big]_{m,l} \\
&=& E(\tilde{\rho})-\sum_{l,m}\Omega_{l,m}\Big(
\sum_p \rho_{m,p} \rho_{p,l}-\rho_{m,l}\Big)\end{aligned}$$ and the matrix $\tilde{\Omega}$ of Lagrange parameters $\Omega_{l,m}$. The minimisation of (\[sfg\]) leads to the matrix equation $$\tilde{H}=\tilde{\rho}\tilde{\Omega}+\tilde{\Omega}\tilde{\rho}-\tilde{\Omega}$$ for the ‘Hamilton matrix’ $\tilde{H}$ with the elements $$H_{\gamma,\gamma'}=
\frac{\partial}{\partial \rho_{\gamma',\gamma}}
E(\tilde{\rho})\;.$$ This equation is satisfied if $\tilde{\rho}^2=\tilde{\rho}$ and $$[\tilde{H},\tilde{\rho}]=0\;.$$ Hence, $\tilde{H}$ and $\tilde{\rho}$ must have the same basis of (single-particle) eigenvectors and, consequently, ${\left| \Phi_0 \right \rangle }$ is the ground state of $$\hat{H}_0^{\rm eff}=\sum_{\gamma,\gamma'}H_{\gamma,\gamma'}
{\hat{c}^{\dagger}}_{\gamma} {\hat{c}^{\phantom{\dagger}}}_{\gamma'}\;.$$
Derivatives of the general energy functional {#ka4}
============================================
In Sect. \[sw1\], we have to calculate the derivative of the ground-state energy and of the constraints with respect to the elements of the local density matrix, see Eq. (\[sdfj\]). Equations (\[qwa\])–(\[8sgdd\]) reveal that, in fact, we only need the derivatives of $m^0_{I,I'}$ (and of $m^{0;\bar{\sigma}}_{I\backslash \bar{\sigma} ,I'}$). For a general density matrix $C_{\sigma,\sigma'}$, their calculation requires an evaluation of determinants such as (\[miiprime\]). However, in Sect. \[sw1\] we work with an orbital basis for which $C_{\sigma,\sigma'}
=\delta_{\sigma,\sigma'}n_{\sigma}$. Hence the derivatives with respect to $C_{\sigma,\sigma'}$ have a much simpler form. For example, for the derivatives of $m^0_{I,I'}$ we find $$\begin{aligned}
\frac{\partial }{\partial C_{\sigma,\sigma}}m^0_{I,I'}=\delta_{I,I'}m^0_{I,I}
\left\{
\begin{array}{cl}
1/n_{\sigma}&{\rm for}\;\; \sigma \in I\\
-1/(1-n_{\sigma})&{\rm for}\;\; \sigma \notin I
\end{array}
\right.\end{aligned}$$ for $\sigma=\sigma'$, and $$\frac{\partial }{\partial C_{\sigma,\sigma'}}m^0_{I,I'}
=\delta_{\bar{I},I \backslash \sigma}
\delta_{\bar{I},I'\backslash \sigma'}\frac{m^0_{\bar{I},\bar{I}}}
{(1-n_{\sigma})(1-n_{\sigma'})}$$ for $\sigma\ne \sigma'$, where $\sigma \in I$ and $\sigma' \in I'$. The derivatives of $m^{0;\bar{\sigma}}_{I\backslash \bar{\sigma} ,I'}$ are given accordingly.
<span style="font-variant:small-caps;"></span> \[1\][\#1]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'S. L. Martell'
- 'S. Duffau'
- 'A. P. Milone'
- 'G. H. Smith'
- 'M. M. Briley'
- 'E. K. Grebel'
nocite:
- '[@CB09]'
- '[@LB11]'
- '[@SM11]'
- '[@CK12]'
- '[@CG10]'
- '[@MS11]'
- '[@PM12]'
- '[@MP12]'
- '[@DVD08]'
- '[@CS11]'
- '[@DMC07]'
- '[@DVD08]'
- '[@DPL09]'
- '[@GN12]'
- '[@KG08]'
- '[@RJ09]'
- '[@KL11]'
- '[@GO97]'
- '[@FZ01]'
- '[@PG07]'
- '[@BK08]'
- '[@KH08]'
- '[@CB10]'
- '[@GP11a]'
- '[@GP11b]'
title: 'Chemical evolution in star clusters: the role of mass and environment'
---
Introduction
============
Although they have long been held up as examples of simple stellar systems, all Galactic globular clusters host significant complexity in light-element abundances (e.g., Carretta et al. 2009; Lardo et al. 2011; Smolinski et al. 2011), and some also have variations in alpha-, iron-peak and neutron-capture element abundances (e.g., Cohen & Kirby 2012; Carretta et al. 2010; Marino et al. 2011). The light-element abundance complexity, and the corresponding dramatic photometric complexity (e.g., Piotto et al. 2012; Milone et al. 2012), are presently explained as a result of a constrained chemical feedback between two closely-spaced stellar generations (e.g., D’Ercole 2008; Conroy & Spergel 2011). In this “stellar-mode” self-enrichment process, material that has been processed through the CN(O), Ne-Na and Mg-Al cycles of hydrogen fusion is ejected in the winds of massive main-sequence stars and AGB stars, and during lossy mass transfer in close binaries (e.g., Decressin et al. 2007; D’Ercole et al. 2008; de Mink et al. 2009). This material mixes with pristine material left over from the formation of the initial population of stars to form a second generation with light-element abundances that fall between the scaled-Solar abundances of the initial population and the highly anticorrelated fusion-processed abundance pattern of the feedback material. Because this process happens quickly, there is not time for supernova winds to contribute enhancements in either alpha-element or iron-peak abundances; indeed, the large amount of kinetic energy deposited by supernova winds clears all remaining gas from the cluster and drives the cluster to expand, causing cluster stars at large radius to become gravitationally unbound.
The stellar-mode self-enrichment process that occurs in globular clusters can be thought of as the low-mass end of a continuum of chemical evolution processes, continuing into ultrafaint dwarf galaxies, which have sporadic star formation that captures snapshots of the galay’s chemical state (e.g., Gilmore et al. 2012); classical dwarf galaxies, which typically have bursty star formation but a smooth progression from SN II-driven chemical evolution to SN Ia-driven chemical evolution (e.g., Koch et al. 2008; Revaz et al. 2009; Kirby et al. 2011); and galaxies at or above the mass of the Milky Way, in which the overall mass, luminosty and metallicity follow well-defined correlations. Stellar feedback processes must contribute to the interstellar medium in all of these systems, but it is only in globular clusters that it produces a noticeable abundance effect: presumably the signature of hot hydrogen fusion is washed out by the dramatically larger mass in heavier elements produced by supernovae.
A minimum mass for self-enrichment?
===================================
The ability of a dark matter-free system like a globular cluster to retain stellar winds is necessarily related to its mass at the time of star formation, with some minimum mass below which star clusters, though their stars produce fusion-processed winds, are not able to convert those winds into new stars. This raises the question of why there are no long-lived, chemically simple star clusters in the Milky Way: were they never formed, or is the minimum mass for self-enrichment quite low, or is the Galaxy such an inhospitable place for low-mass star clusters that any single-generation star clusters that were initially formed have since been destroyed by tidal forces, disk shocking and internal 2-body interactions (e.g., Gnedin & Ostriker 1997)? Efforts to reconstruct the initial cluster mass function based on the present-day log-normal globular cluster mass function and various cluster-dissolution processes (e.g., Fall & Zhang 2001; Parmentier & Gilmore 2007; Baumgardt et al. 2008) do not find consistent results.
The initial masses of globular clusters are not directly observable, but attempts have been made to correlate present-day masses to the presence of light-element abundance complexity (e.g., Kayser et al. 2008; Carretta et al. 2010a). In @CB10 it is pointed out that all Galactic globular clusters with masses above a few $\times 10^{\rm 4} M_{\odot}$ that have been surveyed spectroscopically have been found to host multiple light-element abundance populations, with a wider abundance range found in the more massive clusters. We consider here an environmental explanation for the lack of single-generation globular clusters in the Milky Way. This explanation leaves open the possibility that other environments might be more hospitable to the long-term survival of lower-mass clusters.
Star clusters in the Large Magellanic Cloud
===========================================
As a way to explore the influence of the large-scale environment on chemical complexity in star clusters, we have taken moderate-resolution spectra of individual red giant stars in the intermediate-age populous LMC star clusters NGC 1651 and NGC 1751. Using the FORS2 spectrograph [@S94] on UT1 (Antu) at the VLT, we obtained spectra for 4 RGB stars belonging to NGC 1651 and 5 in NGC 1751 on 17 December 2011 as part of ESO program 088.D-0807. From the literature, it was unclear what result we should expect: although @MO09 found light-element abundance variations among RGB stars in old globular clusters in the LMC, @MC08 found only a narrow O-Na anticorrelation for RGB stars in intermediate-age LMC clusters. Figure 1 shows the CN band strength index $S(3839)$, measured from our FORS2 spectra following the definition of @N81, versus apparent [*HST*]{} ACS/WFC F814W magnitude for all of these stars, with NGC 1651 stars shown as open circles and NGC 1751 stars shown as filled circles. In both clusters, there is a distinct range in $S(3839)$ at fixed luminosity, indicating that stars in both clusters exhibit a range in nitrogen abundance.
Figure 2 shows a generalized histogram of $\delta S(3839)$ (the vertical distance between each star’s CN band strength and the baseline shown in Fig. 1) for both clusters together. Since they have very similar metallicities and ages (\[Fe/H\]$=-0.3$, age$=1.8$ Gyr for NGC 1651 and \[Fe/H\]$=-0.44$, age$=1.5$ Gyr for NGC 1751), data for the two clusters can be combined this way without significant differential effects on the CN band strengths. In this figure there is a distinct width to the distribution, indicating a range of nitrogen abundance in the stars.
This is particularly interesting in light of the recent photometric work by @MB08, @MB09, and Goudfrooij et al. (2011a, 2011b), which found that the main sequence turnoffs in a significant fraction of populous intermediate-age LMC clusters are broadened or split. This has been interpreted as a sign of extended or two-burst star formation in those clusters. In this context, we view our CN band strength data as a sign that the complex star formation histories in intermediate-age populous star clusters in the LMC included a phase of stellar-mode self-enrichment, though further work is required to solidify this claim.
It has been suggested by @CS11 that star clusters moving in the gravitational potential of the LMC will be more able to accrete material from their surroundings than star clusters belonging to the Milky Way because of their lower orbital velocities. This would then lower the effective minimum mass for the formation of a second stellar generation, since (setting aside the problem of maintaining a consistent metallicity) accretion would allow gas from all along the cluster’s orbit, and not just the gas initially associated with the cluster, to contribute to the construction of the second generation.
Conclusions
===========
We find evidence for complexity in the light-element abundances of NGC 1651 and NGC 1751. These two clusters demonstrate that the multiple stellar generations suggested by the broadened or split main sequences in intermediate-age LMC clusters can be accompanied by complex abundances. Both are at or above the minimum present-day mass for self-enrichment for Galactic globular clusters [@BP12], which unfortunately does not provide any information about whether the minimum mass for self-enrichment is lower in the LMC environment. However, the presence of complex abundances in intermediate-age clusters does support the claim of @KM11 that multi-generation star formation may be a standard feature of star cluster formation, and is not restricted to ancient globular clusters.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Today, leveraging the enormous modular power, diversity and flexibility of manycore systems-on-a-chip (SoCs) requires careful orchestration of complex resources, a task left to low-level software, e.g. hypervisors. In current architectures, this software forms a single point of failure and worthwhile target for attacks: once compromised, adversaries gain access to all information and full control over the platform and the environment it controls. This paper proposes [*Midir*]{}, an enhanced manycore architecture, effecting a paradigm shift from SoCs to distributed SoCs. [*Midir*]{} changes the way platform resources are controlled, by retrofitting tile-based fault containment through well known mechanisms, while securing low-overhead quorum-based consensus on all critical operations, in particular privilege management and, thus, management of containment domains. Allowing versatile redundancy management, [*Midir*]{}promotes resilience for all software levels, including at low level. We explain this architecture, its associated algorithms and hardware mechanisms and show, for the example of a Byzantine fault tolerant microhypervisor, that it outperforms the highly efficient MinBFT by one order of magnitude.'
author:
-
bibliography:
- 'midir.bib'
title: 'Behind the Last Line of Defense\'
---
fault and intrusion tolerance, hypervisor, processor architecture
Introduction {#sec:introduction}
============
Practically all activity of modern societies depends on information and communication technologies (ICT). Such dependency obviously hinges on the correctness of these systems, some of them critical, which may fail in a combination of multiple causes and ways [@facebook_privacy; @singapore_health_privacy; @under_armour; @CloudOutage; @ukrain_power; @tesla]. Systems have been progressively pushed to extremes of efficiency through modularity in platform sharing, firstly through virtualization and lately by leveraging the enormous power growth, functional diversity and adaptation flexibility offered by multi- and manycore. This has taken platform sharing to new heights, into the realm of multi-processor systems-on-a-chip (MPSoCs).
The organization of these complex computing resources depends on low-level platform management hardware (e.g., memory-management units (MMUs)) and software (e.g., firmware, hypervisors, management engines). However, current MPSoC architectures are such that these management components, which should form a last line of defense against severe accidental faults or adversaries intruding the system (malicious faults), instead constitute a single point of failure (*SPoF*), for two main reasons. First, the way platform privilege-enforcement mechanisms (e.g. MMUs or hardware-enforced capabilities [@Woodruff:2014:CCM:2665671.2665740]) are designed allows faults in a core/tile to propagate through MPSoC components. Second, faults in this lowest-level management software, e.g., hypervisors configuring these privileges, are bound to propagate across management and managed components, again causing common-mode failure scenarios.
If these SPoFs are compromised by adversaries, the latter gain full authority over the platform’s privilege-enforcement mechanisms and, through them, access to all information and complete control over all platform resources (e.g., cloud-based systems), including, in the case of cyber-physical systems, extended control over the physical environments on which they act (e.g., nuclear power plants or autonomous cars).
Is this a real risk? It is, if the vulnerability rate of these low-level platforms is non-negligible. Recent problems, whether in Intel’s CSME [@me_vulnerability], Xen/Critix [@xen_vulnerabilities] or concerning Spectre [@spectre] and Meltdown [@meltdown], have been repeatedly reminding us of how brittle the assumption of “tamperproof and unattackable low-level platform management assets” is. Even formally verified kernels (e.g., seL4 [@Klein+al:sosp:sel4:2009]) may fail due to model/reality discrepancies or hardware faults violating modeling assumptions [@Biggs_LH_18]. Being the risk real, are there no solutions yet? The solution design space for contemporary hardware platforms dependability and security has been unfolding in two directions: (i) application-specific system-level replication (e.g., triple modular redundancy, mainly in cyber-physical systems (CPS), by means of multiple electronic control units (ECUs)), where the lack of flexibility limits the extension to general systems; (ii) manycore-level replica management and consolidation, which then, if on bare MPSoCs, reintroduces the SPoF concern, now for the low-level replication management component.
At this time, we call the reader’s attention to an interesting fact, which will become crucial to our solution. The current MPSoC architectures’ complexity, modularity and networked interconnectivity, suggests attributes of distributed systems [@SapeMullenderDisSys], albeit imperfect such systems (an example of which is the aforementioned SPoF syndrome). However, distributed systems have been used to mitigate SPoF syndromes and to implement fault and intrusion tolerance schemes [@Powell:Delta4; @MaftiaSecPrivMag]. In consequence, the root of the MPSoC problems just presented may also be an avenue to their solution.
So, in this paper, we start by identifying the gaps from (MP)SoCs to distributed systems, and propose (MP)SoC mechanisms to bridge them, which essentially means achieving: fault independence and fault containment, despite low software-level compromise and while retaining the flexibility (MP)SoCs offer. Having a manycore that behaves as a (closely-coupled) distributed system, should allow us to design a set of efficient and low-overhead distributed systems-inspired modular protection and redundancy management mechanisms, e.g., Byzantine fault tolerant state machine replication (BFT-SMR), for fault and intrusion tolerance (FIT). The remaining problem, how to implement and where to locate all the mechanisms above, is addressed by the [*Midir*]{}[^1] achitecture presented in this paper, which leverages the computing critical mass and flexibility of contemporary tile-based manycore architectures.
[*Midir*]{}constrains the connection of all tiles to the network-on-chip (NoC) through simple and self-contained hardware-based trusted components, which we call [*T2H2*]{}. Exploring the concept of architectural hybridization [@Verissimo:2006:TTW:1122480.1122497], whilst we consider those components to be ultra-reliable and not fail, we are agnostic about the reliability of individual tiles, which may be compromised or fail. The assumption is justified by the simplicity of the former, promoting verifiability.
The [*T2H2*]{}components implement the functionality achieving fault independence, containment, and tolerance mechanisms mentioned above. In consequence, tile-internal software or hardware faults are contained in the tile and the objects the tile can access. Furthermore, the baseline mechanisms for protection and redundancy management provided by [*T2H2*]{}can be extended and recursively applied at any software layer, giving the designer ample latitude for crafting resilience into systems, both “horizontally” (incremental power of defense mechanisms) and “vertically” (depth of defense).
Locating [*T2H2*]{}between the tile and the NoC interconnect not only provides a clear pathway for integration by chip manufacturers and integrators, it also allows drawing from many well-understood building blocks (e.g., region protection, capabilities [@needham:cap], and other chip-level resource management mechanisms [@config_isolation], capable of isolating tiles and the resources they can access). The novelty of [*Midir*]{}lies in their arrangement to avoid SPoFs, even while they are reconfigured.
In a nutshell, contributions of this paper are:
\(1) An analysis of the gaps separating current MPSoC architectures from genuine distributed systems, and gap fixing through measures promoting fault independence and fault containment in tile-based architectures, enforced at the level of the tile-to-NoC interface.
\(2) An architecture ([*Midir*]{}) leveraging the resulting distributed system-on-a-chip (DSoC) to achieve incremental levels of modular fault and intrusion tolerance, through a range of diverse redundancy management techniques implemented by simple hardware-based voting/consensus mechanisms.
\(3) The design of a simple and ultimately [**t**]{}rusted-[ **t**]{}rustworthy [**h**]{}ardware [**h**]{}ybrid, [*T2H2*]{}— the core component of [*Midir*]{}, staged at the tile-to-NoC interface — providing just two generic baseline functions: access control (capability registers) and quorum-based consensus (voters). By configurations and combinations of these two basic functions, [*T2H2*]{}is capable of implementing all the techniques mentioned in (1) and (2).
\(4) As a proof of concept, we give and evaluate an implementation featuring [*Midir*]{}and essential parts of a fault and intrusion tolerant microhypervisor built on top of it. Though the architecture serves several reliability strategies, we chose the most effective, active replication with error masking. Being the most complex and costlier, we believe to have shown the performance and practicality of our concept.
Next, we evaluate the challenges for bridging from SoCs to DSoCs (Sec.\[sec:manycore\]), and present the system and threat model (Sec.\[sec:threat\_model\]). Then, we introduce the [*Midir*]{}architecture (Sec.\[sec:muenchhausen\]) and the [*T2H2*]{}component in Sec.\[sec:T2H2\]. At this point, we are able to show [*Midir*]{}in action, discussing the design of a fault and intrusion tolerant microhypervisor built on top of it (Sec.\[sec:hypervisor\]), as an example of critical low-level management software. Finally, we discuss some relevant implementation matters in Sec. \[sec:implementation\], and in Sec.\[sec:evaluation\], we evaluate [*Midir*]{}on a Zynq ZC702 board, showing how [*Midir*]{}’s hardware voters accelerate BFT-SMR protocols, voted execution of system calls and consensual reconfiguration of [*T2H2*]{}. An analysis of related work (Sec. \[sec:related-work\]) follows, and Sec.\[sec:conclusions\] concludes the paper, pointing to further research and innovation opportunities.
From MPSoCs to Distributed SoCs {#sec:manycore}
===============================
Multi- and manycore systems consolidate in a single chip computing resources that used to reside on multiple chips. Tiles [@raw] are placeholders and instantiation points for resources, typically instantiated with cores and private caches or with slices of shared caches, and connected through the NoC with each other and with memory controllers (to reach out to RAM/IO). It is possible as well to cast accelerators, GPUs and FPGAs, into the tile abstraction.
The modularity and networked interconnection of tiles already suggests attributes of a distributed system and has inspired first steps to hardware-enforced fault containment at tile level, as pioneered by Hive [@hive], Cap [@needham:cap], M3 [@asmussen:m3] and others. Hive introduces MAGIC, a bus-level firewall to confine faults to the individual processors of the Stanford Flash multiprocessor system. M3 follows the same scheme with hardware enforced capabilities, originally introduced in Cap [@needham:cap] to control resource accesses and, thereby, fault containment of heterogeneous processors. Configurable isolation [@config_isolation] leverages dual-mode redundant MMUs to, like M3, confine faults in on-chip resources. Tiles favour functional and non-functional diversity since they can host cores from several makers. This improves fault independence through the implied low likelihood of experiencing the same fault in different tiles. Similarly, different versions of the same code can be used at distinct tiles with the same intent [@avizienis1977implementation; @knight1986experimental; @joseph1988fault].
Note that, emulating the spacial isolation of distributed system nodes, we are agnostic about the semantics and interplay of tile-internal and/or core-level components, e.g., MMUs and their virtualization, copy-on-write, memory protection or recovery functionalities.
A final and subtle gap concerning fault containment and independence affects all previous systems we know of, including those deploying hardware-enforced fault containment [@needham:cap; @hive; @asmussen:m3; @config_isolation]: potentially faulty or compromised low-level kernels still retain control over platform privilege configuration mechanisms. As we explain in Sec.\[sec:muenchhausen\], this is a harmful effect. Our main contribution is to neutralize this effect by imposing that critical platform management operations are performed through consensus of a majority of correct components.
In conclusion, with the enhancements described in this paper, tiles fail like nodes in a distributed system, faults affect only the tile itself and the components (e.g., replicas) executing on it, but they do not propagate to the entire manycore, in particular other components related to the same application or subsystem. This interplay between protection and consensus to achieve fault containment, in particular during platform reconfiguration, including of the fault containment domains themselves, allows hypervisor replicas to retain the flexibility of the MPSoC, even after a minority of hypervisor tiles failed accidentally or have been compromised by an adversary.
System and Threat Model {#sec:threat_model}
=======================
We now describe the system and threat model educating the development of our distributed system-on-a-chip (DSoC).
System Model
------------
We assume a fully connected system, where on-chip network components offer the abstraction of a correct network, interconnecting all tiles to one another. Tiles communicate by messages, and messages sent are eventually delivered, unchanged, to the destination. Network coding [@network-coding], multi-tenant [@multi-tenand] and adaptive routing techniques [@adaptive-noc-routing] substantiate the coverage of this assumption.
We rely on a partially-synchronous model. At first sight, manycores might seem the perfect example of a (closely-coupled) synchronous (distributed) system. However, reality is a bit different, several possibilities for instability in the time domain (speed of tiles throttling for thermal control, cache exceptions, NoC-level bursts, etc.) would prove the strict synchronous model brittle.
However, being a closely-coupled environment, short-term liveness is normally guaranteed, barring delay variations. This has two implications on the design of [*Midir*]{}, for robustness: (i) we absorb possible inter-tile delays, notably by buffering messages (e.g., votes) in [*Midir*]{}’s [*T2H2*]{}; (ii) the structure of the protocols is time-free and, as such, they remain safe in the presence of delay oscillations, provided that the fault assumptions hold.
Threat Model
------------
Our threat model considers software-level compromise at all levels, including hypervisors, firmware and, more generally, in any critical software component. This assumption is consistent with our aim of tolerating an incremental level of threat on tiled manycore systems, up to sophisticated and persistent attacks possibly deployed entirely on-chip. Moreover, we consider a limited set of hardware-level faults and attacks: precisely those whose physical effects are confined to a tile (e.g., trapdoors in a core, but no hardware faults that cause a chip-wide collapse). We consider the tile as a unit of component failure. There is no guaranteed fault containment inside tiles. That is, adversaries (or accidents) will be capable of compromising the whole software in any tile (e.g., but not only, a hypervisor in case the user/supervisor mode isolation failed). Once that happens, we no longer make any assumptions about the correctness of any software in that tile. However, we also consider (and enforce it with the strategy described in Sec.\[sec:manycore\]) that tiles themselves are fault containment domains, such that faults inside a tile do not propagate across the manycore. We enforce the assumption above through architectural hybridization [@Verissimo:2006:TTW:1122480.1122497; @minbft; @cheapbft]. Despite the general system fault model enunciated for tiles, [*T2H2*]{}([*Midir*]{}’s trusted-trustworthy component) follows a more restricted fault model, enforced by construction and, through its simplicity, amenable to verification, failing only by crashing, much like USIG [@minbft] or CASH [@cheapbft]. Thus, [*T2H2*]{}, residing at the tile-to-NoC interface, reliably implements its functions despite faulty tiles.
![Example systems showing the [*Midir*]{}architecture: software in tiles need a capability to authorize access to resources in other tiles (solid lines); capability modifications in a tile (in fact any critical operation) are subject to consensus of a majority of other correct tiles (dashed lines), here the three tiles hosting hypervisor replicas of which one may be faulty. []{data-label="fig:architecture"}](midir_arch){width=".8\columnwidth"}
The [*Midir*]{}Architecture {#sec:muenchhausen}
===========================
As discussed earlier, [*Midir*]{}is an architectural concept based on augmenting manycore systems in a minimally intrusive way through strategically placed, simple and self-contained trusted-trustworthy components ([*T2H2*]{}). In fact, [*T2H2*]{}provides just two generic baseline functions staged *in hardware* at the tile-to-NoC interface: access control (capability registers) and quorum-based consensus (voters).
Fig.\[fig:architecture\] depicts one possible layout, of a stereotypical hypervisor-based system, where the hypervisor is replicated for fault/intrusion tolerance, serving operating system and applications: hypervisor replicas are distributed across tiles, so that each replica executes on a different tile, separate from applications; tiles and software therein interface with each other through the NoC; and [*T2H2*]{}are the “blue dots” performing that interconnection. [*T2H2*]{}interposes such accesses, validating that the invoking tile has sufficient privileges, through the capability registers, which include the logic for privilege enforcement.
As long as the execution in a tile remains within the resources associated to this tile (local caches, memories, accelerators, etc.) no overhead occurs, since [*T2H2*]{}is not involved in authorizing or denying these accesses. In fact, we remind that it is not the purpose of [*Midir*]{}to provide fault containment between software components co-located [*on the same tile*]{}. This is like the internal behavior of nodes in a distributed system, where nodes are the unit of fault containment. Once software components are spread across tiles, they interact through external operations (e.g., via a resource in another tile, via shared on-chip memories or via external memory or IO) and [*T2H2*]{}validates that each such access has been authorized by a capability the tile possesses. Consequently, hardware faults inside a tile or accidential or malicious faults in any part of the software it executes, are limited in propagation to the objects authorized by these capabilities.
Further to capability checking, [*Midir*]{}is capable of subjecting these accesses to voting by distributed components in different tiles. This is especially important for critical operations, be it in application execution or in platform reconfiguration, in order to achieve some form of fault/intrusion tolerance, from error detection, or self-checking by comparison, to error masking by consensus. To vote, tiles must hold a capability to the corresponding voter, which authorizes this tile to make proposals as one of these distributed components. Voting is mandatory to install new or change existing capabilities, in order to prevent faulty hypervisor replicas from bypassing the aforementioned fault containment when reconfiguring the resources a tile can access.
[*Midir*]{}’s concept of controlling the tiles’ lowest-level privilege enforcement mechanism is agnostic of the mechanism used. However, the simpler such a mechanism and the closer it can be implemented to the tile’s NoC interconnect, the more architecture-level faults [*Midir*]{}will be able to tolerate. Hence our choice for capabilities.
Simplicity also governs our voter design. [*Midir*]{}’s voters merely collect and act upon proposals of related operations from different components, letting the voted-upon operation proceed. Because tile-external resources are typically memory mapped, these operations are normally simple writes. The voters themselves implement no error handling or diagnostics functionality, but provide information for the components to perform these tasks. More precisely, voters suspend voting on disagreement, freeze the proposals made by the components and expose them for diagnosis. Moreover, they implement a sequence number $\mathit{seq_i}$ for progress tracking, which they increment after each vote unless the vote gets suspended. A voted upon voter-reset operation resumes voting and as well increments $\mathit{seq_i}$. Sec.\[sec:hypervisor\] shows how we utilize this error handling support and Sec.\[sec:implementation\] details our voter implementations.
[*T2H2*]{}– Midir’s Trusted-Trustworthy Component {#sec:T2H2}
=================================================
In this section, we provide further details about [*T2H2*]{}.
Voted and non-voted operations {#sec:direct_invocation}
------------------------------
\[sec:voted\_invocation\] To retain the flexibility of the software in a manycore system, allowing it to dynamically adapt resource-to-application mappings as needed, [*T2H2*]{}supports direct access to tile-external resources. This way, applications possessing a capability can directly invoke operations on external resources (e.g., to access read-shared or private data in RAM or to interact with non-critical devices). The scenario in Fig.\[fig:invocation\](a) illustrates a non-voted (write) memory access by Tile A, performed by invoking a capability in this tile’s [*T2H2*]{}. Since [*T2H2*]{}’s capability registers hold a read-write capability to the memory region $[p, p+s]$, the operation to write value $val$ in variable $a$ is authorized.
![ (a) Non-voted memory access by tile A through capability invocation. (b) Voted memory access by tiles A-C (tile A faulty) through capability invocation then voting (orange); reconfiguration of a platform capability register in tile A through voting (green). []{data-label="fig:invocation"}](cap_invocation_combined){width="\columnwidth"}
However, [*T2H2*]{}also supports voting, particularly useful when e.g., platform management software or hypervisor replicas, further have to execute critical operations (e.g., privilege change or critical device accesses). These operations are voted upon, within preconfigured detection or tolerance mechanisms, to prevent compromised components from causing harm. Several strategies may be served by [*Midir*]{}, such as self-checking, recovery blocks, or [*f-out-of-n*]{} error masking by majority voting in the presence of $f$ faulty components, but they are all supported by the same baseline voting mechanism. Fig.\[fig:invocation\](b) represents a similar operation as in Fig.\[fig:invocation\](a), but in voted access form. Tiles B and C vote to write value 1, while Tile A, being faulty, votes to write value 0. In order to perform these votes, all tiles invoke a capability on their local [*T2H2*]{}to access the designated voter (in this case, residing on Tile A’s [*T2H2*]{}) . Given that a majority of tiles voted to write 1, value 1 will be written to variable $a$.
[*Midir*]{}does not constrain how systems are configured and hence what faults are tolerated. Instead it provides the means to tolerate an incremental quality of faults, including for highly critical systems up to $f$ faults in system management software (e.g., the hypervisor), by providing $n = 2f+1$ hypervisor replicas and by subjecting all critical operations to voting.
voted operation on the example of consensually installing capability $M:(p,s,R)$ with rights $R$ to the memory region $[p, p + s]$, which is of size $s$ and starts at physical address $p$. To install this capability into capability register $c_2$, the hypervisor replicas invoke their capability to the voter of tile $A$ (e.g., the capability in register $c_1$ for tile $A$). The operation $\mathit{set}_{HV_1}(c_2, M:(p,s,\{r\}))$, invoked by $HV_1$ with capability register $2$ and capability $M:(p,s,\{r\})$, translates (upon agreement) to a write to the interface for configuring capabilities of tile $A$’s [*T2H2*]{}. Whereas the interface to invoke capabilities is accessible only from the tile at which the [*T2H2*]{}is located, the configuration interface is always external (i.e., accessible only through capabilities and a voter). Faulty replica $HV_1$ on tile $A$ aims for read-write privileges ($R=\{r,w\}$), but is outvoted by the remaining two replicas, which grant only read access to this region. [*T2H2*]{}applies this consensual decision and installs the capability $M:(p, s, \{r\})$ in $c_2$. The capability invocation to interact with the voter (in fact all capability invocations) works as usual — the tile specifies the capability register to invoke and the operation to execute, which [*T2H2*]{}validates against the permissions of the capability in this register — with the exception that capabilities to voters identify the replica towards this voter. That is, replica $HV_1$ may only make proposals as replica one at this voter but not as replica two. We denote this as subscript in the voted operation (e.g., $\mathit{set}_1(\ldots)$ for $HV_1$ on tile $A$).
Consensual privilege change {#sec:privilege_reversion}
---------------------------
One particularly relevant scenario for voted access is consensual reconfiguration of the [*T2H2*]{}instances themselves. [*T2H2*]{}’s reconfiguration interface is accessible only through a voter and cannot ever be invoked directly.
Let us understand why this is a relevant innovation. In conventional OS design, any single kernel instance can directly or indirectly enforce modifications on platform resources. So, even in fault tolerant designs, a faulty or compromised kernel instance could still be able to threaten the platform correctness. For example, by manipulating page tables, any low-level OS kernel instance can install virtual-to-physical address mappings to any resource in the platform’s memory map and access it through this mapping. Of course, a trusted underlying layer could solve this issue (e.g., by mediating page-table access). However, whether this layer is software, as in the Inktag kernel [@Hofmann:2013:ISA:2499368.2451146]) or firmware, as in Intel SGX [@intel_sgx]), it becomes a single point of failure for the platform.
[*Midir*]{}provides a further level of protection, whereby the designer can constrain access to the platform reconfiguration, by allowing a particular mechanism, its registers and data structures to be only effected in a consensual manner, through a voter. As with general voting, discussed in Sec. \[sec:voted\_invocation\], these voted accesses will normally correspond to the implementation of detection or tolerance strategies, in this case, directed to the protection against threats on the platform itself. In Fig.\[fig:invocation\](b), in green colour, we represent such a flow of reconfiguration of a platform capability register in tile A’s [*T2H2*]{}. Exemplifying with [*f-out-of-n*]{} error masking in a replicated low-level kernel, several replicas make the reconfiguration request, which is voted (green voter). The result from the voter is wired through a special [*T2H2*]{}capability configuration interface to the concerned capability register, masking the presence of up to $f$ faulty replicas.
One particular case is when a quorum of replicas grants a controlled application (i.e., the resource recipient) access over some resource, but denies the same to all kernel replicas (i.e., the resource managers). We call this situation *privilege reversion* to highlight the fact that recipients may become more privileged than managers.
Privilege reversion has virtuous consequences: it gives rise to interesting design patterns, which were previously available only at application level (assuming trustworthiness of the kernel), but which can now be exploited inside the kernel. For example, by agreeing to grant read access to memory but never direct write access (i.e., writes only through a voter), we obtain the notion of *consensually updateable memory*, which we shall use for constructing read-most shared kernel data structures in ECC memory. Moreover, granting a single application write access to a page while denying the same for all other components (including the kernel), creates an *authenticated buffer*, since only this application can write. We shall use this for communicating system-call requests to the kernel.
Trusted-Trustworthy Hardware Hybrids - T2H2 {#sec:T2H2}
===========================================
Before diving into [*Midir*]{}-aware OS designs, we shall first explain the core details regarding the *T2H2*s. We start by describing the use of these components through their invocation interface for direct and voted resource access and voted upon *T2H2* reconfiguration. Afterwards, we specify the various voter implementation options.
Capability invocation for direct resource accesses {#sec:direct_invocation}
--------------------------------------------------
Privilege enforcement with capabilities works as usual. That is, utilizing *T2H2*’s invocation interface, which is only accessible from within the associated tile, tile soft- or hardware invokes a capability, specifying the operation and the capability register to use. For compatibility reasons, we suggest encoding the capability register identifier in the higher order bits of what tile-internal components consider to be physical addresses[^2].
For example, operation $\mathit{write}(d, a)$ with data $d$ of size $|d|$ and guest physical address $a = [c_0 | \mathit{ofs}]$ invokes the memory capability $M:(p, s,\{r,w\})$ in register $c_0$ of the local *T2H2* to write $d$ to host physical address $p +
\mathit{ofs}$ in the region $[p, p+s]$ of size $s$ starting at $p$. Fig.\[fig:tile\_interaction\] shows this access, assuming $[p, p+s]$ is located in tile $B$. Before authorizing this write, *T2H2* checks (i) whether the capability in $c_0$ is a memory capability, (ii) whether it conveys write permissions ($w$), and (iii) whether the access is within the memory region $\mathit{ofs} + |d| < s$. Only if all these checks succeed, write command $\mathit{write}(d, p + \mathit{ofs})$ is sent over the NoC for $B$ to execute.
![Tile interaction. To access the region $[p, p+s]$ of size $s$ starting at address $p$ in tile $B$, tile $A$ has to invoke its capability $c_0$ through the *T2H2* invocation interface. The reconfiguration interface is never directly accessible.[]{data-label="fig:tile_interaction"}](tile_interaction){height="3.9cm"}
Capability invocation for voted accesses {#sec:voted_invocation}
----------------------------------------
Two types of voted accesses are supported by [*Midir*]{}: voted operations over resources (e.g., updates of critical data structures); and voted reconfiguration of *T2H2*.
[*Midir*]{}gives the designer latitude to use incremental protection, not preventing, in one extreme, configurations where a single instance controls *T2H2*’s privilege enforcement mechanism (by setting $f=0$). On the other extreme, it provides full protection, eliminating all software-level single points of failure, if the system is configured such that the following invariants are preserved during execution (and initially):
- [**Inv.1 *Impersonation prevention:***]{} no correct replica agrees to installing in different tiles capabilities to the same voter with the same replica identifier.
- [**Inv.2 *Bypass prevention:***]{} no correct replica agrees to installing capabilities conveying direct write access to consensually updated resources, which would bypass voting[^3].
- [**Inv.3 *Replica integrity preservation:***]{} under no circumstances correct replicas agree to installing capabilities that convey direct write access to the code and local state of a replica, respectively to critical data structures or resources that have similar integrity compromising effects when misused.
Whereas the architecture allows designer choices for voted or direct access to resources as discussed above, capabilities can never directly reference the interface of any *T2H2* for platform reconfiguration, such as privilege changing. Instead, they must always reference voters, which interpose accesses to this configuration interface. Voted resource accesses work macroscopically in the same way as direct accesses, except that the invoked capability in the local *T2H2* does not point to the resource (or configuration interface) directly, but to a voter on the same or another tile. Moreover, this capability identifies the replica to the voter, thereby preventing one replica from impersonating another. We achieve this by encoding the replica identifier $HV_i$ ($i
\in [1, n]$) into the capability (denoted in Fig.\[fig:reconfiguration\] as subscript in the configuration capability $\mathit{set}[\ldots]_1$ in $c_1$ and in the operations transmitted to the voter).
![Reconfiguration of capability registers (e.g., $c_1$ in the [*T2H2*]{}of tile $A$) is always consensual (requiring agreement of a majority of tile $A$, $B$ and $C$). The voter installs the majority decision (here, the read only region $[p, p+s]$.[]{data-label="fig:reconfiguration"}](mon_reconfiguration){height="3.9cm"}
Fig.\[fig:reconfiguration\] illustrates such a voted access, providing as an example invoking the configuration interface of *T2H2* to change a capability register in tile A’s local *T2H2*. The tile aims at installing a read/writable memory capability $M:(p', s',
\{r,w\})$ for region $[p', p' + s']$ into capability register $c_2$. It therefore invokes the capability in register $c_1$, which refers to its local voter[^4]. Unlike in the direct invocation case above, the operation becomes only effective if a majority of replicas agree. The voter will therefore defer writing $c_2$ until it receives matching votes from a quorum of $|Q| = f+1$ replicas. Different voters proceed independently. In our example ($f=1$), the other two replicas ($HV_2$ on tile $B$ and $HV_3$ on $C$) agree on the memory region, but not on the set of permissions, proposing a read-only capability $M:(p', s', \{r\})$ . Therefore, the voter will disregard the proposal of $HV_1$ on tile $A$ and, following the majority vote, will install a read-only version of this capability.
Towards Fault and Intrusion Tolerant\
Microhypervisors {#sec:hypervisor}
=====================================
We now turn our attention to the construction of [*Midir*]{}-aware FIT microhypervisors, such as suggested in Fig. \[fig:architecture\]. Hypervisor replicas execute on dedicated tiles, from where they remotely configure the privileges of applications executing on other tiles. Most of the other common OS-functionality (e.g., context switching, inter-process communication, (non-critical) device access, etc.) can be left to the application and its kernel-support libraries.
[*Midir*]{}gives the designer latitude to use incremental levels of protection for individual operations or sets thereof. On one extreme, configurations may be allowed where all accesses are direct, and thus unprotected by voting. On the other extreme, the highest level of protection, while retaining the flexibility of a manycore system, eliminates all software-level single points of failure[^5] by subjecting all critical operations to voting. We focus on this facet. The replicated microhypervisor offers a system-call interface executed by its replicas, entering a service loop and maintaining data structures used to handle system call requests, which they receive from applications, other replicas (e.g., requesting a privilege they lack for executing a system call) or from hardware (e.g., triggered by device interrupts).
Remembering that the unit of fault containment in [*Midir*]{}is the tile (equivalent to a node in a distributed system) the essential requirement for a fault tolerant microhypervisor design is that the replicas behind critical operations are placed in different tiles, such that they communicate by messages, are subject to [*T2H2*]{}access control, and converge on the necessary votes as dictated by the algorithm. In order to fully enjoy the baseline functionality provided by [*Midir*]{}, a few additional design principles should be followed:
- [**P.1 *Impersonation prevention:***]{} Correct replicas must deny any operation with a replica identifier that is already in use ([*T2H2*]{}voting relies on identifying the individual replicas through their capability).
- [**P.2 *Bypass prevention***]{} Correct replicas must deny any operation attempting to grant direct write access to a consensual-update-only object (Sec.\[sec:privilege\_reversion\]).
Let us illustrate the design with the example of reallocating the tile to a different application. Signaling the tile, an application-specific library may save the state necessary to resume execution (e.g., utilizing memory assigned for this purpose). The actual switch then proceeds by resetting the tile followed by installing the capabilities the new application’s library needs, in order to load its state. Obviously, reset (and, as we have seen, privilege change) is a critical operation, which must be performed consensually to prevent compromised kernel replicas from prematurely stopping applications. Channeling such critical operations to voters and confining access with capabilities prevents faulty replicas from causing harm, since, as long as no more than $f$ replicas become compromised, a correct majority out of the $n = 2f + 1$ replicas will outvote these operations. This turns system-call execution into updates of replicated state and a sequence of voted operations, which we shall later call *subordinate votes*. This works as well with any other replicated critical software, even firmware such as in SGX (e.g., preventing enclave misconfiguration) or device drivers, when interacting with the physical world. Replies to system calls must also be voted upon, given that hypervisor replicas, by nature, act on behalf of multiple applications, possibly storing information of one that must not be revealed to others.
The above is of course true provided replicas have reached agreement on the system call to execute and on the parameters with which the client application has invoked this call. A further role of the service loop is therefore to reach consensus on system call execution order and parameters. From our evaluation (Sec. \[sec:evaluation\]) we found that [*Midir*]{}’s support for consensually executing critical operations also provides for accelerating the BFT protocol that the kernel replicas must execute to reach this agreement.
Consensual System Calls {#sec:system_calls}
-----------------------
![Read-shared, consensually updated data structures used by the kernel: system calls are recorded in the syscall log, the error log keeps voting error information, a capability space holds an application’s capabilities (Sec.\[sec:evaluation\]).[]{data-label="fig:kernel_memory"}](syscall_error_log){width="0.85\columnwidth"}
Fig.\[fig:kernel\_memory\] provides a more detailed picture of how [*T2H2*]{}’s voters and capability registers contribute to a FIT hypervisor’s service loop reaching consensus on the system call to execute.
The service loop utilizes two data structures: a consensually updated ringbuffer — the *syscall log* — records agreed upon system calls and its parameters to give kernel replicas the opportunity to learn about those agreed upon. Otherwise, this information would only be available to the agreeing quorum of $f+1$ replicas and if faulty replicas participate there, but refuse to execute the system call later on, too few correct replicas would have obtained this knowledge to complete the system call. Similarly, the service loop utilizes an *error log* to protect error information from getting lost in premature resets of the voter. Updates of the syscall and error logs are made through dedicated voters: $v_{\mathit{log}}$ and $v_{\mathit{err}}$, respectively.
Macroscopically, clients place system-call requests in authentic buffers, which the kernel replicas poll[ Sleep/wake protocols can be used in periods where no requests are pending. ]{} for new requests. Consensual privilege change allows creating such buffers by granting write access to a single client, but to no kernel replica. The leading kernel replica proposes one such system call by initiating a vote with $v_{\mathit{log}}$, which followers introspect and agree or deny. Once written to the syscall log, replicas proceed by executing the system call and the votes for its critical operations. We call these *subordinate votes* as they depend on the main vote, logging the system call. That is, no correct replica will engage in a subordinate vote unless the system call has been logged. Subordinate votes include at least replying to the client and advancing the syscall log to the next free slot. They are performed utilizing a set of voters $V = \{v_1, \ldots\}$ that is disjoint from $\{v_{\mathit{log}}, v_{\mathit{err}}\}$.
We make no assumptions on the order in which replicas update their local state (even transactional or speculative updates are imaginable). However, to simplify tracing the progress of the system call (and in turn the code that late or rebooted replicas have to execute to catch up), we require subordinate votes to be executed in the same order by all replicas and assume that this order is completely specified by the system-call parameters.
Our rationale for agreeing on the system call first is to circumvent a fundamental problem of consensus protocols without authenticators: the impossibility to diagnose faults if messages can be altered during multicast operations [@Lamport+Shostak+Pease:1982]. In our setting, cryptographic operations would come at overproportionally high costs relative to the speed of the transport medium (the NoC). We therefore avoid sending unforgeable authentication tokens (e.g., HMACs) and instead exploit the authentication we obtain from a client being the single writer of its request buffer. Additionally, clients maintain write access to their request buffers. Thus, they can change the request after the leader has proposed it, but before followers validate it, which makes it impossible for followers to distinguish whether the leader proposed a wrong system call or whether the leader proposed the client’s original suggestion, but the client changed it afterwards. In consequence, they cannot differentiate faulty clients from faulty leaders to provably identify the leader as faulty. We omit error diagnosis for the system-call vote to regain it when we need it: in the subordinate votes for reaching agreement on critical operations.
The following details the protocols the hypervisor replicas execute to reach consensus on and execute system calls. Leveraging the generic voting pattern in Fig.\[fig:generic\_pattern\], replicas first reach agreement on the system call (Fig.\[fig:consensus\_protocol\_syscall\]) to then consensually perform critical updates during its execution (Fig.\[fig:subordinate\]).
Generic Voting Pattern {#sec:general_voting}
----------------------
1 agreement:
2 $\mathit{seq}_i := v_i.\mathit{seq}$
3 if (replica $= \mathit{seq}_i \mathit{mod}\, n$) {
4 // leader
5 $v_i$.propose($op$, $\mathit{seq}_i$)
6 } else {
7 // follower
8 wait for leader proposal: $op$
9 validate op
10 if (valid) $v_i$.confirm($op$, $\mathit{seq}_i$)
11 else $v_i$.decline($op$, $\mathit{seq}_i$)
12 }
13 // all
14 wait for $f+1$ replicas to
15 agree/disagree/timeout
Fig.\[fig:generic\_pattern\] shows the generic pattern and how replicas interact with voters. Evaluating the sequence number $v_i.\mathit{seq}$ of voter $v_i$, replicas identify the leader as the replica with identifier $v_i.\mathit{seq}\, \mathit{mod}\, n$ in its capability. The leader proposes a request by invoking its vote capability to write operation $op$ to its voter buffer, which the voter prevents from being changed once the leader marks this proposal as complete. Followers wait for the leader to complete its proposal to then validate the operation and express their agreement/disagreement (by submitting the operation they saw or by writing the corresponding value to the agreement vector (see Sec.\[sec:implementation\])).
System Call Vote {#sec:syscalls}
----------------
16 client $c_k$:
17 write $m :=$ syscall opcode + parameters
18 to $c_k$'s request buffer
19 wait for reply in $c_k$'s response buffer@\\[1mm]@
20 hypervisor replica $HV_i$:
21 service loop:
22 poll all client buffers
23 remember new request $(m, c_k)$ as pending@\\[1mm]@
24 on pending request:
25 // leader
26 $(m, c_k) :=$ pending.remove_head
27 if ($m$ is invalid syscall)
28 skip to next pending request
29 $\mathit{VS} := \emptyset$
30 for each voter $v_i$ used to execute $m$
31 // collect voter sequence numbers
32 introspect $v_i$ to read $\mathit{seq}_i := v_i.\mathit{seq}$
33 $\mathit{VS} := \mathit{VS} \cup \{(v_i, \mathit{seq}_i)\}$
34 // follower
35 if (pending requests $\neq \emptyset$)
36 set timeout
37 // all
38 $v_{\mathit{log}}$.agree_on (``write(log, $\langle m, c_k, \mathit{VS}\rangle$)'')
39 with validate $:=$
40 ($\mathit{m} \neq$ request from client $c_k$) ||
41 ($v_{\mathit{log}}.\mathit{seq} \neq \mathit{seq}_{\mathit{log}}$) ||
42 ($\mathit{seq}_v \neq v.\mathit{seq}$, where $(v, \mathit{seq}_v) \in \mathit{VS}$))
43 if (at least one replica disagrees)
44 $v_{\mathit{log}}$.vote_for_reset()
45 if (not $f+1$ agreement)
46 repeat vote
47 execute $m$
In Phase 1, replicas first agree on the system call to execute following the generic pattern above. In Phase 2, they then vote on critical operations. Fig.\[fig:consensus\_protocol\_syscall\] shows the pseudocode for system-call agreement. Lines 16–23 illustrate the client invocation pattern discussed above. The leader selects a pending system call (Line 26) with valid opcode (Line 27) and prepares the entry to log. To prevent equivocation during subordinate votes (e.g., attempts to trick a replica into proposing the next system call without completing the current one), we enforce some additional principles:
- [**P.3 *Coordinated subordinate votes:***]{} correct replicas vote only on subordinate voters ($v_i \in V$) to execute the current system call.
- [**P.4 *Presence of correct replica:***]{} no voted operation succeeds without at least one correct replica.
We enforce P.4 by requiring quorums of at least $f+1$ matching votes, while preventing impersonation (c.f., P.1 in Sec.\[sec:hypervisor\]). In combination, these principles ensure that subordinate voters $v_i
\in V$ will keep their state while in Phase 1 (including their sequence numbers). By agreeing, alongside the system call, on the first sequence number of all voters used in this system call (collected in Lines 29–33 in the set $\mathit{VS}$ and validated in Line 42), we ensure that all replicas know all sequence numbers to start with in subordinate votes, even if they have been lagging behind. In the absence of errors, the $j^{th}$ subordinate vote on $v_i$ will be executed with sequence number $\mathit{seq}_i + j$, assuming $(v_i, \mathit{seq}_i) \in VS$ was the start sequence number of $v_i$. This agreement on the initial sequence number then allows for a simpler progress tracking in Phase 2, when executing subordinate votes.
Because of the impossibility in Sec.\[sec:system\_calls\], system-call votes operate with reduced error diagnostics: replicas reset $v_{\mathit{log}}$ if it got suspended after disagreement (Lines 43, 44) and repeat votes for pending system calls unless they fail for all client-leader combinations, in which case they exclude this client.
Subordinate Votes {#sec:subordinate_votes}
-----------------
48 $HV_i$.vote (log, $v_i$, $\mathit{seq}_i$, req, $m$, dest) {
49 if (syscall_log.log $\neq$ log)
50 return success
51 if ($v_i.\mathit{seq} \neq \mathit{seq}_i$)
52 if ((err[$v_i$].log $\neq$ log) ||
53 (err[$v_i$].req $\neq$ req) ||
54 (err[$v_i$].$\mathit{eseq} > \mathit{seq}_i + 1$))
55 return success
56 push_error_and_reset_voter
57 if (!err[$v_i$].success)
58 repeat vote with $\mathit{seq}_i + 1$
59 // @{\color{red}$HV_i$}@ is up to speed with the others
60 $v_i$.agree_on(``write(dest, $m$)'') with $\mathit{seq}_i$
61 and validate $:=$ ($m$, dest) is valid
62 if (at least one replica disagrees)
63 push_error_and_reset_voter
64 initiate recovery
65 if ($f+1$ agreement)
66 return success
67 repeat vote with $\mathit{seq}_i + 1$
68 } @\\[.1mm]@
69 push_error_and_reset_voter:
70 error $:=$ introspect($v_i$)
71 $v_{\mathit{err}}$.agree_on(``write(err[$v_i$], error)'')
72 with validate $:=$
73 adjust own error information
74 (proposed error $=$ own error)
75 if (error vote fails)
76 $v_{\mathit{err}}$.vote_for_reset($\mathit{eseq}$)
77 repeat pushing the error
78 $v_i$.vote_for_reset($\mathit{seq}_i$)
The code for executing subordinate votes in Fig.\[fig:subordinate\] has to solve two problems: (i) preserve determinism despite errors and (ii) prevent replicas from prematurely resetting voters. From reaching agreement on the system call, we know that the first subordinate vote on $v_i$ starts with $\mathit{seq}_i$ because $(v_i,
\mathit{seq}_i) \in \mathit{VS}$. As such, without errors, the $j^{th}$ subordinate vote on $v_i$ happens with sequence number $\mathit{seq}_i + j$. The same applies to votes with at least one disagreeing replica that all received $f+1$ agreement because, after the voter resets (Line 62), they are not repeated (Line 66). The key for lagging replicas to catch up in case of error is to make sure they learn about all errors, so that they know how many times a vote was repeated and when it was successful. Assume the $k^{th}$ subordinate vote ($k < j$) was the last to fail with $\mathit{seq}_i^k$, then $k$ completed with $\mathit{seq}_i^k+1$ and the system call progressed to subordinate request $j$ if $v_i.\mathit{seq} - \mathit{seq}_i^k = j -
k$.
Solutions to the second problem address the point that all replicas must learn about errors. With $n = 2f+1$ and $|Q| = f+1$, up to $n - |Q| = f$ replicas may lag behind while the remaining $|Q|$ progressed to another subordinate request or even to another system call. In particular, faulty replicas may fail a subordinate vote but agree to reset the voter, which erases the error information about the failed vote from the voter and leaves behind as few as a single correct replica to know about the error. This scenario occurs if $f$ faulty and one correct replica resets the voter before others diagnosed it. Clearly, without costly cryptographic information, the honest replica cannot convince others about what has happened. The following design principle solves this problem by preventing premature resets before error information is pushed to the error log.
- [**P.5 *No reset before error logging:***]{} correct replicas reset subordinate voters only after the error got logged.
This error state contains information about the current system call, i.e.: the system-call entry $\mathit{log}$; the subordinate vote $\mathit{req}$; the sequence number of the voter $v_i$; the point where it failed $\mathit{eseq}$ and which replicas agreed/disagreed. In consequence, lagging replicas can validate if the current subordinate vote succeeded (Lines 52–55) and, if not, who was responsible for it to fail. Voter $v_i$ prevents destructive writes until it is reset, which P.5 and P.4 ensure happens only after error information was written to the log. Non-destructive writes are updates of empty buffers respectively updates of the agreement vector from timeout to agree/disagree and from empty to any of these three.
The argument for why the problem does not recur with the nested vote for logging the error state is as follows: (i) The state to push is held in the voter $v_i$. Therefore, even if a replica lags behind, finding $v_i$ suspended, it knows what information to write to the log. (ii) Because of P.5, and because at least $f+1$ replicas are required (P.4) for votes to succeed, the only way to make progress is by writing correct error information. Therefore, either faulty replicas agree to writing correct error information or eventually correct replicas catch up and write correct information. The exact information seen by the replicas may differ depending on the time they read it, i.e., in late reads, more replicas may have expressed their consent or disagreement. However, it will always contain at least the consensual result of the vote (i.e., whether $f+1$ replicas agree, disagree or timed out) and, in the former two cases, it identifies at least one replica that diverges from the majority (the leader, in case of $f+1$ disagreement). This replica is proven faulty. Followers, reading error information after the leader and finding proposals of additional replicas, downgrade their own information to that of the leader after validating it as described above (Line 73). Repeating the vote while rotating the leader ensures that valid error information is proposed latest after $f$ retries. It then suffices to reset $v_{\mathit{err}}$, whenever it becomes suspended (Line 76). Once error information is pushed, replicas vote to reset the voter $v_i$ for the subordinate vote (Line 78) and continue executing it.
Implementation {#sec:implementation}
==============
The implementation of capability invocation is standard (c.f.[@needham:cap]): [*T2H2*]{}intercepts external operations, looks up the capability in the capability register file, and forwards the operation to the NoC after the privilege check succeeds, silently dropping the operation otherwise. Replica IDs are communicated as labels in the capability [@Hardy:1985:KA:858336.858337], which [*T2H2*]{}inserts as additional parameter into the operation.
Our voter implementation is driven by the following considerations and their impact on functional simplicity.
Buffered vs. Unbuffered Votes
-----------------------------
Perhaps most impactful is the decision to buffer votes to allow replicas to make their proposals without first having to synchronize on the time when the signal for such a vote must be held. Although buffering increases the complexity of the voter, it decouples replicas, allowing them to act in a partially synchronous fashion and, as long as different voters are used, even partially out-of-order[^6]. Buffering votes is ideal in a NoC architecture, since votes are transmitted as normal messages. Tiles can continue executing once the message is sent. We therefore implement voters to contain buffers for storing proposals from the different replicas for the current vote executed with this voter.
![Internal structure of a voter. One, resp. $n$ buffers hold the message of replicas to vote upon and $\mathit{size}$ its length. $f$ defines the fault threshold, $\mathit{seq}$ is a voter maintained sequence number. The agreement and reset vector are described below.[]{data-label="fig:voter"}](voter){width=".75\columnwidth"}
Immediate vs. Deferred Masking
------------------------------
A similarly impactful decision is whether voters should be able to mask faults immediately. Alternatively, voting can be repeated until a valid proposal is made. The consequences, besides time to agreement, are the amount of memory needed for buffering votes vs. the complexity of the voter logic. To mask faults and reach agreement immediately after $|Q| = f + 1$ matching proposals arrive, the voter needs to buffer suggestions from at least $f+1$ replicas. Since up to $f$ such messages may be wrong and because the voter can only find out after receiving $f+1$ matches, buffer space for at least $f+1$ messages is needed to not have to repeat the vote.
We implemented two variants of [*T2H2*]{}voters to evaluate the resource/performance trade-off at the two extremes of this spectrum. Our $n$-buffer variant (Fig.\[fig:voter\] a) implements one message buffer per replica. Each time a message arrives, it is compared against all other stored messages and the operation applied once $f+1$ buffers match. Our single-buffer variant (Fig.\[fig:voter\] b) trades agreement time for a more resource-efficient implementation: there is only one buffer; and only the current leader is granted write access to this buffer. The single-buffer voter follows a leader-follower voting scheme, with the leader proposing a vote and followers validating this proposal. To prevent inconsistency, the voter prevents modification of the leader proposal once the leader marks the proposal as ready. This allows follower replicas to introspect the stored message and express their agreement/disagreement. For this purpose, the single-buffer voter implements an agreement vector with one (initially empty: $-$) tri-state cell for each replica to express agreement $A$ or disagreement $D$. Now, one of three things may happen when replicas propose:
1. a majority of $f+1$ or more replicas disagree with the leader proposal. In this case, the leader proposal is considered invalid and the operation is not applied; or
2. a majority of at least $f+1$ replicas agree. In this case, the proposal is accepted and the voter applies the operation in its buffer.
3. the operation times out without a majority of replicas agreeing / disagreeing. In this case, the replicas record this error and repeat the vote after rotating to the next leader.
The $n$-buffer version requires logic circuits for pairwise buffer comparison whereas in the single-buffer version a 2 data-bit majority gate over the agreement vector suffices.
Internal vs. External Error Handling {#sec:voter_error}
------------------------------------
The third question is whether the voter itself should include provisions for diagnosing errors and for informing replicas about them. Errors are detected when one replica diverges with the majority decision. Voter-initiated error handling translates to the voter tracing back to the voting replicas’ cores to identify where to deliver error-handling interrupts. The expected complexity discourages such a solution. We therefore offload error handling to software and support replicas by a means to track progress (the sequence number $\mathit{seq}$) and by suspending voting after detecting a mismatch. In this situation, $\mathit{seq}$ does not advance but the voter may still apply the operation (in case of $f+1$ agreement). Replicas introspect the voter registers and buffers to diagnose the error, by looking for divergences.
To resume execution of suspended voters, replicas reset the voter, which clears all buffers and the agreement and reset vectors and advances the sequence number by one. Reset itself is a voted operation over the reset vector, which contains one bit per replica. The voter resets once $f+1$ bits in this vector are set. Although this quorum guarantees that at least one correct replica agrees to resetting the voter, it does not prevent faulty replicas from resetting the voter prematurely, that is, before all correct replicas were able to retrieve the error state. P.5 and the protocol in Sec.\[sec:subordinate\_votes\] handles this corner case.
Dimensioning Voters {#sec:voter_dimension}
-------------------
The last question we discuss here is: for how many faults should the voter hardware be laid out. Since we aim at implementing voters in silicon, we have to make this choice at system design time to dimension buffers and vectors large enough for the maximum number of faults to tolerate ($f_{\mathit{max}}$). However, to not always have to execute at this maximum replication degree, a fault threshold $f \le f_{\mathit{max}}$ of voters can be configured at boot time. For instance, if the system should tolerate up to $f_{\mathit{max}} = 3$ faults, it needs to be dimensioned to have $n_{\mathit{max}} = 2 f_{\mathit{max}} + 1 = 7$ fields in the vectors (and $n_{\mathit{max}}$ buffers, assuming $n$-buffer voters). This voter can be operated at any fault threshold $0 \le f \le f_{\mathit{max}}$. The voter design has been kept simple enough, and decoupled enough from the surrounding logic. As such, we can expect with high confidence that [*T2H2*]{}can be implemented and shown correct, as well as stay functional even when the tile it is associated with fails.
Evaluation {#sec:evaluation}
==========
As an early validation of our proposal, we have implemented [*T2H2*]{}in both voter variants in VHDL on a Zynq-7 ZC702 Evaluation Board. We instantiated 3 Microblaze cores as tiles, running at 50 MHz, each with one [*T2H2*]{}, connecting the tiles through [*T2H2*]{}with an AXI interconnect (serving as the NoC). We measured the performance of the service loop (Fig. \[fig:consensus\_protocol\_syscall\]) to agree on and execute client-invoked system calls for granting and priming capabilities. Grant ([ L4.map]{} [@Liedtke:sosp:l4:1995]) copies capabilities between capability spaces and prepares for later revocation. Prime consensually copies a capability from the client’s capability space into a [*T2H2*]{}capability register, where it is ready for invocation. We have measured the performance of grant and prime in two different implementations of capability spaces[^7]: (i) as a private data structure in each replica, requiring, in the case of prime, only the vote to install capabilities and two further to reply to the client and mark the system call as finished; and (ii) as a read-shared, consensually updated data structure, trading off speed for a smaller memory footprint by introducing additional votes for track keeping.
As baselines, we compare to a cross-tile invoked singleton kernel (horizontal line), executing the same system calls on its private state, with 1637 cycles for *grant* (1977 cycles for *prime*) and to a shared-memory variant of MinBFT[^8] requiring 242824 cycles to agree on a system call. Our agreement protocol outperforms MinBFT by one order of magnitude.
### Per-Replica Capability Space {#sec:per_replica_cspace}
![Average execution times of the three consensual system calls — ***null***, ***grant*** and ***prime*** — when executed on a per-replica capability space implementation. System calls are broken down into the individual votes for agreeing on the system call and for performing the critical updates required. Shown are also the Q5 / Q95 percentile and the average costs of executing the respective system calls on a singleton-kernel.[]{data-label="fig:per_replica_cspace"}](per_replica_cspace){width=".8\columnwidth"}
Figure \[fig:per\_replica\_cspace\] shows the average performance of the ***grant*** and ***prime*** system calls in a per-replica capability space implementation relative to the two baselines: ***null*** and a singleton kernel instance performing these system calls in a non-consensual manner. Shown are the system calls broken down into individual votes and the Q5 / Q95 percentiles of the overall measurements.
The minimal costs for learning about a system-call request and executing it are 1571, 1637 and 1977 cycles on average for null, grant and prime, respectively, which is the baseline of the singleton kernel. System calls for the single buffer version have a factor $8.9$ – $9.6$ increase, which can be explained due to the voter not benefiting from caching. Whereas the singleton kernel merely has to copy one request from the memory where the client core places it, missing in all caches in the process, following replicas have to poll the voter to wait for the leader to make a proposal and then confirm (or reject) the proposal made. Each such voter access amounts to costs equivalent to a cache miss.
As can be seen, reaching agreement on the subordinate votes is much faster, which is due to the fact that replicas already align themselves when reaching agreement on the system call to execute.
In the n-buffer version of the voter, higher costs occur during the agreement on the system call, which is due to the writing of the complete request to the voter, not just setting a bit in its agreement vector. However, subordinate votes are much faster, since replicas no longer wait for the leader to make a proposal. Instead, they just propose what should be written as critical operation.
### Consensually Updated Capability Space {#sec:consensual_cspace}
![Average execution times of the three system calls for consensually updated capability spaces.[]{data-label="fig:consensual_cspace"}](consensual_cspace){width="\columnwidth"}
Figure \[fig:consensual\_cspace\] shows a similar diagram as Figure \[fig:per\_replica\_cspace\], this time, however, for consensually updated capability spaces. Granting and priming capabilities now require additional votes to update the data structure.
For grant, these additional votes are the write of the target node into the destination address space, which represents the mapping of this capability from the source, the update of the source node’s next pointer, and the update of the previous pointer of the element that used to be next to the source node. In addition, because destructive updates may lead to inconsistencies with non-faulty, but late replicas, needed values that will later be updated must be stored before advancing with these updates. We do so in the capability space object in the log-destructive operation.
For prime, the additional votes are logging destructive updates, marking the capability node as being loaded into a capability register, creating the back link from capability register to node, and installing the capability into the register. If a valid capability is replaced, one further vote is required to update the replaced node with a copy where the forward pointer to the capability register is cleared.
Again, we observe slightly higher costs for reaching agreement on the system call, which is compensated by the faster subordinate votes when we compare the numbers for n-buffer voters against the single-buffer variant.
As expected, subordinate votes benefit from the determinism obtained from agreeing on the system call first. The average execution time for priming is 12124 cycles for per replica and 15144 cycles for consensual capability spaces (9727 vs. 16847 cycles for grant) for single-buffer voters. The plotted variations (5% and 95% percentile) are due to the replicas not executing in lockstep when proposing votes. The additional votes for consensually updated capability spaces (ii) amount to a 1.2 (1.7) fold increase of these costs, but reduce the memory footprint by $1/n^{th}$ with only a minor overhead for consensual data structures. Subordinate votes in $n$-buffer benefit from the replicas not having to wait for the leader proposal, resulting in overall better performance.
Again, the 6.7 (/ 7.3) times slower performance relative to the singleton kernel can be explained due to the voter not benefiting from caching: *Singleton kernel:* System call execution is triggered by the client writing to shared memory on one core and the kernel (on another core) reading it. From then on, all the operations happen locally in the core of the kernel without any interaction with the outside. Therefore, all memory operations aside from the invocation and reply hit in the core’s cache, which in our setting responds within 1 cycle. The cross-core operations (invocation (1) + reply (2)) dominate these costs.
*Replicated kernel:* System call execution starts as well with invocation (1), but then, the leader needs to propose the request (2), followers validate it and (3) express agreement (4) upon which the voter updates the memory and all replicas wait for the vote to reach agreement (5). In (i), we then execute locally, but for replying (to not introduce storage channels) we have to repeat at least (4) + (5), assuming $n$-buffer voters. As such, even without any delays, we have 7 cache misses vs. 2 in the singleton kernel execution, hence a factor of 3.5. Additionally, more voter accesses are performed to read the sequence number, which we need for flow control.
Despite the overhead of system call execution introduced, we believe the improved resilience outweighs these costs, in particular since they affect privilege changes only. Once started, applications act independently from the kernel.
\
![System calls broken down into individual votes. Shown are the Q5 and Q95 percentile for the main system call vote and each subordinate vote for n-buffer voters. The variations for single-buffer voters are similar.[]{data-label="fig:variations"}](variation_n_buf "fig:"){height="3.5cm"}
To confirm that variations in fact originate from the agreement on the system call to execute, we have broken down system call execution into their individual votes and measured their Q5 and Q95 percentile. Fig. \[fig:variations\] shows these values. As expected, subordinate votes remain close to their average execution times, whereas agreement on the system call varies significantly.
![Code size in lines of C++ / VHDL code (logic/total).[]{data-label="fig:code_size"}](code_size){width="\columnwidth"}
Fig. \[fig:code\_size\] lists the code size (excluding initialization) for the service loop, for consensually executing critical operations and for interfacing with the capability registers. Also shown are the VHDL source lines of code for the logic and the overall design of the voter and capability unit. As can be seen, the amount of code that each replica executes for the above grant and prime system call is well below 1000 lines of code. Faults in this code are masked by the majority of replicas outvoting faulty replicas in critical operations. Similarly, the hardware overhead is just above 400 lines of VHDL code for the logic plus 2411 lines of VHDL for connecting the logic to the AXI interface and for mapping the corresponding internal signals.
Fig.\[fig:performance\] shows the FPGA resources of the (post-synthesis) implementation of our components. LUTs are units with no state, used to implement the combinatorial logic; while registers hold state, e.g, to keep buffer contents, but implement no logic. Each F7 Mux (wide multiplexer) combines the outputs of two LUTs together, while F8 Muxes combine the outputs of two F7 Muxes.
Notice that the absolute resource requirement of T2H2 will not increase significantly if more complex cores are to be controlled. Hence, the relative overhead will shrink when more complex tiles are considered.
![FPGA resources required by [*T2H2*]{}(without / with AXI interface).[]{data-label="fig:performance"}](fpga_resources){width="\columnwidth"}
Related Work {#sec:related-work}
============
In this section, we present several classes of works that motivated [*Midir*]{}: low-level approaches for detection and containment of errors in low-level support software; analyses of the evolution of defects in system support software; attempts at preventing and/or mitigating the resulting errors and potential failures; approaches to replication-based fault/intrusion tolerance and resilience.
Mitigation measures have been studied for detection and containment of errors in OS and manycore support software [@5504713; @Seshadri:2007:STH:1294261.1294294; @romain] through an underlying, assumed-trustworthy layer. However, they still have a non-negligible complexity, and in consequence, even a residual fault or vulnerability rate in these supposedly trusted components may breach the platform’s dependability and security goal.
In fact, as confirmed by [@univis91357728], “simple” components with at least a few KLOCs have a non-negligible statistical fault footprint. Other studies [@Ostrand:2002:DFL:566172.566181; @Ostrand:2004:BUG:1007512.1007524] reveal between 1–16 bugs per 1,000 lines of code go undetected before deployment, even in well-tested software, and operating-system kernels form no exception [@ganapathi; @Matias:2014:EES:2554850.2555021]. Recent insights [@Palix:2014:FL:2642648.2619090] reveal that faults in stateful core subsystems — on which we focus here — outrank driver bugs in severity. Minotaur introduces a toolkit to improve the analysis of software vulnerability to hardware errors by leveraging concepts from software testing [@mahmoud2019minotaur].
Many approaches target operating systems with the goal of improving their resilience against faults. However, typically they protect either applications [@otherworld; @Bolchini:2013:SFT:2467238.2467247; @haft] or specific OS subsystems [@Sundararaman:2010:MOS:1837915.1837919; @Swift:2006:RDD:1189256.1189257; @Zhou:2006:SSR:1267308.1267312; @Shen_Elphinstone] and only from accidental faults. Efforts for providing whole-OS fault tolerance include [@Herder:2006:CHD:1170132.1170290; @Nikolaev:2013:VOS:2517349.2522719; @David:2008:CIR:1855741.1855746; @Lenharth:2009:RDO:1508244.1508251; @osiris; @Govil:1999:CDR:319151.319162; @david:os_attacks]. Furthermore, the complexity of these recovery kernels is comparable to that of a small hypervisor. For example, OSIRIS [@osiris] directs OS recovery to a 29 KLOC reliable computing base (RCB) [@rcb], roughly twice the size of modern microkernels [@Liedtke:sosp:l4:1995; @l4fiasco; @Klein+al:sosp:sel4:2009; @asmussen:m3]. Again, this makes the likelihood of residual faults or vulnerabilities non-negligible.
Several other works have given early steps in the direction of the solutions we advocate in this paper, minimizing the threat surface, or enforcing isolation. Nohype [@Szefer:2011:EHA:2046707.2046754] removes all but a small kernel substrate from application cores, which run functionality-rich OSs in virtual machines (VMs), reducing the threat surface. Cap [@needham:cap] and M3 [@asmussen:m3] exploit hardware capability units and Hive [@hive] a bus-level firewall to isolate VMs at tile granularity. However, although this avoids trusting tile-local kernel substrates for isolation, their configuration interface, which is necessary to retain flexible resource sharing, turns the configuring kernel into a single point of failure. We address this problem in [*Midir*]{}. Cheri [@Woodruff:2014:CCM:2665671.2665740] adds capability protection on top of page-based protection, but includes the MMU and the OS page-table management in the reliable computing base (RCB), which means the former must be trustworthy. The concept behind [*Midir*]{}is independent of the protection model, not being necessarily tied to e.g., capabilities. Also, establishing the fault containment domains at the granularity of tiles, we are agnostic about the semantics and interplay of tile-internal and/or core-level components, e.g., MMUs, memory protection or page-table management. Enforced by [*T2H2*]{}, the protection mechanisms are crafted at inter-tile level, emulating the spacial isolation of distributed system nodes.
Replication has been used before in closely-coupled systems, primarily to tolerate accidental faults in cyber-physical systems (CPS), by replicating controllers to form triple modular redundant (TMR) units, or duplicated self-checking units. An example of the use of TMR in highly critical systems can be seen in the primary flight computers of Boeing 777’s fly-by-wire (FBW) system [@yeh1998triple]. In a similar context, a form of passive redundancy can also be seen in Airbus’ dependability-oriented approach to FBW, where “hot spares” are used in case the active computer interrupts its activity [@traverse2004airbus]. The concept was extended to multi-phase tightly synchronous message-passing protocols still in the CPS domain [@mancini1986modular; @kopetz2003time]. The so-called ’Paxos’ [@schiper2014developing], and ’Byzantine’ [@pbft] Fault-Tolerant State-Machine Replication classes of protocols promote resilience to threats, respectively accidental, and both accidental and malicious, extending the concept to generic classes of applications, namely in loosely-coupled systems. For example, Castro’s seminal BFT-SMR protocol [@pbft] masks the actions of a minority of up to $f$ compromised replicas, by reaching a majority voted consensus of $|Q|=2f+1$ out of $n=3f+1$ replicas. Behind all the categories of techniques above is a baseline voting mechanism amongst the values proposed by a pre-defined number of replicated fault-independent components. [*Midir*]{}offers such baseline mechanism at a low enough level of abstraction to serve essentially any replication-oriented application.
Architectural hybridization [@Verissimo:2006:TTW:1122480.1122497] (i.e., the inclusion of trusted-trustworthy components that follow a differentiated fault model) allows reducing $n$ and $|Q|$ to $2f+1$ and $f+1$, respectively [@1353018; @levin2009trinc; @veronese2013efficient; @cheapbft]. The implementation of [*T2H2*]{}, the [*Midir*]{}hybrid, draws from these quorum reduction results, and further accelerates the BFT-SMR protocol that [*Midir*]{}-enabled FIT microhypervisors use to coordinate system-call execution (Sec. \[sec:hypervisor\]).
Paxos and BFT replication have been attempted as well inside MPSoCs [@bressoud:hypervisor_ft; @kernel_paxos; @paxos; @romain; @barrelfish]. However, all these works were made under the assumption of a trusted low-level kernel (e.g., hypervisor or platform manager), which obviously is a single point of failure (SPoF). One of the key results of [*Midir*]{}lies in the realization of the distributed system-on-a-chip (DSoC) vision, which enables such replication management techniques in MPSoCs, whilst removing the SPoF syndrome of the low-level kernel.
Conclusions and Future Work {#sec:conclusions}
===========================
We have introduced [*Midir*]{}, an architectural concept which breaks new ground and opens promising avenues in the applicability and resilience of manycore architectures (MPSoC). Through minimalist mechanisms integrated in the MPSoC architecture, [*Midir*]{}frees MPSoCs from the SPoF syndrome, fulfilling the vision of *distributed* systems-on-a-chip (DSoC).
In this paper, we show in particular that [*Midir*]{}-enabled DSoCs achieve a quantum step towards off-the-shelf chip resilience, since these mechanisms are generic enough to support, in-chip and with high reliability, a large variety of the protection and redundancy management techniques normally implemented in software at higher layers in ’macro’ systems. To convincingly prove our point, we exemplified and evaluated an implementation, over [*Midir*]{}, of the most complex version of our solution set: a Byzantine fault tolerant microhypervisor. We have shown the practicality of our concept, as having quite satisfying performance, since it outperforms the highly efficient MinBFT protocol by one order of magnitude. The low overhead of our approach shows as well large promise for future full hardware solutions. Furthermore, [*Midir*]{}was intentionally designed as a non-intrusive extension to current chip architectures, being anchored on simple and self-contained hardware extensions. Taken up by a hardware manufacturer or integrator, it allows a backward compatible, non-fracturing evolution. We hope that our findings may be key to enhance general MPSoC architectures towards distributed DSoCs and amongst other avenues, lead to next-generation COTS resilient chips. After this initial work, several questions remain to be answered, namely on kernel design details, rejuvenation and diversification for sustainability, and so forth, which leave ample room for future work.
[^1]: pronounced meedir
[^2]: Following the notion of virtualization, we call tile internal addresses *guest physical addresses* and external addresses *host physical addresses*.
[^3]: [*T2H2*]{}’s reconfiguration interface cannot directly be addressed by a capability; accesses must pass through a voter.
[^4]: From a functional perspective, the location of the voter is irrelevant. However, for performance reasons, voters near replicas are preferable.
[^5]: Modulo [*Midir*]{}’s [*T2H2*]{}, which, justified through its simplicity, we assume will not fail.
[^6]: To simplify monitoring of the progress of a system call, we shall later require that all replicas execute the critical operations of each system call in the same order. Operations of different system calls need not be constrained in this way, and, at the cost of a more complex progress tracking, this requirement can be further relaxed to: same order as far as a single voter is concerned.
[^7]: Container object for an application’s capabilities.
[^8]: We omit client signatures in favor of authentic buffers, but implement UIs with HMACs. USIGs can be accessed without overhead.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we prove that every projectively normal Fano manifold in $\P^{n+r}$ of index $1$, codimension $r$ and dimension $n\geq 10r$ is birationally superrigid and K-stable. This result was previously proved by Zhuang under the complete intersection assumption.'
address: 'Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago'
author:
- Fumiaki Suzuki
title: 'Birational superrigidity and K-stability of projectively normal Fano manifolds of index one'
---
Introduction
============
Birational superrigidity and K-stability of Fano manifolds are two important notions with different backgrounds. The notion of birational superrigidity is motivated by the rationality problem of Fano manifolds. A Fano manifold $X$ with the Picard number $1$ is called [*birationally superrigid*]{} if any birational map from $X$ to the source of another Mori fiber space is an isomorphism. It implies that $X$ is non-rational and $\operatorname{Bir}(X)=\operatorname{Aut}(X)$. On the other hand, the notion of K-stability is motivated by the existence of Kähler-Einstein metric on Fano manifolds. A Fano manifold $X$ is called [*K-stable*]{} if the Donaldson-Futaki invariant is positive for any non-trivial normal test configuration. It is stronger than the K-polystability which is equivalent to the existence of Kähler-Einstein metric [@CDS1; @CDS2; @CDS3; @T]. Birational superrigidity and K-stability are unexpectedtly related according to Odaka-Okada and Stibitz-Zhuang [@OO; @SZ], and it is conjectured by Kim-Okada-Won [@KOW] that every birationally superrigid Fano manifold is K-stable. Both the notions are intensively studied in the case of smooth Fano complete intersections of index $1$: birational superrigidity by Iskovskih-Manin, Pukhlikov, Cheltsov, de Fernex-Ein-Mustaţă, de Fernex, Suzuki, and Zhuang [@IM; @P1; @P2; @P3; @P5; @P6; @C1; @dFEM2; @dF1; @dF2; @S; @Z] (see also the note [@K2] written by Kollár), and K-stability by Fujita and Zhuang [@Fuj; @Z]. Among them, Zhuang [@Z] proves that every smooth Fano complete intersection of index $1$ and small codimension is birationally superrigid and K-stable.
In this paper, we replace the complete intersection assumption of the main theorem of [@Z] by the projective normality:
\[t1\] Every projectively normal Fano manifold in $\P^{n+r}$ of index $1$, codimension $r$ and dimension $n\geq 10r$ is birationally superrigid and K-stable.
So far we do not have any non-complete-intersection examples which satisfy the assumption of Theorem \[t1\]. In fact, the Hartshorne conjecture predicts that every smooth projective variety in $\P^{n+r}$ of codimension $r$ and dimension $n>2r$ is a complete intersection, while the conjecture is widely open.
A key step involves a generalization of multiplicity bounds for cycles on smooth complete intersections due to Pukhlikov [@P4 Proposition 5], Cheltsov [@C3 Lemma 13], and the author [@S Proposition 2.1].
This paper is organized as follows. In Section $2$, we study the ramification locus of the linear projection from a closed point. In Section $3$, we prove multiplicity bounds for cycles. In Section $4$, we prove a stronger version of Theorem \[t1\]. In Section $5$, we discuss the singular case.
Throughout this paper, the base field is the field of complex numbers $\C$.
Let $X$ be a scheme of finite type.
- We denote by $[X]$ the fundamental cycle of $X$.
- For a closed point $x\in X$, we denote by $e_{x}(X)$ be the Samuel multiplicity of $X$ at $x$. For a closed subset $Z\subset X$, let $e_{Z}(X)=\min\left\{e_{x}(X)\mid x\in Z \right\}$. We extend the definition of Samuel multiplicities to arbitrary cycles by linearlity.
- For pure-dimensional cycles $\alpha, \beta$ on $X$ intersecting properly and an irreducible componenet $Z$ of the intersection, we denote by $i(Z,\alpha\cdot \beta; X)$ the intersection multiplicity of $Z$ in $\alpha \cdot \beta$ whenever the intersection product $\alpha\cdot \beta$ is defined at $Z$.
- For a closed subscheme $Z\subset X$, we denote by $s(Z, X)$ be the Segre class of $Z$ in $X$.
- For a vector bundle $E$ on $X$, we denote by $c(E)$ the total Chern class of $E$, and by $c_{i}(E)$ the $i$-th Chern class of $E$.
- We denote by $Z_{i}(X)$ the group of $i$-cycles on $X$, and by $CH_{i}(X)$ (resp. $N_{i}(X)$) the group of those modulo the rational equivalence (resp. the numerical equivalence). When $X$ is smooth, we denote by $N^{j}(X)$ the group of $j$-cocycles on $X$ modulo the numerical equivalence.
For the definitions, we refer the reader to [@F].
The author wishes to thank his advisor Lawrence Ein for constant support and warm encouragement. He is grateful to Ziquan Zhuang for helpful comments.
The ramification locus of the linear projection from a closed point
===================================================================
Let $X\subset \P^{n+r}$ be a non-degenerate smooth projective variety of dimension $n$ and codimension $r$. We take a closed point $p\in \P^{n+r}$ not contained in $X$. The choice of $p$ determines a section $s\in \Gamma(X,N_{X/\P^{n+r}}(-1))$. Let $\pi_{p}\colon X\rightarrow \P^{n+r-1}$ be the restriction of the linear projection from $p$. We define the ramification locus $R(\pi_{p})\subset X$ of $\pi_{p}$ as the zero-scheme of the section $s$ [@F Example 14.4.15]. We consider the following condition $(\star)$: given any closed point $p\in \P^{n+r}$ not contained in $X$, we have $Z\cap R(\pi_{p})\neq \emptyset$ for any closed subset $Z\subset X$ of dimension $\geq r$.
\[l1\] Assume that $X$ is a complete intersection. Then the condition $(\star)$ is satisfied.
The ramification locus $R(\pi_{p})$ is globally defined by $r$ hypersurfaces on $X$. The condition $(\star)$ is obviously satisfied.
\[l2\] Assume $n\geq 3r-2$. Then the condition $(\star)$ is satisfied.
We may assume that $r\geq 2$. We denote by $\operatorname{Tan}(X)$ the tangent variety of $X$. We prove $\operatorname{Tan}(X)=\P^{n+r}$. We denote by $\operatorname{Sec}(X)$ the secant variety of $X$. We have $\operatorname{Tan}(X)=\operatorname{Sec}(X)$ as long as $n\geq r$ by the connectedness theorem of Fulton and Hansen [@L Corollary 3.4.5]. On the other hand, we have $\operatorname{Sec}(X)=\P^{n+r}$ as long as $n\geq 2r$ by Zak’s theorem on linear normality [@L Corollary 3.4.26]. Therefore we have the desired equality of sets under our assumption. It follows that $R(\pi_{p})$ is non-empty, and $\dim R(\pi_{p}) = n-r$ if $p$ is general. We check the condition $(\star)$. We may assume that $p$ is general so that $R(\pi_{p})$ is defined by a regular section. We have $$[R(\pi_{p})]=c_{r}(N_{X/\P^{n+r}}(-1))\cap [X]$$ by [@F Proposition 14.1]. For the right hand side, we have $$c_{r}(N_{X/\P^{n+r}}(-1))\cap [X]=\sum_{i\geq0} (-c_{1}(\mathcal{O}_{X}(1)))^{i}\cdot c_{r-i}(N_{X/\P^{n+r}})\cap [X]$$ by [@F Example 3.2.2]. We have $$c_{r}(N_{X/\P^{n+r}})\cap [X]= \deg X\cdot c_{1}(\mathcal{O}_{X}(1))^{r}\cap [X]$$ by the self-intersection formula [@F Corollary 6.3]. On the other hand, we have $$H^{2i}(X,\Z)=\Z\cdot (c_{1}(\mathcal{O}_{X}(1))^{i}\cap [X]) \text{ for any }0\leq i\leq r-1$$ by the Barth-Larsen theorem [@L Theorem 3.2.1] under the assumption $n\geq 3r-2$. Therefore we have $$[R(\pi_{p})] = a\cdot c_{1}(\mathcal{O}_{X}(1))^{r}\cap [X]\in H^{2r}(X,\Z)$$ for some positive integer $a$. For any closed subset $Z\subset X$ of dimension $\geq r$, we have $$\deg(Z\cdot [R(\pi_{p})]) = a\cdot \deg Z \neq 0,$$ which implies $Z\cap R(\pi_{p})\neq 0$. The proof is done.
\[l3\] Assume $r=2$. Then the condition $(\star)$ is satisfied.
It follows from [@Za Chapter II, Corollary 2.5].
Multiplicity bounds for cycles
==============================
In this section, we prove the following key proposition:
\[p1\] Let $X\subset \P^{N}$ be a non-degenerate smooth projective variety of codimension $r$. Assume that $X$ satisfies the condition $(\star)$. Let $\alpha$ be an effective cycle on $X$ of codimension $s$ such that $$\alpha = m \cdot c_{1}(\mathcal{O}_{X}(1))^{s}\cap [X] \in N^{s}(X).$$ Then we have $e_{x}(\alpha)\leq m$ for any closed point $x\in X$ away from a closed subset of dimension $<rs$, where we use the convention $\dim(\emptyset)=-1$.
The complete intersection case is proved by Pukhlikov [@P4 Proposition 5], Cheltsov [@C3 Lemma 13], and the author [@S Proposition 2.1].
[*Step*]{} $1$: We review the construction of residual intersection classes due to Fulton.
\[t\] We consider a diagram $$\xymatrix{
& R\ar[d]^{b} &\\
D\ar[r]^{a} & W \ar[r]^{j} \ar[d]^{g} & V \ar[d]^{f}\\
& X\ar[r]^{i} & Y
},$$ where the square is a fiber square, $i, j, a, b$ are closed embeddings, and $V$ is a $k$-dimensional variety. Assume that:
1. $i$ is a regular embedding of codimension $r$;
2. $ja$ embeds $D$ as a Cartier divisor of $V$;
3. $R$ is the residual scheme to $D$ in $W$.
Let $N=g^{*}N_{X}Y$ and $\mathcal{O}(-D) = j^{*}\mathcal{O}_{V}(-D)$. We define the residual intersection class $\R\in CH_{k-r}(R)$ by the formula $$\R=\left\{c(N\otimes \mathcal{O}(-D))\cap s(R, V)\right\}_{k-r}.$$ Then we have $$\begin{aligned}
X\cdot_{Y} V = \left\{c(N)\cap s(D, V)\right\}_{k-r} + \R\end{aligned}$$ in $CH_{k-r}(W)$.
In the setting of Theorem \[t\], the class $\R$ is represented by an effective cycle if $\dim R =k-r$. Indeed, let $R_{1},\cdots, R_{t}$ be the irreducible components of $R$. Then we have $$\R = s(R, V)_{k-r} = \sum_{i=1}^{t} e_{i}[R_{i}],$$ where $e_{i}$ is the same as Samuel’s multiplicity $e(q)$ of the primary ideal $q$ determined by $R$ in the local ring $\mathcal{O}_{V, R_{i}}$, which is positive [@F Example 4.3.4].
We construct a residual intersection class associated to the linear projection from a closed point. Let $X\subsetneq \P^{N}$ be a non-degenerate smooth projective variety of codimension $r$. We assume that $X$ satisfies the condition $(\star)$. Let $Z\subsetneq X$ be a closed subvariety with $\dim Z\geq r$. Let $p\in \P^{N}$ be a general closed point. Let $C=\overline{\bigcup_{x\in Z} \langle p,x \rangle}$ be the cone of $Z$ with the vertex $p$. Let $R$ be the residual set to $Z$ in $X\cap C$. We have a diagram with a fiber square $$\begin{aligned}
\label{d1}
\xymatrix{
& R\ar[d] &\\
Z\ar[r] & X\cap C \ar[r] \ar[d] & C \ar[d]\\
& X\ar[r] & \P^{N}
}.\end{aligned}$$ We want to define the residual intersection class $\R\in CH_{\dim Z+1-1}(R)$, but $Z\subset C$ is not a Cartier divisor in general. To remedy this situation, we fix a closed point $q\in \P^{N+1}\setminus \P^{N}$. Let $X^{\ast}$ be the cone of $X$ with the vertex $q$. Let $\widetilde{p}\in \langle p, q\rangle$ be a closed point different from $p$ and $q$. Let $\widetilde{C}$ be the cone of $Z$ with the vertex $\widetilde{p}$. Then $Z\subset \widetilde{C}$ is a Cartier divisor. Let $\widetilde{R}$ be the residual scheme to $Z$ in $X^{\ast}\cap \widetilde{C}$. We have a diagram with a fiber square $$\begin{aligned}
\label{d2}
\xymatrix{
& \widetilde{R}\ar[d] &\\
Z\ar[r] & X^{\ast}\cap \widetilde{C} \ar[r] \ar[d] & \widetilde{C} \ar[d]\\
& X \ar[r] & \P^{N}
}.\end{aligned}$$ Let $\widetilde{\R}\in CH_{\dim Z+1 -r}(\widetilde{R})$ be the residual intersection class. Let $\pi_{q}\colon \widetilde{C}\rightarrow C$ be the restriction of the linear projection from $q$. It is finite. We replace $R$ by $\pi_{q}(\widetilde{R})$ if necessary. We define $\R$ to be $(\pi_{q})_{*}\widetilde{\R}$.
We have $$c_{1}(\mathcal{O}_{X}(1))\cap \R = c_{r}(N_{X/\P^{N}}(-1))\cap [Z].$$
We apply Theorem \[t\] to the diagram (\[d2\]). We have $$X\cdot_{\P^{N}} \widetilde{C} = \left\{c(N)\cap s(Z, \widetilde{C})\right\}_{\dim Z+1-r} + \widetilde{\R}$$ in $CH_{\dim Z +1 -r}(X^{\ast}\cap \widetilde{C})$. The morphism $\pi_{q}$ is generically one-to-one. By pushing-forward, we have $$X\cdot C = \left\{c(N)\cap (\pi_{q})_{*}s(Z, \widetilde{C})\right\}_{\dim Z +1-r} + \R$$ in $CH_{\dim Z+1-r}(X\cap C)$. We have $$\begin{aligned}
c(N)\cap (\pi_{q})_{*}s(Z, \widetilde{C}) &=& c(N)\cap (\pi_{q})_{*}\left(c(\mathcal{O}(1))^{-1}\cap [Z]\right)\\
&=& \sum_{i,j\geq 0} (-c_{1}(\mathcal{O}(1)))^{i}\cdot c_{j}(N)\cap [Z],\end{aligned}$$ where the first equality follows from [@F Proposition 4.1 (a)]. Then we have $$\R = X\cdot C + \sum_{i\geq 1}(-1)^{i}c_{1}(\mathcal{O}_{X}(1))^{i-1}\cdot c_{r-i}(N_{X/\P^{N}})\cap [Z].$$ We apply $c_{1}(\mathcal{O}_{X}(1))\cap$ to the both sides. We have $$\begin{aligned}
c_{1}(\mathcal{O}_{X}(1))\cap \R &=& X\cdot Z + \sum_{i\geq 1}(-c_{1}(\mathcal{O}_{X}(1)))^{i}\cdot c_{r-i}(N_{X/\P^{N}})\cap [Z]\\
&=& c_{r}(N_{X/\P^{N}}(-1))\cap [Z],\end{aligned}$$ where the second equality follows from the self-intersection formula [@F Corollary 6.3]. The proof is done.
As a set, we have $$Z\cap \widetilde{R}= Z\cap R(\pi_{p}).$$
It follows from the argument as in the proof of [@P4 Lemma 3].
By the condition $(\star)$, we have $Z\cap R(\pi_{p})\neq \emptyset$ and $\dim Z\cap R(\pi_{p}) = \dim Z -r$. Therefore we have $Z\cap \widetilde{R}\neq \emptyset$ and $\dim Z\cap \widetilde{R} = \dim Z -r$. It implies that $\dim \widetilde{R} = \dim Z+1-r$. Therefore the class $\widetilde{\R}$ is represented by an effective ($\dim Z +1-r$)-cycle whose support is $\widetilde{R}$, which implies that the class $\R$ is represented by an effective ($\dim Z+1-r$)-cycle whose support is $R$.
There is another possible definition of the residual intersection class $\R'\in CH_{\dim Z+1-r}(R)$ for the diagram (\[d1\]) using the blow-up along $Z$. We refer the reader to [@F Definition 9.2.2] for the details. We prove $\R'=\R$. We apply [@F Corollary 9.2.3] to the diagram (\[d1\]). We have $$X\cdot C = \left\{c(N)\cap s(Z, C)\right\}_{\dim Z+1-r} +\R'.$$ The inverse image scheme $\pi_{q}^{-1}(Z)\subset \widetilde{C}$ is the union of $Z$ and a closed subscheme supported on the inverse image $\pi_{q}^{-1}(Z^{ne})$ of the non-embedding locus $Z^{ne}$ on $Z$ of the linear projection from $p$. We have $$\left\{c(N)\cap s(Z, C)\right\}_{\dim Z+1-r} =\left\{c(N)\cap (\pi_{q})_{*}s(Z, \widetilde{C})\right\}_{\dim Z+1-r}$$ by [@F Proposition 4.2] together with $\dim Z^{ne}\leq \dim Z-r$. Therefore we have the desired equality. The proof is done.
[*Step*]{} $2$: We move cycles by multiple residual intersections.
\[mcbmri\] Let $X\subset \P^{N}$ be a non-degenerate smooth projective variety of codimension $r$. Assume that $X$ satisfies the condition ($\star$). Let $A\subset X$ be a closed subset of codimension $s$. Let $Z\subset A$ be a subvariety of dimension $rs$. Then there is a cycle $\beta$ on $X$ such that
1. $\dim \beta =s$, and $\beta$ intersects $A$ in finitely many points;
2. $|Z\cap \operatorname{Supp}(\beta)|\geq \deg \beta$.
The proof is essentially the same as in [@P4 Proposition 5] (see also [@S Proposition 2.1]). We only give a sketch. We take a general closed point $p\in \P^{N}$ and construct $R$ and $\R$ as in Step $1$. We define $p_{1}=p$, $R_{1}=R$ and $\R_{1}=\R$. We replace $Z$ by $R_{1}$, and repeat the same procedure to define $p_{j}$, $R_{j}$ and $\R_{j}$ for $j=2, \cdots, s$. We have $\dim R_{j}=rs-(r-1)j$ and $$\deg \R_{j} = \deg\left(\left(c_{r}(N_{X/\P^{N}}(-1)\right)^{j}\cap [Z]\right).$$ In particular, we have $\dim R_{s}= s$ and $$\deg \R_{s} = \deg\left(\left(c_{r}(N_{X}/\P^{N}(-1)) \right)^{s}\cap [Z]\right).$$ By careful dimension count using the ramification locus and joins of varieties as in [@P4 Lemma 1] (see also [@S Lemma 2.3] and [@S Lemma 2.4]), we have $$\dim A\cap R_{j} =rs-rj.$$ In particular, we have $$\dim A\cap R_{s}=0.$$ Moreover we have $$Z\cap R_{s}\supseteq Z\cap R_{1}\cap \cdots \cap R_{s} \supseteq Z\cap R(\pi_{p_{1}})\cap \cdots \cap R(\pi_{p_{s}})$$ and $$\begin{aligned}
|Z\cap R(\pi_{p_{1}})\cap \cdots \cap R(\pi_{p_{s}})|&=&Z\cdot [R(\pi(p_{1}))]\cdot \ldots \cdot [R(\pi(p_{s}))]\\
&=& \deg\left(\left(c_{r}(N_{X}/\P^{N}(-1)) \right)^{s}\cap [Z]\right).\end{aligned}$$ Therefore $\beta = \R_{s}$ satisfies the condition.
[*Step*]{} $3$:
We take $\alpha$ as in the statement. It is enough to prove that $e_{Z}(\alpha)\leq m$ for any closed subvariety $Z\subset X$ of dimension $rs$. We may assume that $\alpha \neq 0$ and $Z\subset \operatorname{Supp}(\alpha)$ since otherwise $e_{Z}(\alpha)=0$. Then there is an effective cycle $\beta$ on $X$ such that
1. $\dim \beta = s$, and $\beta$ intersects $\alpha$ in finitely many closed points;
2. $|Z\cap \operatorname{Supp}(\beta)|\geq \deg \beta$.
We have $$\begin{aligned}
m\cdot\deg(\beta) &=&\alpha\cdot \beta \\
&=& \sum_{x\in \operatorname{Supp}(\alpha)\cap\operatorname{Supp}(\beta)}i(x, \alpha\cdot\beta; X)\\
&\geq& \sum_{x\in Z\cap \operatorname{Supp}(\beta)}i(x, \alpha\cdot\beta; X)\\
&\geq& \sum_{x\in Z\cap \operatorname{Supp}(\beta)}e_{x}(\alpha)\cdot e_{x}(\beta)\\
&\geq& e_{Z}(\alpha)\cdot \deg(\beta),\end{aligned}$$ where the second inequality follows from [@F Corollary 12.4]. We divide the both sides by $\deg(\beta)$. The proof is done.
Proof of Theorem \[t1\]
=======================
We prove a stronger version of Theorem \[t1\]:
\[t1’\] Let $X\subset \P^{n+r}$ be a Fano manifold of index $1$, dimension $n$ and codimension $r$. Assume that $X$ is $2r$-normal, that is, the restriction map $$H^{0}(\P^{n+r},\mathcal{O}_{\P^{n+r}}(2r))\rightarrow H^{0}(X,\mathcal{O}_{X}(2r))$$ is surjective, and $n\geq 10 r$. Then $X$ is birationally superrigid and K-stable.
\[l\] Let $X\subset \P^{N}$ be a Fano manifold with the anti-canonical class a multiple of the hyperplane section class. Let $Y\subset X$ be a positive-dimensional linear section. If $X$ is $k$-normal, so is $Y$.
It is enough to prove that, if $Y\subset X$ is a positive-dimensional linear section, the restriction map $$H^{0}(X,\mathcal{O}_{X}(k))\rightarrow H^{0}(Y,\mathcal{O}_{Y}(k))$$ is surjective for any $k\in \Z$. It is enough to prove that, if $Y\subset X$ is a linear section, we have $$H^{i}(Y,\mathcal{O}_{Y}(j))=0 \text{ for any }0<i<\dim Y\text{ and }j\in \Z.$$ It is enough to prove $$H^{i}(X, \mathcal{O}_{X}(j))=0 \text{ for any }0<i<n \text{ and }j\in\Z.$$ It follows from the Kodaira vanishing theorem. The proof is done.
The proof is essentially the same as in [@Z Theorem 1.2] and [@Z Lemma 3.6]. Let $X$ as in the statement. We have $-K_{X}=c_{1}(\mathcal{O}_{X}(1))\cap [X]$ and $r\geq 1$. We have $\operatorname{Pic}(X)=\Z(\mathcal{O}_{X}(1)\cap [X])$ as long as $n\geq r+2$ by the Barth-Larsen theorem [@L Theorem 3.2.1]. The inequality $n\geq r+2$ follows from $r\geq 1$ and the assumption $n\geq 10r$. We may assume that $X$ is non-degenerate. The condition $(\star)$ is satisfied by Lemma \[l2\] together with the assumption $n\geq 10r$ (or by Lemma \[l3\] in the case $r=2$).
We prove that $X$ is birationally superrigid. By the Noether-Fano enequality [@CS Theorem 1.26], it is enough to prove that for any movable linear system $\mathcal{M}\subset |-m K_{X}|$ the pair $\left(X, \frac{1}{m}\mathcal{M}\right)$ is canonical. [*Step*]{} $1$: We prove that
1. there exists a closed subset of dimension $Z\subset X$ of dimension $\leq r-1$ such that the pair $\left(X, \frac{1}{m}\mathcal{M} \right)$ is canonical away from $Z$;
2. there exists a closed subset $Z'\subset X$ of dimension $\leq 2r-1$ such that the pair $\left(X,\frac{2}{m}\mathcal{M}\right)$ is log canonical away from $Z'$.
For $(i)$, we take $D\in \mathcal{M}$. Then $$[D]=m\cdot c_{1}(\mathcal{O}_{X}(1))\cap [X]\in N^{1}(X).$$ By Proposition \[p1\], there exists a closed subset $Z$ of dimension $\leq r-1$ such that $e_{x}(D)\leq m$ for any closed point $x\in X$ away from $Z$. Therefore the pair $\left(X, \frac{1}{m}D \right)$ is canonical away from $Z$ by [@K1 3.14.1]. For (ii), we take $D_{1}, D_{2} \in \mathcal{M}$ intersecting properly. Let $B=D_{1}\cdot D_{2}$. Then $$[B]=m^{2}\cdot c_{1}(\mathcal{O}_{X}(1))^{2}\cap [X]\in N^{2}(X).$$ By Proposition \[p1\], there exists a closed subset $Z'\subset X$ such that $e_{x}(B)\leq m^{2}$ for any closed point $x\in X$ away from $Z'$. For any such $x$, let $S\subset X$ be a general linear section of dimension $2$ through $x$. Then the pair $\left(S, \frac{2}{m}B|_{S}\right)$ is log canonical at $x$ by [@dFEM1 Theorem 0.1]. By inversion of adjunction [@EM Theorem 1.1], the pair $\left(X, \frac{2}{m}B\right)$ is log canonical at $x$.
[*Step*]{} $2$: We prove that for any closed point $x\in X$ and any general linear section $Y\subset X$ of codimension $2r-1$ through $x$ the pair $\left(Y, \frac{1}{m}\mathcal{M}|_{Y}\right)$ is Kawamata log terminal (klt) at $x$. We have $K_{Y}=2(r-1)\cdot c_{1}(\mathcal{O}_{Y}(1))\cap[Y]$. Let $L=\mathcal{O}_{X}(2r)$. Then $L\sim_{\Q}K_{Y}+\frac{2}{m}\mathcal{M}|_{Y}$. The pair $\left(Y, (1-\epsilon)\frac{2}{m}\mathcal{M}|_{Y}\right)$ is klt away from a finite set for all $0<\epsilon\ll 1$, and we have $$h^{0}(Y, L)\leq h^{0}(\P^{n-r+1}, \mathcal{O}_{\P^{n-r+1}}(2r))=\binom{n+r+1}{2r}<\frac{(n-2r+1)^{n-2r+1}}{(n-2r+1)!},$$ where the first (resp. second) inequality follows from Lemma \[l\] together with the $2r$-normality of $X$ (resp. the assumption $n\geq 10r$). Then the pair $\left(Y, \frac{1}{m}\mathcal{M}|_{Y}\right)$ is klt by [@Z Corollary 1.8].
[*Step*]{} $3$: Assume that the pair $\left(X, \frac{1}{m}\mathcal{M} \right)$ is not canonical at $x\in Z$. Let $Y\subset X$ be a general linear section of codimension $2r-1$ throught $x$. Then the pair $\left(Y, \frac{1}{m}\mathcal{M}|_{Y}\right)$ is not log canonical at $x$ by inversion of adjunction, a contradiction.
We prove that $X$ is $K$-stable. Let $$\alpha(X)
=\sup\left\{t \mid \left(X, t D\right) \text{ is log canonical for any }D\in |-K_{X}|_{\Q}\right\}$$ be the alpha invariant. By [@SZ Theorem 1.2], it is enough to prove that $\alpha(X)>\frac{1}{2}$. By [@B Theorem 1.5], it is enough to prove that the pair $(X, \frac{1}{2}D)$ is klt for any $D\in|-K_{X}|_{\Q}$. The proof is similar using Proposition \[p1\] and [@Z Corollary 1.8]. The proof is done.
By analyzing the proof of Theorem \[t1’\], we can further strengthen the statement:
\[t1”\] Let $X\subset \P^{n+r}$ be a Fano manifold of index $1$, dimension $n$ and codimension $r$. Assume $$\sum_{i=0}^{2r-1}(-1)^{i}\cdot \binom{2r-1}{i} \cdot h^{0}(\mathcal{O}_{X}(2r-i)) < \frac{(n-2r+1)^{n-2r+1}}{(n-2r+1)!},$$ and $n\geq 3r-2$. Then $X$ is birationally superrigid and K-stable.
We have $$h^{0}(Y,\mathcal{O}_{Y}(2r))=\sum_{i=0}^{2r-1}(-1)^{i}\cdot \binom{2r-1}{i} \cdot h^{0}(\mathcal{O}_{X}(2r-i))$$ for any linear section $Y\subset X$ of codimension $2r-1$ by the similar argument as in Lemma \[l\]. Therefore the first assumption is equivalent to $$h^{0}(Y,\mathcal{O}_{Y}(2r))<\frac{(n-2r+1)^{n-2r+1}}{(n-2r+1)!}$$ for any such $Y$. We have $n>2r$ after all. We omit the rest of the proof.
The singular case
=================
Due to Liu and Zhuang [@LZ], the result of Zhuang [@Z] is generalized to the singular case (the notions of birational superrigidity and K-stability can be defined for $\Q$-Fano varieties). Replacing the complete intersection assumption by the locally complete intersection and projective normality, we prove:
\[t2\] For integers $\delta\geq -1$ and $r\geq 1$, there exists a positive integer $n_{0}(r, \delta)$ depending only on $\delta$ and $r$ such that, if $X\subset \P^{n+r}$ is a locally complete intersection projectively normal Fano variety of index $1$, codimension $r$ and dimension $n\geq n_{0}(r, \delta)$ such that
1. $\dim\operatorname{Sing}(X)\leq \delta$;
2. every projective tangent cone of $X$ is a Fano complete intersection of index at least $4r+2\delta +2$ and is smooth in dimension $r+\delta$,
then $X$ is birationally superrigid and K-stable.
Let $X\subset \P^{n+r}$ be a non-degenerate projective variety of dimension $n$ and codimension $r$. For a closed point $p\in \P^{n+r}$ not contained in $X$, we define the ramification locus $R(\pi_{p})$ of the restriction $\pi_{p}\colon X\rightarrow \P^{n+r-1}$ of the linear projection from $p$ as the zero scheme of the section of the twisted normal sheaf $\mathcal{N}_{X/\P^{n+r}}(-1)$ associated to $p$. We consider the following condition $(\star\star)$: given any closed point $p\in \P^{n+r}$ not contained in $X$, we have $Z\cap R(\pi_{p})\neq \emptyset$ for any closed subset $Z\subset X$ of dimension $\geq r$ disjoint from $\operatorname{Sing}(X)$.
\[l4\] Assume that $X$ is a complete intersection. Then the condition $(\star\star)$ is satisfied.
The proof is the same as in Lemma \[l1\].
\[l5\] Assume that $X$ is locally complete intersection and $$n\geq \max\left\{3r-2, 2r-1+\delta\right\},$$ where $\delta=\dim \operatorname{Sing}(X)$. Then the condition $(\star\star)$ is satisfied.
We denote by $\operatorname{Tan}'(X)$ the variety of tangent stars of $X$ (see [@Z Chapter I, Definition 1.2] for the definition). We prove $\operatorname{Tan}'(X)=\P^{n+r}$. As a consequence of the connectedness theorem of Fulton and Hansen, we have $\operatorname{Tan}'(X)=\operatorname{Sec}(X)$ as long as $n\geq r$ [@Z Chapter I, Theorem 1.4]. On the other hand, we have $\operatorname{Sec}(X)=\P^{n+r}$ as long as $n\geq 2r-1+\delta$ by Zak’s theorem on linear normality for singular varieties [@Z Chapter II, Theorem 2.1]. Therefore we have the desired equality of sets under our assumption.
For a closed point $p\in \P^{n+r}$ not contained in $X$, we denote by $R'(\pi_{p})$ be the J-ramification locus of the restriction of the linear projection from $p$ (see [@Z Chapter II, Section 1] for the definition and property of unramified morphisms in the sense of Johnson, or J-unramified morphisms). Then $R'(\pi_{p})\neq \emptyset$, and $\dim R'(\pi_{p})\leq n-r$ for general $p$. We prove that $R(\pi_{p})\neq \emptyset$, and $\dim R(\pi_{p})=n-r$ for general $p$. By definition of $R(\pi_{p})$ and $R'(\pi_{p})$, we have $$R(\pi_{p})\supseteq R'(\pi_{p}),\, R(\pi_{p})\cap X^{sm}=R'(\pi_{p})\cap X^{sm}.$$ Thus the complement in $R(\pi_{p})$ of $R'(\pi_{p})$ is supported on $\operatorname{Sing}(X)$, while we have $\dim \operatorname{Sing}(X)=\delta\leq n-r$ by the assumption. Now it is enough to observe that $R(\pi_{p})$ is locally defined by $r$ equations in $X$ and we have $\dim R(\pi_{p})\geq n-r$. It follows that $R(\pi_{p})$ is defined by a regular section for general $p$. We have $$[R(\pi_{p})]=c_{r}(N_{X/\P^{n+r}}(-1))\cap [X]$$ by [@F Proposition 14.1].
To check the condition $(\star\star)$, it is enough to prove that $$c_{r}(N_{X/\P^{n+r}}(-1))\cap [Z]\neq 0$$ for any closed subvariety $Z\subset X$ of dimension $r$. By the Barth-Larsen theorem for locally complete intersection varieties [@L Corollary 3.5.13], we have $$H_{2i}(X,\Z)=H_{2i}(\P^{n+r}, \Z) \text{ for any }0\leq i\leq r-1.$$ Therefore the homology class of a subvariety of $X$ of dimension $\leq r-1$ is uniquely determined by its degree. Combined with the self-intersection formula [@F Corollary 6.3], it follows that the function $$Z_{r}(X)\rightarrow \Z,\, Z \mapsto c_{r}(N_{X/\P^{n+r}}(-1))\cap [Z]$$ is $a\cdot \deg Z$, where $a$ is a constant not depending on $Z$. Taking a general linear section of dimension $r$, we have $a\neq 0$. The proof is done.
\[l6\] Assume $r=2$. Then the condition $(\star\star)$ is satisfied.
It follows from [@Z Chapter II, Corollary 2.5].
\[p2\] Let $X\subset \P^{N}$ be a non-degenerate projective variety of codimension $r$. Assume that $X$ satisfies the condition $(\star\star)$. Let $\alpha$ be an effective cycle on $X$ such that $$\alpha= m \cdot c_{1}(\mathcal{O}_{X}(1))^{s}\cap [X] \in N_{N-r-s}(X).$$ Then we have $e_{x}(\alpha)\leq m$ for any closed point $x\in X$ away from a closed subset of dimension $\leq rs+\delta$, where $\delta = \dim \operatorname{Sing}(X)$ and we use the convention $\dim(\emptyset)=-1$.
The complete intersection case is proved by Pukhlikov [@P4 Proposition 5] and the author [@S Proposition 2.1].
We take $\alpha$ as in the statement. It is enough to prove that $e_{Z}(\alpha)\leq m$ for any closed subvariety $Z\subset X$ of dimension $rs$ disjoint from $\operatorname{Sing}(X)$. We move $Z$ by multiple residual intersections so that the residual intersections avoid $\operatorname{Sing}(X)$. The rest of the proof is similar.
The proof is essentially the same as in [@LZ Theorem 1.3]. We use the Barth-Larsen theorem for locally complete intersection varieties [@L Corollary 3.5.13] combined with the universal coefficient theorem to compute the Picard group. The condition $(\star\star)$ is satisfied by Lemma \[l5\] when $n$ is large enough for fixed $\delta$ and $r$ (or by Lemma \[l6\] in the case $r=2$). Then Proposition \[p2\] is used to bound the dimension of the non-canonical, non-klt or non-lc locus of certain pairs on $X$ as Proposition \[p1\] in the proof of Theorem \[t1\]. We refer the reader to [@LZ] for the details.
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---
abstract: 'The Su-Schrieffer-Heeger (SSH) model lays the foundation of many important concepts in quantum topological matters. Since it tells one that topological states may be distinguished by abelian geometric phases, a question naturally arises as to what happens if one assembles two topologically distinct states. Here, we show that a spin-dependent double-well optical lattice allows one to couple two topologically distinct SSH chains in the bulk and realise a glided-two-leg SSH model that respects the glide reflection symmetry. Such model gives rise to intriguing quantum phenomena beyond the paradigm of a traditional SSH model. It is characterised by Wilson line that requires non-abelian Berry connections, and the interplay between the glide symmetry and interaction automatically leads to charge fractionalisation without jointing two lattice potentials at an interface. Our work demonstrates the power of ultracold atoms to create new theoretical models for studying topological matters.'
author:
- 'Shao-Liang Zhang, Qi Zhou'
title: 'A two-leg Su-Schrieffer-Heeger chain with glide reflection symmetry'
---
The beauty of the Su-Schrieffer-Heeger(SSH) model[@ssh; @shen] is reflected by its extremely simple form that well captures a variety of deep concepts lying at the heart of modern condensed matter physics. Such model describes a one-dimensional chain, which is characterised by two tunnelling amplitudes $t_1$ and $t_2$ between two sublattices A and B, as shown in figure (1a). SSH model serves as a textbook example for discussing the Zak phase, an abelian geometric phase that characterises distinct topological phases in one dimension, and zero energy end states in a finite system with open boundaries[@zak; @zak2; @zak3; @zak4]. It is also a prototypical model for studying fractionalised charges, one of the most exotic phenomena in quantum systems, if interfaces exist in the lattice potential to separate topologically distinct chains into multiple domains in the real space[@rm; @cf].
Ultracold atoms have emerged as a highly controllable platform for simulating topological models that are difficult to access in solids in the past a few years[@topo1; @topo2; @topo3; @zhangj; @Sengstock; @ChenS]. Among these studies, the double-well optical lattice, which is composed of a long and a short lattice, has been demonstrated as a powerful tool. Since the wave vector of the long lattice is half of the short one, each lattice site contains a left and right well, as shown in figure (1b)[@do; @do2; @do3]. Such lattice is naturally described by the SSH model with two tunnelling amplitudes. I. Bloch’s group has applied an elegant Ramsey interferometry to a double-well optical lattice and measure the difference of the Zak phases between the two distinguished topological phases of the SSH model for the first time in laboratories[@bloch1]. Double-well lattices have also been used by both I. Bloch’s[@bloch2] and Y. Takahashi’s[@takahashi] groups to realise topological charge pumping. However, charge fractionalisation has not been explored in double-well optical lattices yet, since it is difficult to joint two double-well lattices at an interface. This is not a challenge specific to optical lattices, as it is a renowned difficult task to directly observe charge fractionalisation in a generic many-body system in condensed matter physics[@frac; @frac2; @frac3; @frac4].
Despite of the aforementioned exciting progresses, a question naturally arises on whether physicists could use ultracold atoms to explore new theoretical models other than simulating those readily in the literature. In this Article, we show that a spin-dependent optical double-well lattice[@spinlattice] allows one to realise a glided-two-leg SSH model, which is composed of two one-dimensional SSH chains shifted from each other by half of the lattice spacing, as shown in figure (1b). Unlike the conventional means of linking two topologically distinct chains at an interface as shown in figure (1a), the two chains here are coupled in the bulk, and provide one a unique playground to explore the interplay between topology, symmetry and interaction. The theoretical description of the system is fundamentally different from that for traditional SSH. Because of band touching points, which are protected by the glide symmetry[@glide1; @glide2; @glide3], in the Brillouin zone(BZ), the conventional abelian geometric phase is no longer capable for capturing the topological phases. Non-abelian Berry connections and Wilson line are are inevitably required[@wilsonline1] . Such Wilson line can be measured using a simple Bloch oscillation, as shown by the recent experiment done by I. Bloch’s group[@wilsonline2]. Introducing interaction to the system, even more interesting phenomena arise. Repulsive interaction gives rise to ferromagnet at half filling. Without resorting to producing domains in the lattice potential, doping the ferromagnet naturally leads to the splitting of an extra particle into two deconfined domain walls, each of which carries half of the charge of the extra particle. Such fractionalised charge can be easily manipulated as mobile or localised ones, and are directly observable using standard in-site density images of atoms.\
[**Results**]{}
[**Spin-dependent double-well lattice**]{} We consider the Hamiltonian of a two hyperfine spin states of fermions in a spin-dependent double-well lattice potential in the presence of a rf coupling[@rf], $$\hat{H}=\int d x\big[\hat{\psi}^\dagger_{\sigma}(x)\hat{H}_\sigma \hat{\psi}_\sigma(x)+\Omega(\hat{\psi}^\dagger_{\uparrow}(x)\hat{\psi}_{\downarrow}(x)+h.c.)\big]\label{Hr}$$ where $$\hat{H}_{\sigma}(x)=\frac{\hat{p}^2}{2m}-V_S \cos^2(\frac{2\pi x}{d})+2V_L\sigma_z\sin (\frac{2\pi x}{d}),$$ $\sigma=\uparrow, \downarrow$ characterise the hyperfine spin, $\sigma_z=\pm 1/2$, $d$ is the lattice spacing, $V_S>0$ and $V_L$ are the lattice depths of the short and long lattices respectively, and $\Omega$ is the rf coupling strength. Such a lattice potential can be realised by choosing a spin-independent short lattice and spin-dependent long lattice, resembling polyacetylene with an opposite dimerisation between nearest neighbour chains[@poly]. The frequency of the long lattice potential is red and blue detuned for the spin-up and spin-down atoms respectively. Apparently, the rf coupling represents a $\sigma_x$ term for the spin. If a spin rotation is applied so that $\sigma_x\leftrightarrow \sigma_z$, one sees that the transformed Hamiltonian $\hat{H}'$ describes spin-independent lattices $-V_S \cos^2(\frac{2\pi x}{d})$ in the presence of a spatially variant coupling $2V_L\sigma_x\sin (\frac{2\pi x}{d})$ and a uniform Zeeman field $\sim\Omega\sigma_z$. In practise, $\hat{H}'$ can be realised using a number of techniques(Supplementary Note 1). Theoretically, these two descriptions are equivalent. In this Article, we focus on the analysis of $\hat{H}$ and all results can be directly applied to $\hat{H}'$ upon simple variable transformations.
For both spin-up and spin-down atoms, $\hat{H}_{\sigma}(x)$ describes a standard double-well lattice, each of which shifts from the other by half of the lattice spacing $d/2$. A tight binding model can be constructed straightforwardly, $$\begin{split}
\hat{H}_L&=\sum_j\Big[t_1 (\hat{a}^\dag_{j\uparrow}\hat{b}_{j\uparrow}+\hat{b}^\dag_{j\downarrow}\hat{a}_{j+1\downarrow})+t_2 (\hat{b}^\dag_{j\uparrow}\hat{a}_{j+1\uparrow}\\&+\hat{a}^\dag_{j\downarrow}\hat{b}_{j\downarrow})\Big]
+t\sum_j\big(\hat{a}^\dag_{j\uparrow}\hat{a}_{j\downarrow}+\hat{b}^\dag_{j\uparrow}\hat{b}_{j\downarrow}\big)+h.c.
\label{dssh},
\end{split}$$ where $\hat{a}^\dagger_{j\sigma}$ and $\hat{b}^\dagger_{j\sigma}$ are the creation operator for spin-up or spin-down atoms at left and right well on site $j$, $t_1$ and $t_2$ are the intra-leg tunnelling, and $t$ is the inter-leg tunnelling. In this Article, $j$ is reserved for the site index of the double-well lattice, each of which corresponds to two wells. Apparently, each leg is a conventional SSH model. $t_1$ and $t_2$ switch positions in these two legs, due to the relative shift of half of the lattice spacing. All parameters can be calculated from the exact numerical solutions of the band structure of the Hamiltonian in equation (\[Hr\]).
In the extreme case where $t=0$, this glided-two-leg SSH model reduces to two independent SSH chains. Since this model has readily included both the two topologically distinct configurations of a single SSH model, it is rather clear that regardless of the location of the boundary, there always exists one zero energy end state at each end of a finite system, as shown in figure (1b). However, for a finite $t$, the results are far more from obvious, since now two topologically distinct SSH chains are coupled to each other. We first consider a finite system, and the end states can be solved numerically. As shown in figure (1c), for a small inter-leg tunnelling, $t<|t_1+t_2|$, the zero energy end states exist. With increasing the inter-leg tunnelling $t$, the localisation length of the end states increases and eventually become divergent, which signifies the absence of the end states in the strong inter-leg tunnelling limit where $t>|t_1+t_2|$. Thus, $t_c=|t_1+t_2|$ represents a topological phase transition point.
To understand the nature of the topological phase transition, we solve bulk spectrum. The Fourier transform of the Hamiltonian to the momentum space is written as $H=\sum_k\hat{\Psi}^\dag_kM_k\hat{\Psi}_k$ where $\Psi^\dag_k=(\hat{a}^\dag_{k\uparrow},\hat{b}^\dag_{k\uparrow},\hat{a}^\dag_{k\downarrow},\hat{b}^\dag_{k\downarrow})$ and, $$M_k=\left(\begin{array}{cccc} 0 & t_1+t_2e^{-ikd} & t & 0 \\ t_1+t_2e^{ikd} & 0 & 0 & t \\ t & 0 & 0 & t_2+t_1e^{-ikd} \\ 0 & t & t_2+t_1e^{ikd} & 0 \end{array}\right).
\label{hamk}$$ Such Hamiltonian can be block diagonalised so that the Hamiltonian can be rewritten as $H=\sum_{k\pm} \hat{\phi}^\dag_{k\pm} h_{k,\pm} \hat{\phi}_{k\pm}$, where $\hat{\phi}^\dag_{k\pm}=(\hat{s}^\dag_{k\pm},\hat{p}^\dag_{k\pm})$, $$h_{k,\pm}= \left(\begin{array}{cc} [ t\pm(t_1+t_2)\cos(\frac{kd}{2})] & \mp i(t_1-t_2)\sin (\frac{kd}{2}) \\ \pm i(t_1-t_2)\sin (\frac{kd}{2}) & -[ t\pm(t_1+t_2)\cos(\frac{kd}{2})] \end{array}\right),\label{Hsp}$$ which satisfy $h_{k,\pm}=h_{k+\frac{2\pi}{d,}\mp}$, and $$\begin{split}
\hat{s}^\dag_ {k\pm}=\frac{1}{2}\big[(\hat{a}^\dag_{k,\uparrow}+\hat{a}^\dag_{k,\downarrow})\pm e^{{ikd}/{2}}(\hat{b}^\dag_{k,\uparrow}+\hat{b}^\dag_{k,\downarrow})\big],\\
\hat{p}^\dag_{k\pm}=\frac{1}{2}\big[(\hat{a}^\dag_{k,\uparrow}-\hat{a}^\dag_{k,\downarrow})\mp e^{{ikd}/{2}}(\hat{b}^\dag_{k,\uparrow}-\hat{b}^\dag_{k,\downarrow})\big]. \\
\end{split}$$ The block diagonalised Hamiltonian can be solved straightforwardly, as $h_{k,\pm}$ corresponds to a model describing the hybridisation of the $s$ and $p$ bands in a lattice, which has been well studied in the literature[@spmodel; @spmodel2; @spmodel3]. It is known that $t=|t_1+t_2|$ characterises a topological phase transition, across which the Zak phase of a single band in the BZ, which corresponds to $k\in[-2\pi/d, 2\pi/d]$ here due to that $h_{k,\pm}=h_{k+\frac{4\pi}{d},\pm}$, changes by $\pi$. Here, we have a four-band model with a periodicity $2\pi/d$, half of that of $h_{k,\pm}$. This fact leads to intriguing band touching points in our system, as discussed below.
The Bloch wave function of lowest two bands $|\psi_{k,\pm}\rangle$ satisfy $h_{k,\pm}|\psi_{k,\pm}\rangle=E_{k,\pm}|\psi_{k,\pm}\rangle$, where $$E_{k,\pm}=-\sqrt{t^2+t^2_1+t^2_2+2t_1t_2\cos kd\pm2t(t_1+t_2)\cos\frac{kd}{2}}.$$ The energies of upper two bands are simply $-E_{k,\pm}$. Typical band structures are shown in figure (2a). Without loss of generality, we have chosen $0>t_2\ge t_1$. For the lattice potential considered in equation(\[Hr\]), $t_1$ and $t_2$ have the same sign. In the extreme limit $t=t_2=0$, one observes that both the ground and the excited bands are flat and two-fold degenerate. This simply comes from the fact that in both the spin-up and spin-down chain, the eigen states of the Hamiltonian are the localised orbitals in the atomic limit, i.e., $(\hat{a}^\dagger_{j\uparrow}\pm \hat{b}^\dagger_{j\uparrow})/\sqrt{2}$ and $(\hat{b}^\dagger_{j\downarrow}\pm \hat{a}^\dagger_{j+1\downarrow})/\sqrt{2}$ respectively. Turning on a finite $t$ and $t_2$, one expects that the two-fold degeneracy is lifted. This is certainly true for a general $k$ away from the zone boundary $k=\pm \pi/d$. However, the double degenerate band touching points at $k=\pm \pi/d$ remains stable. In particular, such band touching point exist regardless of the value of $t$. As discussed before, $t=t_c$ signifies the disappearance of the zero energy end state in a finite system. In the bulk spectrum, when $t=t_c$, the lowest two and the highest two bands touch at $k=0$. When $t>t_c$, a gap reopens to separate the lowest two bands from the highest two. Nevertheless, the band touching points between the lowest(highest) two bands remain.
The band touching point at the zone boundary can first be understood from that the periodicity of $h_{k,\pm}$, the block diagonalised one, is actually $4\pi/d$, doubles that of $H_k$. In particular, the relation that $h_{k,\pm}=h_{k+\frac{2\pi}{d,}\mp}$ allows one to extend the dispersion $E^{o}_{k,+}$($E^{o}_{k,-}$) in the first BZ $k\in [-\pi/d, \pi/d]$ to the extended zone $k\in [-2\pi/d, 2\pi/d]$ so that it becomes $E^o_{k-}$($E^o_{k+}$), since $E^o_{k-}=E^o_{k+2\pi/d,+}$, where the superscript $o=g,e$ represent the ground($g$) and excited($e$) bands of $h_{k,\pm}$ respectively. In other words, the energy bands of our Hamiltonian $H_k$ is obtained from folding the one of either $h_{k,+}$ or $h_{k,-}$, which inevitably gives rise to the band touching at $k=\pm \pi/d$. As the Bloch wave function must have a periodicity $2\pi/d$, one obtains, $$\begin{split}
|\psi_{k,1}\rangle=|\psi^g_{k,-}\rangle, |\psi_{k,2}\rangle=|\psi^g_{k,+}\rangle, \, \frac{(4n-1)\pi}{d}\le k< \frac{(4n+1)\pi}{d}\\
|\psi_{k,1}\rangle=|\psi^g_{k,+}\rangle, |\psi_{k,2}\rangle=|\psi^g_{k,-}\rangle, \, \frac{(4n+1)\pi}{d}\le k< \frac{(4n+3)\pi}{d},
\end{split}\label{gl}$$ where $n$ is an integer, and $1, 2$ are the indices for the lowest two bands. Similar relations hold for the wave functions of the highest two bands $|\psi_{k,3}\rangle$ and $|\psi_{k,4}\rangle$.
More deeply, such degenerate points originate from the glide symmetry of the Hamiltonian. Apparently, if one combines the spin rotation $\uparrow\leftrightarrow \downarrow$ and a spatial translation of a distance $d/2$, half of the lattice spacing, the Hamiltonian in (\[Hr\]) is invariant. If one treats the spin as a synthetic dimension along the $y$ direction, this invariance exactly corresponds to a glide symmetry. It was realised recently that such symmetry is crucial for certain types of topological superfluids and crystalline insulators[@glide1; @glide2; @glide3]. Here, the glide symmetry naturally emerges from the spin-dependent lattice. Indeed, the energy eigenstates $|\psi_{k,\pm}\rangle$ are also the eigenstates of the glide operator, $\hat{G}=\hat{T}_{d/2} \hat{R}$, where $\hat{T}_{d/2}$ is the spatial translation of a distance $d/2$, and $\hat{R}$ is a spin flip $\uparrow\leftrightarrow \downarrow$. As $ \hat{G}^2=e^{ikd}$ is satisfied here, one concludes that the eigenvalue of $\hat{G}$ is $ \pm e^{ikd/2}$. We use $\eta=\pm$ to distinguish these two different eigenvalues and the corresponding eigenstates. From $\hat{H}\hat{G}=\hat{G}\hat{H}$, one classifies the energy eigenstates using $\hat{G}_k|\psi^g_{k,\pm}\rangle= \pm e^{ikd/2}|\psi^g_{k,\pm}\rangle$ and $ \hat{G}_k|\psi^e_{k,\pm}\rangle= \pm e^{ikd/2}|\psi^e_{k,\pm}\rangle$. The explicit expression of the glide operator is written as $$\hat{G}_k={ \pm} e^{\frac{ikd}{2}}\big(\cos\frac{kd}{2}\sigma_1\tau_1+\sin\frac{kd}{2}\sigma_1\tau_2\big) \label{GO},$$ where $\tau$ is the pseudospin representing the sublattice $A$ and $B$. Clearly, when $k\rightarrow k+2\pi/d$, $\eta$ changes sign. Thus there must exists a band crossing point. As pointed out in reference[@glide1], in the presence of an additional symmetry, the mirror reflection with respect to the centre of $A-B$ bond here, such a band crossing point must appear at the zone boundary $\pm \pi/d$, as shown in figure 2. It is worth pointing out that even in the absence of the mirror symmetry, such band crossing point could still appear at the zone boundary, as discussed later.
[**Wilson line**]{} The glide symmetry protected band touching points tell one that the abelian geometric phase is no longer applicable to describe the topological states in the system, unlike the traditional single SSH chain. [ Wilson line must be required to characterise the topological properties]{}[@wilsonline1]. Using the periodic Bloch wave function $|u_{k,1}\rangle=e^{-ikx}|\psi_{k,1}\rangle$ and $|u_{k,2}\rangle=e^{-ikx}|\psi_{k,2}\rangle$, the Wilson line that describes the lowest two bands is written as $$\hat{W}_{k\rightarrow k+\frac{2\pi}{d} }=\hat{\textsf{P}}\exp\left(i \int_{k}^{k+\frac{2\pi}{d}}d{ q} \hat{A}({ q}) \right),$$ where $\hat{\textsf{P}}$ is the path ordering operator and the matrix representation of $\hat{A}(k)$ is written as $$A_{mn}{(k)}=i\langle u_{k,m}|\partial_k|u_{k,n}\rangle.$$ $m,n=1,2$ here. It has been shown both theoretically[@wilsonline3] and experimentally[@wilsonline2] that such a Wilson line can be measured using Bloch oscillations of ultracold atoms in the limit $w\ll Fd\ll E_G$, where $F$ is the strength of the effective electric field, $w$ is the total band width of the lowest two bands, and $E_G$ is the energy separation between the lowest and the highest two bands. In such adiabatic limit, the transition to the highest two bands, as well as the dispersions of the the lowest two bands, $E_{k,1}$ and $E_{k2}$, is negligible, so that the dynamics is well characterised by the $\hat{W}_{k\rightarrow k+\frac{2\pi}{d} }$. Under the effective electric field $Fx$, the time evolution of the momentum follows $\hbar d q/dt =F$, and $|W^{mn}_{k\rightarrow k+\frac{2\pi}{d}}|^2\equiv|\langle u_{k,m}| \hat{W}_{k\rightarrow k+\frac{2\pi}{d} }|u_{k,n}\rangle|^2$ describes the probability of having the particle in the $m$th state after an evolution circle $k\rightarrow k+2\pi/d$ if the partial is initially prepared in the $n$th state.
Equation (\[gl\]) tells one that $\hat{W}_{k\rightarrow k+\frac{2\pi}{d}}$ may be computed using $|u_{k,\pm}\rangle$, instead of $|u_{k,1}\rangle$ and $|u_{k,2}\rangle$. A key point is that $\eta$ is conserved in the Bloch oscillation, i.e., $\langle u_{k,\mp}|\partial_k|u_{k,\pm}\rangle\equiv 0$. In the extreme case $t=0$, where $u_{k,+}$ and $u_{k,-}$ contain only one hyperfine spin state, such result can be seen easily from the fact that the spin-independent effective electric field does not coupling two different hyperfine spin states. For a finite $t$, the approval is provided in Supplementary Note 2. One concludes that $\eta$, the sign of the eigenvalue of the glide operator as aforementioned, is conserved in the Bloch oscillation. A particle initially in a state $|u_{k,\eta}\rangle$ always stays in a single band with the same $\eta$. As shown in figure 3(a), this simply corresponds to a Bloch oscillation governed by $h_{k,\eta}$ with a vanishing inter-band transition between the $+$ and $-$ bands. In the adiabatic limit, where $Fd\ll E_G$, the wave function accumulates a phase in such oscillation, i.e., $|u_{k\pm}\rangle\rightarrow e^{i\varphi_{\pm}}|u_{k'\pm}\rangle$, when $k\rightarrow k'$. Whereas $\varphi_{\pm}$ is gauge dependent if $k-k'\neq 0 \mod 4\pi/d$, it gives rise to the well known Zak phase $\varphi_\mathrm{Zak}$when $k\rightarrow k+4\pi/d$, which is $\pi$ or $0$ depending on whether $t$ is smaller or larger than $|t_1+t_2|$, as that in a standard hybridised $s$-$p$ model with a lattice spacing $d/2$[@spmodel; @spmodel2; @spmodel3].
Now return to the question on the form of $\hat{W}_{k\rightarrow k+\frac{2\pi}{d} }$, the matrix form of which needs to be evaluated in the basis $|u_{k,1}\rangle$ and $|u_{k,2}\rangle$ so that $|\psi_{k,1}\rangle$ and $|\psi_{k,2}\rangle$ have the periodicity of $H_k$, which is $2\pi/d$. From the above discussions, one obtains the Wilson line for $k\rightarrow k+2\pi/d$. $$\Big(W^{mn}_{k\rightarrow k+\frac{2\pi}{d}}\Big)=\Big(\begin{array}{cc}0 & e^{i\varphi_{-}} \\e^{i\varphi_{+}} & 0\end{array} \Big)$$ Though neither $\varphi_{+}$ nor $\varphi_-$ is well defined individually, since $k\rightarrow k+2\pi/d$ finishes only half of the BZ of $h_{k,\pm}$, due to the relation $h_{k+2\pi/d, \pm}=h_{k,\mp}$, we conclude $$\varphi_++\varphi_-=\varphi_\mathrm{Zak},$$ which can be easily understood from the fact that both $|u_{1,k}\rangle\rightarrow e^{i\varphi_-}|u_{2,k}\rangle$ and $|u_{2,k}\rangle\rightarrow e^{i\varphi_+}|u_{1,k}\rangle$ are satisfied when $k\rightarrow k+2\pi/d$ across the band touching point. Thus, we obtain $$\Big(W^{mn}_{k\rightarrow k+\frac{2\pi}{d}}\Big)=e^{i \varphi_\mathrm{Zak}/2}\Big(\begin{array}{cc}0 & e^{-i\varphi_{r}} \\e^{i\varphi_{r}} & 0\end{array} \Big)\label{Wl},$$ where $\varphi_{r}=(\varphi_{+}-\varphi_{-})/2$. Equation (\[Wl\]) clearly shows the non-abelian nature of the geometric phase here, since $|u_{1,k}\rangle$ and $|u_{2,k}\rangle$ have to exchange with each other when $k\rightarrow k+2\pi/d$, resembling a Möbius strip[@glide1; @glide3; @RLiu]. It also tells one that $\hat{W}_{k\rightarrow k+\frac{2\pi}{d}}$ can be decomposed to a $U(1)$ phase $e^{i \varphi_\mathrm{Zak}/2}$ and a $SU(2)$ transformation corresponding to rotating a pseudo-1/2 formed by the lowest two bands. Thus, it topologically corresponds to a Möbius strip, i.e, when $k$ finishes a full circle, the state does not come back to the original one but transform to an orthogonal one. Alternatively, if considering $k\rightarrow k+4\pi/d$, i.e., the momentum finishes two circles, one concludes, $$\Big(W^{mn}_{k\rightarrow k+\frac{4\pi}{d}}\Big)=\Big(\begin{array}{cc} e^{i\varphi_\mathrm{Zak}} & 0 \\0 & e^{i\varphi_\mathrm{Zak}} \end{array} \Big)=e^{i \varphi_\mathrm{Zak}}\mathcal{I}\label{Wl2},$$ i.e., the Wilson line becomes an identity matrix $\mathcal{I}$.
Both equations (\[Wl\]) and (\[Wl2\]) are verified by numerical simulations of the dynamics in the four-band model(Method). The populations in different bands are shown in figure 3(b), if a particle is initially prepared at state $|u_{k,1}\rangle$ or $|u_{k,2}\rangle$ . The populations approach the step functions and acquire sudden jumps at $k=\pi/d$ and $k=3\pi/d$, which are direct approvals of equations (\[Wl\]) and (\[Wl2\]). The phases $\varphi_+$ and $\varphi_-$ can also be measured directly in experiments using the same interferometric method that has been applied by I.Bloch’s group[@wilsonline2]. It is worth pointing out that, compared with the Wilson line measured in a two-dimensional honeycomb lattice, the one discussed here has a few new features. First, the Wilson line in our system originates from the glide symmetry of the glided-two-leg SSH model, unlike that in a honeycomb lattice produced by the structure factor of the lattice, $(e^{i{\bf G}\cdot {\bf r}_A}, e^{i{\bf G}\cdot {\bf r}_B}) $, where ${\bf G}$ is the reciprocal lattice vector and ${\bf r}_A$(${\bf r}_B$) is the position of A(B) sublattice site in a unit cell. Second, this Wilson line describes the lowest two bands in a four-band system, unlike the honeycomb lattice where a two-band model is sufficient. As aforementioned, the inter-leg tunnelling provides one additional degree of freedom to control the topological properties, since total phase $\varphi_++\varphi_-$ has a $\pi$ difference across the topological transition point $t_c=|t_1+t_2|$. On both sides of the transition point, the [ $\mathrm{SU}(2)$]{} part of the Wilson line exists, and the difference comes from the $\mathrm{U}(1)$ part, i.e., a $\pi/2$ difference in the total phase. This is half of the $\pi$ difference in Zak phases of an ordinary one-dimensional topological system where an abelian description is sufficient.
Whereas we have been focusing on the lowest two bands, all the above discussions also apply to the highest two bands, provided that the gap $E_G$ remains finite. Near the transition point $t_c=|t_1+t_2|$, the gap becomes small, and $Fd\ll E_G\ll w$ is satisfied. Since the adiabatic criterion is still satisfied, in the sense that the excitation to the highest two bands is negligible the above discussion on [Wilson line]{} still holds. In particular, $\eta$, the sign in front of the eigenenergy of the glide operator remains as a good quantum number. The only quantitative difference is that the dispersions $E_{1k}$ and $E_{2k}$ cannot be ignored any longer, so that the trivial dynamical phase factor $\int dk E_{1 k}$ and $\int dk E_{2 k}$ also contribute to the dynamics. At the critical point, the lowest two bands touch the highest two bands at $k=0$. It thus requires a full description including all the four bands(Supplementary Note 3).
[**Charge fractionalisation**]{} We have seen that the two-leg SSH model has readily given rise to interesting topological physics in non-interacting systems. Introducing interaction to such model shall provide one even more intriguing quantum phenomena. We here consider repulsive interaction, $$\hat{V}=U\sum_j (\hat{n}_{j,a\uparrow}\hat{n}_{j,a\downarrow}+\hat{n}_{j,b\uparrow}\hat{n}_{j,b\downarrow}),$$ where $U>0$ is the onsite interaction strength. From the previous discussions on single particle physics, we have learnt that flat bands rise in the extreme case $t_2=t=0$ where the localised orbitals $$\begin{split}
& \hat{c}^\dag_{j\uparrow}|0\rangle=(\hat{a}^\dag_{j\uparrow}+\hat{b}^\dag_{j\uparrow})|0\rangle/\sqrt{2}\\
& \hat{c}^\dag_{j\downarrow}|0\rangle=(\hat{b}^\dag_{j\downarrow}+\hat{a}^\dag_{j+1\downarrow})|0\rangle/\sqrt{2}\label{lo}
\end{split}$$ are the degenerate eigenstates of this flat band with energy $t_1$. Since $t_1<0$ is chosen, the high energy states $(\hat{a}^\dag_{j\sigma}-\hat{b}^\dag_{j\sigma})|0\rangle/\sqrt{2}$ is not relevant in the low energy limit, provided that $|t|$, $|t_2|$, and $U$ are much smaller than $|t_1|$. In such flat band limit, ferromagnet naturally emerges at half filling, i.e., all atoms fill one of the two lowest degenerate bands, either the one for spin-up or spin-down atoms, in figure 4(a), since it saves interaction energy and meanwhile does not cost extra kinetic energy in a flat band. In other words, repulsive interaction lifts the single-particle degeneracy. The emergent ferromagnet has a clear interpretation in the real space. As shown in figure 4(a), all atoms occupy one of double-well lattices. Clearly, such ferromagnet has a two-fold degeneracy, and the ground state can be $$|G\rangle_1=\prod_j \hat{c}^\dagger_{j\uparrow}|0\rangle, \,\,\,\,\,\,\,\,\, |G\rangle_2=\prod_j \hat{c}^\dagger_{j\downarrow}|0\rangle$$
In the presence of small $t_2$ and $t$, it is expected that the ferromagnet protected by the gap given by the repulsive interaction. To verify this fact, we use time-evolving block decimation (TEBD) algorithm[@tebd; @tebd2] to numerically obtain the ground state of the state at half filling. For wide range of realistic lattice parameters, we have found that ferromagnet emerges in the parameter regime $|t_2|, |t|\ll U\ll |t_1|$. For instance, figure 4(b) shows that for $V_S=8E_R$, $V_L=4E_R$, and $\Omega=0.01E_R$, which correspond to $t_1=0.2E_R$, $t_2=0.002E_R$, $t=0.006E_R$, the critical value of the interaction strength is $U_c=0.03E_R$. In terms of temperature, the gap is about $5\mathrm{nK}$, which is accessible in current experiments.
Due to the two-fold degeneracy of the ground state at half filling, doping the ferromagnet leads to intriguing phenomena. Consider adding one more atom to one of the spontaneous symmetry breaking ground states $|G\rangle_1$, in the limit that $U\ll |t_1|$, an extra particle prefer to occupy the spin-down chain to avoid the large kinetic energy penalty, which is of the order of $|t_1|$, caused by occupying an atomic orbital $(\hat{a}^\dagger_{j\uparrow}- \hat{b}^\dagger_{j\uparrow})|0\rangle/{\sqrt{2}}$. As shown in figure 4(b) and (c), such an extra particle creates two domain walls. A natural question is then, whether these two domain walls are confined with each other or they are deconfined? Two extreme cases are rather simple. When $t_2=0$ and $t\neq 0$, it is clear that either of these two spin-up atoms that have spatial overlap with the extra spin-down atoms can tunnel to the spin-down chain to gain the kinetic energy from the inter-leg tunnelling. Interestingly, such a tunnelling does not cost any interaction energy, since the number of domain walls remains to be 2. Such progress continuously occurs, and these two domain walls become deconfined so that the length of the spin-down domain becomes arbitrary, as shown in figure 4(b). Since the separation between the two domain walls can be infinity, one conclude that each domain wall carries $1/2$ of the charge of the extra particle. Such fractionalisation is naturally induced by the interplay between interaction and the glide symmetry of the non-interacting Hamiltonian, so that it is not required to create an interface in the lattice potential to separate topologically distinct phases. In contrast, if $t=0$ and $t_2\neq 0$, what is relevant is the tunnelling of a single spin-down atom in the spin-down chain. Clearly, the two domain walls are always confined with each other, as shown in figure 4(c). In such a confined state, charge is not fractionalised.
For a generic case with finite both $t_2$ and $t$, we explore how the confinement of the domain walls evolves to the deconfinement. Whereas such question can be answered by numerically solving the problem, we first consider adding one more particle in a finite system with periodic boundary condition, where the exact analytical solution available. For $N$ lattice sites with $N+1$ atoms, the Hilbert space composed of states with two and only two domain walls can be spanned using the Fock states, which can be written as $$|l_1l_2\rangle=
\Bigg\{
\begin{array}{ll}
\prod_{1\le j\le l_1} \hat{c}^\dagger_{j\uparrow}\prod_{l_1\le j\le l_2}\hat{c}^\dagger_{j\downarrow}\prod_{l_2<j\le N}\hat{c}^\dagger_{j\uparrow}|0\rangle, & l_1\le l_2 \\\\
\prod_{1\le j\le l_2} \hat{c}^\dagger_{j\downarrow}\prod_{l_2< j\le l_1}\hat{c}^\dagger_{j\uparrow}\prod_{l_1\le j\le N}\hat{c}^\dagger_{j\downarrow}|0\rangle, &l_2<l_1\\ \label{Fs1}
\end{array}$$ where $l_1$ and $l_2$ specify the locations of the two domain walls, since $\hat{c}^\dag_{j\uparrow}|0\rangle$ and $\hat{c}^\dag_{j\downarrow}|0\rangle$ are shifted from each other by half of the lattice spacing $d/2$ as shown by equation (\[lo\]). $l_1$( $l_2$) is defined such that spin-up(down) and spin-down(up) atoms are on the left(right) and right(left) hand side of the domain wall respectively. If the system has periodic boundary condition, one could further recast the Fock states in terms of the center of mass and relative motion of the domain walls, $|Mm\rangle=\hat{D}^\dag_{Mm}|0\rangle$ where $M=(l_1+l_2)/2$ and $m=l_2-l_1$ when $l_1\le l_2$, $M=(l_1+l_2+N)/2\,\,(\mathrm{mod\,\,N})$ and $m=l_2+N-l_1$ when $l_1>l_2$, $\hat{D}^\dag_{Mm}$ is the corresponding creation operator of the pair of domain walls. The physical meanings of $M$ and $m$ were shown in Supplementary Note 4. For such definitions, $0\le m\le N-1$ is an integer. If $m$ is even(odd), $1/2\le M\le N$ is an integer(half-integer). Projecting the Hamiltonian to these Fock states $|Mm\rangle$ results in an effective two-dimensional lattice model $H_\mathrm{eff}$ as shown in figure 5(a). Each site of this square lattice represents a Fock state $|Mm\rangle$. Such model contains two tunnelling amplitude, $J$ and $J_2$. $J_2=|t_2|/2>0$ characterises the tunnelling along the edge highlighted using blue colour(Supplementary Note 4). Such tunnelling corresponds to the increase of the center of mass coordinate by one lattice spacing $d$, and the distance between the two domain walls is fixed as $d/2$. $J=t/2$ is the tunnelling between two nearest neighbour sites in the bulk, which corresponds to the inter-leg tunnelling in the original two-leg SSH model and increases the distance between the two domain walls by $d$. The effective Hamiltonian is written as $$\begin{split}
&\hat{H}_\mathrm{eff}=J\sum_{Mm}\big(\hat{D}^\dag_{M,m}\hat{D}_{M+\frac{1}{2},m+1}+\hat{D}^\dag_{M,m}\hat{D}_{M-\frac{1}{2},m+1}\big)\\
&+J_2\sum_M\big(\hat{D}^\dag_{M,0}\hat{D}_{M+1,0}+\hat{D}^\dag_{M-\frac{1}{2},N-1}\hat{D}_{M+\frac{1}{2},N-1}\big)+h.c.
\end{split}$$ More details on each term in this Hamiltonian are provided in Supplementary Note 4. If one applies the periodic boundary condition, it is rather clear that the center of mass momentum is a good quantum number, which is denoted as $Q$. Define a $m$-dependent Fourier transform, $$\hat{D}^\dag_{Q,m}=\frac{1}{\sqrt{N}}\sum_M\hat{D}^\dag_{M,m}e^{iQMd},$$ the two-dimensional lattice problem reduces to a series of one-dimensional one as $\hat{H}_\mathrm{eff}=\sum_Q\hat{H}_{Q}$ where, $$\begin{split}
\hat{H}_Q&=2J\cos(Qd/2)\sum^N_{m=1}\big(\hat{D}^\dag_{Q,m}\hat{D}_{Q,m+1}+h.c.\big) \\
&+2J_2\cos(Qd)\big(\hat{D}^\dag_{Q,0}\hat{D}_{Q,0}+\hat{D}^\dag_{Q,N-1}\hat{D}_{Q,N-1}\big)
\end{split}$$ For any value of $Q$, $\hat{H}_Q$ is a simple one-dimensional lattice Hamiltonian describing a particle confined in a box potential which contains two impurity potential at the edge. Whereas the $Q$-dependent tunnelling replies on $J$, the impurity potential purely depends on $J_2$. The eigenenergy of the two-dimensional lattice model is then written as $$E_Q=\left\{\begin{array}{cc} -4|J|\cos(Qd/2) & J_2\cos(Qd)>-|J|\cos(Qd/2) \\ 2J_2\cos(Qd) & J_2\cos(Qd)<-|J|\cos(Qd/2) \end{array}\right.$$ For any $Q$, the ground state wave function $|\psi_Q\rangle$ of $\hat{H}_Q$ can be obtained. Consider two special cases, $Q=0$ and $Q=\pi/d$. When $J_2=0$, textbook results tell one that the ground state wave function $|\psi_Q\rangle$ is maximized in the middle of the box potential, which corresponds to the the largest separation of the two domain walls in the two-leg SSH model with the periodic boundary condition. Not surprisingly, two domain walls are deconfined in this case. $Q=\pi/d$ is a special case. In the effective two-dimensional lattice model, $|\psi_{\pi/d}\rangle$ describes a localised edge state, which can be seen from perfect destructive interference. For instance, as shown in figure 5(a), if the wave function at the edge (black dots), i.e, along the blue lines corresponding to $m=0$ or $3$, has alternative signs in the nearest neighbour sites, the weight of the wave function in the bulk, say the lattice sites corresponding to $m=1$ or $m=2$(red dots), must vanish. Compare $E_Q$ for all possible $Q$, we obtain the ground state of the effective two-dimensional lattice model. A first order transition $J_2=2|J|$ is identified. Figure 5(b) shows the average distance $D$ between the two domain walls as a function of $J_2/|J|$. Whereas for small $J_2$, $D$ is proportional to the size of the system, it abruptly decreases to $d/2$, signifying a first order transition to the confined phase.
We now discuss how to probe the fractionalised charge carried by the deconfined domain walls. It is crucial to detect the location of the domain walls. Using equations (\[Fs1\]), one sees that $$\langle l_1l_2| \hat{n}_{j}|l_1l_2\rangle=1+\frac{1}{2}(\delta_{j, l_1}+\delta_{j, l_2+1}),$$ where $\hat{n}_{j}=\sum_\sigma \hat{a}^\dagger_{j\sigma} \hat{a}_{j\sigma}+\hat{b}^\dagger_{j\sigma} \hat{b}_{j\sigma}$ is the total density operator on the $j$th lattice site. The above equation tells one that the quality $\tilde{n}_j=n_j-1$ directly traces the location of the domain walls, where $n_j=\langle l_1l_2| \hat{n}_{j}|l_1l_2\rangle$ is the total particle number per site. We have performed both numerical simulation for the exact model $\hat{H}+\hat{V}$ using TEBD and exact diagonalisation for the effective lattice model $\hat{H}_\mathrm{eff}$ with open boundary condition. As shown in figure [ 6(a)]{}, both methods confirm that, in the deconfined phase, the two domain walls move freely and the only constraint is that they cannot penetrate each other. As a result, $\tilde{n}_j$ resembles the density distribution of two free hard core particles in one dimension. In contrast, in the confined phase, the two domain walls are tightly bound with each other, and $\tilde{n}_j$ resembles the density distribution of a molecule, whose size is $d$.
An alternative method to detect the fractionalised charge is to introduce local potential to pin down the domain walls in certain lattice sites. This can be realised by applying localised laser beam so that the lattice potential becomes deeper at two lattice wells, say the left well of $j_L$ and the right well of $j_R$. Whereas the localised potential may also change the onsite interaction strength at site $j_L$ and $j_R$, the leading contribution is the potential energy gained $\epsilon$. Each domain wall, which corresponds to some extra particle numbers, prefers to occupy these two sites to gain the energy $\epsilon$, the potential energy produced by the deep local potentials $V_L$ and $V_R$. Define $\Delta n_j=n_j-n_j^0$, where $n_j^0=1$ is the particle number per lattice site (including two wells) of the ferrromagnet at half filling. Both TEBD and the exact diagonalisation show that $\Delta n_j$ is indeed peaked around $j_L$ and $j_R$, [ as shown in figure 6(b)]{}. The width of the peak $\xi$ depends on the ratio $J/\epsilon$. Choosing the distance between the two localised potential $|j_L-j_R|\gg \xi$, one could compute the total extra charge in the left and right side of the system, $$\Delta N_L=\sum_{i=1}^{N/2} \Delta n_j, \,\,\,\,\,\,\,\,\, \Delta N_R=\sum_{i=N/2+1}^{N} \Delta n_j,$$ we indeed find out that $\Delta N_L=\Delta N_R=1/2$. In the strong localisation limit, $J\ll \epsilon$ and $\xi \sim d$, $\Delta N_{L}\approx \Delta n_{j_L}$ and $\Delta N_{R}\approx \Delta n_{j_R}$ and the fractionalised charge-1/2 localised at sites $j_L$ and $j_R$. To further confirm such fractionalised charge-1/2, we compute the number fluctuation in the left and right half of the system, and have found out that the number fluctuation is zero. In the strong localisation limit, this is equivalent to the number fluctuation at the site $j_L$ or $j_R$. Such observation distinguishes the fractionalised charge-1/2 from the trivial one produced by a single particle hopping between two lattice sites, where the average occupation in each site is also 1/2 and the charge fluctuation is of the same order. Whereas we have been focusing on well localised potentials $V_L$ and $V_R$, which is achievable in current experiments, in practise, a potential with a width of a few lattice spacing also works, since it only quantitatively affects the width of the density peaks.
[**Discussions**]{}
In previous discussions, we have been focusing on the symmetric double well lattice, in which the left and right well in each single lattice site is symmetric. We now consider the effects of a number of perturbations. The first one is a mismatch of the phases of the long and short lattice, which produces a tilt in the double-well lattice potential, so that the Hamiltonian becomes $$\hat{H}_{\sigma}(x)=\frac{\hat{p}^2}{2m}-V_S \cos^2(\frac{2\pi x}{d})+2V_L\sigma_z\sin (\frac{2\pi x}{d}+\phi).\label{aL}$$ A finite $\phi$ thus produces an energy difference between the left and right wells. Correspondingly, the lattice model becomes $$\begin{split}
\hat{H}'_L&=t_1\sum_j\big(\hat{a}^\dag_{j,\uparrow}\hat{b}_{j,\uparrow}+\hat{b}^\dag_{j,\downarrow}\hat{a}_{j+1,\downarrow}\big)
+t_2\sum_j\big(\hat{b}^\dag_{j,\uparrow}\hat{a}_{j+1,\uparrow} \\
&+\hat{a}^\dag_{j,\downarrow}\hat{b}_{j,\downarrow}\big)
+t\sum_j\big(\hat{a}^\dag_{j,\uparrow}\hat{a}_{j,\downarrow}+\hat{b}^\dag_{j,\uparrow}\hat{b}_{j,\downarrow}\big)+h.c. \\
&+\frac{\Delta}{2}\sum_j\big(\hat{a}^\dag_{j,\uparrow}\hat{a}_{j,\uparrow}-\hat{b}^\dag_{j,\uparrow}\hat{b}_{j,\uparrow}
-\hat{a}^\dag_{j,\downarrow}\hat{a}_{j,\downarrow}+\hat{b}^\dag_{j,\downarrow}\hat{b}_{j,\downarrow}\big)
\end{split}$$ Such tilt breaks the mirror symmetry but not the gilde symmetry, which can be seen from that $\hat{H}_{\uparrow}(x)=\hat{H}_{\downarrow}(x+d/2)$ is still valid. Performing an exact band structure calculation, we find out that the band touching points remain. Interestingly, with the broken mirror symmetry, such band touching points are still located at the zone boundary $\pm\pi/d$. In the presence of a finite $\Delta$, the system still respects a symmetry, that is a combination of exchanging $A$ and $B$ sublattices and the inversion, i.e., $\phi\rightarrow -\phi$ in Eq.(\[aL\]) and $x\rightarrow -x$. The corresponding operation, which is denoted as $\mathcal{CI}$, satisfies $\mathcal{CI} G T_d=G \mathcal{CI} $, where $T_d$ is the translation for a lattice spacing $d$, similar to the mirror operation $\mathcal{M}$ satisfying $\mathcal{M}GT_d=G \mathcal{M}$. Thus, $\mathcal{CI}$ anticommutes with the glide operation $G$ at $k=\pm \pi/d$, and gives rise to the band touching points at the zone boundary[@glide1].
The band touching points can also be understood from the explicit form of the glide operator at $k=\pi/d$, which becomes $\hat{G}_{\pi/d}={ \pm}i\sigma_1\tau_2$. For a real lattice potential $V(r)$, its Fourier transform must satisfy $V_k=V_{-k}^*$. At the zone boundary, one then has $V_{\pi/d}=V_{-\pi/d}^*=V_{\pi/d}^*$, which tells one that $V_{\pm \pi/d}$ must be real. Thus one is always able to find out real eigenstates at $k=\pm \pi/d$. For an arbitrary eigenstate $|\Psi\rangle=(\alpha_1, \alpha_2,\alpha_3,\alpha_4)$, where $\alpha_i$ are all real, one has $ \hat{G}_{\pi/d}|\Psi\rangle=\pm(\alpha_4, -\alpha_3, \alpha_2,-\alpha_1)$, and $\langle \Psi|\hat{G}_{\pi/d}|\Psi\rangle=0$. Since $\hat{H}'_{\pi/d}\hat{G}_{\pi/d}=\hat{G}_{\pi/d}\hat{H}'_{\pi/d}$, one concludes that $\hat{G}_{\pi/d}|\Psi\rangle$ must be orthogonal eigenstates and thus there is at least a double degeneracy at the zone boundary. In particular, equation (\[Hsp\]) becomes $$h'_{k,\pm}= \left(\begin{array}{cc} \big[t\pm(t_1+t_2)\cos\frac{kd}{2}\big] & \frac{\Delta}{2}\mp i(t_1-t_2)\sin\frac{kd}{2} \\ \frac{\Delta}{2}\pm i(t_1-t_2)\sin\frac{kd}{2} & -\big[t\pm(t_1+t_2)\cos\frac{kd}{2}\big] \end{array}\right),\label{Hsptilt}$$ and $h'_{k,\pm}=h'_{k+\frac{2\pi}{d,}\mp}$ is still satisfied with a finite $\Delta$. The discussions on Wilson line can therefore be generalised straightforwardly to such a tilted glided-two-leg SSH model. Another type of perturbation is that the short lattices for spin-up and spin-down atoms may not be exactly the same, i.e., $V_{S\uparrow}=V_S+\Delta V_S$, $V_{S\downarrow}=V_S-\Delta V_S$. This gives rise to different tunnelling amplitudes in the lattice model $\hat{H}_L''$, i.e., $t_{1\uparrow}=t_{1}+\delta t_1$, $t_{2\uparrow}=t_{2}+\delta t_2$, $t_{1\downarrow}=t_{1}-\delta t_1$, $t_{2\downarrow}=t_{2}-\delta t_2$. Such perturbation breaks the glide symmetry and opens a small gap $\delta$ at the zone boundary. However, in the strong field limit, where $\delta\ll Fd$, the previous discussions on Wilson line is still valid, since the details of the dispersions are not relevant in such strong field limit. If one uses $|u_{1,k}\rangle$ and $|u_{2,k}\rangle$ as the basis, the matrix forms $(W^{mn}_{0\rightarrow 2\pi/d})$ and $(W^{mn}_{0\rightarrow 4\pi/d})$ remain unchanged. Alternatively, if one uses $|u''_{1,k}\rangle$ and $|u''_{2,k}\rangle$ as the basis, $(W^{''mn}_{0\rightarrow 2\pi/d})$ is a simply unitary transformation of $(W^{mn}_{0\rightarrow 2\pi/d})$, and $(W^{''mn}_{0\rightarrow 4\pi/d})=(W^{mn}_{0\rightarrow 4\pi/d})$ (Supplementary Note 5). It is worth pointing out that both types of perturbations do not affect the results of the charge fractionalisation, since the ferromagnet is provided by a gap $\sim U$. We have verified from numerical simulations that introducing either type of perturbation leads only quantitative changes in the results.
The search for new topological matters is one of the main themes in the current frontier of condensed matter physics[@newtopo1; @newtopo2; @newtopo3; @newtopo4; @newtopo5; @newtopo6]. In this Article, we have shown that a simple spin-dependent optical lattice allows one to construct new theoretical models for exploring very rich physics regarding the interplay among the glide symmetry, topology and interaction. We hope that our work may stimulate more studies on realising novel topological matters and its interplay with interaction and symmetry using the highly controllable ultracold atomic samples.
[**Method**]{}
The time-dependent Schrödinger equation for the system subject to an effective electric field ${F} x$ is written as $$i\partial_t |\Psi(t)=(\hat{H}(t)-F x |\Psi(t)\rangle,$$ where $\hbar=1$. Projecting the above equation to the basis of instantaneous eigenstates of $\hat{H}(t)$, which satisfy $\hat{H}(t)|\psi_n(t)\rangle=E_n^0|\psi_n(t)\rangle$, one obtains $$\begin{split}
&i\partial_t \alpha_m(t) +i\sum_{n=1}^4\alpha_n(t) \langle \psi_m(t)| \partial_t |\psi_n(t)\rangle\\
=&E_m^0\alpha_m(t) - \sum_{n=1}^4\alpha_n(t) \langle \psi_m(t)|F x |\psi_n(t)\rangle
\end{split}$$ Using $|\psi_n(t)\rangle=|\psi_{nk(t)}\rangle=e^{ik(t)x}|u_{nk(t)}\rangle$ and the equation of motion $\dot{ k}={ F}$, the above time-dependent Schrödinger equation is solved numerically and the populations at different bands are computed for an initial state occupying the first or the second band. In the limit $w\ll Fd\ll E_G$, one focuses on the lowest two bands and standard approaches show that the adiabatic evolution is described by the Wilson line, as discussed in the main text.
[**Acknowledgments**]{} We acknowledge useful discussions with I. Bloch, E. Mueller and C.X. Liu. We credit J. Zhang for asking the question on the Hamiltonian of Raman dressed lattice that inspired us to consider the equivalent Hamiltonian $\hat{H}'$. This work is supported by National Natural Science Foundation of China (NSFC)/ Research Grants Council (RGC) Joint Research Scheme(NCUHK453/13)
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**Supplementary Figures**
[**Supplementary Note 1**]{}
[**A few schemes to realise $\hat{H}'$**]{}
The model considered in the main text is equivalent to $$\hat{\tilde{H}}=\int dx\big[\hat{\psi}^\dag_\sigma(x)\big(\frac{\hat{p}^2}{2m}-V_S\cos^2(\frac{2\pi x}{d})+{ \Omega}\sigma_z\big)\hat{\psi}_\sigma(x)+V_L\sin(\frac{2\pi x}{d})(\hat{\psi}^\dag_\uparrow(x)\hat{\psi}_\downarrow(x)+h.c.)\big].
\label{tworaman}$$ The simplest scheme to realised such model is to apply a spatially dependent [ rf]{} field, the strength of which is written as $V_L\sin(\frac{2\pi x}{d})$.
An alternative approach is to dress an ordinary optical lattice by an additional laser along the perpendicular direction, as shown in figure S1. The electric field is written as $${\bf E}=E_0\vec{\sigma}_0 (e^{i k_0 x}e^{i\omega t}+e^{-i k_0 x}e^{i\omega t})+E_\bot\vec{\sigma}_\bot e^{i\beta} e^{i k_\bot y} e^{i\omega' t}.$$ $\beta$ is the phase difference between laser 1 and laser 2. $\vec{\sigma}_0$ and $\vec{\sigma}_\bot$ are the polarisations. The two counter-propagating blue detuning lasers $1$ and $2$ with the same frequency $\omega$ form a spin-independent optical lattice, $$V_\mathrm{OL}=V_S\cos^2(\frac{2\pi x}{d}),$$ where $d=2\pi/k_0$ and $V_S>0$. Laser $3$ with different frequency $\omega'$ gives rise two Raman processes in this system, one with laser $1$ and the other with $2$, respectively. Using $V_L$ to denote the Raman coupling strength, we obtain a periodically modulated Raman coupling in the real space, $$\begin{split}
\hat{V}_R&=\frac{V_L}{2} e^{i k_0 x}e^{-i k_\bot y} e^{-i\beta}+\frac{V_L}{2}e^{-i k_0 x}e^{-i k_\bot y} e^{-i\beta}+h.c.\\
& =V_L \cos(k_1x)e^{-i k_2 y} e^{-i\beta}+h.c.
\end{split}$$ The two-photon detuning $\delta=\omega-\omega'$ contributes to the effective Zeeman energy $\Omega \sigma_z$ together with a magnetic field applied to the system.
Since we consider a one-dimensional system here, the phase $e^{ik_\perp y}$ may be replaced by a constant $e^{ik_\perp y^*}$, where $y^*$ is the center of the Wannier wave function along the $y$ direction. Such phase factor and $e^{-i\beta}$ be gauged away. By simple transformation, $x \rightarrow x-\frac{\pi}{2d}$, we get the effective Hamiltonian in equation (\[tworaman\]).
[**Supplementary Note 2**]{}
[**Absence of [ transition between the $+$ and $-$ branches of the eigenstates of the glide operator]{}**]{}
The periodic Bloch wave functions of the lowest two bands can be written as $$\begin{split}
u^g_{k,+}(x)&=\alpha_{k,+}u_{k,s+}(x)+\beta_{k,+}u_{k,p+} \\
u^g_{k,-}(x)&=\alpha_{k,-}u_{k,s-}(x)+\beta_{k,-}u_{k,p-}, \\
\end{split}$$ where $(\alpha_{\pm}, \beta_{\pm})$ is the ground state of $h_{k,\pm}$ in equation (5) of the main text. Equation (6) of the main text tells on the relation between $u_{k,s\pm}(x)$ and $u_{k, A\sigma}(x)$($u_{k, B\sigma}(x)$), the periodic Bloch wave functions of the A and B sublattices defined as $$u_{k,A\uparrow}(x)=\frac{1}{N_{cell}}\sum_{\bf R_{i}} W_{A\uparrow}({ x-R_{i}}) e^{-i {k}\cdot{(x-R_{i})}},\,\,\,\,\,\,\,\, u_{k,B\uparrow }(x)=\frac{1}{N_{cell}}\sum_{\bf R_{i}} W_{B\uparrow}({x-R_{i}}) e^{-i { k}\cdot{(x-R_{i})}},$$ where $W_{A\sigma}({ x-R_{i}})$ and $W_{B\sigma}({ x-R_{i}})$ are the Wannier wave functions of the $A$ and $B$ sublattices for $\sigma=\uparrow, \downarrow$ respectively. We thus obtain $$\begin{split}
u^g_{k,+}(x)&=\frac{\alpha_{k,+}+\beta_{k,+}}{2}u_{k,A\uparrow}(x)|\uparrow\rangle
+e^{-ikd/2}\frac{\alpha_{k,+}-\beta_{k,+}}{2}u_{k,B\uparrow}|\uparrow\rangle\\
&+\frac{\alpha_{k,+}-\beta_{k,+}}{2}u_{k,A\downarrow}(x)|\downarrow\rangle
+e^{-ikd/2}\frac{\alpha_{k,+}+\beta_{k,+}}{2}u_{k,B\downarrow}|\downarrow\rangle \\
u^g_{k,-}(x)&=\frac{\alpha_{k,-}+\beta_{k,-}}{2}u_{k,A\uparrow}(x)|\uparrow\rangle
-e^{-ikd/2}\frac{\alpha_{k,-}-\beta_{k,-}}{2}u_{k,B\uparrow}|\uparrow\rangle\\
&+\frac{\alpha_{k,-}-\beta_{k,-}}{2}u_{k,A\downarrow}(x)|\downarrow\rangle
-e^{-ikd/2}\frac{\alpha_{k,-}+\beta_{k,-}}{2}u_{k,B\downarrow}|\downarrow\rangle.
\label{pbloch}
\end{split}$$
Because of the glide symmetry, the Wannier wave functions have the relations, $$W_{A\uparrow}({x -R_{i}}) =W_{B\downarrow}({x-R_{i}+ d/2}) , \,\,\,\, W_{A\downarrow}({x-R_{i}})=W_{B\uparrow}({x-R_{i}+d/2}),$$ and we obtain $$u_{k,A\uparrow}(x)=e^{ikd/2}u_{k,B\downarrow}(x+d/2),\,\,\,\,\,\,\,\,\,\,u_{k,A\downarrow}(x)=e^{ikd/2}u_{k,B\uparrow}(x+d/2).$$
The periodic Bloch wave function in equation (\[pbloch\]) can be written as: $$\begin{split}
u^g_{k,+}(x)&=\frac{\alpha_{k,+}+\beta_{k,+}}{2}\Big(u_{k,A\uparrow}(x)|\uparrow\rangle+u_{k,A\uparrow}(x-d/2)|\downarrow\rangle\Big)
+\frac{\alpha_{k,+}-\beta_{k,+}}{2}\Big(u_{k,A\downarrow}(x)|\downarrow\rangle+u_{k,A\downarrow}(x-d/2)|\uparrow\rangle\Big) \\
&=\frac{\alpha_{k,+}+\beta_{k,+}}{2}\Big(u_{k,A\uparrow+}(x)\Big)
+\frac{\alpha_{k,+}-\beta_{k,+}}{2}\Big(u_{k,A\downarrow+}(x)\Big) \\
u^g_{k,-}(x)&=\frac{\alpha_{k,-}+\beta_{k,-}}{2}\Big(u_{k,A\uparrow}(x)|\uparrow\rangle-u_{k,A\uparrow}(x-d/2)|\downarrow\rangle\Big)
+\frac{\alpha_{k,-}-\beta_{k,-}}{2}\Big(u_{k,A\downarrow}(x)|\downarrow\rangle-u_{k,A\downarrow}(x-d/2)|\uparrow\rangle\Big) \\
&=\frac{\alpha_{k,-}+\beta_{k,-}}{2}\Big(u_{k,A\uparrow-}(x)\Big)
+\frac{\alpha_{k,-}-\beta_{k,-}}{2}\Big(u_{k,A\downarrow-}(x)\Big)
\end{split}$$
To simplify the notations, we use delta functions to describe the Wannier wave functions, $$u_{k,A\uparrow}(x)=\frac{e^{ik s}}{N_{cell}}\sum_{\bf R_{i}}\delta(x-R_i+s)\,\,\,\,\,\,\,\, u_{k,B\uparrow }(x)=\frac{e^{ -ik s}}{N_{cell}}\sum_{\bf R_{i}}\delta(x-R_i-s)$$ $$u_{k,A\downarrow}(x)=\frac{e^{ik (d/2-s)}}{N_{cell}}\sum_{\bf R_{i}}\delta(x-R_i+(d/2-s))\,\,\,\,\,\,\,\, u_{k,B\downarrow }(x)=\frac{e^{-ik (d/2-s)}}{N_{cell}}\sum_{\bf R_{i}}\delta(x-R_i-(d/2-s)).$$ These four wave functions are orthogonal to each other. [ We have verified that using realistic Wannier wave functions with finite widths do not change the conclusions.]{}
To evaluate $\int dx \Big(u^{g *}_{k,\mp}(x)\Big)\partial_k\Big(u^g_{k,\pm}(x)\Big)$, there are two contributions to [ $\partial_ku^g_{k,\pm}(x)$ , one from the derivatives of the coefficients $\alpha_{k,\pm}$ and $\beta_{k,\pm}$, the other from the derivatives of the wave functions like $\partial_k u_{k,A\uparrow}(x)$. ]{}The derivatives of the coefficients do not contribute to the overlap intergrals, due to the orthogonal conditions of the wave functions.[ One then only needs to compute the contribution from the derivatives of the wave functions. It is straightforward to show that ]{} $$\begin{split}
&\int dx\Big(u^*_{k,A\uparrow-}(x)\Big)\partial_k\Big(u_{k,A\uparrow+}(x)\Big) \\
=&\int dx\Big(u^*_{k,A\uparrow}(x)|\uparrow\rangle-u^*_{k,A\uparrow }(x-d/2)|\downarrow\rangle\Big) \partial_k\Big(u_{k,A\uparrow}(x)|\uparrow\rangle+u_{k,A\uparrow }(x-d/2)|\downarrow\rangle\Big)\\
=&\int dx\Big(u^*_{k,A\uparrow}(x)\Big) \partial_k\Big(u_{k,A\uparrow}(x)\Big)-\int dx\Big(u^*_{k,A\uparrow }(x-d/2)\Big) \partial_k\Big(u_{k,A\uparrow }(x-d/2)\Big)\\
=&0.
\end{split}$$ Similarly, we obtain $$\begin{split}
&\int dx\Big(u^*_{k,A\downarrow-}(x)\Big)\partial_k\Big(u_{k,A\uparrow+}(x)\Big) =\int dx\Big(u^*_{k,A\uparrow-}(x)\Big)\partial_k\Big(u_{k,A\downarrow+}(x)\Big)=\int dx\Big(u^*_{k,A\downarrow-}(x)\Big)\partial_k\Big(u_{k,A\downarrow+}(x)\Big) =0.
\end{split}$$ We thus conclude that, $${\int dx \Big(u^{g *}_{k,-}(x)\Big)\partial_k\Big(u^g_{k,+}(x)\Big)=0.}$$ [ Similarly, we have $\int dx \Big(u^{g *}_{k,+}(x)\Big)\partial_k\Big(u^g_{k,-}(x)\Big)=0$, and thus conclude that there is no transition between the $+$ and $-$ branches of the eigenstates of the glide operator after an external electric field is applied.]{}\
[**Supplementary Note 3**]{}
[**At the critical point $t_c=|t_1+t_2|$**]{}
At the critical point $t_c=|t_1+t_2|$, the lowest two bands touch the highest two bands at both $k=0$ and $k=\pm\pi/d$, [ as shown by Figure S2]{}. It thus requires a full description including all the four bands. The periodic Bloch wave function of highest two bands can also be calculated by diagonalizing equation (5) using the same method as that in supplementary note 2, $$\begin{split}
u^e_{k,+}(x)&=\frac{-\beta^*_{k,+}+\alpha^*_{k,+}}{2}\Big(u_{k,A\uparrow+}(x)\Big)
+\frac{-\beta^*_{k,+}-\alpha^*_{k,+}}{2}\Big(u_{k,A\downarrow+}(x)\Big) \\
u^e_{k,-}(x)&=\frac{-\beta^*_{k,-}+\alpha^*_{k,-}}{2}\Big(u_{k,A\uparrow-}(x)\Big)
+\frac{-\beta^*_{k,-}-\alpha^*_{k,-}}{2}\Big(u_{k,A\downarrow-}(x)\Big).
\end{split}$$ One can also conclude that $$\int dx \Big(u^{\eta *}_{k,+}(x)\Big)\partial_k\Big(u^{\eta'}_{k,-}(x)\Big)=0,$$ where $\eta,\eta'=g,e$ characterise the lowest and highest bands. It means the external electric field can not couple the $+$ and $-$ branches of the eigenstates of the glide symmetry not only in the two-band approximation, which has been discussed in Supplementary Note 2, but also the complete four-band description. Nevertheless, it can couple $u^g_{k,+}(x)$ and $u^e_{k,+}(x)$ so that both $$A^{eg}_{++}=i\int dx \Big(u^{e *}_{k,+}(x)\Big)\partial_k\Big(u^g_{k,+}(x)\Big),$$ and $A^{eg}_{--}$ are finite. Whereas both $A^{eg}_{++}$ and $A^{eg}_{--}$ can be computed straightforwardly, in the limit where $w_T\ll Fd$, where $w_T$ is the total width of the four bands, the Wilson line can be directly evaluated using $W_{mn;G}=U_k^\dag D_GU_k$ where $D_G=\mathrm{diag}[e^{i G s_{A,\uparrow}}, e^{i G s_{B,\uparrow}},e^{i G s_{A,\downarrow}},e^{i G s_{B,\downarrow}}]$ and $s_{A,\sigma}$($s_{B,\sigma}$) is the center of the Wannier wave function of the $A$($B$) sublattice sites for the spin-$\sigma$ ($\sigma=\uparrow,\downarrow$) chain. $U_k$ is the unitary matrix which can diagonalize the matrix in equation (4) in main text($D_k=U^\dag_kM_kU_k$ is a diagonal matrix). Away from the critical point, a finite gap $E_G$ opens so as to separate the lowest two hands from the highest two ones. In the limit $Fd\ll E_G$, discussions in the main text then apply.\
[**Supplementary Note 4**]{}
[**Effective lattice model and the sign of $J_2$**]{}
[ Supplementary figure 3 shows a few representative Fock states. In S2(b), $l_1=5$ and $l_2=1$, where $l_{1,2}$ have been defined in equation (19) of the main text. Equivalently, this state can be written as $|M=7,m=4\rangle$. Whereas an intra-leg tunnelling $t_2$ increases the numbers of domain walls and costs extra interaction energy, as shown in figure S3(a), the inter-leg tunnelling $t$ changes the value of $l_2$ without the penalty of the interaction energy. As shown in figure S3(c), this leads to $|M=13/2,m=3\rangle$. For Fock states corresponding to the edge of the two-dimensional lattice model in figure 5 of the main text, i.e, those with $m=0$ and $m=N-1$, the $t_2$ tunnelling fixed the value of $m$ and changes $M$ by $1$. As shown in figure S3(d) and (e) which lead to $|M=3,m=0\rangle$ and $|M=2,m=0\rangle$ respectively.]{}
For the lattice potential in equation (3) of the main text, $t_1$ and $t_2$ in the tight binding model always have the same sign. Whereas this can be directly verified by numerical simulations, one could also understand it from considering the extreme case $V_L=0$, so that $t_1=t_2$. For convenience, we set $t_1<0$ and $t_2<0$ in the main text.
When $|t_1|\gg |t_2|, |t|$, we construct the localised eigen states for the ground bands $c^\dag_{j\sigma}|0\rangle$ as defined in the main text. A finite $t_2$ leads to the coupling between $c^\dag_{j\downarrow}|0\rangle=\frac{1}{\sqrt{2}}(b^\dag_{j\downarrow}+a^\dag_{j+1\downarrow})|0\rangle$ and $c^\dag_{j+1\downarrow}|0\rangle=\frac{1}{\sqrt{2}}(b^\dag_{j+1\downarrow}+a^\dag_{j+2\downarrow})|0\rangle$ so that the energy bands becomes dispersive. Consider the $t_2$ term in the Hamiltonian, $$H_{t_2}=t_2\sum_j\big(b^\dag_{j\uparrow}a_{j+1\uparrow}+a^\dag_{j\downarrow}b_{j\downarrow}+h.c.\big)
\label{hamt2}$$ we obtain the coupling between the localised orbitals, such as, $c^\dag_{j\downarrow}|0\rangle$ and $c^\dag_{j+1\downarrow}|0\rangle$, $$H_{t_2' \downarrow}=\frac{t_2'}{2}\sum_j\big(c^\dag_{j\downarrow}c_{j+1\downarrow}+h.c.\big),
\label{hamt2}$$ where $t_2'=t_2<0$. Alternatively, one could set $t_1>0$ and $t_2>0$. A straightforward calculation shows that $t_2'=-t_2<0$. One then concludes that $t_2'$ is always negative regardless of the choices of the signs of $t_1$ and $t_2$.
As explained in the main text, adding one extra spin-down particle on the top of the fully filled spin-up chain, two domain walls are created. The wave function is written as $$|M0\rangle=D^\dag_{Mm}|\tilde{0}\rangle=\left(\prod_{1\le j\le M} c^\dagger_{j\uparrow}\right)c^\dagger_{M\downarrow}\left(\prod_{M<j\le N}c^\dagger_{j\uparrow}\right)|0\rangle$$ where $M$ and $m$ are the coordinates of the center of mass and the relative motion, respectively, and $|\tilde{0}\rangle$ and $|{0}\rangle$ are the vacua of the domain walls and particles, respectively.
It is clear that $H_{t_2'}$ changes $M$ by 1 and leaves $m$ unchanged, as seen from $$\begin{split}
H_{t_2' \downarrow}|M0\rangle&=\frac{t_2'}{2}\big(\sum_j c^\dag_{j\downarrow}c_{j+1\downarrow}+h.c.\big)\Big(\prod_{j\le M} c^\dagger_{j\uparrow}c^\dagger_{M\downarrow}\prod_{j>M}c^\dagger_{j\uparrow}|0\rangle\Big) \\
&=\frac{t_2'}{2}\Big(\prod_{j\le M} c^\dagger_{j\uparrow}c^\dagger_{(M-1)\downarrow}\prod_{j>M}c^\dagger_{j\uparrow}|0\rangle\Big)+\frac{t_2'}{2}\Big(\prod_{j\le M} c^\dagger_{j\uparrow}c^\dagger_{(M+1)\downarrow}\prod_{j>M}c^\dagger_{j\uparrow}|0\rangle\Big) \\
&=-\frac{t_2'}{2}\Big(\prod_{j\le M-1} c^\dagger_{j\uparrow}c^\dagger_{(M-1)\downarrow}\prod_{j>M-1}c^\dagger_{j\uparrow}|0\rangle\Big)-\frac{t_2'}{2}\Big(\prod_{j\le M+1} c^\dagger_{j\uparrow}c^\dagger_{(M+1)\downarrow}\prod_{j>M+1}c^\dagger_{j\uparrow}|0\rangle\Big) \\
&=-\frac{t_2'}{2}|(M-1)0\rangle-\frac{t_2'}{2}|(M+1)0\rangle. \label{Ht2'}
\end{split}$$ The minus sign comes from the anticommutors of Fermi operators. If one considers the counterpart $H_{t_2' \uparrow}$, a similar result can be obtained straightforwardly.
In the effective two-dimensional [ lattice model describing the motion of two domain walls, as shown by figure 5 of the main text]{}, the term $$H_{J_2}=J_2\sum_M\big(D^\dag_{M0}D_{(M+1)0}+h.c.\big),\label{HJ2}$$ changes the center of mass of the two domain walls changes by one lattice spacing, and the relative motion remains unchanged.
Compare equations (\[Ht2’\]) and (\[HJ2\]), we conclude that $$J_2=-t_2'/2=|t_2|/2>0$$
[**Supplementary Note 5** ]{}
[**Dynamics in the presence of a small gap with broken glide symmetry** ]{}
The Hamiltonian in momentum space is $\hat{H}_L''=\sum_k\Psi^\dag_kM_k''\Psi_k$ where $\Psi^\dag_k=(a^\dag_{k\uparrow},b^\dag_{k\uparrow},a^\dag_{k\downarrow},b^\dag_{k\downarrow})$ and, $$M_k''=\left(\begin{array}{cccc} & t_{1\uparrow}+t_{2\uparrow}e^{ikd} & t & \\ t_{1\uparrow}+t_{2\uparrow}e^{-ikd} & & & t \\ t & & & t_{2\downarrow}+t_{1\downarrow}e^{ikd} \\ & t & t_{2\downarrow}+t_{1\downarrow}e^{-ikd} & \end{array}\right).
\label{hamktilt}$$ [ Whereas no simple analytical resolutions are available, we solve the quantum dynamics numerically. As shown in Supplementary Figure 2(a), $|W_{11}|^2$, which tells on the probability of a particle staying in the ground band, is no longer a step function when the glide symmetry is broken such that a band gap $E_G'$ opens between the lowest two bands, as shown in Supplementary Figure 2(b) . With increasing the band gap, $|W_{11}|^2$ gradually approach 1, consistent with the expectation that the particle remains at the ground band if $E_G'\gg Ed$. When [ $t_{1\uparrow}=t_{1\downarrow}$, $t_{2\uparrow}=t_{2\downarrow}$]{}, the glide symmetry restores, $E_G'=0$, and results in the main text are recovered.]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Mobile phones provide a powerful sensing platform that researchers may adopt to understand proximity interactions among people and the diffusion, through these interactions, of diseases, behaviors, and opinions. However, it remains a challenge to track the proximity-based interactions of a whole community and then model the social diffusion of diseases and behaviors starting from the observations of a small fraction of the volunteer population. In this paper, we propose a novel approach that tries to connect together these sparse observations using a model of how individuals interact with each other and how social interactions happen in terms of a sequence of proximity interactions. We apply our approach to track the spreading of flu in the spatial-proximity network of a 3000-people university campus by mobilizing 300 volunteers from this population to monitor nearby mobile phones through Bluetooth scanning and to daily report flu symptoms about and around them. Our aim is to predict the likelihood for an individual to get flu based on how often her/his daily routine intersects with those of the volunteers. Thus, we use the daily routines of the volunteers to build a model of the volunteers as well as of the non-volunteers. Our results show that we can predict flu infection two weeks ahead of time with an average precision from 0.24 to 0.35 depending on the amount of information. This precision is six to nine times higher than with a [random guess]{} model. At the population level, we can predict infectious population in a two-week window with an r-squared value of 0.95 (a [random-guess]{} model obtains an r-squared value of 0.2). These results point to an innovative approach for tracking individuals who have interacted with people showing symptoms, allowing us to warn those in danger of infection and to inform health researchers about the progression of contact-induced diseases.'
author:
- Wen Dong
- Tong Guan
- Bruno Lepri
- Chunming Qiao
bibliography:
- 'sample.bib'
title: 'PocketCare: Tracking the Flu with Mobile Phones using Partial Observations of Proximity and Symptoms'
---
<ccs2012> <concept> <concept\_id>10003120.10003130.10011762</concept\_id> <concept\_desc>Human-centered computing Empirical studies in collaborative and social computing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003120.10003138.10003140</concept\_id> <concept\_desc>Human-centered computing Ubiquitous and mobile computing systems and tools</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003120.10003138.10011767</concept\_id> <concept\_desc>Human-centered computing Empirical studies in ubiquitous and mobile computing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257</concept\_id> <concept\_desc>Computing methodologies Machine learning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010341.10010349.10010354</concept\_id> <concept\_desc>Computing methodologies Discrete-event simulation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction
============
Mobile phones provide a powerful sensing platform that researchers in social science, epidemiology, social psychology, marketing, etc. may adopt to understand proximity interactions among people and the diffusion, through these interactions, of diseases [@madan2010social; @dong2012modeling; @dong2012graph; @farrahi2014epidemic], eating habits [@madan2010eating], opinions [@madan2010eating], behaviors [@pan2011composite], emotions, etc. However, it remains a challenge to track the proximity-based interactions of a whole community or organization and model the social diffusion of behaviors, diseases, and opinions starting from the observations of a small fraction of the volunteer population, and it is extremely hard to provide the right incentives to the whole organization and community for participating to the data collection. Thus, being unable to observe or infer the individual-level interactions at large-scale imposes serious limitations in the living lab-based studies because the solution of many societal problems involves the observation of not only the volunteers but also the non-volunteers. For example, being able to control epidemics involves knowing diseases spreading among the observed as well as the unobserved individuals in the social network.
Our proposed solution tries to connect together the sparse data points, obtained from volunteers’ mobile phones, using a model of how individuals interact with each other and how social interactions happen in terms of a sequence of elementary events (e.g., a sequence of proximity interactions). Each of these elementary events involves only few individuals but in sequence the events let emerge complex behaviors. More specifically, we proceed inferring the latent state of the complex system and calibrating the probabilities of the elementary events as a function of the latent state, using the sparse data points contributed by the volunteers as observations of the latent process. In this way, we combine techniques from generative machine learning, signal processing, and agent-based modeling to reveal the dynamics of the whole organization or community.
In the current paper, we describe the application of this approach to track the spreading of flu in the spatial-proximity network of a 3000-people university campus by mobilizing 300 volunteers from this population to monitor nearby mobile phones through Bluetooth scanning every 5 minutes and to daily report flu symptoms about and around them. Our aim is to predict the likelihood for an individual to get flu based on how often her/his daily routine intersects with those of the volunteers. Specifically, while the non-volunteers are not observable, their routines will share some similar patterns with the ones of the volunteers from the same community, and their spatial proximity with one another is detected through the Bluetooth scanning of the volunteers’ mobile phones. Thus, we use the daily routines of the volunteers to build a model of the volunteers as well as of the non-volunteers among the population: In particular, we model when they stay at the apartments; how they move from one class to the next one; when they take their meals and go to the gym; how many people spend time in proximity of apartments, classes, food court and gym, and so on. Then, we use the observations of the volunteers’ behaviors and proximity interactions as constraints for the model to unroll over time: When, where and how large the classes are for individuals to mix, how common cold and flu spread among the individuals to give rise to the daily symptoms reported by the volunteers, and how the unobserved part of the proximity network is wired to fit the reported symptoms and the observed routines.
Our work contributes to the research of predicting the spatial-proximity interactions among individuals in a community from the observations of a small set of volunteers. More specifically, we derive machine learning algorithms for a discrete-event model to infer the latent interactions from isolated observations. The premise of introducing the discrete-event model is that many complex social-interaction dynamics can be factored into a sequence of elementary events which individually involve only a few individuals but together bring the complex behavior of the system (e.g., the dyadic interactions and infections monitored by the volunteers). We provide evidence that this approach of stitching together isolated observations by volunteers with high-fidelity social interaction models may build an affordable “microscope" for computational social scientists and policy researchers to study our social systems.
The remainder of this paper is organized as follows. In Section 2, we discuss other research on capturing proximity interactions through mobile phones and wearable devices, and on modeling the spreading of behaviors, diseases, opinions, and psychological states. In Section 3 we introduce PocketCare, the Android sensing platform we developed to track proximity interactions, locations, and behaviors, and describe the data sets collected. In Section 4, we present our discrete-event model, the so-called stochastic kinetic model used to capture the spatial-proximity network dynamics, and the inference and learning algorithms. Section 5 reports the results obtained in tracking the spread of the flu, while in Section 6 we discuss the implications and limitations of our work and draw some conclusions.
Literature Review
=================
Over the past decade, mobile phones [@eagle2006reality; @eagle2009inferring; @aharony2011social; @laurila2012mobile; @stopczynski2014measuring; @wang2014studentlife; @centellegher2016mobile] and other wearable devices [@olguin2009sensible; @lepri2012sociometric; @alshamsi2015beyond] have been increasingly used as tools for closely observing individual and community behaviors. In one of the earliest studies, Eagle and Pentland analyzed the routines [@eagle2006reality] and the social interactions [@eagle2009inferring] of various research teams at the MIT Media Laboratory and the MIT Sloan School of Management by means of 84 Nokia 6600 smart phones. Similarly, using the multimodal data (i.e., infra-red, accelerometer, Bluetooth, and audio) captured by sociometric badges [@olguin2009sensible], Olguin *et al.* studied the information flow, efficiency, and employee satisfaction of various organizations. Aharony *et al.* observed the daily lives of 130 adult members of a young-family residential living community for 15 months using the FunF Android app [@aharony2011social]. In Nokia data-collection challenge, 185 participants shared five months of data about their daily activities and social interactions for scientific research [@laurila2012mobile]. More recently, Stopczynski *et al.* [@stopczynski2014measuring] conducted the Copenhagen Networks Study on a densely connected population of 1000 individuals for multiple years, using mobile phones to collect data on face-to-face and communication interactions, social networks, locations, and background information such as demographics and personality traits. Table \[tab:studies\] summarizes the previous research on capturing spatial-proximity interactions and the adopted mobile sensing and wearable platforms. Our approach stands out for monitoring the social interactions of a significantly larger community of individuals by mobilizing only a small number of volunteers.
\[tab:studies\]
[|>p[0.135]{}|>p[0.2]{}|>p[0.19]{}|>p[0.09]{}|>p[0.27]{}|]{} Research Study & Subjects & Observed Behavior & Platform & Signals[\
]{} Reality Mining [@eagle2006reality] & Loosely coupled teams of 2-10 people each & Routines and social interactions & Nokia 6600 & Sensor and survey data about participants for 9 months[\
]{} Sociometric Badges [@olguin2009sensible; @lederman2017open] & Teams of 10-20 people & Work-space dynamics & Proprietary hardware & Sensor and survey data about participants for 1-14 days[\
]{} Social Patterns [@genois2015compensating] & Communities of 100 people & Spatial-proximity interactions & Proprietary hardware & Sensor and survey data about participants for 1-7 days[\
]{} Social Evolution [@madan2010social] & 84 undergraduates in a student dormitory & Network dynamics and social diffusion & Android & Sensor and survey data about participants for 9 months[\
]{} Friends & Family [@aharony2011social] & 130 adult members of a dormitory & Influence and intervention & Android & Sensor and survey data about participants for 12 months[\
]{} Nokia Data Collection [@laurila2012mobile] & 185 participants & Mobile crowd sensing & Android & Sensor and survey data about participants for two years[\
]{} Copenhagen Networks [@stopczynski2014measuring] & 1000 incoming undergraduates & Spatial-proximity interactions & Android & Sensor and survey data about participants for multiple years[\
]{} AWARE [@ferreira2015aware; @van2016measuring] & 15 students in one study & Spatial-proximity interactions & Android, iOS & Sensor and survey data about participants for an hour[\
]{} Student Life [@lane2011bewell; @wang2014studentlife] & 48 students in a single class & Students’ well-being and performance & Android & Sensor and survey data about participants for 5 months[\
]{} PocketCare & 3000 people in a university department & Spatial-proximity interactions & Android & Sensor and survey data from 300 volunteers for multiple years[\
]{}
Modeling and predicting the dissemination of opinions and diseases in a community is emerging as an important reason to capture face-to-face, proximity, and communication interactions by means of mobile phones and wearable devices. In particular, three main approaches have been proposed to deal with social dissemination: 1) a data-driven approach for predicting diffusion from a set of features, 2) a simulation approach for specifying high-fidelity diffusion dynamics based on Monte Carlo methods, and 3) an analytical approach for specifying diffusion dynamics by means of a tractable generative model and inferring the latent diffusion process from the observations.
Among the data-driven approaches, Pan *et al.* demonstrated the influence of social interactions on mobile-phone app installation by fitting multimodal social network data with an exponential random graph model [@pan2011composite]. Madan *et al.* showed that long-time exposure to obese and inactive people influences gaining weight [@madan2010social]. Lane *et al.* found that the prediction of behaviors, habits, and psychological states — such as transportation choice, hours of sleep, diet, and mood — are improved by also considering the behaviors, habits, and psychological states of the social network’s neighbors [@lane2014connecting]. Along this line, Sandstrom *et al.* found that emotional well-being is dependent on not only friends and family but also weaker social ties [@sandstrom2014social]. Finally, the results of a study by Wu *et al.* [@wu2008mining] offered evidence that employee productivity in a workspace is dependent on the productivity of the face-to-face network’s neighbors. However, it is worth noting that data-driven approaches require the collection of a sufficient amount of training data regarding the state of a network’s neighbors in order to identify predictive features and train a classifier. Thus, these approaches become difficult to apply when the training data is scarce — for example, in non-recurrent situations.
Simulation modeling is widely used in policy research, where insights and optimal policies are identified by running computer simulation and Monte Carlo integration. State-of-the-art simulators of social diffusion include EpiSims [@eubank2004modelling] and STEM [@edlund2010spatiotemporal]. However, a big problem with simulating social diffusion through high-resolution spatial-proximity information collected with wearable sensors is that the network is often incomplete. G[é]{}nois *et al.* demonstrated a method to alleviate the issue of using incomplete data about human interaction networks in social diffusion simulations by resampling the network [@genois2015compensating]. Fournet and Barrat proposed another method to augment an incomplete network by resampling new edges using surveyed friendship information [@fournet2014]. Farrahi *et al.* [@farrahi2014epidemic] developed an approach using phone communication as a proxy for spatial-proximity in epidemic tracing. Barrat *et al.* showed that the spreading pathways of the epidemic process are strongly affected by the temporal structure of the network data [@barrat2013empirical]. Finally, Lee *et al.* demonstrated that immunization protocol based on a temporal contact network outperforms a static network, and therefore that the temporal contact structure has more information to exploit for developing a vaccination protocol [@lee2012exploiting].
Another approach for modeling dissemination through proximity interaction is to define a generative model of infection at the individual level as a hidden Markov process and infer the latent infection process from the observations of self-reported symptoms. With this approach, Dong *et al.* developed a graph-coupled hidden Markov model to predict the spread of infection from one individual to another using the susceptible-infectious-susceptible (SIS) dynamics and the dynamics of the spatial-proximity network captured by mobile-phone Bluetooth scanning [@dong2012graph; @dong2012modeling]. They adopted this approach to model the spatial-proximity network dynamics of more than 80% of residents in an undergraduate student dormitory, where the residents spent more than 10 hours interacting with other people in the community. They dealt with the incompleteness of the observed dynamic proximity network by introducing an event to represent an infection from outside the network. Fan *et al.* then developed a variational inference algorithm for graph-coupled hidden Markov models to make faster infection predictions [@fan2016unifying]. More recently, Xu *et al.* introduced the stochastic kinetic model to generalize the graph-coupled hidden Markov model and derived a variational inference algorithm to accelerate infection prediction [@xu2016using].
A common element of these previous works is that they require a high-quality spatial-proximity network to be able to predict the spreading of an infection from one node to another in the network. A useful process is to infer both the social diffusion and the network dynamics from incomplete information, such that diffusion can be tracked among not only the observed volunteers but also the non-volunteers.
In the remainder of this paper, we describe our efforts to mobilize a small fraction of a population to monitor the presence of other mobile phones (and their owners) through Bluetooth scanning and signs of social diffusion (symptoms of the common cold and flu), and use machine learning algorithms and generative models to stitch together these isolated observations.
{width="0.24\columnwidth"} {width="0.24\columnwidth"} {width="0.24\columnwidth"} {width="0.24\columnwidth"}
\[fig:pocketcare\]
Mobile Sensing Platform and Collected Data
==========================================
We developed the Android app *PocketCare* (see screenshots in Figure \[fig:pocketcare\]) to monitor flu propagation and help users track and improve their health. To this end, PocketCare continuously 1) tracks a user’s proximity network by periodically scanning for other Bluetooth devices within a few meters, including other mobile phones running PocketCare, 2) records the approximate location from GPS or nearby WiFi access points, depending on the user’s settings, and 3) determines the user’s activities (i.e., stationary, walking, or in a moving vehicle). In addition, PocketCare asks users to voluntarily report if they personally have flu symptoms and or have observed flu symptoms in people nearby, and provides users with useful health tips.
As an incentive for the study participants to provide information on flu symptoms, they accumulate reward points by giving the app permission to collect data. These points unlock additional useful features as well as prizes. The app runs in the background with negligible impact on normal phone usage or battery consumption.
Data are collected anonymously and do not reveal any personal information about the user. The collected data are stored on a secure server that can be accessed only by authorized researchers (although study participants also have the option to store such data locally on the phone). PocketCare adheres strictly to the privacy policies reviewed and approved by the university’s Social and Behavioral Sciences IRB (SBSIRB), which can be found online.
Using PocketCare, we have now monitored flu propagation and provided flu information on a university campus for over a year. Approximately 300 users continuously operate PocketCare on campus: 108 graduate students, 144 undergraduate students, and 24 faculty/staff members from one university department with a total population of around 3000, as well as 64 users not from the department but from the same university. We selected 80% of participants from the same department to capture a sufficient number of interactions and establish the ground truth. We chose the other 20% of participants from the wider university population to evaluate the potential for tracking epidemics at larger scale from incomplete information about both social networks and symptoms using a mixture of generative machine learning models and agent-based models.
During the study, we ask participants daily whether they personally are experiencing local symptoms (such as sore throat, sneezing, runny nose, or cough) or systemic symptoms (such as headache, fever, or muscle pains), and whether they have observed others sneezing or coughing. Different symptoms give us meaningful information about cold and flu development and progression, as the cold and flu syndromes typically involve early symptoms of headache, sneezing, and chilliness that are distinct from later symptoms such as coughing [@jackson1958transmission]. The survey is a conventional tool used by epidemiologists — well-studied and widely applied in discovering epidemic progression at the population level.
{width="0.45\columnwidth"} {width="0.45\columnwidth"}
\[fig:sick\]
About 70% of study participants fill out symptom surveys, and approximately 4% of surveyed reports of symptoms say either that the participants have symptoms or that they have observed symptoms in others (Figure \[fig:sick\]). When participants observe symptoms nearby, they are twice as likely to have symptoms themselves. We collected a sufficient number of symptom cases in our data set because people have on average two episodes of cold or flu per year, each lasting around one week, and these numbers are higher in younger people. Moreover, the symptoms recorded by the surveys are also the vector for spreading cold and flu, since they facilitate transmission [@monto2000clinical]. A cold or flu that causes a sub-clinical infection is unlikely to succeed in transmission, and the most successful common cold viruses are generally those that cause the most runny noses, coughs, and sneezes [@eccles2005asymptomatic].
We use the Bluetooth interface with PocketCare to monitor nearby mobile phones, including those running PocketCare, and to establish a dynamic spatial-proximity network of mobile phones. The idea behind tracking spatial proximity is for one device to transmit signals through its Bluetooth transmitter while another measures the received signal strength indicator (RSSI) through its receiver. If the RSSI value is above a certain threshold, the app will consider the two mobile phone users to be in spatial proximity. After some initial calibration, the app can also estimate the distance between the two phones based on this RSSI value.
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\[fig:proximity\]
PocketCare scans nearby Bluetooth devices every five minutes, and collects the anonymized PocketCare user ID, the Unix time when the scan was made, the media access control (MAC) addresses of nearby devices, and the RSSI values. The Bluetooth scanning records show three macro clusters among PocketCare users, corresponding to the graduates, the undergraduates, and the faculty/staff of the department, with each macro cluster consisting of smaller clusters according to where people live, which classes they take, which labs they belong to, and so on (Figure \[fig:proximity-users\]). PocketCare-scanned mobile devices account for approximately 40% of the department’s population (Figure \[fig:proximity-others\]) and about 15% of the total population on campus, where we have used Bluetooth class of device/service and the organizationally unique identifier (OUI) in the MAC address to infer whether a device is a mobile phone, and the location of the PocketCare scanner to infer the location of scanned devices. Thus, the 300 PocketCare users monitor a spatial-proximity network far bigger than the network of the participants themselves. This is a sufficient level of coverage of a community to validate our modeling and machine learning algorithms.
Additionally, we have used the WiFi interface with PocketCare to learn the structure of users’ daily activities and their interactions in these activities. Scanning nearby WiFi access points every five minutes, PocketCare collects the anonymized PocketCare user ID, the Unix time when a scan was made, the MAC addresses of the WiFi access points, and the RSSI values. When combined with the geographic locations of the scanned WiFi access points (e.g., from the Wireless Geographic Logging Engine or <https://wigle.net/>) and information about those locations (e.g., from OpenStreetMap), the device scanning reveals a meaningful structure that can be used to synthesize agent activities and interactions for studying the spread of epidemics at the individual level. We can also train a classifier to identify from these WiFi scanning records whether two PocketCare users are in spatial proximity. For example, two users are likely to be in proximity if their RSSIs to the same access points are close and they have often seen the same access points at the same times in the past. Given the locations of the WiFi access points (e.g., their latitudes and longitudes, or their relative positions on a floor plan), we can use as few as nine WiFi scan records to calibrate the signal-decay models and compute users’ locations with an average of two-meter accuracy.
The WiFi access-point scanning records show three macro clusters and subclusters of PocketCare users, according to how long the users share spaces with one another (Figure \[fig:proximity-wlan\]). In the data set, users on average frequent the same four to eight places per day, including labs, lecture and residence buildings, university libraries, and the student center, at similar hours of the day. However, some users can frequent as many as 40 places at the 90th percentile. Some places, such as student dormitories and office buildings, see the same set of PocketCare users, while others such as libraries and the student center see almost all users. This behavior agrees with previous findings made at a university dormitory [@dong2011modeling; @guan2017fine], and provides the basis for us to specify a generative model of the dynamic spatial-proximity network through how people move around and perform different activities throughout the day (Figures \[fig:wifi-weekday\] and \[fig:wifi-weekend\]).
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\[fig:wifi\]
To establish the ground truth of the campus-wide spatial-proximity network, we used syslog records containing which users are connected to which university WiFi access points at what times. These data contain the anonymized user ID, the anonymized device MAC address connecting to an access point, the OUI of the connecting device MAC address (which has information about device class), the MAC address of the connected access point, and the starting and ending times of the connection. This data set contains the interactions of about 60,000 devices from 30,000 users with 3,000 WiFi access points distributed to approximately 500 building floors. An average user frequents the same four to eight places per day at similar hours of a day, but a few users can frequent as many as 140 places on a specific day. A floor is generally frequented by the same two people to several hundreds of people every day, but some floors (for example, at the student center or libraries) can see different thousands of people daily. These behaviors are compatible with those in the PocketCare WiFi-scan data (Figures \[fig:cit-weekday\] and \[fig:cit-weekend\]).
Finally, we also monitored user activities every five minutes with the ActivityRecognitionApi Google API for Android in PocketCare. The device is still 86% of the time, in use for 5%, on a user who is walking or running for 3%, and in a vehicle for 2%. Thus, for a significant fraction of time users are likely indoors, and their locations and interactions are better captured by WiFi and Bluetooth than using GPS.
While Bluetooth and Wi-Fi scanning are supposedly energy-intensive, battery-life issues caused by scanning have not been reported by the volunteers. The battery drain appears to be negligible.
Methodology
===========
In this section, we introduce a discrete event model called the stochastic kinetic model to capture the dynamics of a complex social system, then we formulate social diffusion and spatial-proximity network dynamics with a discrete event model, and finally we offer inference and learning algorithms.
Modeling
--------
We introduce the stochastic kinetic model to capture the dynamics of a complex social system driven by a set of events. A *stochastic kinetic model* is a biochemist’s way of describing the temporal evolution of a biological network with $M$ species driven by $V$ mutually independent events [@gillespie2007stochastic; @wilkinson2011stochastic], where the stochastic effects are particularly prevalent (e.g., a transcription network or a signal transduction network). Let $\mathbf X=(\mathbf X^{(1)},\cdots,\mathbf X^{(M)})$ denote individuals belonging to the $M$ species in the network. An event (chemical reaction) $v$ is specified by a production $$\begin{aligned}
& {\alpha_{v}^{(1)}\mathbf X^{(1)}+\cdots+\alpha_{v}^{(M)}\mathbf X^{(M)}\overset{c_v}\to\beta_{v}^{(1)}\mathbf X^{(1)}+\cdots+\beta_{v}^{(M)}\mathbf X^{(M)}}.\label{eq:reaction}\end{aligned}$$ The production is interpreted as having *rate constant* $c_{v}$ (probability per unit time, as time goes to 0), $\alpha_{v}^{(1)}$ individuals of species $1$, $\alpha_{v}^{(2)}$ individuals of species $2$ ... interact according to event $v$, thus resulting in their removal from the system; and $\beta_{v}^{(1)}$ individuals of species $1$, $\beta_{v}^{(2)}$ individuals of species $2$ ... are introduced into the system. Hence, event $v$ changes the populations by $\Delta_{v}=(\beta_{v}^{(1)}- \alpha_{v}^{(1)},\cdots,\beta_{v}^{(M)}-\alpha_{v}^{(M)})$. The species on the left side of the production are *reactants*, the species on the right side of the production are *products*, and the species $m$ with $\alpha_v^{(m)}=\beta_v^{(m)}$ are *catalysts*.
At the system level, let $x_{t}=(x_{t}^{(1)},\dots,x_{t}^{(M)})$ be the populations of the species in the system at time $t$. A stochastic kinetic process initially in state $x_{0}$ at time $t=0$ can be simulated through the Gillespie algorithm [@gillespie1976] (Algorithm \[alg:Gillespie\]). In this algorithm, event rate $h_{v}(x_t,c_{v})$ is the rate constant $c_{v}$ multiplying a total of $\prod_{m=1}^{M}(x^{(m)}_t)^{\alpha_{v}^{(m)}}$ different ways for individuals to interact in the system, assuming homogeneous populations; and event rate takes a more complex form if the populations are not homogeneous. Exponential distribution is the maximum entropy distribution given the rate constant, and consequently is most likely to occur in natural reactions [@gillespie2007stochastic]. The stochastic kinetic model thus assigns a probabilistic measure to a sample path induced by a sequence of events $v_{1},\dots,v_{n}$ happening between times $0$ and $T$, $0<t_{1}<\dots<t_{n}<T$, which is $$\begin{aligned}
& P(v_{1:n},t_{1:n},x_{1:n}) = \prod\limits _{i=1}^{n}h_{v_{i}}(x_{t_{i-1}},c_{v_{i}})\exp(-\sum\limits _{i=1}^{n}h_{0}(x_{t_{i-1}},c)(t_{i}-t_{i-1})),\label{eq:CTSKM} \\
& \text{where }h_{v}(x,c_{k})=c_{v}g_{v}(x)\text{ for }v=1,\cdots,V,\text{ and }h_{0}(x,c)=\sum\limits_{v=1}^{V}h_{v}(x,c_{v}). \label{eq:rate}\end{aligned}$$
**Input**: A stochastic kinetic process specified by a set of independent events (Eq. \[eq:reaction\] for $v=1,\cdots,V$) with initial populations $x_{t}=(x_{t}^{(1)},\dots,x_{t}^{(M)})$ at time $t=0$, and event rate $h_v(x_t,c_v)$ for each event $v$. Termination condition which is often simulating upper time $T$.
**Output**: A sequence of events $v_1,\cdots,v_n$, the times of these events $t_0=0, t_1,\cdots,t_n$ as well as the corresponding populations $x_0, x_1,\cdots,x_n$.
**Procedure**: Iterate the following steps until the termination condition is satisfied.
1. Sample the time $\tau$ to the next event according to exponential distribution $\tau\sim\mbox{Exponential}(h_{0}(x_t,c))$, where $h_{0}(x,c)=\sum_{v=1}^{V}h_{v}(x_t,c_{v})$ is the rate of all events.
2. Sample the event $v$ according to categorical distribution $v\sim(\frac{h_{1}}{h_{0}},\dots,\frac{h_{V}}{h_{0}})$ conditional on event time $\tau$.
3. Update the system time $t\leftarrow t+\tau$ and populations $x\leftarrow x+\Delta_{v}$.
The stochastic kinetic model is one way to define a discrete event process, and its equivalents in other fields include the stochastic Petri net [@marsan1994modelling; @goss1998quantitative], the system dynamics model [@forrester1961w], the multi-agent model specified through a flow chart or state chart [@borshchev2013big], and the production rule system [@newell1972human]. These equivalent models have the same power in capturing the dynamics of a system, but are different in their representations. As such, the stochastic kinetic model can also be used in the fields where there are equivalent models.
Dynamics of social diffusion and the proximity network
------------------------------------------------------
In this subsection, we formulate the dynamics of epidemic spread and the activity-based spatial-proximity network, building on previous research.
The common cold is the most common ubiquitous infectious disease, generally occurring in the winter and spring. Average adults get 2 3two or three colds per year, and average children 6-8six to eight. The common cold is caused by infection of the throat, sinuses, and larynx by one of over 200 virus strains through close contact with infected people or indirectly through contact with objects in the environment, followed by transfer to the mouth or nose. Symptoms include coughing, sore throat, runny nose, sneezing, headache, and fever, which can appear within two days after of exposure to the virus and last for seven to ten days [@centers2016common]. Risk factors include going to daycare, not sleeping well, and psychological stress [@allan2014prevention].
We use SIS dynamics to capture the dynamic spreading of cold and flu through spatial proximity. In SIS dynamics, each individual is either infectious (I) or susceptible (S), and the system has three events: 1) an infectious individual in the network infects a susceptible individual in the network and turns that person infectious with rate constant $c_1^{(d)}$ (probability per unit time), 2) an infectious individual recovers and becomes susceptible again with rate constant $c_2^{(d)}$, and 3) a susceptible individual becomes infectious by contacting an infectious individual from outside the system with rate constant $c_3^{(d)}$.
$$\begin{aligned}
& I +S \to 2\times I, \text{infection, rate constant } = c_1^{(d)},\\
& I \to S, \text{recover, rate constant} = c_2^{(d)},\\
& S \to I, \text{infection from outside, rate constant} = c_3^{(d)}.\end{aligned}$$
To model SIS dynamics at the individual level, we mark the infectious and susceptible individuals with person indexes: at any time, a person $p$ is either an infectious individual denoted as $I^{(p)}$ or a susceptible individual denoted as $S^{(p)}$, but not both. We further define $i^{(p)}$ as a binary random variable that is $1$ when person $p$ is infectious and $0$ when the person is susceptible, and similarly define $s^{(p)}$ as another binary random variable that is $1$ when person $p$ is susceptible and $0$ when the person is not susceptible. Thus $i^{(p)}\in\{0,1\}$, $s^{(p)}\in\{0,1\}$ and $i^{(p)} + s^{(p)} = 1$. The probability for a susceptible person $p$ to become infectious through one unit time of contact with an infectious person $q$ is therefore $h(I^{(p)}+S^{(q)}\to I^{(p)}+I^{(q)}; c_1^{(d)}) = c_1^{(d)}\cdot s^{(p)} i^{(q)} = c_1^{(d)}$.
Conditioned on the instantaneous spatial-proximity network, the reported symptoms of the PocketCare users and of their local proximity network neighbors serve as independent observations.
Trip generation has been a well-studied field since H[ä]{}gerstrand’s constraint-based modeling [@hagerstrand1987human] and Chapin’s activity-based modeling [@chapin1968activity]. In a trip-generation model, people engage in different activities and trips are undertaken to fulfill these activities. These include major activities such as being at home, working, shopping, being at school, eating out, socializing, enjoying recreation, and serving passengers (i.e., picking up and dropping off), as well as numerous other activities that people engage in on a less-than-daily or even weekly basis, such as going to the doctor or the bank. The trips and activities are dependent on many factors and are traditionally simulated from surveyed results. However, with the ability to survey the travels of a large population with mobile phones, and the availability of large databases, a new method of trip generation combining machine learning and signal processing is possible.
To model the human dynamics in performing a sequence of activities (e.g., taking courses, performing research, staying in an apartment, dining in a cafeteria, etc.) throughout the day, we characterize people with their locations. Let $l_i$, $l_j$, and $p_k$ denote locations $i$ and $j$ and person $k$. The system is driven by a single type of event, $p_{k}\circ l_{i}\to p_{k}\circ l_{j}$ — person $k$ moving from location $i$ to location $j$ with rate constant $c_{l_{i},l_{j}}^{(t)}$ (number of events per unit time). Here we use “$\circ$” to represent a bond: person $k$ binds to location $i$ before the event and binds to location $j$ after the event. Let latent state $x_t^{(l)}$ be the population at location $l$ at time step $t$. Then, this event changes the population at location $l_{i}$ from $x_{t}^{(l_{i})}$ to $x_{t+1}^{(l_{i})}=x_{t}^{(l_{i})}-1$, and changes the population at location $l_{j}$ from $x_{t}^{(l_{j})}$ to $x_{t}^{(l_{j})}+1$. According to this model, an individual stays at location $i$ for an average duration $1/\sum_{j}c_{l_{i},l_{j}}^{(t)}$ and on exit chooses the next location with a probability proportional to the rate constant $c_{l_{i},l_{j}}^{(t)}/\sum_{j'}c_{l_{i},l_{j'}}^{(t)}$.
We use the WiFi scanning trajectories from PocketCare users and syslog to construct activity durations at each WiFi access point, and the transition probabilities from one access point to another. A number of researchers have characterized the structure of daily trips on university campuses [@guan2017fine; @dong2011modeling].
Then, we use the number of smartphones scanned by PocketCare users as independent noisy observations of the populations at different locations, assuming that both the Bluetooth-discoverable devices and the PocketCare users are uniformly sampled from the system. Let $x_0$ be the total population in the system and $y_0$ the total population with Bluetooth-discoverable phones. The probability of observing $y_{t}^{(l_{j})}$ population at location $j$ conditioned on there being $x_{t}^{(l_{j})}$ population in total is $p(y_{t}^{(l_{j})}|x_{t}^{(l_{j})})=\footnotesize\left(\begin{matrix}x_{t}^{(l_{j})}\\
y_{t}^{(l_{j})}
\end{matrix}\right)\left(\begin{matrix}x_0-x_{t}^{(l_{j})}\\
y_0-y_{t}^{(l_{j})}
\end{matrix}\right)\bigg/\left(\begin{matrix}x_0\\
y_0
\end{matrix}\right)$. When the total population in the system is large, the percentage of Bluetooth-discoverable population at a given location is roughly the percentage in the system.
To summarize, we capture the latent social network dynamics with three social diffusion events characterized by rate constants $c_1^{(d)}$, $c_2^{(d)}$, and $c_3^{(d)}$, and activity events characterized by rate constants $c^{(t)}_{l_i,l_j}$. We use volunteers’ Bluetooth-scanning of the local proximity network and their surveyed symptom reports as noisy observations to infer the latent dynamics. These two types of rate constant are treated the same, which demonstrates the versatility of the discrete event model in capturing diverse dynamics in a unified framework.
Inference and learning
----------------------
Although the stochastic kinetic model is a continuous time model, we work with a discrete time stochastic model in the rest of this paper because our goal is to track stochastic kinetic dynamics from observations of populations or individuals with countably many computational steps. To this end, we approximate the continuous time process with a discrete time process on a countable set of equally spaced time points $0,\tau,2\tau,\dots$, with a time interval so small that the probability of more than one event happening in the interval $\tau$ is negligible. This approximation works because the state transition kernel from time $0$ to time $\tau$ is
$$p(x_{0}\to x_{\tau})=\sum\limits_{n=0}^{\infty}(I+\frac{Q}{\gamma})^{n}\exp(-\gamma\tau)\frac{(\gamma\tau)^{n}}{n!}.\label{eq:pf-uniformization}$$
according to the uniformization method [@grassman77uniformization], where $\gamma$ is a uniformization rate, $I$ the identity matrix and $Q$ the infinitesimal generator defined by $h_{k},k=1,\dots,V$. With $\gamma\to\infty$ and $\gamma\tau=1$, we get a first-order approximation of the state transition kernel $I+Q\cdot\tau$.
Specifically, let $v_{1},\dots,v_{T}$ be a sequence of events in the discrete time stochastic kinetic system, $x_{1},\dots,x_{T}$ a sequence of states (populations of species), and $y_{1},\dots,y_{T}$ a set of observations about the populations. Our goal is to make inferences about $\{v_{t},x_{t}:t=1,\dots T\}$ from $\{y_{t}:t=1,\dots,T\}$ according to the following probability measure, where indicator function $\delta(x_{t}-x_{t-1}=\Delta_{v_{t}})$ is 1 if the previous state is $x_{t-1}$ and the current state is $x_{t}=x_{t-1}+\Delta_{v_{t}}$, and 0 otherwise. $$\begin{aligned}
& P\left(v_{1,\dots,T},x_{1,\dots,T},y_{1,\dots,T}\right)=\prod_{t=1}^{T}P(x_{t},y_{t},v_{t}|x_{t-1}),\label{eq:DTSKM}\\
& \mbox{where }P(x_{t},y_{t},v_{t}|x_{t-1}) = P(v_{t}|x_{t-1})\delta(x_{t}-x_{t-1}=\Delta_{v_{t}})P(y_{t}|x_{t}),\label{eq:DTSKMTransition}\\
& \mbox{and }\hspace*{1em}P(v_{t}|x_{t-1})=\begin{cases}
c_{k}\tau\cdot g_{k}\left(x_{t-1}\right) & \mbox{if }v_{t}=k\\
1-\sum_{j}c_{j}\tau g_{j}\left(x_{t-1}\right) & \mbox{if }v_{t}=\emptyset
\end{cases}.\label{eq:DTSKMP}\end{aligned}$$
We apply a particle filter to track the dynamics of this process (Figure \[fig:pfpm\]). The particle filter maintains a collection of particles $x_{t}^{k}$ for $k=1,\cdots,N$ and $t=1,\cdots,T$ to represent the likelihood of the latent state of a stochastic process $x_{t}$ at different regions of the state space with each particle representing a system state, given noisy and partial observations $y_{1,\cdots,T}$. It tracks the evolution of a stochastic process by alternately mutating/sampling the collection of particles according to stochastic process dynamics $P(x_{t}|x_{t-1})$ and selecting/resampling the particles according to observations $P(y_{t}|x_{t})$. Here the particle mutation and selection metaphors are from statistical physics [@del2004feynman], and refer to particle sampling and resampling respectively. In comparison with a particle filter, a simulation run uses only one particle and does not use observations to perform particle selection.
![A graphical illustration of particle filtering and particle smoothing algorithms.[]{data-label="fig:pfpm"}](PF_scheme){width="0.8\columnwidth"}
Specifically, let $x_{t}^{k}$ for $k=1,\cdots,N$ be the collection of particle positions and $v_t^k$ the corresponding events from particle mutation, and $i_{t}^{k}\in\{1,\cdots,N\}$ be the collection of particle indexes from particle selection. To make inferences about the latent state $x_{t}$ of a stochastic process starting at state $x_{0}$ from observations $y_{1:t}$, we initialize particle positions and indexes as $x_{0}^{1},\cdots,x_{0}^{N}=x_{0}$ and $i_{0}^{1}=1,\cdots,i_{0}^{N}=N$, and iteratively sample the next event $v_{t}^{k}$ according to how likely it is that different events will occur conditioned on system state $x_{t-1}^{i_{t-1}^{k}}$ for $k=1,\cdots,N$ (Eq. \[eq:pf-v\]), then update $x_{t}^{k}=x_{t-1}^{k}+\Delta_{v_{t}^{k}}$ accordingly (Eq. \[eq:pf-x\]) and resample these events per their likelihoods with regard to the observation $y_{t}$ for $t=1,\cdots,T$ (Eq. \[eq:pf-i\]). $$\begin{aligned}
& v_{t}^{k}|x_{t-1}^{i_{t-1}^{k}}\sim\text{Categorical}(1-\frac{h_{0}(x_{t-1}^{i_{t-1}^{k}})}{\gamma},\frac{h_{1}(x_{t-1}^{i_{t-1}^{k}})}{\gamma},\cdots,\frac{h_{V}(x_{t-1}^{i_{t-1}^{k}})}{\gamma}),\label{eq:pf-v}\\
& x_{t}^{k}=x_{t-1}^{i_{t-1}^{k}}+\Delta_{v_{t}^{k}},\label{eq:pf-x}\\
& i_{t}^{k}|(x_{t}^{1:N},y_{t})\sim\text{Categorical}(p(y_{t}|x_{t}^{1}),\cdots,p(y_{t}|x_{t}^{N})).\label{eq:pf-i}\end{aligned}$$
To determine a particle trajectory from the posterior distribution of a stochastic kinetic process with respect to observations, we trace back the events that led to the particles $x_{T}^{i_{T}^{k}}$ for $k=1,\cdots,N$: $$\begin{aligned}
& x_{0},v_{1}^{j_{1}^{k}},x_{1}^{j_{1}^{k}},\cdots,v_{T}^{j_{T}^{k}},x_{T}^{j_{T}^{k}},\text{where }{\scriptstyle j_{T}^{k}=i_{T}^{k},j_{T-1}^{k}=i_{T-1}^{j_{T}^{k}},\cdots,j_{1}^{k}=i_{1}^{j_{2}^{k}}}.\label{eq:pf-path} \end{aligned}$$ To learn the rate constants (the parameters) of a stochastic kinetic model, we sample from the posterior distribution of these parameters conditioned on the particle trajectory, using a beta distribution as conjugate prior. Let $v_{1},\cdots,v_{T}$ be the sequence of events in a particle trajectory (Eq. \[eq:pf-path\]). The posterior probability distribution of a rate constant is a beta distribution that matches the expected number of events in a sample path with the number of events that actually occur, where $a_v$ and $b_v$ are hyper-parameters: $$\begin{aligned}
&c_{v}|v_{1:T},x_{1:T}\sim\text{Beta}(a_{v}+\sum\nolimits_{i=1}^{T}\delta_{v}(v_{t}),b_{v}+\sum\nolimits_{t=0}^{T}g_{v}(x_{t})-\sum\nolimits_{t=1}^{T}\delta_{v}(v_{t}) ).\label{eq:pf-learn}\\\end{aligned}$$
Overall, we have developed a particle-based algorithm to make inferences about a complex system using a stochastic kinetic model and noisy observations (Algorithm \[alg:PF\]). This algorithm gives social scientists a way to track real-world complex social systems from continued noisy observations with their simulator, as long as they can evolve the system state with the simulator and compute the likelihood of the system state with respect to observations. When the complex system has an extremely high dimension, variational inference algorithms [@xu2016using; @fang2017expectation; @yang2019optimal] should be considered to avoid particle degeneracy issues. However, as our experiment shows, the particle-based algorithm works well with social systems under general considerations.
**Input**: Observations $y_{1},\cdots,y_{T}$ of a stochastic kinetic process (Eq. \[eq:DTSKM\]) specified by a set of events (Eq. \[eq:reaction\] and \[eq:DTSKMP\] for $v=1,\cdots,V$).
**Output**: Resampled particles $(v_{t}^{i_{t}^{k}},x_{t}^{i_{t}^{k}})_{t=1:T}^{k=1:N}$ from particle filter, particle trajectories $(v_{t}^{j_{t}^{k}},x_{t}^{j_{t}^{k}})_{t=1:T}^{k=1:N}$ from particle smoother.
**Procedure**:
- Initialize $x_{0}^{1}=\cdots=x_{0}^{N}=x_{0}$, $i_{0}^{1}=1,\cdots,i_{0}^{N}=N$.
- (Filtering) For $t$ in $1,\cdots,n$ and $k$ in $1,\cdots,N$, sample $v_{t}^{k}$ and $i_{t}^{k}$ according to Eq. \[eq:pf-v\], \[eq:pf-x\] and \[eq:pf-i\], where $p(y_{t}|x_{t})$ is defined in Eq. \[eq:DTSKM\].
- (Smoothing) Back-track particle trajectory from $x_{T}^{i_{T}^{k}}$ according to Eq. \[eq:pf-path\], for $k=1,\cdots,N$.
- (Parameter Learning) Sample rate constants according to Eq. \[eq:pf-learn\].
Results
=======
In this section, we establish the statistical significance of observing symptom propagation in a physical-proximity network, then we evaluate the performance of predicting infection at both the individual level and the population level.
Feasibility of tracking a proximity network with a small number of volunteers
-----------------------------------------------------------------------------
In this data set, interactions happens at places of with high population densityies. For example, students take classes at lecture halls, research teams meet in department buildings, and people take meals together at the commons and or do work outs at the gym. Hence, Iit is possible to track the activities and interactions of a large population with a small number of volunteers using their mobile phones to Bluetooth-scan nearby mobile phones because p. People performed similar activities daily at their regular times and locations,. which allowsHence it is possible to modeling the activities and interactions of non-volunteers from those of the volunteers of the same types (e.g., students and the facultyies of different departments).
In Figure \[Fig:survey-community-network\], we show that we can randomly sample 5% of the population as observers to detect the presence of up to 80% of the population who use the same WiFi access points at the same time on a university campus of around 30 30,000thousand people. If we randomly sample 10% of the population as observers, the presence and activities of 90% of the population can be observed. The data to generate this figure is are the syslog from the university-owned Wi-Fi access points containing which anonymized device IDs are connected to which Wi-Fi access points for what times and for how long.
![The probabilities for infectious and healthy individuals to report nearby infectious cases[]{data-label="Fig:infection:net_effect"}](observing-large-population-through-few-participant){width="1\linewidth"}
![The probabilities for infectious and healthy individuals to report nearby infectious cases[]{data-label="Fig:infection:net_effect"}](infection-network_effect){width="1\linewidth"}
Statistical significance of symptom propagation
-----------------------------------------------
In our data, a PocketCare user with a symptom has 2 two times higher odds of seeing other users in his spatial-proximity network with the same symptom (Figure \[Fig:infection:net\_effect\]). As such, it makes sense to fit the time-tested infection model with real-world data of symptom reports and proximity observations, and infer how people infect one another through their contacts. In order to determine whether the higher odds could somehow be due to chance, we have conducted the following permutation test to reject the null hypothesis that “the spatial-proximity network is unrelated to symptoms,”, and we can reject that null hypothesis with $p<0.01$. The permutation test shuffles the mapping between the users and the nodes in the proximity network and estimates the probability distribution of the number of neighbors with the same symptom among all possible shuffling outcomes. If proximity networks are not related to the timing of when a user exhibits a symptom, then all mappings between the users and the nodes would be equally likely, and the number of friends with the same symptom would take the more likely values.
Further, the probability of exhibiting a symptom increases approximately linearly with the number of spatial -contacts reporting the same symptom. The base probability of having a symptom is 4 cases per day per 100 persons, and every additional friend exhibiting the symptom adds increases the probability by about 1%. This relationship again agrees with the theory of epidemic dynamics, which predicts that the rate of contagion will be proportional to the likelihood of contact with an infected individual.
The epidemic model says that an infected/infectious individual recovers with a constant rate $\gamma$, and therefore the duration of the infection follows an exponential distribution. We fit the observed symptom durations with an exponential distribution using maximum likelihood estimation to check the compatibility of the epidemic model with the collected data on recovery rate. The average duration of the symptoms in the fitted exponential distributions is about 3 three days. As such, we have satisfactory goodness-of-fit statistics in Kolmogorov-Smirnov hypothesis testing.
In Figure \[Fig:epidemic-progression\], we show the stochastic matrix of common cold and flu progression generated from the surveyed daily symptoms reports about from the volunteers, with the states ordered according to their times of occurrence in a disease episode. The progression of a common cold/flu episode starts from with runny nose, advances to coughing, soare throat, and fever, and ends with normal state. In the experiment, we have asked the subjects to report daily whether they have fever, runny nose, coughing, sore throat and nausea, and we code their responses as a 5-bit string, with 1 bit per symptom. For example, if a subject reports runny nose and coughing, we code the response as “01100.”. To generate the stochastic matrix, we tabulate the responses of the volunteers on the current day and the next day, and from the table we estimate the probability distributions of the response in the next day conditioned on the response of the current day. To order the responses according to their relative positions in a common cold/flu episode, we simulate common cold/flu progression in an episode from the stochastic matrix and estimate the average times for the responses to occur. Thus, each row of the stochastic matrix is identified by the response of the current day, and the entries in the row are the probabilities of the responses in the next day.\
Configuration for evaluating epidemic-tracking performance
----------------------------------------------------------
Using the noisy observations of local proximity networks through volunteers’ Bluetooth scanning of nearby mobile phones and the reported symptoms about and around the volunteers, we are interested in the following predictions: 1) the probability for individuals to be infected in the near future, given their positions relative to the volunteers in the spatial-proximity network, 2) the smoothing probability distribution of a sample path given the observations, and 3) the total number of infectious people in the near future. Prediction (1) is important for healthcare policy researchers in order to optimally allocate limited resources for controlling epidemic spreading. Prediction (2) is important for calibrating model parameters, while (3) is important for researchers to predict outbreaks ahead of time.
We use 10-fold cross-validation to evaluate the contributions of the observed spatial-proximity network and of the symptom reporting in predicting whether an individual is infectious two weeks ahead of time, and in inferring whether an individual is infectious on a given day using the available data before and after that day. In other words, we split the 300 volunteers into 10 equal partitions and report the average performance, making predictions on a single partition of held-out volunteers using the other 9 partitions of volunteers. The ground truth is whether the participants reported sick from their daily health surveys. To evaluate the contribution of symptom reports by the volunteers, we remove the daily health survey records (self-sick, nearby-sick, and symptoms) of the held-out volunteers to be predicted, use these as ground truth, and keep the spatial-proximity networks of these volunteers in the prediction. To evaluate the contribution of symptoms as well as the proximity-network structure reported by the volunteers, we take out not only their daily health survey records but also their spatial-proximity networks. So, in the latter case the proximity information about a held-out volunteer is only as complete as what can be observed by the other 9-fold volunteers.
We draw the precision-recall curve in different configurations, and report the average precision as a performance metric. [A precision-recall curve shows the capability of a binary classifier to detect the positive cases as the discrimination threshold of this classifier is varied.]{} Here, precision is the probability for an individual to actually be infectious (positive) when we predict infectiousness, recall is the fraction of the actually infectious individuals we can identify in the system, and the usage of different thresholds gives different precision-recall pairs. In a precision-recall plot, the [no-skill (random-guess)]{} line is the total number of positive cases divided by the total number of positive (infectious) and negative (non-infectious) cases, which is precision = 0.04. Perfect skill corresponds with precision = 1 and recall = 1. [The average precision summarizes the skill of the binary classifier over the whole precision-recall curve, and is computed as the area under the precision-recall curve when recall spans from zero to one.]{} A precision-recall curve is preferred to a receiver operating characteristic curve because we are less interested in predicting the negative cases correctly when there is an overwhelmingly large fraction of them. [Another way to show the ability of a binary classifier is to plot error (i.e. $1 - \text{precision}$) versus recall, resulting a curve that is visually similar to a receiver operating characteristic (ROC) curve. We will use the more well-known precision-recall curve in the following. The precision-recall curve is a stochastic process with probability dependence on both the discriminant function of the binary classifier, and the training/testing data sampled from an underlying probability distribution. The curves could take rugged shapes when they are drawn from insufficient amount of data. Depending on many factors, sometimes they can cross one another.]{}
We use bootstrapping to evaluate the performance of predicting daily infectious-population size two weeks ahead of time, because it is not feasible to mobilize an entire population of several thousands to establish the ground truth. To this end, we construct a sample path within the dynamic proximity network of the whole community and epidemic diffusion from the partial observations using the particle smoothing algorithm. More specifically, we sample 10% of the population as volunteers, and then replay the sample path to predict the infectious populations in a two-week time window using only the partial observations up to the current time at each time point. Then we repeat this procedure, obtaining many synthesized ground-truth sample paths and estimated daily infectious populations from partial observations, from which we get the sample averages.
By means of the Monte Carlo integration just described, we identify the contribution of the partially observed proximity network in predicting the total infectious population two weeks ahead of time by comparing our discrete event model particle filter with a scaling-based algorithm and a support vector regression algorithm, where the particle filter algorithm uses the information contained in the partially observed proximity network and the latter two baseline approaches do not. Instead, the scaling-based algorithm uses the percentage of the respondents reporting symptoms as a surrogate for the community-wide percentage of infection, as is commonly used by healthcare researchers. The support vector regression algorithm predicts the community-wide percentage of infection in a two-week time window from the percentage of respondents reporting symptoms about themselves and the people around them in the previous seven days. On the other side, the discrete event model particle filter algorithm simulates network and social diffusion dynamics at the individual level in agreement with the reported symptoms and the local proximity-network structure.
In the following section, we discuss the r-squared statistics and visually compare the predicted infectious populations from noisy observations using these three algorithms (the discrete event model, a scaling-based algorithm, and support vector regression) with the ground truth in the same plot. Let $f_{t}$ be the predicted infectious population at time $t$ calculated from the information until two weeks before time $t$, $y_{t}$ the ground truth, and $\bar{y}$ the average of $y_{t}$. We define $R^{2}=1-\sum_{t}(f_{t}-y_{t})^{2}/\sum_{t}(y_{t}-\bar{y}_{t})^{2}$. A higher $R^{2}$ indicates a better fit between the estimated time series and the ground truth, with $R^{2}=1$ indicating a perfect fit and $R^{2}<0$ a fit worse than using the average.
We also use a stress test to calibrate the minimum requirement to predict infection at the individual and system levels from volunteer-reported local spatial-proximity networks and symptoms of infection. To this end, we randomly select 150, 100, and 30 volunteers from the set of 300 and evaluate prediction performance at the individual and aggregate levels. Asymptotically, removing volunteers is equivalent to having volunteers not contributing data.\
Performance in tracking an epidemic
-----------------------------------
In this section, we report the performance in predicting infection at both the individual level and the aggregate level from the signs of infection and the local spatial-proximity network structure observed by the volunteers.
Figure \[fig:performance-roc\] shows in different configurations the precision-recall curves of predicting whether an individual is infectious from the reported symptoms and the local proximity network contributed by PocketCare users. The goal of this comparison is to establish the contributions of the various pieces of observations in predicting whether an individual is infectious. In this plot, the *prediction* curves correspond to prediction of probability of infection two weeks ahead of time, the *smoothing* curves correspond to estimating whether an individual is infectious based on observations from other individuals, and the *unreported* curve corresponds to the prediction from only the reported symptoms without using the proximity network. The *participant* curves correspond to the prediction/smoothing of infectious status using the subject’s own local proximity network through Bluetooth scanning, and the *non-participant* curves correspond to the computation using the proximity networks of other people. The curves marked with 150, 100, and 30 volunteers correspond to the usage of observations from 5%, 3.3%, and 1% of the population to make predictions.
The [random-guess]{} curve corresponds to precision = 0.04, and we can infer whether a volunteer reported being sick in the health survey with an average precision = 0.70 given all observations before and after the day of prediction (*participant + smoothing*). Given only the fraction of volunteers who reported symptoms, we can achieve an average precision of 0.09 in predicting infection in a two-week time window (*unreported*), which is twice the [random-guess]{} baseline. Given the most complete information about an individual’s local spatial-proximity network — which is the situation where the individual contributes to Bluetooth-scanning the data but not to symptom surveys — we can achieve an average precision of 0.35 (*participant + prediction*), which is nine times the baseline. With incomplete proximity-network information observed by volunteers, we can achieve an average precision of 0.24 (*non-participant + prediction*), which is six times the baseline, and we can predict the 10% most likely infection cases (where recall $\le$ 0.1) with 60% precision. When we drop the fraction of volunteers who report spatial-proximity networks and symptoms of infection to 5%, 3.3%, and 1%, the average precision first drops slightly to 0.22, then collapses to 0.17 and 0.10. The observations about the proximity network improve the performance of individual-level prediction significantly, because infectious individuals make different contributions to epidemic progression. Indeed, infectious diseases from randomly infected individuals go first to the hubs of a social network then spread to the other nodes starting from those hubs.
Figure \[fig:inference-snapshot\] shows a snapshot of particle-smoothing inference from partial observations of volunteer symptoms and proximity-networks. In this figure, we use points of bigger and smaller sizes to represent the volunteers and the non-volunteers in the community, and embed the proximity network in a two-dimensional space according to the distance needed for a disease to travel from one person to another person. We use a heat-map in the background to represent the likelihood for a person in the area to be infectious — that is, there are bigger fractions of infectious people in “hotter” regions.
While we do not have the complete proximity network in the inference of epidemic diffusion, we can nevertheless construct structurally equivalent networks concerning epidemic diffusion dynamics. Similarly, in inferring the likelihood for a volunteer or non-volunteer to be infectious, all we need is that individual’s relative positions with respect to other volunteers in the proximity network. In other words, two people sharing the same position in the network will have the same likelihood of being infectious. As a result, the PocketCare framework can predict infection for both the volunteers and the non-volunteers, as long as they can specify to what extent they interact with the volunteers.
![\[fig:performance-aggregate\]Predicting percentage of population being infectious in 2-week window](symptoms-progression){width="1\linewidth"}
![\[fig:performance-aggregate\]Predicting percentage of population being infectious in 2-week window](precision-recall){width="100.00000%"}
![\[fig:performance-aggregate\]Predicting percentage of population being infectious in 2-week window](Snapshot){width="100.00000%"}
![\[fig:performance-aggregate\]Predicting percentage of population being infectious in 2-week window](infectious-population-estimation){width="100.00000%"}
Finally, Figure \[fig:performance-aggregate\] compares the performance of predicting the infectious population in a two-week time window every day, starting from recruiting 10% of the population as volunteers to report daily partial observations about the spatial-proximity network and epidemic diffusion. In other words, each prediction is based on observations up to two weeks before. The partial observation of the dynamic proximity network is made through Bluetooth-scanning of nearby mobile phones, and the partial observation of symptom diffusion through the symptoms of the volunteers themselves and of the other people spotted by the volunteers. The discrete event model particle filter has an r-squared value of 95%, in comparison with 60% for support vector regression and 20% for a scaling-based method. When we drop the volunteer population to 150, 100, or 30 (corresponding to 5%, 3.3%, and 1% of the total population), we can still achieve an r-squared value of 90%, 80%, or 70%, respectively. So, even 1% of the population can capture the structure of the spatial-proximity network and the dynamics of epidemic spreading reasonably well at the aggregate level. Conversely, a scaling-based method has a large deviation from the ground truth because the respondent population is small in comparison with the population to monitor. A model-free approach such as support vector regression still has a large deviation from the ground truth. This is because not all infectious individuals contribute to epidemic progression in the same way.
Discussion and Conclusions
==========================
In the current paper, we have described a multi-disciplinary approach of monitoring the spatial-proximity interactions among individuals in a community by mobilizing a small fraction of the community as volunteers to detect other people with mobile-phone Bluetooth-scanning and to report signs of epidemic diffusion with mobile-phone surveys. To stitch together the isolated observations of the volunteers and the latent interactions involving the non-volunteers with high-fidelity social interaction models, we derived inference and learning algorithms for a discrete event model, based on the premise that many complex social-interaction dynamics can be factored into a sequence of elementary events that individually involve only a few people but together describe the complex behavior of the system. We have applied our approach to the prediction of common cold and flu propagation in a community of 3000 people using the noisy observations of spatial-proximity interactions from 300 volunteers randomly selected from the community. The results show that we can predict common cold and flu infection two weeks ahead of time with an average precision from 0.24 to 0.35 depending on the amount of information. This is six to nine times the precision that a [random-guess]{} model can achieve. Moreover, we can predict infectious population in a two-weeks window with an r-squared value of 0.95, in comparison with an r-squared value of 0.2 for a [random-guess]{} model.
A fundamental challenge in applying ubiquitous-computing technologies to sense social dynamics is estimating diverse statistics from heterogeneous data sources, including for example 1) location observations of mobile phone users, 2) various dynamic networks defined by spatial proximity, phone calls, and SMSs, and 3) attribute assessments about individuals. Usually, estimation is achieved either by training a machine learning model to map hand-crafted features to the desired statistics or by running a simulator calibrated with data to generate trajectories for calculating statistics. Is there a principled method for estimating these statistics from a common set of data, hypotheses, or machine learning algorithms? In this paper, we have proposed an approach that infers the posterior distribution of complex system dynamics with high resolution and fidelity by connecting the dots of isolated observations using simulation models of social dynamics and machine learning, and applies the inferred individual-level dynamics to calculate the desired variables from cross-domain observations and dynamics and to identify actionable insights and solutions. The proposed approach is desirable because it provides an affordable microscope with which researchers can conduct accountable and repeatable experiments in living-lab settings.
Our multi-disciplinary approach is based on several working hypotheses. Theoretically, we hypothesized that complex system dynamics could be factorized into a sequence of known elementary events with each involving the interaction of only a few individuals. This hypothesis is strongly supported by our results and by the wide application of various modeling languages in capturing discrete event dynamics, including the stochastic Petri-net, stock and flow diagram, continuous time probabilistic model, and compound Poisson process. In an uncharted domain where the discrete event dynamics can be unknown, automatically learning a sparse discrete event representation of the unknown dynamics poses an interesting problem and parallels the trend of machine learning research, where novel sources of data demand more powerful algorithms to learn representations that approach the expressiveness of a simulation model. Practically, we hypothesized that a significant fraction of the community setting their mobile phones to be Bluetooth-discoverable and their Wi-Fi to be constantly scanning nearby access points. This hypothesis is also strongly supported by our results discussed.
[We thank the anonymous reviewers for improving the manuscript. Research reported here was supported by a University at Buffalo Innovative Micro-Programs Accelerating Collaboration in Themes (IMPACT) award.]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present optical images and spectroscopy for a dozen of BL Lac objects. Most of these objects were not previously studied and we give for the first time the properties of their host galaxies. The properties of the new host galaxies are generally consistent with those derived in previous optical studies. We found a case (1101-23) where the external isophotes of the galaxy are clearly boxy.
In addition we gathered spectroscopy for several BL Lac objects with unknown redshift and for companion galaxies. This allowed us to derive a tentative redshift for two new BL Lacs and to investigate the environment around PKS 0829+04. These data complement existing data available in the literature on host galaxies of BL Lacs and their (close) environments.
author:
- Renato Falomo
- 'Marie-Helene Ulrich'
date: 'Received / Accepted'
title: ' Optical imaging and spectroscopy of BL Lac objects. '
---
=cmbx10
Introduction
============
In the past decade BL Lac objects have been actively investigated in direct imaging and spectroscopy using ground based telescopes and HST.The imaging effort has been directed towards detecting the host galaxy, and when possible towards measuring its absolute luminosity and colors and determining its morphological properties. The aim of the spectroscopy has been to measure the redshift of the host or to measure the redshift of companions galaxies in order to assess a possible group or cluster membership.
Apart from studies on individual objects a number of papers have presented optical images for samples or lists of objects.
Twenty three objects have been imaged with the William Herschel Telescope in the R filter and 14 are resolved (Abraham 1991). However due to either unknown redshift or poorly detected nebulosity only for 6 sources absolute quantities are derived. Some cases of disc dominated host galaxies are proposed.
Sixteen objects in the southern sky have been studied by Falomo (1996) using sub-arcsec images obtained at the ESO 3.5m New Technology Telescope (NTT). Eleven sources were resolved and the hosts found to be luminous ellipticals (M$_R \sim$ –23.5). For a number of objects close companion galaxies are detected. Due to their small projected distance it is likely that they are associated with the BL Lac but spectroscopy is needed to assess this point.
A larger sample but with poorer average resolution was investigated using the 3.6m CFHT (Wurtz et al. 1996). Fifty objects have been observed and 36 well resolved. For another ten objects the host galaxy has been only marginally detected. No difference of host properties is found between objects discovered in radio surveys (i.e. 1Jy sample) and those derived from X-ray surveys (i.e. EMSS). With very few exceptions all the BL Lac objects investigated are classified as ellipticals based on the surface brightness profiles.
More recently a study of the host galaxies in a large sample of X-ray selected (high frequency peaked) BL Lacs have been presented by (Falomo & Kotilainen 1999). They used high resolution images in the R filter at the Nordic Optical Telescope (NOT) to image 52 targets from EMSS and Einstein Slew samples. All the 45 objects resolved are well represented by elliptical models. On average the hosts are found 1 magnitude more luminous than M$^*$ (M$^*_R \sim$ -22.5; Mobasher et al. 1993; assuming R-K = 2.7).
In addition to ground based studies several 0.1 arcsec resolution short exposure images have been obtained with WFPC2 camera on board of HST during a snapshot survey (Scarpa 2000; Urry 2000). Objects from various samples, and in the redshift interval 0.05 $< z <$ 1.3, were observed and 69 out of 110 observed are resolved. The highest redshift host galaxy detected is $z=0.664$ for 1823+568. For 80% of the resolved host galaxies an elliptical model is clearly preferred over a disc galaxy. The median absolute magnitude of these host galaxies ($M_R \sim -23.7$) is at least one magnitude brighter than $M^*$. The nuclei are always well centered over the body of the galaxy and have luminosity similar to that of its host galaxy. From the point of view of the optical morphology the hosts of BL Lacs appear indistinguishable from “normal” (non active) ellipticals.
The main aim of all these observations outlined above was to detect the host galaxies and to determine their structural and photometric properties. The knowledge of the kind of galaxies that host a BL Lac phenomenon in the nucleus is of importance not only for understanding/studying the nuclear activity vs galaxy connection (see e.g. Lawrence 1999) but also as a probe to test unified models of radio loud AGN. In particular if BL Lacs are FR I radio galaxies whose jet is aligned along the line of sight (e.g. Urry & Padovani 1995; Ulrich 1989) their host galaxies should exhibit exactly the same photometrical and morphological properties as the hosts of FR I. The properties of the BL Lacs hosts can also be compared with those of related beamed objects such as FSRQ and HPQ (see e.g. Kotilainen 1998a).
The aim of this work is to complement the existing data on BL Lac host galaxies and close environment with new imaging and spectroscopy for a dozen of (previously not well studied) objects. A general discussion and comparison of the properties of BL Lacs and radio galaxies will be presented elsewhere.
In this paper we therefore present results from optical images of BL Lac objects collected at the NTT with mostly sub-arcsec resolution. Most of the objects presented here were not previously investigated with adequate capabilities. These observations therefore complement the existing data on BL Lac host galaxies.
We also present spectroscopic observations for some of the objects performed with the aim of deriving the redshift of the host galaxies and of some nearby companion galaxies. When no spectroscopic redshift is available we give an estimate of the photometric redshift derived by assuming that the host has M$_R$ = –23.85 and R$_e$ = 9 kpc (the typical median values found in previous studies of BL Lacs hosts; e.g. Falomo & Kotilainen 1999).
In Sect. 2 we describe the observations and data analysis. Section 3 reports the results obtained for each individual objects. Section 4 gives a summary of the results and discussion.
Observations and data analysis
==============================
Optical observations were obtained using the 3.5m New Technology Telescope (NTT) at the European Southern Observatory (ESO), operated via remote control from the ESO headquarters in Garching (Germany). We acquired images using the Superb Seeing Imager (SUSI; Melnick 1992) which is installed at one of the Nasmyth foci of the NTT. Configuration used was R-band filter and a CCD (TK 1024) with 24$\mu$m pixel size corresponding to 0.13on the sky. Conditions were photometric and seeing was ranging from 0.55 to 1.2 arcsec (FWHM), and in most cases $<$ 1. Observations of standard stars (Landolt 1992) were used to set the photometric zero point.
We obtained images centered on the BL Lac object with exposure times ranging from 10 to 30 min (see Table 1). For many objects we also secured one short (2 minutes) exposure in order to be sure to get unsaturated images of the nucleus of the targets and to enable us to use bright stars in the field to study the PSF.
The images were processed in the standard way (bias subtracted, trimmed, flat fielded, and cleaned of cosmic rays) using the Image Reduction and Analysis Facility (IRAF) procedures. A journal of the observations is given in Table 1.
[lllllll]{}\
\
Name & z & Class$^{a)}$ & Date & T(int) & Seeing & A$_R$\
& & & & (sec) & (arcsec) &\
\
PKS 0138-097 & 0.733 & LBL & 1996 Jan 19 & 600 & 1.2 & 0.14\
0301-243 & 0.26 & HBL & 1996 Jan 19 &1200 & 0.8 & 0.10\
0338-21 & (0.45) & LBL & 1996 Jan 19 & 600 & 0.6 & 0.12\
REX 0353-36 & (0.40) & HBL & 1996 Jan 19 & 600 & 0.6 & 0.04\
PKS 0735+178 & $>$0.424 & LBL & 1996 Jan 19 &1800 & 0.7 & 0.26\
PKS 0736+017 & 0.191 & FSRQ & 1996 Jan 19 &1200 & 0.55 & 0.72\
PKS 0818-128 & ... & LBL & 1996 Jan 19 & 600 & 0.7 & 0.36\
H 1101-23 & 0.186 & HBL & 1996 Jan 19 &1200 & 0.9 & 0.24\
MS 1312.1-422 & 0.108 & HBL & 1996 Jan 19 & 600 & 1.4 & 0.36\
MS 1332.6-293 & 0.25 & HBL & 1996 Jan 19 & 600 & 1.5 & 0.22\
\
\
Name & z & Slit/PA$^{b)}$ & Date & T(int) & &\
\
0301-243 & 0.26 & 2off/90 &1996 Jan 20 &3600 & &\
PKS 0548-322 & 0.068 & 2/90 &1996 Jan 20 &3600 & &\
PKS 0754+10 & 0.28 & 2off/0 &1996 Jan 20 &3600 & &\
PKS 0818-128 & ... & 2/0 &1996 Jan 20 &1800 & &\
PKS 0829+046 & 0.18 & 2off/109 &1996 Jan 20 &3600 & &\
\
\
\
\
Spectroscopy of the the objects and/or of galaxies in the field were obtained for some targets in order to determine the redshift of BL Lacs and/or nearby companion galaxies. For this purpose the ESO multi mode instrument EMMI (Melnick 1992) was used with red arm and grism elements. In general the slit has been oriented in order to obtain in a single observation both the BL Lac object and one or more galaxies around the source.
All the images have been analyzed following the methods and procedure described in Falomo (1996). In particular surface photometry analysis was performed down to the surface brightness magnitude $\mu_R \sim$ 26 mag./arcsec$^2$ in order to derive the properties of the host galaxies. A fit of the radial brightness profile was performed assuming a simple two model components: a point source plus a elliptical galaxy described by a de Vaucouleurs law $$I(r)=I_0exp\{-7.67[(r/r_e)^{1/4}-1]\}$$ where $I(r)$ is the surface brightness and $r_e$ the effective radius.\
Also disc galaxies models were attempted but in no cases they gave a better fit than the elliptical model. This is consistent with what was found in previous studies on a larger number of sources ( Falomo & Kotilainen 1999; Urry 1999; Scarpa 2000).
To obtain absolute quantities we applied correction for Galactic extinction and redshift (K-correction). The former was determined using the Bell Lab Survey of neutral hydrogen N$_H$ converted to E$_{B-V}$ (Stark et al. 1992; Shull & Van Steenberg 1985), while the latter was computed from the model of Coleman (1980) for elliptical galaxies. Throughout this paper, are adopted.
Results
========
In Fig. 1 we report the observed radial brightness profile of the objects together with the best fit with the two components (point source plus elliptical galaxy) for the objects resolved. Parameters of the fit and absolute quantities for host galaxies and the nuclei are given in Table 2. In this Table columns 4–8 we give the results from this paper. The redshift in column 2 is drawn from literature except that for 0301-24 and two cases where a photometric redshift (given in parenthesis) is derived from the observed host properties. In the following discussion absolute quantities are given including corrections for galactic extinction and redshift (K-correction). Optical spectra of the BL Lacs or companion galaxies are reported in Fig. 2 together with the main identifications of observed spectral features.
[lllllllll]{}\
\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)\
Name & z$^{a)}$ & K$_R$ & R(nucleus) & R(host) & M$_R$(nuc)& M$_R$(host) & R$_e$(kpc) & resolved$^{b)}$\
\
PKS0138-097 & 0.733 & 1.46 & 17.64 & \* & -26.39 & \* & \*& N\
0301-24 & 0.26 & 0.28 & 15.96 & 17.46 & -25.36 & -24.14 & 23.1 & Y\
0338-21 & (0.45)& \* & 16.44 & 18.86 & \* & \* & \* & Y+\
REX0353-36 & (0.40)& \* & 18.05 & 18.92 & \* & \* & \* & Y+\
PKS0735+17 & $>$0.424 & \* & 15.22 & \* & \* & \* & \* & N\
0736+01 & 0.191 & 0.20 & 16.34 & 17.08 & -24.87 & -24.33 & 12.0 & Y\
0818-12 & ? & \* & 16.17 & \* & \* & \* & \* & N\
1101-23 & 0.186 & 0.19 & 16.80 & 16.41 & -23.87 & -24.45 & 22.3 & Y\
MS1312-42 & 0.108 & 0.10 & 18.64 & 16.25 & -20.89 & -23.38 & 5.3 & Y+\
MS1332-29 & 0.25 & 0.27 & 19.44 & 20.36 & -21.92 & -21.27 & 4.2 & Y+\
\
\
\
Continue.
Comments for individual objects
-------------------------------
This object was observed under 1.2seeing and it looks unresolved. Heidt 1996 have presented deep sub-arcsec images of this source that indicate the presence of close companion objects. These could be responsible for the intervening absorption system at z = 0.501 (Stickel 1993) seen in the spectrum of the BL Lac object. Our image was taken under relatively poor seeing but nevertheless some evidence of the southern feature at $\sim$1.5 from the center of the source is present in our image. This object has also been imaged by HST and found to be unresolved (Scarpa 2000) but the presence of a companion galaxy at 1.5South from the nucleus is clearly apparent.
Recent spectroscopy (Stocke & Rector 1997) detects for the first time the emission-line redshift of z=0.733 based upon weak Mg II and \[O II\] emission features. At this relatively high redshift our image result is consistent with this object being in a luminous (not detected) host galaxy at z = 0.733.
We took a 20 minute image under good seeing ( 0.8 ) of this BL Lac object that clearly shows an extended nebulosity (ellipticity $\varepsilon$ = 0.3; $\epsilon$ = 1- $b/a$ ) with a complex close environment (see Fig. 3).
The immediate region around the object is rich with faint galaxies and there is a marked enhancement of the galaxy density within $\sim$60from the BL Lac object.
The spectra of two galaxies (G1 and G2; see Fig. 3) at $\sim$ 6and 20 from 0301-243 indicate that they are at $z = 0.263$ suggesting a cluster of galaxies of Abell richness class 0 might be associated with the BL Lac source at this redshift (Pesce 1995). The radial profile is adequately well represented by a point source plus the elliptical model while the fit with an exponential disk is not acceptable. Fig. 3 (right panel) shows the field after subtraction of the BL Lac model (nucleus plus host galaxy) revealing the faint galaxy $\sim$ 3.5 South of the nucleus. After masking out the companion from the image we find that the surrounding nebulosity is very well centered on the nucleus within an accuracy of 0.2.
We took three optical spectra of the nebulosity with the slit off the nucleus by 2. They are still dominated by the signal from the non-thermal source but all three show one weak emission line at $\lambda$ = 6303 Å (see Figure 2). The most plausible identification for this emission is \[ O III\] 5007 Å that yields a redshift of 0.26. Fainter emissions like \[ O III\] 4959 Å or H$_\beta$ could be present at this $z$ but not detectable in our spectrum because the features are lost in the noise. Other possible identifications like MgII 2800 (at z = 2.25) are not acceptable because the host galaxy would be too luminous (M $<$ –30). The redshift of 0.26 is very similar to the redshift of the companion galaxies G1 and G2 (respectively of M$_R$ = -20.7 and M$_R$ = -22.3) and supports the idea that the host of the BL Lac is the dominant member of a cluster of galaxies. We note that few other examples have been reported in the literature of BL Lacs in clusters whose membership has been proved spectroscopically. H0414+00 (Falomo 1993a) is in a cluster of Abell class 0; PKS 0548-32 (Falomo 1995) is in a cluster of Abell class 1-2.
At this redshift (z = 0.26) the absolute magnitude of the host galaxy of 0301-24 is M$_R$ = -24.1
Our image obtained with 0.6seeing shows the object to be resolved. This is the first detection of the nebulosity for the source. However its magnitude is not consistent with the redshift of the object published twenty years ago ( z = 0.048 ; Wright 1977). The strongest absorption line identified in Wright is indeed a telluric band at 6870 Å. Subsequent spectroscopy of the source has failed to confirm this redshift and a pure featureless optical spectrum has been observed (Falomo 1994). In fact at this redshift of z = 0.048 the nebulosity would correspond to an unreasonably faint and small host galaxy (M$_R \sim $ -18.5). Assuming a typical host galaxy (see Sect. 1) we can well fit the radial brightness profile with a nucleus plus host galaxy obtaining a photometric redshift z $\sim$ 0.45.
The source was identified as a BL Lac object in the REX survey of AGN (Wolter 1997). Its optical spectrum is featureless (Wolter 1998). We obtained an image under very good seeing (0.6 arcsec) and are able to detect the surrounding nebulosity and measure its luminosity and R$_e$. This is the first detection of its host. There is no spectroscopic redshift but we can estimate a photometric redshift from the image decomposition assuming the host galaxy has average properties for BL Lacs hosts. The value of the photometric redshift so obtained is z $\sim$ 0.4.
This is a very well know BL Lac object at z = 0.068 (Fosbury & Disney, 1976) with a very large host galaxy in a rich environment (Falomo et al 1995). We took one relatively short exposure but good signal-to-noise spectrum centered in the nucleus to search for possible emission lines as have been reported in a number of nearby BL Lacs (e.g. BL Lac itself, Vermeulen 1995 ).The spectrum, shown in Fig. 2, exhibits a substantial contribution from the stellar population of the host galaxy. The MgI 5175 Å and Na blend 5892 Åare well detected with equivalent widths of 12 Å and 6.5 Å, respectively. We could not find any emission down to a limit of equivalent with of 2 Å. This limit corresponds to H$_\alpha$ line luminosity L(H$_\alpha$) $\sim$ 5 $\times$ 10$^{40}$ erg s$^{-1}$ which is about a factor 10 lower than the line detected in BL Lac (Vermeulen 1995).
This BL Lac object is bright and strongly variable. It has been extensively studied in the radio range and several moving components have been detected in VLBI. The optical spectrum shows the absorption line due to an intervening system at 3980 Å, which if identified with Mg II gives z $>$ 0.424 (Carswell 1974) Our images were obtained under seeing of 0.8 but the source remains unresolved. Previous images were presented by Hutchings (1988) who also found this source unresolved. There is no sign in our image (see Fig. 4) that the galaxy 7NW is distorted by interaction with 0735+178 as suggested by previous lower resolution images (Hutchings 1988). The object was also imaged by Stickel (1993) who are not able to detect the surrounding nebulosity. They obtained a spectrum of the galaxy 7NW and found z = 0.645. This BL Lac object is unresolved also in a short exposure image obtained with HST (Scarpa 2000). In addition to the two well resolved companion galaxies we detect a faint emission at $\sim$ 3.5East from the BL Lac (see Fig. 4). Given its projected distance from the BL Lac (25 kpc at z = 0.424) it could be related to the intervening absorption at z = 0.424 but we cannot exclude that it is just a faint background source.
From our image we can set a lower limit to the redshift (again assuming the typical properties for the host) of z $>$ 0.5, consistent with the limit derived from intervening absorption.
The excellent (seeing 0.55 arcsec) image (see Fig. 1) shows the flat spectrum radio quasar PKS 0736+01 (z = 0.191) as well as two close resolved faint companions that are embedded in the nebulosity of the object. The radial luminosity profile (see Fig. 1 ) is very well represented by an elliptical galaxy with a bright point source in the nucleus. It is found that the galaxy has M$_R$ = -24.3 and effective radius of $\sim$ 12 kpc.
This host galaxy was previously detected in the optical with lower resolution by Wright (1998). They derive M$_R$ = -22.0, which is substantially fainter than our value. We note that this discrepancy could be due to a problem in the Wright image calibration as their surface brightness goes unbelievably faint. At 5from the nucleus their surface brightness is about $\mu_R$ =28 while our value at the same radius is $\mu_R$ = 24.
The object has been also resolved in the NIR by Taylor (1996) who found M$_K$ = -26.3, and by Kotilainen (1998a) who found M$_H$ = -26.2. The R-H color turns out to be $\sim$ 2.0, consistent with the range of values reported by Kotilainen 1998b for a number of BL Lacs.
We took two spectra of this BL Lac object for which no firm value of the redshift is available but whose host galaxy had already been detected (Abraham 1991; Falomo 1996). The tentative redshift ($z=0.66$) proposed by Persic & Salucci (1986) based on inspection of the photographic spectrum reported by Wilkes (1983) is unlikely as the host galaxy would be extremely luminous (($M_R \sim -26$ mag). Our spectra were obtained positioning the slit 2from the nucleus in order to reduce the contamination of light from the bright nucleus. Therefore the spectrum (see Fig. 2) is noisy and it is still dominated by the nuclear non thermal emission. We are not able to unambiguously identify spectral lines but some hint of the CaII break signature from the host galaxy is possibly apparent at $\lambda$ = 5045 Å which corresponds to z = 0.28. At this redshift the detected surrounding nebulosity would be M$_R$ $\sim$ -23. We note that this is consistent with the value of the redshift of the companion galaxy (see Fig. 5) 13.6north-east of the BL Lac object ($z$=0.27; Pesce 1995) and could be another case of a companion galaxy physically associated with a BL Lac object. A definitive redshift determination is however still needed for this BL Lac object.
There is no redshift for this object and its optical spectrum is featureless (Falomo 1994). Our optical images, obtained with seeing of 0.7, are not able to detect the host galaxy. The radial brightness profile is well matched by that of a scaled PSF (see Fig. 1). We can set a lower limit to the redshift assuming its nucleus is hosted by a standard luminous ( M$_R$ $\sim$ -23.8) elliptical. The limit of redshift we found for such a galaxy to be undetected in our image is z $>$ 0.5.
In order to search for emission or intervening absorption line we gathered spectra in a wide wavelength range. Our spectrum ( see Fig. 2) is still dominated by the non-thermal featureless emission. The only feature (in addition to telluric bands) we can detect is an absorption at 6284 Å (e.w. 0.7 Å). The most likely identification of this feature is with an interstellar diffuse absorption band at the same wavelength. This is consistent with the low galactic latitude (b$_{II} \sim$ 13$^{o}$) of the source. Alternatively the absorption line could be identified with MgII 2800 Å and this would yield approximately z $>$ 1.2 and, consequentially, the object would be extremely luminous (M$_R < --29$).
Previous images obtained at sub-arcsec resolution showed that the host galaxy (z = 0.18) has M$_R$ $\sim$ -23 (Falomo 1996). There is also an excess density of galaxies around this object (Pesce 1994). But our spectroscopy shows that only some of them may be physically associated with the BL Lac object. Pesce 1994 obtained the redshifts of galaxies G1 and G2 (see Fig. 6) at respectively z =0.24 and z = 0.204. We took additional spectra of two other galaxies (G3 and G4, see Fig.s 2 and 6). We found that G4 is at significantly higher redshift (z = 0.29) while G3 is at z = 0.175, consistent with being associated with PKS 0829+04 at projected distance of $\sim$ 120 kpc. In fact G3 is the only galaxy which is at the same redshift as the BL Lac. On one hand this is another case of similar redshift of a companion galaxy and its BL Lac. On the other hand the environment of 0829+04 must be less rich than what can be estimated from galaxy counts.
This is a BL Lac discovered from X-ray survey and is surrounded by a conspicuous rather elongated nebulosity (see Fig. 7 ) at the proposed $z = 0.186$ (Remillard 1989) confirmed by Falomo (1994). The radial brightness profile extends to 15 arcsec along the major axis. We found that the luminosity profile is well fitted by an elliptical galaxy model plus a point source. The luminosity of the host galaxy is very high. The absolute magnitude, M$_R$ = –24.45, sets this galaxy among the brightest hosts of BL Lac objects (Falomo & Kotilainen 1999).
For this object (see Fig. 8) we performed detailed surface photometry analysis using the AIAP package (Fasano 1990) in order to study the structural properties of the galaxy. From this analysis we derived photometric and structural parameters (surface brightness, ellipticity, position angle and Fourier coefficient C$_4$ describing the deviation of isophotes from the ellipse) as a function of the equivalent radius $r = a\times(1-\epsilon)^{1/2}$ where $a$ is the semi-major axis and $\epsilon$ is the ellipticity of the ellipse fitting a given isophote. We found the ellipticity profile is increasing from the center outwards up to $\epsilon$ = 0.45. The profile of the C$_4$ (see Fig. 9) shows [*disky*]{} (positive C$_4$) trend in the inner region while the external isophotes are substantially [*boxy*]{} (negative C$_4$), possibly due to merging processes ( e.g. Bender 1988). This is the only clear evidence of significantly [*boxy*]{} isophotes ever found in a BL Lac host.
Another example of a very luminous host galaxy (M$_{\rm R}$= –24.8, or -24.45 if de Vaucouleurs law is fitted) was reported by Heidt (1999) for 1ES 1741+196. Also in this case the host galaxy isophotes have high ellipticity ( $\sim$ 0.35). There is no information, however, about the detailed shape of the isophotes and the amount of possible boxiness.
This source, drawn from the EMSS of BL Lacs (Maccacaro . 1994) was observed during bad seeing conditions (seeing of 1.4) but since it is at relatively low redshift ( z = 0.108; Morris 1991) it is rather well resolved. The host galaxy is indeed dominant with respect to the nuclear source ( ratio nucleus/host = 0.1). Our fit of the brightness profile yields M$_R$ = -23.4. No other detection of this host galaxy can be found in the literature.
Note that in the calculation of $\alpha_{OX}$ it is usually the luminosity of the whole object (nucleus + host) which is used in the calculation. Such a procedure if applied to 1312-42 would overestimate the optical flux by a factor $\sim$ 10.
The target belongs to the EMSS sample of BL Lacs although its classification is uncertain. Optical spectra showed either emission lines at z=0.256 or strong CaII break (Stocke 1991 ) due to a substantial contribution from stellar emission.
Our image shows this object is only marginally resolved. This is in part due to the bad seeing ($\sim$ 1.5) and also because the host galaxy is substantially under-luminous (M$_R$ = -21.3) with respect to the average of the host galaxies of BL Lacs (M$_R$ = -23.8 Falomo & Kotilainen 1999).
We note that the same object (1ES1322-297) is listed in the Einstein Slew sample of BL Lacs (Perlman 1996) and has a redshift z = 0.512 quite different from the previous finding. Since in neither cases there are spectra published we are not able to make our own judgment of the validity of the redshift values. However, the latter value seems confirmed by another optical spectrum (albeit noisy) reported by Rector (1998). At z = 0.512 the host galaxy and point source would be much more luminous (M$_{nuc}$ = -23.5 and M$_{host}$ = –23.3) and well within the averages of these types of objects.
Summary and conclusions
=======================
We have presented optical images of a number of BL Lacs that were not previously well studied. For several of these objects the first detection of the host galaxy is presented here. The properties of the hosts are consistent with them being luminous ellipticals as found in previous similar studies. For two of the resolved objects that have not a spectroscopic redshift we derive a photometric redshift based on the observed properties of the surrounding nebulosity.
[*A bright and boxy elliptical*]{}
We find that the external isophotes of the luminous host galaxy of 1101-23 are significantly [*boxy*]{} while the inner [*disky*]{} region suggests the presence of a small disc component. This is the first clear example of a [*boxy*]{} galaxy hosting a BL Lac object. Boxy isophotes are observed in a fraction of luminous ellipticals (Bender 1988) and could be ascribed to merging events from equal mass galaxies (e.g. Naab 1999). It would be interesting to know what fraction of hosts of BL Lacs exhibit boxy isophotes as compared with non active ellipticals. Very little data are, however, available on isophote shapes of BL Lacs host galaxies because the presence of the bright nucleus and the quality of data often hinder a reliable estimation of this parameter especially at high redshift. For relatively low redshift objects with high resolution images it should be possible to investigate the isophote shape in a systematic way.
[*The immediate environment of BL Lacs*]{}
Our spectroscopy has allowed us to derive a redshift for 0301-24 (z = 0.26) and possibly for 0754+10 (z=0.28). Both objects have companion galaxies at redshifts very similar to that of the BL Lacs. The companions and the BL Lacs are thus very likely to be gravitationally bound. A third case is PKS 0829+04 for which we took the spectra of two galaxies in the immediate environment and found that one is at the same redshift as the BL Lac object. These spectroscopic results improve the scanty data on redshifts of companion galaxies of BL Lacs. Together with previous findings ( Falomo 1993a,b; Pesce et al 1994,1995; Heidt 1999) our new results yield convincing evidence that galaxies around BL Lacs are (often) gravitationally bound with the BL Lacs. On the other hand only in very few cases do these interactions lead to significantly (observable) disturbed morphology (see e.g. Falomo 1995, Heidt 1999).
This work was partly supported by the Italian Ministry for University and Research (MURST) under grant Cofin98-02-32. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We thank A. Wolter for providing coordinates and finding charts of REX 0353-36 before publication. RF thanks the ESO visitor program for hospitality during several visits at the ESO headquarter.
References
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A specific class of partially entangled states known as Knill-Laflamme-Milburn states (or KLM states) has been proved to be useful in relation to quantum information processing [@knill01]. Although the usage of such states is widely investigated, considerably less effort has been invested into experimentally accessible preparation schemes. This paper discusses the possibility to employ a tunable controlled phase gate to generate an arbitrary Knill-Laflamme-Milburn state. In the first part, the idea of using the controlled phase gate is explained on the case of two-qubit KLM states. Optimization of the proposed scheme is then discussed for the framework of linear optics. Subsequent generalization of the scheme to arbitrary $n$-qubit KLM state is derived in the second part of this paper.'
address: |
RCPTM, Joint Laboratory of Optics of Palacky University and Institute of Physics of Academy of Sciences of the Czech Republic, Faculty of Science, Palacky University\
17. listopadu 12, 77146 Olomouc, Czech Republic
author:
- Karel Lemr
title: 'Preparation of Knill-Laflamme-Milburn states using tunable controlled phase gate'
---
Introduction
============
Important developments have been demonstrated in quantum information processing (QIP) in the past few decades [@alber01; @bowmeester00; @nielsen02]. Several outcomes of this scientific field such as quantum cryptography [@benett84; @benett92; @eckert91; @gisin02] or random number generation [@inoue83; @peres92; @jennewein00; @stefanov00; @katso08] have already found their industrial applications. In other cases a lot of effort has yet to be invested into the research. Mainly the lack of some experimental tools (e.g. strong optical non-linearity [@turchette95]) prevents from developing efficient quantum devices. An important discovery has been achieved by Knill, Laflamme and Milburn [@knill01], when they have derived that a specific class of partially entangled states (so called Knill-Laflamme-Milburn states, or simply KLM states) can be used to significantly improve the efficiency of quantum computing. They have proposed a nearly deterministic teleportation based protocol for quantum computation using the KLM states as ancillas. In this protocol the overall success probability of quantum computation goes asymptotically to unity with growing number of photons in the ancillary KLM state. Their work has been followed by several other related proposals and experiments [@franson02; @okamoto07; @grudka08]. Franson *et al.[@franson02] have generalized the original KLM scheme so that the success probability of quantum computing scales better with growing number of photons, but at the expense of lower fidelity of the output states. Several schemes for preparation of KLM states have also already been proposed. The general preparation idea has been mentioned in the original KLM paper [@knill01] though there was no specific recipe. The first explicit scheme for preparation of the KLM states was proposed by Franson *et al.and it uses non-deterministic controlled sign gates and single photon interference to generate arbitrary photon-number KLM states [@franson04]. Another scheme limited only to 2-photon KLM states, but not requiring any post-selection, was also proposed [@lemr08] and subsequently experimentally implemented [@lemr10].**
This paper investigates yet another approach for experimentally accessible preparation of KLM states using the controlled phase gate (c-phase gate). The advantage in using this gate is the fact that the c-phase gate is considered an important part of the QIP toolbox [@sleator95; @barenco95]. The Franson *et al. scheme also employs the controlled phase gates (or in their case controlled sign gates) but with constant phase shift set to $\pi$. In this paper a fully tunable controlled phase gate is considered and a scheme for it’s usage as a resource for KLM state generation is developed. By this strategy the overall success probability of the KLM state preparation can be increased considerably for some KLM states as it is shown in this paper. The presented scheme is fully general and allows to prepare KLM states of arbitrary number of qubits. Also no previous entanglement between the input qubits is required as the entangling capability of the gate itself is sufficient. The fully tunable controlled phase gate capable of imposing any phase shift in the range from 0 to $\pi$ has already been both proposed theoretically [@kieling10] and implemented experimentally [@lemr11] on the platform of linear optics and thus can be considered experimentally accessible.*
Basic 2-qubit scheme
====================
Using the qubit representation, one can express the $n$-qubit KLM state in the form of $$\label{eq:multiKLM}
|\psi\rangle_{{\mbox{\tiny KLM}}} = \sum_{j=0}^n \alpha_j |{1}\rangle^j |{0}\rangle^{n-j}.$$ The original definition by Knill, Laflamme and Milburn sets $\alpha_j = \frac{1}{\sqrt{n+1}}$ for $j = 0,...,n$, but the subsequent research carried out by Franson *et al. [@franson02] indicates, that additional benefits can be found in using general amplitudes $\alpha_j$. Their research revealed that one can increase the efficiency of teleporation based quantum computing for instance by choosing triangular shaped amplitudes $\alpha_j$ (that is $\alpha_0 = \alpha_n = 0$ and alpha linearly growing towards maximum at $\alpha_{n/2}$ and then decreasing). This improvement is obtained at the expense of lower fidelity of the output state. (For more details please consult [@franson02]).*
In the first part of this paper let us consider the preparation of two-qubit KLM states (see figure \[fig:scheme\]). The generalization to an arbitrary number of qubits would be presented later. Using the general definition for the KLM states (\[eq:multiKLM\]) one can find that the two-qubit KLM states are in the form of $$\label{eq:klm_qubit}
|\psi\rangle_{{\mbox{\tiny 2-QUBIT KLM}}} = \alpha_0 |{0}{0}\rangle + \alpha_1 |{1}{0}\rangle + \alpha_2 |{1}{1}\rangle,$$ where $\alpha_j$ (for $j=0,1,2$) are arbitrary complex amplitudes following the normalization condition $\sum_{j=0}^2|\alpha_j|^2 = 1$. Having the target state well defined let us now inspect the properties of the c-phase gate.
The c-phase gate is a two-qubit quantum gate whose action in the gate’s computational basis reads $$\begin{aligned}
\label{eq:cphase}
|00\rangle &\rightarrow & |00\rangle \nonumber\\
|01\rangle &\rightarrow & |01\rangle \nonumber\\
|10\rangle &\rightarrow & |10\rangle \nonumber\\
|11\rangle &\rightarrow & \mbox{e}^{i\varphi} |11\rangle \end{aligned}$$ with numbers in brackets denoting first and second qubit state. General c-phase gate can be set to impose an arbitrary phase shift $\varphi$ to the two-qubit state $|{1}{1}\rangle$.
![Scheme of the proposed procedure for generation of two-qubit KLM states. The signal and control input qubit undergo a c-phase gate with tunable phase shift $\varphi$ yielding the two-qubit KLM state.[]{data-label="fig:scheme"}](figScheme)
Any signal and control qubit can be expressed in terms of the gate’s computational basis $$\label{eq:cphase_input}
|\psi_{c,s}\rangle = \cos{\theta_{c,s}} |{0}_{c,s}\rangle + \mbox{e}^{i\phi_{c,s}} \sin{\theta_{c,s}} |{1}_{c,s}\rangle,$$ where indexes $c$ and $s$ denote the control and signal qubit. Please note that this state can always be prepared with high fidelity using only single qubit transformations (e.g. wave-plates in the case of photon polarization encoding). The separable input state $|\psi_c\psi_s\rangle$ is transformed by the gate yielding $$\begin{aligned}
\label{eq:cphase_output}
|\psi\rangle_{{\mbox{\tiny OUT}}} & = & \cos{\theta_c} \cos{\theta_s} |{0}{0}\rangle + \mbox{e}^{i\phi_{s}} \cos{\theta_c}\sin{\theta_s} |{0}{1}\rangle + \nonumber\\
& & + \mbox{e}^{i\phi_{c}} \sin{\theta_c}\cos{\theta_s} |{1}{0}\rangle + \nonumber\\
& & + \mbox{e}^{i(\phi_{c}+\phi_{s}+\varphi)} \sin{\theta_c}\sin{\theta_s} |{1}{1}\rangle.\end{aligned}$$ Using the expression for signal qubit (\[eq:cphase\_input\]), the output state can be rewritten to the following form $$\label{eq:cphase_output_alter}
|\psi\rangle_{{\mbox{\tiny OUT}}} = \cos{\theta_c} |{0}\psi_s\rangle + \mbox{e}^{i\phi_{c}} \sin{\theta_c} \left( \tau |{1}\psi_s\rangle + \epsilon |{1}\psi^\bot_s\rangle\right),$$ where $|\psi^\bot_s\rangle$ is the orthogonal state to $|\psi_s\rangle$ so that $\langle\psi^\bot_s|\psi_s\rangle = 0$ and the parameters $\tau$ and $\epsilon$ are defined as $$\begin{aligned}
\label{eq:tau_epsilon}
\tau & = & \langle\psi_s|\left(\cos{\theta_s}|{0}\rangle + \mbox{e}^{i(\phi_{s}+\varphi)} \sin{\theta_s} |{1}\rangle \right) = \nonumber\\
& = & \cos^2{\theta_s} + \mbox{e}^{i\varphi} \sin^2{\theta_s}, \nonumber\\
\epsilon & = & \langle\psi^\bot_s|\left(\cos{\theta_s}|{0}\rangle + \mbox{e}^{i(\phi_{s}+\varphi)} \sin{\theta_s} |{1}\rangle \right) = \nonumber\\
& = & \mbox{e}^{i\phi_s} \sin{\theta_s} \cos{\theta_s} \left(1-\mbox{e}^{i\varphi}\right).\end{aligned}$$ After performing the single qubit transformation $$\label{eq:signal_transform}
|\psi_s\rangle \rightarrow |{0}\rangle, |\psi^\bot_s\rangle \rightarrow |{1}\rangle$$ in the signal mode, one can clearly recognize the two-qubit KLM state in the output state of the gate $$\label{eq:cphase_output_klm}
|\psi\rangle_{{\mbox{\tiny OUT}}} = \cos{\theta_c} |{0}{0}\rangle + \mbox{e}^{i\phi_{c}} \tau \sin{\theta_c} |{1}{0}\rangle + \mbox{e}^{i\phi_{c}} \epsilon \sin{\theta_c} |{1}{1}\rangle,$$ The remaining task is to map the complex amplitudes in (\[eq:cphase\_output\_klm\]) to the original amplitudes $\alpha_j$ and to show that any two-qubit KLM state is achievable.
First let us consider the relative amplitude ratio and phase between $\alpha_0$ and $(\alpha_1 + \alpha_2)$. Any amplitude ratio can easily be set just by the choice of the $\theta_c$ parameter of the input control state $$\label{eq:a_bg}
\frac{|\alpha_1|^2 + |\alpha_2|^2}{|\alpha_0|^2} = \tan^2\theta_c.$$ As for the phase, the freedom in setting any value of $\phi_c$ assures that any phase shift between $\alpha_0$ on one side and $\alpha_1$ and $\alpha_2$ on other side is achievable.
The relation between $\alpha_1$ and $\alpha_2$ is also simple. For instance setting the phase shift $\varphi = \pi$ simplifies the amplitude ratio to $$\label{eq:g_b}
\frac{|\alpha_2|}{|\alpha_1|} = \frac{|\epsilon|}{|\tau|} = \tan 2\theta_s$$ and an arbitrary phase shift between $\alpha_1$ and $\alpha_2$ can be set by the choice of $\phi_s$. Please note that setting $\varphi = \pi$ allows to cover the whole class of KLM states. This fact will be used for the discussion in section 5. The equations (\[eq:a\_bg\] and \[eq:g\_b\]) manifest that any amplitude ratio between $\alpha_0$, $\alpha_1$ and $\alpha_2$ is achievable since $\tan$ goes from 0 to $\infty$.
Success probability optimization
================================
One may conclude that the tunability of the gate in the phase shift $\varphi$ is a redundant feature. However this parameter can be used for optimization of the procedure. One of the most promising platforms for QIP is linear optics [@munro05; @obrien07; @walmsley08; @aspelmeyer08; @politi08]. For this reason let us now focus on the optimization of the proposed procedure for linear optics. Recently Kieling *et al. [@kieling10] have identified the maximum success probability of a c-phase in the framework of linear optics as $$\begin{aligned}
P_{C}(\varphi) =
\left(1+2\left|\sin\frac{\varphi}{2}\right|+2^{3/2}\sin\frac{\pi-\varphi}{4}{\left|\sin\frac{\varphi}{2}\right|^{1/2}}\right)^{-2},
\label{eq:cphase_psucc}\end{aligned}$$ which does not depend on the input state. The optimization of the proposed scheme seeks to maximize the success probability of the c-phase gate used for KLM state preparation. With respect to that a numerical simulation (or optimization) has been carried out to reveal the maximum achievable success probability for several KLM states. The target KLM state of presented numerical simulation is the mono-parametric class of two-qubit KLM state motivated by Franson’s *et al. definition [@franson02] (triangular-shaped amplitude function) $$\label{eq:klm_franson}
|\psi\rangle_{{\mbox{\tiny KLM}}} = \alpha_0 |{0}{0}\rangle + \alpha_1 |{1}{0}\rangle + \alpha_0 |{1}{1}\rangle.$$**
![Maximum achievable $|\alpha_0/\alpha_1|$ ratio for a given phase shift of the c-phase gate. The success probability of the optimal linear optical c-phase gate as a function of its phase shift is also depicted for reference.[]{data-label="fig:maxGBratio"}](figMaxGBratio)
The amplitudes $\alpha_0$ and $\alpha_1$ are now considered to be real numbers as it has been shown above that the phase can always be set by the choice of $\phi_c$ and $\phi_s$. These phases are independent of the gate phase shift $\varphi$ and therefore have no effect on the success probability. The presented optimization will focus on the amplitude ratio $|\alpha_0/\alpha_1|$ and investigate the corresponding success probability. First numerical simulation has been performed to determine the maximum achievable $|\alpha_0/\alpha_1|$ ratio for a given phase shift. Results of this simulation are presented in figure \[fig:maxGBratio\]. One can observe that maximum achievable $|\alpha_0/\alpha_1|$ ratio grows monotonously with the phase shift $\varphi$. For reference the success probability (\[eq:cphase\_psucc\]) as a function of the phase shift $\varphi$ is also depicted along with the reference ratio $|\alpha_0/\alpha_1| = 1$ corresponding to the original KLM state definition.
![Maximum achievable success probability and corresponding optimal $\theta_s$ and $\varphi$ parameters are plotted as a function of $|\alpha_0/\alpha_1|$ ratio. Please note that the optimal setting of $\varphi$ for $|\alpha_0/\alpha_1| > 0.54$ is $\varphi = \pi$ (this explains the step of $\varphi$ at $|\alpha_0/\alpha_1| = 0.54$).[]{data-label="fig:GBratioPsucc"}](figGBratioPsucc)
The second numerical simulation has been carried out to determine the maximum achievable success probability for a given $|\alpha_0/\alpha_1|$ ratio (see figure \[fig:GBratioPsucc\]). Also the setting of the phase shift $\varphi$ and the parameter of the signal qubit $\theta_s$ are depicted to illustrate the optimal strategy. This strategy is different in two regions separated by the amplitude ratio $|\alpha_0/\alpha_1| \approx 0.54$. In the first region ($|\alpha_0/\alpha_1| \leq 0.54$) setting $\theta_s = \frac{\pi}{4}$ and the phase shift $\varphi$ accordingly is the optimal way. One tries to minimize the phase shift used for the KLM state preparation, because the success probability is a decreasing function of the phase shift. To keep the phase shift minimal, one has to set $\theta_s = \frac{\pi}{4}$, because for a given phase shift $\varphi$ the setting $\theta_s = \frac{\pi}{4}$ maximizes the $|\alpha_0/\alpha_1|$ ratio.
On the other hand, in the second region ($|\alpha_0/\alpha_1| > 0.54$) the previously mentioned strategy will not yield optimal results. This is because of the success probability not being monotonous in this region. Setting $\varphi = \pi$ and adjusting the $\theta_s$ instead is the optimal way here.
Both this and the original Franson *et al. scheme requires $n-1$ times using the c-phase gate in order to generate $n$-qubit KLM state. This leads to the overall success probability for $n$-qubit KLM state $$P_\mathrm{KLM} = \prod_{i=1}^{n-1} P_C(\varphi_i),$$ where $n$ denotes the number of qubits and $P_C(\varphi_i)$ is the success probability of the controlled phase gate set for the phase shift $\varphi_i$ used in the $i^{\mathrm{th}}$ repetition of the c-phase gate. The Franson *et al. proposal considers only $\varphi_i = \pi$ for all values of $i$. So for example in the 2-qubit case, the success probability of Franson scheme would yield a constant value of $0.11$ (based on the optimal linear optical controlled phase gate). To emphasize the improvement achieved by the tunability of the phase gate, let us consider an example of $|\alpha_0/\alpha_1| = 0.25$. For this particular choice the success probability of the scheme proposed in this paper would be $0.18$, which is a 60% improvement. This improvement in success probability varies with the particular choice of the target KLM state (see figure \[fig:GBratioPsucc\]).**
Generalization to $n$-qubit KLM states
======================================
![Generalization of the two-qubit scheme to an arbitrary number of qubits. Input $n$-qubit KLM state is combined with a new qubit initially in $|{0}\rangle$ state. $H$ denotes the Hadamard gate and $C$ denotes the c-phase gate (this time set to impose the phase shift $\varphi = \frac{\pi}{2}$).[]{data-label="fig:scheme_multi"}](figSchemeMulti.eps)
The proposed two-qubit scheme can be generalized to prepare KLM states of an arbitrary number of qubits. For simplicity let us now presume all complex amplitudes of the $n$-qubit KLM state being equal (original KLM state definition). To illustrate the generalization procedure the step from two-qubit to three-qubit KLM state is explained and also illustrated in figure \[fig:scheme\_multi\]). Going from two to three qubit KLM state means to perform the following transformation $$\begin{aligned}
\label{eq:2to3}
|{0}_1{0}_2\rangle & \rightarrow & |{0}_1{0}_2{0}_3\rangle \nonumber \\
|{1}_1{0}_2\rangle & \rightarrow & |{1}_1{0}_2{0}_3\rangle \nonumber \\
|{1}_1{1}_2\rangle & \rightarrow & |{1}_1{1}_2{0}_3\rangle + |{1}_1{1}_2{1}_3\rangle,\end{aligned}$$ where indexes 1 and 2 denote the first and second original qubits of the two-qubit KLM state and the index 3 denotes the newly added qubit. This transformation can be implemented by addition of a new qubit initially in the state $|{0}\rangle$. This new qubit is firstly subjected to the Hadamard gate $$\label{eq:hadamard}
|{0}\rangle \rightarrow |{0}\rangle + |{1}\rangle.$$ After that it is propagated through the c-phase gate set to phase $\varphi = \frac{\pi}{2}$ along with the last of the original KLM qubits. At the end an inverse Hadamard gate is placed in the new qubit mode. One can see that in the case of the last original qubit being $|{0}\rangle$, the phase shift imposed to the new qubit is zero and the new qubit leaves the scheme in the state $|{0}\rangle$. On the other hand if the last original qubit is in the state $|{1}\rangle$ the new qubit gets a $\frac{\pi}{2}$ phase shift and yields $|{0}\rangle + |{1}\rangle$ after leaving the inverse Hadamard gate.
The generalization to an arbitrary number of qubits is straightforward. To generate an $(n+1)$-qubit KLM state from an $n$-qubit KLM state ($n \leq 2$) a new qubit is added at the end of the original qubits and subjected to the procedure described in previous paragraph. The general scheme is depicted in figure \[fig:scheme\_multi\].
Optimization of the generalized scheme
======================================
The previous section is just a proof of the scalability of the scheme, but does not give optimal setting with respect to the success probability. A similar optimization as for the two-qubit KLM states can be considered to maximize the yield of the scheme. Hadamard gates can be replaced by more general single qubit transformations and together with the tunability of the phase shift imposed by every controlled phase gate the overall success probability can be optimized with respect to the selected target KLM state.
One can use the iterative procedure starting from $n$-qubit KLM state with amplitudes $\alpha_j^\mathrm{[n]}$, $j = 0...n$ and going to $(n+1)$-qubit KLM state with amplitudes $\alpha_j^\mathrm{[n+1]}$, $j = 0...n+1$. Here the upper index denotes the $n$-qubit starting KLM state and $(n+1)$-qubit target KLM state. Note that in this case the c-phase gate is applied to the last of the original qubits ($n^\mathrm{th}$ qubit) and a newly added ($n+1)^\mathrm{th}$ qubit. This new qubit can be expressed in the form of $|\psi_s\rangle$ as defined by (\[eq:cphase\_input\]) and the last original qubit takes effectively the form similar to $|\psi_c\rangle$ with
$$\begin{aligned}
\cos\theta_c & = & \sqrt{\sum_{j=0}^{n-1}|\alpha_j^\mathrm{[n]}|^2} \quad \mbox{(corresponding to the $|0\rangle$ state)}\nonumber\\
\sin\theta_c & = & |\alpha_{n}^\mathrm{[n]}| \quad \mbox{(corresponding to the $|1\rangle$ state)} \nonumber\\
\phi_c & = & \arg\left(\alpha_{n}^\mathrm{[n]}\right)
\label{eq:nmapping}\end{aligned}$$
also following the original definition (\[eq:cphase\_input\]). With this mapping one can proceed in the similar way as explicitly described in the second section. The resulting amplitudes $\alpha_j^\mathrm{[n+1]}$ are then in the form $$\begin{aligned}
\alpha_j^\mathrm{[n+1]} & = & \alpha_j^\mathrm{[n]}, \mathrm{for\,} j = 0...n-1 \nonumber\\
\alpha_{n}^\mathrm{[n+1]} & = & |\alpha_{n}^\mathrm{[n]}|\mathrm{e}^{i\phi_c} \tau \nonumber\\
\alpha_{n+1}^\mathrm{[n+1]} & = & |\alpha_{n}^\mathrm{[n]}|\mathrm{e}^{i\phi_c} \epsilon,\end{aligned}$$ where $\phi_c$ is defined by (\[eq:nmapping\]) and $\tau$ and $\epsilon$ by (\[eq:tau\_epsilon\]). The equations become increasingly complicated with the growing number of qubits. For this reason one can seek the solution numerically.
As a result of such a numerical optimization, one can for example prepare a 4-qubit KLM state of the triangular-shaped amplitudes in the form of $$\begin{aligned}
|\psi\rangle_{{\mbox{\tiny KLM}}} = \frac{1}{N}\sum_{j=0}^4 \alpha_j |{1}\rangle^j |{0}\rangle^{n-j} \\
\alpha_0 = \alpha_4 = 1, \alpha_1 = \alpha_3 = 3, \alpha_2 = 6\end{aligned}$$ ($N = \sqrt{\sum_{j=0}^n |\alpha_j|^2}$) with the success probability of 0.19% while the original proposal would give only 0.14% success probability (40% improvement). Note that this improved success probability would allow almost 1.5 times higher rate of preparation of KLM states for the “nearly deterministic” protocol proposed by Knill, Laflamme and Milburn [@knill01]. The reason for the improvement in the success probability is the fact that using a tunable phase shift, one can operate the controlled phase gate at optimal phase shift. Because one can always set the gate to operate at the phase $\pi$ and set single qubit operations accordingly, the proposed scheme would never give lower success probability as the one proposed by Franson *et al. The optimal strategy for setting the phase shift imposed by the gate in every step of the generalized procedure is similar to the strategy discussed in the Sec. III for the 2-qubit case. This can be summarized by an inequality $$\begin{aligned}
P_\mathrm{KLM_Franson} = \prod_{i=1}^{n-1} P_C(\pi) = \nonumber \\
= P_C(\pi)^{n-1} \leq P_\mathrm{KLMnew} = \prod_{i=1}^{n-1} P_C(\varphi_i),\end{aligned}$$ where the left-hand side corresponds to the success probability of the Franson *et al. proposal and the right-hand side corresponds to the success probability of the scheme described in this paper. In the worst case scenario hereby proposed scheme allows to set $\varphi = \pi$ to generate any KLM state and in this case the inequality would be saturated.**
Conclusions
===========
The scheme presented in this paper shows how a tunable controlled phase gate can be used to generate arbitrary $n$-qubit KLM states. In comparison with the Franson *et al. proposal, this scheme gives higher success probability depending of the requested KLM state. It can offer a significant improvement in generation of ancillary states for efficient quantum computing. Please note that this paper discusses the improved generation success probability (rate) for the KLM ancillary states. It should not be confused with the success probability of the teleportation based KLM scheme that employs these ancillary states and considers them as already prepared. Several specific KLM states are discussed in this paper and their preparation success probabilities shown to demonstrate this improvement.*
Acknowledgement
===============
The author would like to thank his colleagues Jan Soubusta and Antonín Černoch for fruitful discussion on the subject of this paper.
The author gratefully acknowledge the support by the Operational Program Research and Development for Innovations - European Regional Development Fund (project CZ.1.05/2.1.00/03.0058 and the Operational Program Education for Competitiveness - European Social Fund (project CZ.1.07/2.3.00/20.0017 of the Ministry of Education, Youth and Sports of the Czech Republic, by Palacky University (internal grant PrF-2011-009).
This paper is dedicated to my girlfriend Barborka.\
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|
---
abstract: 'We consider a chemotaxis-Navier-Stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account. This phenomenon results in an inhomogeneous Robin-type boundary condition. Moreover, the system is studied without the logistic growth of the bacteria population. We prove that in two dimensions, the system has a unique global classical solution, while the existence of a global weak solution is shown in three dimensions. In the latter case, we show that the energy is bounded uniformly in time. A key idea is to utilise a boundary energy to derive suitable [*a priori*]{} estimates. Moreover, we are able to remove the convexity assumption on the domain.'
address:
- 'Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria'
- 'Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria'
author:
- Marcel Braukhoff
- Bao Quoc Tang
title: 'Global solutions for chemotaxis-Navier-Stokes system with Robin boundary conditions'
---
Introduction and Main results
=============================
In recent years the analysis of pattern formation in biology has become a thriving field, especially in the analysis of chemotaxis describing bacteria cells and their interaction with a chemical. In this paper, we study the following chemotaxis-Navier-Stokes system with signal consumption $$\label{C-NS}\left\{
\begin{aligned}
\partial_t n + u\cdot {\nabla}n - \Delta n &= {\nabla}\cdot (n{\nabla}c), &x\in\Omega, \; t>0,\\
\partial_t c+ u\cdot {\nabla}c - \Delta c &= -nc, &x\in\Omega, \; t>0,\\
\partial_t u - \mu \Delta u + {\nabla}\cdot(u\otimes u) &=\nabla P -n{\nabla}\varphi, &x\in\Omega, \; t>0,\\
{\nabla}\cdot u &= 0, &x\in\Omega, \; t>0,
\end{aligned}\right.$$ subject to boundary and initial data conditions $$\label{boundary}
\begin{aligned}
{\nabla}c \cdot \nu = \kappa(x) (\gamma(x) - c), \qquad {\nabla}n \cdot \nu = n{\nabla}c\cdot \nu, \qquad u = 0, \qquad x\in\Gamma, \; t>0\\
n(x,0) = n_0(x), \qquad c(x,0) = c_0(x), \qquad u(x,0) = u_0(x), \qquad x\in\Omega.
\end{aligned}$$ Here $\Omega\subset \mathbb R^d$, $d=1,2,3$, is a bounded domain with smooth boundary $\Gamma:= \partial\Omega$, $\mu > 0$ is the viscosity, $\nu(x)$ is the unit outward normal vector at $x\in \Gamma$, and $\varphi$ is the gravitational potential. The explanation of the importance and the role of $\kappa ,\gamma:\Gamma\to \mathbb R_{\geq0}$ in the boundary condition is explained in full detail in the next subsection.
System (with slightly different boundary conditions) was introduced in [@tuval] (see also, e.g., [@tao_bdoxygenconsumption], or Sections 4.1 and 4.2 of the survey [@BBTW]). In this model, the population density of the bacteria is denoted by $n$, whereas $c$ stands for the chemical concentration. Assuming that the bacteria and the chemical are solved in an incompressible liquid like water, we use the Navier-Stokes equation for velocity $u$ to model its flow. Due to the gravitational potential $\varphi$, the bacteria cells influence the liquid flow through their weight.
As an example, this model may be applied to describe the density $n$ of the species *Bacillus subtilis* in a drop of water given by $\Omega$. Their otherwise random motion is known to be directed towards higher concentration $c$ of oxygen contraction, which they consume. In [@dombrowskietal; @tuval], one can experimentally observe that large coherent patterns emerge after some time, which became an interesting research topic in the mathematical community [@win_transAMS]. However, the rigorous results were devastating with respect to this matter: In order to facilitate the problem, usually the system was analyzed for homogeneous boundary conditions, i.e., $\kappa\equiv0$. On the one hand, it was shown that solutions subject to small initial data in a three dimensional domain combined with homogeneous Neumann boundary condition converged to the stationary, constant state $(\frac1{|\Omega|}\int_{\Omega}n_0,0,0)$. On the other hand, also every classical solution in two spacial dimensions converges to the same stationary state [@fan_zhao; @jiang_wu_zheng; @win_fluid_konvergenzresultat; @zhang_li_decay]. Finally, the case was settled in [@win_transAMS] that even “eventual energy solution” converge to the constant state.
There are also different versions of the system of porous-medium type (see e.g. [@difrancesco_lorz_markowich]) or where the chemotaxis term is given in a more general form [@win_CalcVarPDE]. However, the long term behavior remains qualitatively the same - also without involving a fluid, see [@fan_jin; @lswx; @tao_win].
The boundary conditions
-----------------------
All the previous mentioned articles have in common to use homogeneous boundary conditions. Nevertheless, in the experiments [@tuval], the drop of water is surrounded by air which leads inevitable to an oxygen exchange between the drop and the surroundings [@atkins]. Actually, already in the original paper [@tuval] introducing the model , the authors already use inhomogeneous Dirichlet conditions.
Therefore, let us have a closer look how to model the oxygen exchange and why this is crucial for the experiment. We follow the derivation of [@Bra17; @BrLa19]. Assume that water is an ideal solvent for oxygen. The oxygen exchange at the boundary can be modeled using Raoul’s law: On the one hand the amount of solving oxygen at $x\in\Gamma$ is proportional to the vapor pressure of the gaseous oxygen around $x$. On the other hand, the outgoing rate of oxygen is proportional to the concentration on the boundary, i.e., the rate of oxygen molecules leaving the drop at $x\in\Gamma$ is proportional to the number of molecules at $x$ (see [@atkins Section 5.3, page 144]). In order to have a closed system, we suppose that the oxygen vapor pressure is a given function. This is reasonable, because the oxygen-diffusion coefficient in air is three orders of magnitude larger than that in the fluid [@tuval page 2279]. Moreover, the negligibility of the influence of the drop to the gaseous oxygen implies that the vapor pressure is constant in time. Adding both effects, we see that the oxygen-flux at the boundary is an affine function of the concentration, which we write in the form $$\label{boundary2_0}
\begin{aligned}
{\nabla}c(x,t) \cdot \nu = \kappa(x) (\gamma(x) - c(x,t)), \qquad
x\in\Gamma, \; t>0
\end{aligned}$$ for $\gamma,\kappa:\Gamma\to\mathbb{R}_{\geq0}$. This condition is also known as Henry’s law in the context of sorption of chemicals to surfaces [@atkins]. Note that we do not want to assume that the drop is entirely surrounded by air, but also part of it can by connected to a solid exterior where there is no oxygen exchange. Therefore for on the solid–water interface we assume that $\kappa$ vanishes, which does not have to be the case on the water–air boundary. For function $\gamma(x)$ (as in [@BrLa19]), one can interpret it as the maximal saturation of oxygen in the fluid. Note that for $\Omega$ being the ball and $\gamma$ and $\kappa$ being radially symmetric, one obtains Dirichlet boundary conditions (see e.g. [@tuval]) $$c(x)=\gamma(x)\qquad \text{for }x\in \partial \Omega$$ as a limit of for $\kappa\to\infty$, see [@BrLa19 Proposition 5.3] for a proof of the stationary problem.
Having changed the boundary condition for the oxygen concentration $c$, we need to adjust the boundary conditions for $n$ as well in order to preserve the mass of bacteria. For this we choose the no-flux conditions for $n$. In addition, we close the Navier-Stokes system with Dirichlet boundary conditions. Therefore the set of boundary conditions are given by $$\label{boundary2}
\begin{aligned}
{\nabla}c \cdot \nu = \kappa(x) (\gamma(x) - c), \qquad {\nabla}n \cdot \nu = n{\nabla}c\cdot \nu, \qquad u = 0, \qquad x\in\Gamma, \; t>0.
\end{aligned}$$ In [@BrLa19], the system combined with this boundary conditions is treated without the flow, i.e., $u=\nabla P = \nabla \varphi \equiv0$. Therein it is shown that if $\kappa\not\equiv0$ and $\gamma=const$ then and admit a unique stationary state for a given mass $\int_{\Omega} n dx$. Moreover $n$ and $c$ are positive but not constant. In the radial symmetric case, $n$ and $c$ are even strictly convex. Up to the best of our knowledge, this is the only qualitative result for the system showing a non-trivial steady state.
Let us mention related works on chemotaxis systems involving inhomogeneous boundary conditions. The articles [@chertock_etal_numeric; @lee_kim_numerical_bioconvection; @tuval] show numerically that models with inhomogenous boundary conditions match the experimental results. In [@Lorz], a chemotaxis-fluid system with an inhomogeneous Dirichlet condition for $c$ on parts of the boundary is treated on a bounded two dimensional domain, and the local existence of weak solutions is shown therein. Recently, [@preprint_zhaoyin] imposes inhomogeneous Dirichlet condition on one side of the domain $\mathbb R^2\times(0,1)$. Under stronger technical assumptions on the consumption term, [@preprint_zhaoyin] proves the existence and convergence of solutions for initial data being close to $(0,\gamma,0)$. Moreover, in spatial dimension one [@knosalla_global; @knosalla_nadzieja_stationary] treat the related chemotaxis system $$\begin{cases}
n_t=n_{xx} - (n E(c)_x)_x,\\
c_t=c_{xx} - n E(c)
\end{cases}$$ for either a inhomogeneous Dirichlet or Neumann conditions. Here, $E$ satisfies $E(c)\to 0$ for $c\to 0$ and $c\to \infty$. The existence of global, bounded solutions is proved in [@knosalla_global], whereas [@knosalla_nadzieja_stationary] proves the existence and uniqueness of the stationary state.
Global existence vs. logistical source
--------------------------------------
The first analytical results for system – with logistic growth of the density, i.e. the equation for $n$ is replaced by $$\label{logistic}
\partial_t n + u\cdot {\nabla}n - \Delta n = {\nabla}\cdot(n{\nabla}c) + n(1-n)$$ were delivered in a paper of the first author [@Bra17], in which the global existence of classical and weak solutions was shown in two and three dimensions, respectively.
The global existence of solutions to with homogeneous boundary conditions crucially depends on the energy functional $$\label{normal-energy}
S(t) = \int_{\Omega}n(t)\log n(t)dx + a\int_{\Omega}\left|{\nabla}\sqrt{c(t)} \right|^2dx + b\int_{\Omega}|u(t)|^2dx$$ which is decreasing for suitable constants $a, b>0$, see e.g. [@Win12; @Win16]. This gives the necessary a-priori estimates to start the bootstrapping, which eventually leads to global (strong, weak) solutions. In the case of Robin-type boundary conditions (for the oxygen $c$), this strategy is not directly applicable since the functional $S(t)$ fails to decrease in time because of the boundary terms in the estimate. This problem was solved in [@Bra17], firstly by transforming into homogeneous Neumann boundary conditions, and secondly, to cope with the extra terms coming from the transformation, by introducing the logistical growth term as in . The logistic term gives a bound in $L^2(0,T;L^2(\Omega))$ for free just by integrating on $\Omega\times (0,T)$. This estimate can then be used in an essential way in a bootstrap argument to get global solutions.
The logistic nonlinearity acts as a damping term and therefore it usually helps in the analysis of chemotaxis systems. For instance, under homogeneous boundary conditions, system with a logistic growth is very well studied in [@lankeit_m3as] in which global weak solutions were shown to be smooth after some positive time. Moreover, convergence of solutions to the steady state $(1,0,0)$ was also proved. Similar results were obtained in [@lankeit_wang] for the case without fluids and in [@win_nutrienttaxis] in the case of food-supported proliferation. A recent study [@Miz19] demonstrates well the effect of logistic growth (together with nonlinear diffusion) to the well-posedness of chemotaxis systems. We however remark that a logistic growth term might lead to interesting new effects in chemotaxis [@lankeit_thresholds; @hillenpainter_spatiotemporalchaos; @win_transient]. For example, one can easily see that the mass of the bacteria is no longer conserved if a logistic source term is added to the first equation.
The global well-posedness of the chemotaxis-Navier-Stokes system without logistic growth together with the inhomogeneous boundary conditions is therefore a challenging problem, and it is the main aim of the present paper.
Key ideas
---------
As mentioned in the previous subsection, due to inhomogeneous boundary conditions , the usual energy is not decreasing in time along a trajectory of . Moreover, the lack of the logistic growth also seems to break the strategy of transforming – into a system with homogeneous boundary conditions. Our key idea to deal with this issue is first to introduce [*a boundary energy*]{} of the form $$S^{\text{boundary}}(t):= \int_{\Gamma} \kappa(x)\left[\gamma(x)\log\frac{\gamma(x)}{c(x,t)} - \gamma(x) + c(x,t)\right]{d\mathcal{H}_x^{d-1}},$$ and then to look at the evolution of the [*total energy*]{} $$\mathcal F(t) = S(t) + S^{\text{boundary}}(t)
$$ with $a = 2$ and $b = K$, for a sufficiently large constant $K$. We will show that this total energy satisfies $$\label{en1}
\frac{d}{dt}\mathcal F(t) \leq p\mathcal F(t) + q$$ for some constant $p, q>0$, which consequently leads to a set of a-priori estimates. These estimates are enough in two dimension to start a bootstrap argument to obtain global classical solutions, while they ensure an approximating procedure in three dimensions to get global weak solutions.
As one can see from that though the solution is global, the energy might grow exponentially. To show that the total energy $\mathcal{F}(t)$ is in fact bounded uniformly in time, we introduce yet another energy functional $$S^{\text{add}}(t):= \int_{\Omega}\left[c(x,t)\log \frac{c(x,t)}{\widehat{\gamma}(x)} - c(x,t) + \widehat{\gamma}(x)\right]dx$$ where $\widehat{\gamma}$ is a smooth extension of $\gamma$ to $\overline{\Omega}$. Now by considering $\mathcal{F}^{\text{new}}(t) = \mathcal{F}(t) + LS^{\text{add}}(t)$ for some suitable constant $L>0$, we obtain $$\label{en2}
\frac{d}{dt}\mathcal{F}^{\text{new}}(t) + \lambda \mathcal{F}^{\text{new}}(t) \leq C$$ for some $\lambda, C>0$. This inequality gives the uniform-in-time bound for $\mathcal{F}^{\text{add}}$ and eventually the desired bound for the total energy $\mathcal{F}$.
We also would like to emphasize that we do not assume the domain $\Omega$ to be convex. The convexity of $\Omega$ was very useful in the literature when dealing with the analysis of , see e.g. [@Win12; @Win16]. Though it is natural to assume that a fluid drop has a convex shape, there exist situations when it is not the case, for instance, when the drop is in contact with an uneven surface. In [@lankeit_m3as; @MS14], the authors were also able to remove this technical condition on the convexity of $\Omega$ by using the boundedness of the domain curvature (see [@MS14 Lemma 4.2]). Our main idea is to go one step further and use the full power of the dissipation terms arising from the diffusion of the oxygen (see the proof of Lemma \[lem:energy2\]).
Main Results
------------
We begin with definitions of classical and weak solutions.
A quadruplet $(n,c,u,P)$ is called a classical solution to – on $(0,T)$ if $$\begin{aligned}
n,c&\in C^{2+2\delta,1+\delta}\left(\overline\Omega\times(0,T)\right)\cap C^{0}\left(\overline\Omega\times[0,T)\right),
\\
u&\in C^{2+2\delta,1+\delta}\left(\overline\Omega\times(0,T)\right)\cap C^{0}\left(\overline\Omega\times[0,T)\right),
\\
P&\in C^{1+\delta,\delta}\left(\Omega\times(0,T)\right),
\end{aligned}$$ for some $\delta>0$, and the equations in – are satisfied pointwise.
\[weak\_sol\] A triple $(n,c,u)$ is called a global weak solution of – if $$n\in L^1_{loc}([0,\infty);W^{1,1}(\Omega)), \quad c\in L^1_{loc}([0,\infty); W^{1,1}(\Omega)), \quad u \in L^1_{loc}([0,\infty); W^{1,1}_0(\Omega;\mathbb R^3))$$ such that $n\geq 0$ and $c\geq 0$ a.e. in $\Omega\times (0,\infty)$, $$nc \in L^1_{loc}(\Omega\times [0,\infty)), \quad u\otimes u \in L^1_{loc}(\Omega\times[0,\infty); \mathbb R^{3\times 3}), \qquad \text{ and }$$ $$n{\nabla}c, \quad nu, \quad cu \quad \text{ belong to } \quad L^1_{loc}(\Omega\times [0,\infty); \mathbb R^3),$$ that ${\nabla}\cdot u = 0$ a.e. in $\Omega\times (0,\infty)$, and that $$\begin{aligned}
-\int_0^\infty\int_{\Omega} n\partial_t \psi dxdt = \int_{\Omega}n_0\psi(\cdot,0)dx\\ -\int_0^\infty\int_{\Omega}{\nabla}n\cdot {\nabla}\psi dxdt + \int_0^\infty\int_{\Omega}n{\nabla}c \cdot {\nabla}\psi dxdt + \int_0^\infty\int_{\Omega}nu\cdot {\nabla}\psi dxdt,
\end{aligned}$$ $$\begin{aligned}
-\int_0^\infty\int_{\Omega}c\partial_t\psi dxdt = \int_{\Omega}c_0\psi(\cdot,0)dx
-\int_0^\infty\int_{\Omega}{\nabla}c\cdot {\nabla}\psi dxdt\\ + \int_0^\infty\int_{\Gamma}\kappa (\gamma - c)\psi {d\mathcal{H}_x^{d-1}}dt - \int_0^\infty\int_{\Omega}nc\psi dxdt + \int_0^\infty\int_{\Omega} cu\cdot {\nabla}\psi dxdt
\end{aligned}$$ for all $\psi \in C_0^\infty(\Omega\times [0,\infty))$, and $$\begin{aligned}
-\int_0^\infty\int_{\Omega} u\cdot \xi_t dxdt = \int_{\Omega}u_0\cdot \xi(\cdot,0)dx\\
-\int_0^\infty\int_{\Omega} {\nabla}u\cdot {\nabla}\xi dxdt + \int_0^\infty\int_{\Omega} u\otimes u\cdot {\nabla}\xi dxdt - \int_0^\infty\int_{\Omega}n{\nabla}\varphi \cdot \xi dxdt
\end{aligned}$$ for all $\xi \in C_0^\infty(\Omega\times[0,\infty); \mathbb R^3)$ satisfying ${\nabla}\cdot \xi \equiv 0$.
As usual, we denote by $$L^2_\sigma(\Omega):=\overline{D_\sigma(\Omega)}^{\|\cdot\|_{L^2(\Omega)}}, \quad \text{ where } \quad D_\sigma(\Omega):= \{u\in C_0^\infty(\Omega)^d:\nabla\cdot u =0\},$$ and let $\mathcal P^\infty$ be the Helmholz projection $L^2(\Omega)^d \to L^2_\sigma(\Omega)^d$. We denote by $$A: D(A) \subset L_\sigma^2(\Omega) \to L_\sigma^2(\Omega), \quad Au:= -\mathcal P^\infty \Delta u$$ the Stokes operator with Dirichlet boundary conditions, where the domain of $A$ is given by $$D(A) = L^2_\sigma(\Omega)^d\cap H_0^1(\Omega)^d \cap H^2(\Omega)^d.$$ The main results of this paper are the following two theorems.
\[thm:main2D\] Let $d \leq 2$ and assume that the data satisfies $$0< \kappa, \gamma \in C^1(\Gamma), \quad \varphi \in C^1(\overline{\Omega}).$$ Then for any initial data $(n_0, c_0, u_0)$ satisfying $$\left\{\begin{aligned}
0< n_0&\in C^0(\overline{\Omega})\cap H^1(\Omega),\\
0< c_0&\in W^{1,10}(\Omega),\\
u_0&\in D(A^{\alpha}) \qquad \text{for some } \frac d4<\alpha<1.\end{aligned}\right\},$$ there exists a unique global classical solution to –.
\[thm:main3D\] Let $d=3$, and assume that $$\label{varphi}
\varphi \in W^{1,\rho}(\Omega), \quad \text{ for some } \quad \rho > 6,$$ and $$\label{cond_data}
\sqrt{\kappa}\in H^1(\Gamma)\cap L^\infty(\Gamma), \quad 0 < \underline{\gamma} \leq \gamma, \quad \text{ and } \quad \sqrt{\gamma} \in H^1(\Omega)\cap L^\infty(\Gamma).$$ Then for any initial data $(n_0, c_0, u_0)$ satisfying $$\begin{aligned}
& n_0 >0 \quad \text{ and } \quad \int_{\Omega}n_0\log n_0 dx < +\infty,\\
& 0 < c_0 \in L^\infty(\Omega) \quad \text{ and } \quad \sqrt{c_0} \in H^1(\Omega),\\
& u_0 \in L^2_{\sigma}(\Omega),
\end{aligned}$$ the system – has a global weak solution. Moreover, the global energy is bounded uniformly in time, i.e. $$\sup_{t\in [0,\infty)}\left(\int_{\Omega}n(t)\log n(t)dx + \| {\nabla}\sqrt c(t)\|_{L^2(\Omega)}^2 + \|u(t)\|_{L^2(\Omega)}^2\right) \leq C$$ where $C$ depends only on initial energy, on the data $\mu, \kappa, \gamma, \varphi$, and on the domain $\Omega$.
We believe that our approach is extendable to a more general system than –, for instance $$\label{C-NS-extension}
\left\{
\begin{aligned}
\partial_t n + u\cdot {\nabla}n - \Delta n &= {\nabla}\cdot (n\chi(c){\nabla}c), &x\in\Omega, \; t>0,\\
\partial_t c+ u\cdot {\nabla}c - \Delta c &= -nf(c), &x\in\Omega, \; t>0,\\
\partial_t u - \mu \Delta u + {\nabla}\cdot(u\otimes u) &=\nabla P -n{\nabla}\varphi, &x\in\Omega, \; t>0,\\
{\nabla}\cdot u &= 0, &x\in\Omega, \; t>0,
\end{aligned}
\right.$$ for some functions $\chi$ and $f$ satisfying suitable conditions (see e.g. [@Win16] for the case with homogeneous boundary conditions), though non-trivial modifications need to be carried out. We leave this interesting open issue for the interested reader.
In the next section, we consider approximate systems of and derive necessary *a priori* estimates. Using these estimates, we prove the main theorems in Section \[proofs\].
[**Notation:**]{} In this paper, we will use the following notation:
- We will denote by $C$ a generic constant [*independent of time*]{}, which can be different from line to line, or even in the same line. When a constant depends on the time horizon $T>0$, we will write $C_T$ instead.
- For any $T>0$ and $1\leq p\leq \infty$, we denote by $Q_T:= \Omega\times (0,T)$ and $$L^p(Q_T):= L^p(0,T;L^p(\Omega))$$
with the usual norm $$\|f\|_{L^p(Q_T)}:= \left(\int_0^T\int_{\Omega}|f|^pdxdt \right)^{\frac 1p}$$ when $p<\infty$ and $$\|f\|_{L^\infty(Q_T)}:= \text{ess sup}_{t\in (0,T)}\|f(t)\|_{L^\infty(\Omega)}.$$
Approximate systems and a-priori estimates
==========================================
If the evolution equation for the density $n$ in – is replaced by $$\partial_t n + u\cdot \nabla n - \Delta n = \nabla\cdot(n\nabla c) + n(1-n),$$ then the local existence of a classical solution was done in [@Bra17 Proposition 2.6] by a standard fixed point argument. It is remarked that the proof of this result does not use any structural of the logistic growth $n(1-n)$, and it is therefore also applicable to –. For the reader’s convenience we recall Proposition 2.6 from [@Bra17] (without the logistic term).
\[a-prop-local-solution-hom\] Let $d\in\{1,2,3\}$ and $\Omega\subset\mathbb{R}^d$ be a bounded domain with smooth boundary.
Then there exists a maximal $T_{\max} \in(0,\infty]$ such that - possesses a classical solution on $(0,T)$ for every $0<T<T_{\max}$ with $n \geq 0$ and $c\geq 0$. Furthermore, if $$\label{a-formula-explosion-condition-transformed}
\limsup_{t\uparrow T_{\max}}\left( \|n(t)\|_{L^\infty(\Omega)}+\|{\nabla}n (t)\|_{L^2(\Omega)}+\|c(t)\|_{W^{1,4}(\Omega)}+\|A^\alpha u(t)\|_{L^2(\Omega)}\right) < +\infty$$ then $T_{\max} = \infty$. The solution $(n,c,u,P)$ is unique up to a constant for $P$.
In case $d=3$, as we do not expect to prove the existence of a global classical to the Navier-Stokes equation, we aim for weak solutions. Therefore, we consider in this case the following approximating sequence for ${\varepsilon}\geq0$ and $m\in \mathbb N\cup\{\infty\}$. $$\label{Smod}\left\{
\begin{aligned}
&{\partial}_t n^{{\varepsilon},m} +u^{{\varepsilon},m}\cdot\nabla n^{{\varepsilon},m}- \Delta n^{{\varepsilon},m} = {\nabla\cdot}(n^{{\varepsilon},m}{\nabla}c^{{\varepsilon},m})+{\varepsilon}n^{{\varepsilon},m}(1-(n^{{\varepsilon},m})^2), &x\in\Omega,\\
&{\partial}_t c^{{\varepsilon},m} +u^{{\varepsilon},m}\cdot\nabla c^{{\varepsilon},m}- \Delta c^{{\varepsilon},m} = -n^{{\varepsilon},m}c^{{\varepsilon},m}, &x\in\Omega,\\
&{\partial}_tu^{{\varepsilon},m} =-Au^{{\varepsilon},m}-\mathcal P^m[\nabla (u^{{\varepsilon},m}\otimes u^{{\varepsilon},m})+n^{{\varepsilon},m}\nabla\varphi], &x\in \Omega,\\
&{{\partial}_\nu}c^{{\varepsilon},m} = \kappa(x) (\gamma(x)-c^{{\varepsilon},m}), &x\in\Gamma,\\
&{{\partial}_\nu}n^{{\varepsilon},m} = n^{{\varepsilon},m}{{\partial}_\nu}c^{{\varepsilon},m}, &x\in\Gamma,\\
&c^{{\varepsilon},m}(x,0) = c_0^{\varepsilon,m}(x), \; n^{{\varepsilon},m}(x,0) = n_0^{\varepsilon,m}(x), \; u^{{\varepsilon},m}(0)=u_0^{\varepsilon,m} &x\in\Omega,
\end{aligned}\right.$$ where $$\label{a-initial-functions}
\left\{\begin{aligned}
0< n_0^{\varepsilon,m}&\in C^0(\overline{\Omega})\cap H^1(\Omega),\\
0< c_0^{\varepsilon,m}&\in W^{1,10}(\Omega),\\
u_0^{\varepsilon,m}&\in D(A^{\alpha}) \qquad \text{for some } \frac d4<\alpha<1,\end{aligned}\right.$$ and $$\lim_{\varepsilon\to 0}\sup_{m\in \mathbb N\cup \{\infty\}}\left(\|n_0^{{\varepsilon},m} - n_0\|_{L^1(\Omega)} + \|c_0^{{\varepsilon},m} - c_0\|_{L^\infty(\Omega)} + \|u_0^{{\varepsilon},m} - u_0\|_{L^2(\Omega)}\right) = 0,$$ $$\sup_{\varepsilon>0, m\in \mathbb N\cup \{\infty\}}\left[\int_{\Omega}n_0^{\varepsilon,m}\log n_0^{\varepsilon,m}dx + \|\sqrt{c_0^{\varepsilon,m}}\|_{H^1(\Omega)}\right] <+\infty.$$ Here, $\mathcal P^m$ denotes the Leray projection onto the space of the first $m$ eigenvectors of $A$. For any fixed $\varepsilon>0$ and $\mathbb N\ni m < \infty$, there exists a global classical solution $(n^{\varepsilon,m}, c^{\varepsilon,m}, u^{\varepsilon,m})$ to with $n^{\varepsilon,m}\geq 0$ and $c^{\varepsilon,m} \geq 0$ (see [@Bra17 Proposition 4.6]).
\[rem.equivalent\] The systems – and are equivalent for ${\varepsilon}=0$ and $m=\infty$ (see [@kato Theorems 1.7, 7.5 and 7.6]).
The general strategy to study global existence of solutions for is the following: when $d\in \{1,2\}$, we show that the local solution obtained in Proposition \[a-prop-local-solution-hom\] satisfies the criterion , whence its global existence; while in the case $d = 3$, we prove that as $\varepsilon\to 0$ and $m \to \infty$, the global classical solution to the approximate system converges to a global weak solution of . In both cases, we will use the same [*a priori*]{} estimates for either the local solution in Proposition \[a-prop-local-solution-hom\] or the global solution of the approximate system . Therefore, for the rest of this paper, we use a fixed (but arbitrary) time horizon $T$ with $0<T<T_{\max}$ when dealing with the former solution, while $0<T<\infty$ when dealing with the latter. To avoid complicated notation we will, in this section, suppress the superscript ${\varepsilon}$ and $m$ in the solution of , and write it simply $(n,c,u)$. Moreover, the generic constants $C>0$, which we use frequently, do not depend on ${\varepsilon}$ nor on $m$.
We start with the following immediate estimates, which will be useful in the sequel analysis.
\[L1Linf\] For all $t\in (0,T)$, $$\|n(t)\|_{L^1(\Omega)} \leq C \quad \text{ and } \quad \|c(t)\|_{L^\infty(\Omega)} \leq \max\left\{\|\gamma\|_{L^\infty(\Gamma)}; \|c_0\|_{L^\infty(\Omega)}\right\}.$$
The $L^\infty$-estimate of $c$ follows from the maximum principle. For the estimate of $n$ we integrate the equation of $n$ in and use the incompressibility ${\nabla}\cdot u = 0$ as well as the boundary condition $\partial_\nu n = n\partial_\nu c$ to get $$\label{nl}
\partial_t \int_{\Omega} n dx + {\varepsilon}\int_{\Omega}n^3dx = {\varepsilon}\int_{\Omega}ndx \leq {\varepsilon}\int_{\Omega}\left(n^3 - n + \frac{4\sqrt{6}}{9}\right)dx$$ thanks to the non-negativity of $n$. Thus $$\partial_t \int_{\Omega} n dx + {\varepsilon}\int_{\Omega} n dx \leq \frac{4\sqrt{6}}{9}|\Omega|{\varepsilon}.$$ Hence $$\int_{\Omega}n(x,t)dx \leq e^{-{\varepsilon}t}\int_{\Omega}n_0(x)dx + \frac{4\sqrt{6}}{9}|\Omega|(1 - e^{-{\varepsilon}t}) \leq \|n_0\|_{L^1(\Omega)} + \frac{4\sqrt{6}}{9}|\Omega|.$$ which gives the desired estimate for $n$ since $n$ is non-negative.
We are going to use the following functions $$\label{entropy}
s(y):=y\log y -y+1 \qquad \text{ and } \qquad s^\infty(y|z):=y\log \frac{y}{z} -y+z.$$
\[lem.ee.n\] We have the following identity for all $t\in (0,T)$ $$\frac d{dt}\int_{\Omega}s(n)dx+4\int_{\Omega}\left|\nabla\sqrt{n}\right|^2 dx+{\varepsilon}\int_{\Omega}n(1-n^2)\log ndx
=\int_{\Omega}\nabla c\cdot \nabla ndx$$
Using the equation for $n$ and integration py parts, we directly see that $$\begin{aligned}
\frac d{dt}\int_{\Omega}s(n)dx &=
\int_{\Omega}\partial_tn\log{n}dx
\\&= \int_{\Omega}(\Delta n-\nabla\cdot (n\nabla c))\log ndx-\int_{\Omega}u\cdot\underbrace{\nabla n\log{n}}_{\nabla s(n)}dx-{\varepsilon}\int_{\Omega}n(1-n^2)\log ndx
\\&= -\int_{\Omega}(\nabla n-n\nabla c)\cdot \nabla\log{n}dx+\int_{\Omega}\underbrace{\nabla\cdot u}_{=0} s(n)dx-{\varepsilon}\int_{\Omega}n(1-n^2)\log ndx
\\&=- \int_{\Omega}\frac{|\nabla{n}|^2}n dx-{\varepsilon}\int_{\Omega}n\log n(1-n^2)dx
+\int_{\Omega}\nabla c\cdot \nabla ndx,
\end{aligned}$$ where we have used $\nabla n\cdot \nu = n\nabla c \cdot \nu$ and $u = 0$ on $\Gamma$.
\[lem.entropy.equality.part2\] The following identity holds $$\begin{gathered}
\label{p0}
\partial_t\int_{\Omega}|\nabla\sqrt{c}|^2dx
+\frac12 \int_{\Omega}|\nabla^2\log c|^2cdx
+\int_\Omega|\nabla \sqrt{c}|^2ndx
\\=\int_{\Omega}\Delta(|\nabla\sqrt{c}|^2)dx
- \frac12\int_{\Omega}\nabla n\cdot\nabla cdx-2\int_\Omega \nabla\sqrt c\cdot\nabla ^Tu\nabla \sqrt{c}dx.\end{gathered}$$
To prove Lemma \[lem.entropy.equality.part2\], we first derive the equation of $\sqrt{c}$. In $\Omega$ we have $$\begin{aligned}
\partial_t \sqrt c+u\cdot\nabla \sqrt{c}-\Delta \sqrt c &=
\frac{1}{2\sqrt c}(\partial_t c+u\cdot \nabla c-\Delta c )+\frac{|\nabla c|^2}{4\sqrt{c}^3}
\\&=-\frac{1}{2\sqrt{c}}nc+\frac{|\nabla\sqrt c|^2}{\sqrt c},
\end{aligned}$$ and on $\Gamma$ $$\begin{aligned}
\label{boundary_sqrt_c}
\partial_\nu\sqrt c=\frac{1}{2\sqrt c}\partial_\nu c=\frac{\kappa }{2}\left(\frac{\gamma}{\sqrt{c}}-\sqrt{c}\right)\end{aligned}$$ From that, we can calculate $$\label{p1}
\begin{aligned}
\partial_t|\nabla\sqrt{c}|^2
&=
2\nabla \partial_t \sqrt{c}\cdot \nabla \sqrt{c}
\\&=
2\nabla \Delta \sqrt{c}\cdot \nabla \sqrt{c}
+2\nabla \frac{|\nabla\sqrt{c}|^2}{\sqrt{c}}\cdot \nabla \sqrt{c}
-\nabla(\sqrt{c} n )\cdot \nabla \sqrt{c} -2\nabla (u\cdot \nabla\sqrt c)\cdot\nabla \sqrt{c}
\\&=
\Delta |\nabla\sqrt{c}|^2
-2 |\nabla^2\sqrt{c}|^2
+2\nabla \frac{|\nabla\sqrt{c}|^2}{\sqrt{c}}\cdot \nabla \sqrt{c}
\\&
\qquad
-n |\nabla \sqrt{c}|^2-\nabla n\cdot\sqrt{c}\nabla \sqrt{c}-2\nabla\sqrt c\cdot\nabla ^Tu\nabla \sqrt{c}-2u\cdot \nabla^2\sqrt c\nabla \sqrt{c}
\end{aligned}$$ using $$\begin{aligned}
2\nabla \Delta \sqrt{c}\cdot \nabla \sqrt{c}
&=\Delta |\nabla\sqrt{c}|^2-2|\nabla^2\sqrt{c}|^2.
\end{aligned}$$ For the third term on the right hand side of , we compute $$\begin{aligned}
&2\nabla \frac{|\nabla\sqrt{c}|^2}{\sqrt{c}}\cdot \nabla \sqrt{c}
\\&=
2 (\nabla(\sqrt{c})^{-1}|\nabla\sqrt{c}|^2
)\cdot \nabla \sqrt{c}
+2 (\frac{1}{\sqrt{c}}\nabla|\nabla\sqrt{c}|^2
)\cdot \nabla \sqrt{c}
\\&=-\frac{2}{c}|\nabla\sqrt{c}|^4
+\frac{4}{\sqrt{c}}\nabla\sqrt{c}\cdot \nabla^2\sqrt{c}
\nabla \sqrt{c} .
\end{aligned}$$ Moreover, $\nabla\cdot u=0$ implies $$\begin{aligned}
-2u\cdot \nabla^2\sqrt c\nabla \sqrt{c} &= -u\cdot \nabla|\nabla \sqrt{c}|^2=-\nabla\cdot (u|\nabla\sqrt c|^2)
\end{aligned}$$ Inserting these computations into leads to $$\begin{aligned}
\partial_t&|\nabla\sqrt{c}|^2
+2 |\nabla^2\sqrt{c}|^2-\frac{4}{\sqrt{c}}\nabla\sqrt{c}\cdot \nabla^2\sqrt{c}
\nabla \sqrt{c} +\frac{2}{c}|\nabla\sqrt{c}|^4
\\&=
\Delta |\nabla\sqrt{c}|^2-\nabla\cdot (u|\nabla\sqrt c|^2)
-|\nabla\sqrt{c}|^2 n-\frac12 \nabla n\cdot\nabla c-2\nabla\sqrt c\cdot\nabla ^Tu\nabla \sqrt{c}.
\end{aligned}$$ From the binomial formula for matrices, it follows that $$\begin{gathered}
|\nabla^2\sqrt{c}|^2-2\nabla\sqrt{c}\cdot\nabla^2\sqrt{c}\nabla\sqrt{c} +\frac{1}{c}|\nabla\sqrt{c}|^4
\\=
\left|\nabla^2\sqrt{c}-\frac{1}{\sqrt{c}}\nabla\sqrt{c}\otimes\nabla\sqrt{c}\right|^2
=\left|\sqrt{c}\nabla\left(\frac{\nabla \sqrt{c}}{\sqrt{c}}\right)\right|^2=c|\nabla^2\log\sqrt{c}|^2 .
\end{gathered}$$ Therefore, after an integration over $\Omega$, we have $$\begin{aligned}
& \partial_t\int_{\Omega}|\nabla\sqrt{c}|^2dx
+\frac12 \int_{\Omega}c|\nabla^2\log c|^2dx
+\int_\Omega|\nabla \sqrt{c}|^2ndx
\\&\qquad=\int_{\Omega}\Delta(|\nabla\sqrt{c}|^2)dx -\int_{\Omega}\nabla\cdot (u|\nabla\sqrt c|^2)dx
- \frac12\int_{\Omega}\nabla n\cdot\nabla cdx-2\int_\Omega \nabla\sqrt c\cdot\nabla ^Tu\nabla \sqrt{c}dx.
\end{aligned}$$ We observe that the second term on the r.h.s. vanishes, because of the Gauß formula and the fact that $u=0$ on $\Gamma$.
To estimate the last term on the right-hand side of , we need the following lemma.
\[lem.bernstein\] The inequality $$\begin{aligned}
\frac14{\int_{\Omega}}\frac{|{\nabla}c|^4}{ c^3}\leq \int_{{\partial}\Omega}|{\nabla}\log c| ^2\kappa \left(\gamma- c\right)d\mathcal{H}^{d-1}_x+(2+d){\int_{\Omega}}c|\nabla^2\log c|^2,
\end{aligned}$$ holds for any smooth function $c$ satisfying $\partial_\nu c = \kappa (\gamma - c)$ on $\Gamma$.
We follow the ideas from [@Win12 Lemma 3.3]. We compute $$\begin{aligned}
{\int_{\Omega}}\frac{|{\nabla}c|^4}{c^3}dx &={\int_{\Omega}}|{\nabla}\log c| ^2{\nabla}\log c\cdot {\nabla}c\, dx.
\end{aligned}$$ Using integration by parts and the boundary conditions for $c$, we have $$\begin{aligned}
{\int_{\Omega}}|{\nabla}\log c| ^2{\nabla}\log c\cdot {\nabla}c\,dx
&=
\int_{{\partial}\Omega}|{\nabla}\log c| ^2{\partial}_\nu c\, d\mathcal{H}^{d-1}_x
\\&\quad -
{\int_{\Omega}}{\nabla}|{\nabla}\log c| ^2\cdot({\nabla}\log c) c\,dx
-
{\int_{\Omega}}|{\nabla}\log c| ^2(\Delta \log c) c\,dx
\\&=
\int_{{\partial}\Omega}|{\nabla}\log c| ^2 \kappa \left(\gamma- c\right)\mathcal{H}^{d-1}_x
\\&\qquad-2{\int_{\Omega}}\frac{1}{ c}({\nabla}^2\log c{\nabla}c)\cdot{\nabla}c
dx
-{\int_{\Omega}}\frac{|{\nabla}c|^2}{ c}\Delta \log c\,dx
\end{aligned}$$ using $\nabla |\nabla \log c|^2=2\nabla^2 \log c\nabla \log c$. By Young’s inequality, we see that $$\begin{aligned}
-2{\int_{\Omega}}\frac{1}{ c}({\nabla}^2\log c{\nabla}c)\cdot{\nabla}c
dx
&\leq \frac12{\int_{\Omega}}\frac{|{\nabla}c|^4}{ c^3}+2{\int_{\Omega}}c|\nabla^2\log c|^2
\end{aligned}$$ and
$$\begin{aligned}
-{\int_{\Omega}}\frac{|{\nabla}c|^2}{ c}\Delta \log c\,dx
&\leq \frac14{\int_{\Omega}}\frac{|{\nabla}c|^4}{ c^3}+{\int_{\Omega}}c|\Delta\log c|^2.
\end{aligned}$$
Note that we have the fundamental estimate $|\Delta \log c|^2=|\mathrm{trace}({\nabla}^2\log c)|^2\leq d|{\nabla}^2\log c|^2$. Collecting all the previous calculations and estimates yields $$\begin{aligned}
\frac14{\int_{\Omega}}\frac{|{\nabla}c|^4}{ c^3}\leq \int_{{\partial}\Omega}|{\nabla}\log c| ^2\kappa \left(\gamma- c\right)d\mathcal{H}^{d-1}_x+(2+d){\int_{\Omega}}c|\nabla^2\log c|^2,
\end{aligned}$$ which implies the assertion.
With the help of Lemma \[lem.bernstein\], the identity in Lemma \[lem.entropy.equality.part2\] is estimated further in the next lemma.
\[cor.entropy.equality.part2\]There exists a constant $\xi>0$ such that, for all $t\in (0,T)$, $$\begin{aligned}
&\partial_t\int_{\Omega}|\nabla\sqrt{c}|^2dx
+\frac18 \int_{\Omega}c|\nabla^2\log c|^2dx
+\xi \int_\Omega|\nabla \sqrt[4]{c}|^4dx
+\int_\Omega|\nabla \sqrt{c}|^2ndx
\\&\leq\int_{\Omega}\Delta(|\nabla\sqrt{c}|^2)dx
- \frac12\int_{\Omega}\nabla n\cdot\nabla cdx+
\frac{1}{2}\int_{\Gamma}|{\nabla}\sqrt c|^2 \kappa \left(\frac{\gamma}c- 1\right)d\mathcal{H}^{d-1}_x
+4\|c\|_{L^\infty(\Omega)}\|u\|_{H^1(\Omega)}^2.
\end{aligned}$$
We apply Young’s inequality and obtain $$\begin{aligned}
-2\int_\Omega \nabla\sqrt c\cdot\nabla ^Tu\nabla \sqrt{c}dx &\leq \frac14\int_\Omega \frac{|\nabla\sqrt c|^4}{c}dx+4\int_\Omega c|\nabla ^Tu|^2dx
\\&=\frac1{72}\int_\Omega \frac{|\nabla c|^4}{c^3}dx+4\int_\Omega c|\nabla ^Tu|^2dx
\end{aligned}$$ Then by Lemma \[lem.bernstein\], we have $$\begin{aligned}
&-2\int_\Omega \nabla\sqrt c\cdot\nabla ^Tu\nabla \sqrt{c}dx\\
&\leq
\frac{1}{16}\int_{\Gamma}|{\nabla}\log c| ^2\kappa \left(\gamma- c\right)d\mathcal{H}^{d-1}_x+\frac{2+d}{16}{\int_{\Omega}}c|\nabla^2\log c|^2
+4\int_\Omega c|\nabla ^Tu|^2dx
\\&
\leq
\frac{1}{4}\int_{\Gamma}|{\nabla}\sqrt c|^2 \kappa \left(\frac{\gamma}c- 1\right)d\mathcal{H}^{d-1}_x+\frac{5}{16}{\int_{\Omega}}c|\nabla^2\log c|^2
+4\|c\|_{L^\infty(\Omega)}\|u\|_{H^1(\Omega)}^2
\end{aligned}$$ since $d\leq 3$. Using this for the estimate from Lemma \[lem.entropy.equality.part2\] entails $$\begin{gathered}
\partial_t\int_{\Omega}|\nabla\sqrt{c}|^2dx
+\frac18 \int_{\Omega}c|\nabla^2\log c|^2dx
+\int_\Omega|\nabla \sqrt{c}|^2ndx
\\\leq\int_{\Omega}\Delta(|\nabla\sqrt{c}|^2)dx
- \frac12\int_{\Omega}\nabla n\cdot\nabla cdx+
\frac{1}{4}\int_{\Gamma}|{\nabla}\sqrt c|^2 \kappa \left(\frac{\gamma}c- 1\right)d\mathcal{H}^{d-1}_x
+4\|c\|_{L^\infty(\Omega)}\|u\|_{H^1(\Omega)}^2
\end{gathered}$$ Finally, we use again Lemma \[lem.bernstein\] to yield the desired assertion.
Looking at Lemma \[cor.entropy.equality.part2\], in the case of homogeneous Neumann boundary condition ${{\partial}_\nu}c = 0$ and convex domain, as considered in [@Win12], we have ${{\partial}_\nu}|{\nabla}\sqrt{c}|^2 \leq 0$ and therefore the term $\int_{\Omega}\Delta(|\nabla \sqrt{c}|^2)dx$ can be eliminated immediately. In the present paper, since the boundary condition is inhomogeneous and the domain is possibly not convex, we will have to deal with this term differently. Our key idea is to consider the boundary energy (see Lemma \[boundary\_energy\]). Before that we derive some useful estimates.
\[lem.1.bd.integral\] It holds for any smooth function $c$ satisfying $\partial_\nu c = \kappa(x)(\gamma(x) - c)$ on $\Gamma$ that $$\begin{gathered}
\label{eq.1.bd.integral}
\int_{\Omega}\Delta(|\nabla\sqrt c|^2)+\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+\frac{\gamma}{2c} \right)d\mathcal H^{d-1}_x
+2\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x
\\\leq\int_{\Gamma} \partial_{\nu}|\partial_\nu\sqrt{c} |^2d\mathcal{H}^{d-1}_x
+{
\int_{\Gamma}\frac{4|\nabla_\Gamma \sqrt{\kappa} |^2}{\gamma} \left(\gamma-c\right)^2d\mathcal H^{d-1}_x}.
\end{gathered}$$
First, we see by Gauß’ theorem that $$\begin{aligned}
\frac12\int_{\Omega}\Delta(|\nabla\sqrt{c}|^2) dx &=\frac12\int_{\Gamma}\nu\cdot\nabla (|\nabla\sqrt{c}|^2) d\mathcal{H}^{d-1}_x
\end{aligned}$$ Notice that $$\begin{aligned}
\frac12\nu\cdot\nabla(|\nabla \sqrt{c}|^2)
&=\nu\cdot \nabla^2 \sqrt{c}\nabla\sqrt{c}
\end{aligned}$$ Let $x\in \Gamma$ and $\vartheta:\mathbb R\to \mathbb R^d$ be differentiable such that $\vartheta(0)=x$ and $\vartheta'(0)=\nabla\sqrt{c(x)}$. Then $$\begin{aligned}
\nu(x)\cdot &\nabla^2 \sqrt{c(x)}\sqrt{c(x)}
=
\nu(x)\cdot \nabla^T(\nabla \sqrt{c})\circ\vartheta(0)\vartheta'(0)
\\&=\nu\circ\vartheta(x)\cdot\frac{d}{ds}\bigg|_{s=0}(\nabla \sqrt{c})\circ\vartheta(s)
\\&=\frac{d}{ds}\bigg|_{s=0}(\nu\cdot\nabla \sqrt{c})\circ\vartheta(s)
-\vartheta'(0)\cdot \nabla^T\nu(x)\nabla \sqrt{c(x)}
\\&=\vartheta'(0)\cdot \nabla (\partial_{\nu}\sqrt{c}(x))-\vartheta'(0)\cdot \nabla^T\nu(x)\vartheta'(0)
\end{aligned}$$ Now, let $\tau=(1-\nu\otimes\nu)\vartheta'(0)=(1-\nu\otimes\nu)\nabla\sqrt{c} $ be the projection of $\vartheta'(0)$ onto $T_x\Gamma$. It holds (at $x$) $$\begin{aligned}
\vartheta'(0)\cdot \nabla (\partial_{\nu}\sqrt{c})
&=\partial_\tau( \partial_{\nu}\sqrt{c})+\nu \cdot \nabla\sqrt{c} \nu\cdot \nabla ( \partial_{\nu}\sqrt{c})
\\
&=\partial_\tau( \partial_{\nu}\sqrt{c})+ \partial_\nu\sqrt{c} \partial_{\nu} ( \partial_{\nu}\sqrt{c})
\\
&=\partial_\tau( \partial_{\nu}\sqrt{c})
+ \partial_\nu\sqrt{c} \partial_{\nu}^2\sqrt{c}
\end{aligned}$$ Using the fact that $\tau\in T_x\Gamma$ and the boundary condition for $\sqrt{c}$ in , we obtain $$\begin{aligned}
\partial_\tau ( \partial_\nu \sqrt{c} )
&=\partial_\tau \left(\frac{\kappa }{2 } \left(\frac{\gamma}{\sqrt{c}} -\sqrt{c} \right) \right)
\\&=\partial_\tau \kappa \left(\frac{1}{2} \left(\frac{\gamma}{\sqrt{c}} -\sqrt{c} \right) \right)
-\partial_\tau\sqrt{c} \frac{\kappa }{2 } \left(1+\frac{\gamma}{c} \right)
\\&=\partial_\tau \log \kappa \partial_\nu \sqrt{c}
-\partial_\tau\sqrt{c} \frac{\kappa }{2 } \left(1+\frac{\gamma}{c} \right).
\end{aligned}$$ Thus, by inserting $\tau=(1-\nu\otimes\nu)\nabla\sqrt{c} $, this entails $$\begin{aligned}
\partial_\tau \partial_\nu \sqrt{c}
&=
\sqrt{c_\infty}\nabla\sqrt{c} \cdot\nabla\log\kappa\partial_\nu \sqrt{c}
-\partial_\nu\sqrt{c} \partial_\nu\log \kappa\partial_\nu \sqrt{c}
\\&\qquad
-\nabla\sqrt{c} \cdot\nabla\sqrt{c} \frac{\kappa}{2} \left(1+\frac{\gamma}{c} \right)
+\partial_\nu\sqrt{c} \partial_\nu\sqrt{c} \frac{\kappa}{2} \left(1+\frac{\gamma}{c} \right)
\\&=
\nabla\sqrt{c} \cdot\nabla\log\kappa\partial_\nu \sqrt{c}
- |\partial_\nu\sqrt{c} |^2\partial_\nu\log\kappa
\\&\qquad
- (\underbrace{ |\nabla\sqrt{c} |^2- |\partial_\nu\sqrt{c} |^2 }_{=|\nabla_{\Gamma}\sqrt{c}|^2})\frac{\kappa}{2} \left(1+\frac{\gamma}{c} \right).
\end{aligned}$$ Thus, combining these calculations yields $$\begin{aligned}
\frac12\nu\cdot\nabla(|\nabla \sqrt{c}|^2)
&=\vartheta'(0)\cdot \nabla (\partial_{\nu}\sqrt{c})-\vartheta'(0)\cdot \nabla^T\nu\vartheta'(0)
\\&=
\partial_\tau\partial_{\nu}\sqrt{c}+ \frac12\partial_{\nu}|\partial_\nu\sqrt{c} |^2
-\vartheta'(0)\cdot \nabla^T\nu\vartheta'(0)
\\&=
\nabla\sqrt{c} \cdot\nabla\log\kappa\partial_\nu \sqrt{c}
- |\partial_\nu\sqrt{c} |^2\partial_\nu\log\kappa-|\nabla_{\Gamma}\sqrt{c}|^2\frac{\kappa}{2} \left(1+\frac{\gamma}{c} \right)
\\&\qquad + \frac12\partial_{\nu}|\partial_\nu\sqrt{c} |^2-\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}
\\&=
\nabla_{\Gamma}\sqrt{c} \cdot\nabla_\Gamma\log\kappa\partial_\nu \sqrt{c}
-|\nabla_{\Gamma}\sqrt{c}|^2\frac{\kappa}{2} \left(1+\frac{\gamma}{c} \right)
\\&\qquad + \frac12\partial_{\nu}|\partial_\nu\sqrt{c} |^2-\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}
\end{aligned}$$ Using $$\begin{aligned}
\nabla_{\Gamma}\sqrt{c} \cdot\nabla_\Gamma\log \kappa \partial_\nu \sqrt{c}
=\frac{\nabla_{\Gamma}c}{2\sqrt c} \cdot\nabla_\Gamma\log \kappa \frac{\kappa }{2}\left(\frac{\gamma}{\sqrt{c}}-\sqrt{c}\right)
=\frac{\nabla_{\Gamma}c}{4 c} \cdot\nabla_\Gamma \kappa \left(\gamma-c\right),
\end{aligned}$$ this shows $$\begin{gathered}
\int_{\Omega}\Delta(|\nabla\sqrt c|^2)+\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+\frac{\gamma}{c} \right)d\mathcal H^{d-1}_x
+2\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x
\\=\int_{\Gamma} \partial_{\nu}|\partial_\nu\sqrt{c} |^2d\mathcal{H}^{d-1}_x
+
\int_{\Gamma}\frac{\nabla_{\Gamma}c}{2 c} \cdot\nabla_\Gamma \kappa \left(\gamma-c\right)d\mathcal H^{d-1}_x.
\end{gathered}$$ Finally, we use Young’s inequality to see that $$\begin{aligned}
\int_{\Gamma}\frac{\nabla_{\Gamma}c}{2 c} \cdot\nabla_\Gamma \kappa \left(\gamma-c\right)d\mathcal H^{d-1}_x
&\leq
\int_{\Gamma}\underbrace{\frac{|\nabla_{\Gamma}c|^2}{8 c^2} \gamma}_{\frac12 |\nabla_{\Gamma}\sqrt c|^2\frac{\gamma}{c}} \kappa d\mathcal H^{d-1}_x
+
\int_{\Gamma}\frac{4|\nabla_\Gamma \sqrt{\kappa}|^2}{\gamma}\left(\gamma-c\right)^2 d\mathcal H^{d-1}_x,
\end{aligned}$$ which finishes the proof.
To control the first term on the right hand side of , our key idea is to introduce a “boundary energy" of the form $\int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}$ (where $s^\infty$ defined in ), whose time derivative produces the first term on the right-hand side of with an opposite sign (see the last term on the right-hand side of ).
\[boundary\_energy\] For all $t\in (0,T)$, it holds $$\label{p2}
\begin{aligned}
\frac{d}{dt}\int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}&+
\frac34\int_{\Gamma}\gamma|\nabla_{\Gamma}c|^2\frac{\kappa }{c^2}d\mathcal{H}^{d-1}_x
+
4\int_{\Gamma}\nabla\cdot\nu |\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
\\&+
2\int_{\Gamma}(\partial_{\nu}\sqrt c)^2c\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x
\\
&\leq
{8}
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\sqrt{\kappa} |^2}{ \gamma}\left(c-\gamma\right)^2d\mathcal{H}^{d-1}_x
+{
2\int_{\Gamma}\frac{ |\nabla_{\Gamma}\gamma|^2}{\gamma}\kappa d\mathcal{H}^{d-1}_x}
\\ &+
\int_{\Gamma}n\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x
-
2\int_{\Gamma}\partial_{\nu}|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
.
\end{aligned}$$
Thanks to the relation between the Laplace operator and Laplace-Beltrami operator $
\Delta_{\Gamma}=\Delta-\nabla\cdot\nu \partial_\nu-\partial_\nu^2$ on $\Gamma$, and the fact that $u = 0$ on $\Gamma$, we have $$\begin{aligned}
\label{eq4}
\partial_t{c}&=
\Delta_{\Gamma}{c}+\partial_\nu^2{c}+ \nabla\cdot\nu \partial_\nu {c}-{ c}n \quad \text{ on } \Gamma.
\end{aligned}$$ We can therefore compute $$\begin{aligned}
\frac{d}{dt}\int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}&=\int_{\Gamma}\partial_t c\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x
\\&=
\int_{\Gamma}\Delta_{\Gamma }c\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x
+
\int_{\Gamma}\nabla\cdot \nu\partial_{\nu}c\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x
\\&\qquad+
\int_{\Gamma}\partial_\nu^2 c\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x-
\int_{\Gamma}cn\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x
\end{aligned}$$ Using integration by parts, we have $$\begin{aligned}
\int_{\Gamma}\Delta_{\Gamma }c\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x&=
-
\int_{\Gamma}\nabla_{\Gamma}c\cdot \nabla_{\Gamma}\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x-
\int_{\Gamma}\gamma|\nabla_{\Gamma}c|^2\frac{\kappa }{c^2}d\mathcal{H}^{d-1}_x
\\&\qquad{+
\int_{\Gamma}\frac{\kappa }{c}\nabla_{\Gamma}c\cdot \nabla_{\Gamma}\gamma d\mathcal{H}^{d-1}_x}.
\end{aligned}$$ We calculate $$\begin{aligned}
\partial_\nu^2 c\kappa \left(1-\frac\gamma{c}\right)&=4\sqrt c \partial_{\nu}^2\sqrt c\frac{\kappa }2\left(1-\frac\gamma{c}\right)+2(\partial_{\nu}\sqrt c)^2\kappa \left(1-\frac\gamma{c}\right)
\\&=
4\sqrt c \partial_{\nu}^2\sqrt c\frac{\kappa }2\left(1-\frac\gamma{c}\right)+2(\partial_{\nu}\sqrt c)^2\kappa \left(1-\frac\gamma{c}\right)
\\&=
-4\partial_{\nu}\sqrt c \partial_{\nu}^2\sqrt c+2(\partial_{\nu}\sqrt c)^2\kappa \left(1-\frac\gamma{c}\right)
\\&=
-2\partial_{\nu}|\partial_{\nu}\sqrt c|^2+2(\partial_{\nu}\sqrt c)^2\kappa \left(1-\frac\gamma{c}\right)
\end{aligned}$$ using $\partial_{\nu}\sqrt{c}=\frac{\kappa }{2}\left(\frac{\gamma}{\sqrt{c}}-\sqrt{c}\right)=-\frac{\kappa }{2}\sqrt{c}\left(1-\frac{\gamma}{c}\right)$. Combing these equalities yields $$\begin{aligned}
\frac{d}{dt}\int_{\Gamma}\kappa s^\infty(\gamma|c)dx&=
-
\int_{\Gamma}\nabla_{\Gamma}c\cdot \nabla_{\Gamma}\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x-
\int_{\Gamma}\gamma|\nabla_{\Gamma}c|^2\frac{\kappa }{c^2}d\mathcal{H}^{d-1}_x
\\&\qquad
+
\int_{\Gamma}\nabla\cdot \nu\partial_{\nu}c\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x-
2\int_{\Gamma}\partial_{\nu}|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
\\&\qquad+
2\int_{\Gamma}(\partial_{\nu}\sqrt c)^2\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x
-
\int_{\Gamma}cn\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x\\
&\qquad + \int_{\Gamma}\frac{\kappa }{c}\nabla_{\Gamma}c\cdot \nabla_{\Gamma}\gamma d\mathcal{H}^{d-1}_x.
\end{aligned}$$ It remains to estimate the first, the third and the last terms on the right-hand side. Finally Young’s inequality implies $$\begin{aligned}
-
\int_{\Gamma}\nabla_{\Gamma}c\cdot \nabla_{\Gamma}\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x
&\leq {\frac18}
\int_{\Gamma}\gamma|\nabla_{\Gamma}c|^2\frac{\kappa }{c^2}d\mathcal{H}^{d-1}_x
+{8}
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\sqrt{\kappa} |^2}{\gamma}\left(c-\gamma\right)^2d\mathcal{H}^{d-1}_x
\end{aligned}$$ [and]{} $$\begin{aligned}
{
\int_{\Gamma}\frac{\kappa }{c}\nabla_{\Gamma}c\cdot \nabla_{\Gamma}\gamma d\mathcal{H}^{d-1}_x}&{\leq
\frac18
\int_{\Gamma}\gamma|\nabla_{\Gamma}c|^2\frac{\kappa }{c^2}d\mathcal{H}^{d-1}_x+2
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\gamma|^2}{\gamma}\kappa d\mathcal{H}^{d-1}_x.}
\end{aligned}$$ These two estimates and the identity $$\begin{aligned}
\int_{\Gamma}\nabla\cdot \nu\partial_{\nu}c\kappa \left(1-\frac\gamma{c}\right)d\mathcal{H}^{d-1}_x
&=
\int_{\Gamma}\nabla\cdot\nu \frac{\partial_{\nu}c}{c}\kappa \left(c-\gamma\right)d\mathcal{H}^{d-1}_x
\\&=-
\int_{\Gamma}\nabla\cdot\nu \frac{|\partial_{\nu}c|^2}{c}d\mathcal{H}^{d-1}_x
\\&=-
4\int_{\Gamma}\nabla\cdot\nu |\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x\end{aligned}$$ imply the assertion of Lemma \[boundary\_energy\].
By combining Lemmas \[lem.1.bd.integral\] and \[boundary\_energy\], we have the following lemma.
\[lem.entropy.equality.part3\] It holds for all $t\in (0,T)$ that $$\begin{gathered}
\frac{d}{dt}\int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}+
2\int_{\Omega}\Delta(|\nabla\sqrt c|^2)dx+2\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+2\frac{\gamma}{c} \right)d\mathcal H^{d-1}_x
\\ +4\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x
+
2\int_{\Gamma}\left(2\nabla\cdot\nu +c\kappa (\gamma-c)\right)|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
\\
\leq
{16
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\sqrt\kappa |^2}{ \gamma}\left(c-\gamma\right)^2d\mathcal{H}^{d-1}_x
+
2\int_{\Gamma}\frac{ |\nabla_{\Gamma}\gamma|^2}{\gamma}\kappa d\mathcal{H}^{d-1}_x}
+
\int_{\Gamma}n\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x.
\end{gathered}$$
We add the following two estimates $$\begin{gathered}
\int_{\Omega}\Delta(|\nabla\sqrt c|^2)dx+\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+\frac{\gamma}{2c} \right)d\mathcal H^{d-1}_x
+2\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x
\\\leq\int_{\Gamma} \partial_{\nu}|\partial_\nu\sqrt{c} |^2d\mathcal{H}^{d-1}_x
+
4\int_{\Gamma}\frac{|\nabla_\Gamma \sqrt\kappa |^2}{ \gamma} \left(\gamma-c\right)^2d\mathcal H^{d-1}_x
\end{gathered}$$ and $$\begin{gathered}
\frac12\frac{d}{dt}\int_{\Gamma}\kappa s^\infty(\gamma|c)dx+
\frac32\int_{\Gamma}\gamma|\nabla_{\Gamma}\sqrt c|^2\frac{\kappa }{c}d\mathcal{H}^{d-1}_x
\\+
2\int_{\Gamma}\nabla\cdot\nu |\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
+
\int_{\Gamma}(\partial_{\nu}\sqrt c)^2c\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x
\\\leq
4\int_{\Gamma}\frac{ |\nabla_{\Gamma}\sqrt\kappa |^2}{ \gamma}\left(c-\gamma\right)^2d\mathcal{H}^{d-1}_x
+
{
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\gamma|^2}{\gamma}\kappa d\mathcal{H}^{d-1}_x}
\\ +
\frac12\int_{\Gamma}n\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x
-
\int_{\Gamma}\partial_{\nu}|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
\end{gathered}$$ to obtain $$\begin{gathered}
\frac12\frac{d}{dt}\int_{\Gamma}\kappa s^\infty(\gamma|c)dx+
\int_{\Omega}\Delta(|\nabla\sqrt c|^2)+\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+2\frac{\gamma}{c} \right)d\mathcal H^{d-1}_x
\\ +2\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x
+
\int_{\Gamma}\left(2\nabla\cdot\nu +c\kappa (\gamma-c)\right)|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
\\
\leq
8
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\sqrt\kappa |^2}{\gamma}\left(c-\gamma\right)^2d\mathcal{H}^{d-1}_x
+
{
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\gamma|^2}{\gamma}\kappa d\mathcal{H}^{d-1}_x}
+
\frac12\int_{\Gamma}n\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x.\qedhere
\end{gathered}$$
From Lemmas \[lem.ee.n\], \[cor.entropy.equality.part2\], \[lem.entropy.equality.part3\] we have the preliminary energy estimates for $n$ and $c$.
\[energy\] For all $t\in (0,T)$ we have $$\label{energy_1}
\begin{aligned}
\frac d{dt}\left(\int_{\Omega}s(n)dx+ 2\int_{\Omega}|\nabla\sqrt{c}|^2 dx+\int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}\right)
\\+4\int_{\Omega}\left|\nabla\sqrt{n}\right|^2 dx
+{\varepsilon}\int_{\Omega}n(n^2-1)\log ndx
\\ +\frac14 \int_{\Omega}c|\nabla^2\log c|^2dx
+\xi \int_\Omega|\nabla \sqrt[4]{c}|^4dx
+2\int_\Omega|\nabla \sqrt{c}|^2ndx
\\+2\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+\frac{\gamma}{c} \right)d\mathcal H^{d-1}_x
+4\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x
\\+
2\int_{\Gamma}\left(2\nabla\cdot\nu +c\kappa (\gamma-c)\right)|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
\\
\leq
{16
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\sqrt\kappa |^2}{\gamma}\left(c-\gamma\right)^2d\mathcal{H}^{d-1}_x
+
2\int_{\Gamma}\frac{ |\nabla_{\Gamma}\gamma|^2}{\gamma}\kappa d\mathcal{H}^{d-1}_x}
\\
+
\int_{\Gamma}n\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x
+8\|c\|_{L^\infty(\Omega)}\|u\|_{H^1(\Omega)}^2.
\end{aligned}$$
We recall Lemma \[lem.ee.n\] $$\frac d{dt}\int_{\Omega}s(n)dx+4\int_{\Omega}\left|\nabla\sqrt{n}\right|^2 dx+{\varepsilon}\int_{\Omega}n\log n(1-n^2)dx
=\int_{\Omega}\nabla c\cdot \nabla ndx$$ and Lemma \[cor.entropy.equality.part2\] $$\begin{gathered}
2\partial_t\int_{\Omega}|\nabla\sqrt{c}|^2dx
+\frac14 \int_{\Omega}|\nabla^2\log c|^2cdx
+\xi \int_\Omega|\nabla \sqrt[4]{c}|^4dx
+2\int_\Omega|\nabla \sqrt{c}|^2ndx
\\\leq2\int_{\Omega}\Delta(|\nabla\sqrt{c}|^2)dx
- \int_{\Omega}\nabla n\cdot\nabla cdx+
\int_{\Gamma}|{\nabla}\sqrt c|^2 \kappa \left(\frac{\gamma}c- 1\right)d\mathcal{H}^{d-1}_x
+8\int_\Omega c|\nabla ^Tu|^2dx\end{gathered}$$ for some $\xi>0$ and Lemma \[lem.entropy.equality.part3\] $$\begin{gathered}
\frac{d}{dt}\int_{\Gamma}\kappa s^\infty(\gamma|c)dx+
2\int_{\Omega}\Delta(|\nabla\sqrt c|^2)dx+2\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+2\frac{\gamma}{c} \right)d\mathcal H^{d-1}_x
\\ +4\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x
+
2\int_{\Gamma}\left(2\nabla\cdot\nu +c\kappa (\gamma-c)\right)|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x
\\
\leq
{16
\int_{\Gamma}\frac{ |\nabla_{\Gamma}\sqrt\kappa |^2}{\gamma}\left(c-\gamma\right)^2d\mathcal{H}^{d-1}_x
+
2\int_{\Gamma}\frac{ |\nabla_{\Gamma}\gamma|^2}{\gamma}\kappa d\mathcal{H}^{d-1}_x}
+
\int_{\Gamma}n\kappa \left(\gamma-{c}\right)d\mathcal{H}^{d-1}_x.\end{gathered}$$ By adding these three relations, we obtain the assertion of Lemma \[energy\].
The form of energy estimate in Lemma \[energy\] is particularly suited for convex domains as then $\nabla\cdot \nu\geq0$ and $\nabla^T\nu$ is positive semi-definite on $\Gamma$. Therefore, the terms involving $\nabla\cdot\nu$ and $\nabla^T\nu$ on the l.h.s. in the energy estimate can be neglected as they are non-negative.
When the domain is not convex, we show in the next lemma that these terms can be controled using the higher order terms $\int_{\Omega}c|{\nabla}^2\log c|^2dx$ and $\int_{\Omega}|{\nabla}\sqrt[4]{c}|^4dx$.
\[lem:energy2\] There exists a $\lambda>0$ and a $C>0$ depending on $\kappa,\gamma,\|c\|_{L^\infty}$ and the curvature of $\Gamma$ such that $$\begin{aligned}
\frac d{dt}\left(\int_{\Omega}s(n)dx+ 2\int_{\Omega}|\nabla\sqrt{c}|^2 dx+\int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}\right)
\\+4\int_{\Omega}\left|\nabla\sqrt{n}\right|^2 dx
+{\varepsilon}\int_{\Omega}n(n^2-1)\log ndx
\\ +\lambda\int_{\Omega}|\nabla^2\sqrt{c}|^2dx+\lambda \int_\Omega|\nabla \sqrt[4]{c}|^4dx
+2\int_\Omega|\nabla \sqrt{c}|^2ndx
\\+2\int_{\Gamma}|\nabla_{\Gamma}\sqrt{c}|^2
\kappa \left(1+\frac{\gamma}{c} \right)d\mathcal H^{d-1}_x
\\
\leq
C\left(1+\int_{\Omega}|\nabla\sqrt{c}|^2dx\right)+8\|c\|_{L^\infty(\Omega)}\|u\|_{H^1(\Omega)}^2.
\end{aligned}$$
From Lemma \[energy\], we need to control the tenth and eleventh terms on the left-hand side, and the first, second and third terms on the right-hand side of .
The first step is to show that we can estimate the $H^2(\Omega)$ norm of $\sqrt c$. We need to have a closer look at the integral involving $ |\nabla^2\log c|^2c$. Using the chain rule and $|a+b|^2\geq \frac 12|a|^2- |b|^2$, we have $$\begin{aligned}
|\nabla^2\log c|^2c
&=
\left|\sqrt{c}\nabla\left(\frac{\nabla {c}}{{c}}\right)\right|^2
=
4\left|\sqrt{c}\nabla\left(\frac{\nabla \sqrt{c}}{\sqrt{c}}\right)\right|^2
\\&=4
\left|\nabla^2\sqrt{c}-\frac{1}{\sqrt{c}}\nabla\sqrt{c}\otimes\nabla\sqrt{c}\right|^2
\\&
\geq 2|{\nabla}^2\sqrt{c}|^2-4\frac{|\nabla\sqrt{c}|^4}{c}=2|{\nabla}^2\sqrt{c}|^2-4\cdot{2^4}{|\nabla\sqrt[4]{c}|^4}.
\end{aligned}$$ This directly implies $$\begin{aligned}
\int_{\Omega}|\nabla^2\sqrt{c}|^2 dx\leq \frac12 \int_{\Omega} |\nabla^2\log c|^2c\,dx + 32\int_{\Omega}{|\nabla\sqrt[4]{c}|^4}dx.
\end{aligned}$$ We use the fact that (see e.g. [@Gri85 Theorem 1.5.1.10] for any $\theta>0$, there exists $C_\theta > 0$ such that $$\label{trace}
\int_{\Gamma}|f|^2{d\mathcal{H}_x^{d-1}}\leq \theta\int_{\Omega}|\nabla f|^2dx + C_{\theta}\int_{\Omega}|f|^2dx.$$
The tenth term on the left hand side of can be estimated as, for any $\theta>0$, $$\begin{aligned}
\left|4\int_{\Gamma}\nabla \sqrt{c}\cdot \nabla^T\nu\nabla \sqrt{c}d\mathcal H^{d-1}_x\right|
&\leq 4\|\nabla^T\nu\|_{L^\infty(\Gamma)}\|\nabla\sqrt c\|_{L^2(\Gamma)}^2
\\&=\theta \int_{\Omega}|\nabla^2\sqrt{c}| dx +C_{\theta,\Gamma}\int_{\Omega}|\nabla\sqrt{c}|^2 dx
\end{aligned}$$ for some $C_{\theta,\Gamma}>0$ depending on $\theta$. Likewise for the first part of the eleventh term, for any $\theta > 0$, $$\begin{aligned}
\left|4\int_{\Gamma}\nabla\cdot\nu|\partial_{\nu}\sqrt c|^2d\mathcal{H}^{d-1}_x\right|
&\leq
\theta \int_{\Omega}|\nabla^2\sqrt{c}| dx +C_{\theta,\Gamma}\int_{\Omega}|\nabla\sqrt{c}|^2 dx,
\end{aligned}$$ using that $\kappa$ and $\gamma$ are uniformly bounded. The second part of the eleventh term on the left-hand side of is estimated as follows $$\begin{aligned}
&\left|2\int_{\Gamma}c\kappa(\gamma - c)|{{\partial}_\nu}\sqrt{c}|^2{d\mathcal{H}_x^{d-1}}\right|\\
&\leq 2\|\kappa\|_{L^\infty(\Gamma)}\|\gamma\|_{L^\infty(\Gamma)}\int_{\Gamma}c|{{\partial}_\nu}\sqrt c|^2{d\mathcal{H}_x^{d-1}}+ 2\|\kappa\|_{L^{\infty}(\Gamma)}\int_{\Gamma}c^2|{{\partial}_\nu}\sqrt c|^2{d\mathcal{H}_x^{d-1}}\\
&\leq C\int_{\Gamma}|{{\partial}_\nu}c|^2{d\mathcal{H}_x^{d-1}}+ C\int_{\Gamma}c|{{\partial}_\nu}c|^2{d\mathcal{H}_x^{d-1}}\\
&= C\int_{\Gamma}\kappa^2(\gamma - c)^2{d\mathcal{H}_x^{d-1}}+ C\int_{\Gamma}c\kappa^2(\gamma - c)^2{d\mathcal{H}_x^{d-1}}\\
&\leq C\left(1 + \int_{\Gamma}|c|^3{d\mathcal{H}_x^{d-1}}\right)\\
&\leq C\left(1 + \int_{\Omega}c^{\frac 32}|{\nabla}\sqrt c|^2dx + \|c\|_{L^3(\Omega)}^3\right)\\
&\leq C\left(1+\int_{\Omega}|{\nabla}\sqrt c|^2dx\right).
\end{aligned}$$ The second term on the right-hand side of is bounded by $$\left|2\int_{\Gamma}\frac{|{\nabla}_\Gamma \gamma|^2}{\gamma}\kappa {d\mathcal{H}_x^{d-1}}\right| \leq \|\kappa\|_{L^\infty(\Gamma)}\|{\nabla}\sqrt{\gamma}\|_{L^2(\Gamma)}^2 \leq C.$$ The third term on the right-hand side of is estimated as $$\begin{aligned}
\int_{\Gamma}n\kappa(\gamma - c){d\mathcal{H}_x^{d-1}}&\leq \|\kappa\|_{L^\infty(\Gamma)}\|\gamma\|_{L^\infty(\Gamma)}\int_{\Gamma}|n|{d\mathcal{H}_x^{d-1}}\\
&\leq C\int_{\Gamma}|\sqrt{n}|^2{d\mathcal{H}_x^{d-1}}\\
&\leq 2\int_{\Omega}|{\nabla}\sqrt n|^2dx + C\int_{\Omega}|\sqrt n|^2dx\\
&\leq 2\int_{\Omega}|{\nabla}\sqrt n|^2dx + C
\end{aligned}$$ thanks to the Trace inequality and the fact that $\|n\|_{L^1(\Omega)} \leq C$ in Lemma \[L1Linf\]. We estimate the first term on the right hand side of as $$\begin{aligned}
\label{b0}
\left|16\int_{\Gamma}\frac{|{\nabla}_\Gamma \sqrt\kappa|^2}{\gamma}(c-\gamma)^2{d\mathcal{H}_x^{d-1}}\right| &\leq 32\|{\nabla}\sqrt{\kappa}\|_{L^2(\Gamma)}^2\|1/\gamma\|_{L^\infty(\Gamma)}\left(\|\gamma\|_{L^\infty(\Gamma)}^2 + \|c\|_{L^\infty(\Gamma)}^2\right).
\end{aligned}$$ We now show that $$\label{b1}
\|c\|_{L^\infty(\Gamma)}^2 \leq C\left(1+ \|\nabla \sqrt{c}\|_{L^2(\Omega)}^2\right).$$ Indeed, for any $p>2$, we have $$\begin{aligned}
\|c\|_{L^p(\Gamma)}^p &=\int_{\Gamma}\left(|\sqrt{c}|^p\right)^2{d\mathcal{H}_x^{d-1}}\\
&\leq C\left(\frac{p^2}{4}\int_{\Omega}|c|^{p-1}|\nabla \sqrt c|^2dx + \int_{\Omega}|c|^pdx \right)\\
&\leq C\left(p^2\|c\|_{L^\infty(\Omega)}^{p-1}\|\nabla \sqrt c\|_{L^2(\Omega)}^2 + \|c\|_{L^p(\Omega)}^p\right)\\
&\leq C\left(\|\nabla \sqrt{c}\|_{L^2(\Omega)}^p + p^{\frac{2p}{p-2}}\|c\|_{L^\infty(\Omega)}^{\frac{p(p-1)}{p-2}} + \|c\|_{L^p(\Omega)}^p \right).
\end{aligned}$$ By taking root with order $p$ of both sides and letting $p\to\infty$, we get $$\|c\|_{L^\infty(\Gamma)} \leq C\left(\|\nabla \sqrt{c}\|_{L^2(\Omega)} + \|c\|_{L^\infty(\Omega)}\right),$$ hence thanks to the boundedness of $\|c\|_{L^\infty(\Omega)}$. Inserting into , we have controlled the first term on the right-hand side of , and thus completes the proof of Lemma \[lem:energy2\].
\[energy-NS\] For any $\delta > 0$, there exists $C_\delta$ depending on $\delta$ and $\|\varphi\|_{W^{1,\rho}(\Omega)}$ such that $$\label{a}
\frac{d}{dt}\int_{\Omega}|u|^2dx + C(\mu)\|u\|_{H^1(\Omega)}^2 \leq C + \delta\int_{\Omega}|{\nabla}\sqrt n|^2dx.$$
From the well-known energy estimate for the approximate Navier-Stokes equations in and the Poincaré inequality $\|\nabla u\|_{L^2(\Omega)} \geq C\|u\|_{L^2(\Omega)}$, we have $$\label{aa}
\frac{d}{dt}\|u\|_{L^2(\Omega)}^2 + C(\mu)\|u\|_{H^1(\Omega)}^2 \leq 2\left|\int_{\Omega}n{\nabla}\varphi \cdot u dx\right|.$$ We now show that for any $\delta_0, \delta_1>0$, $$\label{aaa}
\left|2\int_{\Omega}n\nabla \varphi \cdot udx\right| \leq C + \delta_0\|n\|_{L^{3}(\Omega)} + \delta_1\|u\|_{H^1(\Omega)}^2.$$ By Hölder’s inequality and the continuous embedding $H^1(\Omega) \hookrightarrow L^6(\Omega)$ (since $d\leq 3$) we have $$\left|\int_{\Omega}n\nabla \varphi \cdot udx\right| \leq \|u\nabla \varphi \|_{L^{\frac 65}(\Omega)}\|u\|_{L^6(\Omega)} \leq C\|u\nabla \varphi \|_{L^{\frac 65}(\Omega)}\|u\|_{H^1(\Omega)} \leq C\|u\nabla \varphi\|_{L^{\frac 65}(\Omega)}^2 + \delta_1\|u\|_{H^1(\Omega)}^2.$$ Let $\eta = \frac{5\rho}{6} > 5$ and $\beta = \frac{\eta}{\eta - 1} < \frac 54$ (recalling $\rho>6$ in ). By Hölder’s inequality again, it follows that $$\begin{aligned}
\label{bb}
\|n\nabla \varphi\|_{L^{\frac 65}(\Omega)}^2
\leq \|{\nabla}\varphi\|_{L^{\frac 65 \eta}(\Omega)}^2\|n\|_{L^{\frac 65 \beta}(\Omega)}^2.
\end{aligned}$$ By using the interpolation inequality with $$\begin{aligned}
\label{bbb}
\|n\|_{L^{\frac 65\beta}(\Omega)}^2 \leq \|n\|_{L^1(\Omega)}^{2\theta}\|n\|_{L^3(\Omega)}^{2(1-\theta)} \quad \text{ with } \quad \frac{5}{6\beta} = \frac{\theta}{1} + \frac{1-\theta}{3}.
\end{aligned}$$ From that $\theta = \frac{5-2\beta}{4\beta}$ and therefore $$2(1-\theta) = \frac{6\beta - 5}{2\beta} < 1$$ since $\beta < \frac 54$. From , and $\|n\|_{L^1(\Omega)}\leq C$ we obtain $$C\|n{\nabla}\varphi\|_{L^{\frac 65}(\Omega)}^2 \leq C\|\varphi\|_{W^{1,\rho}(\Omega)}^2C^{2\theta}\|n\|_{L^3(\Omega)}^{2(1-\theta)} \leq C + \delta_0\|n\|_{L^3(\Omega)}$$ where we used Young’s inequality at the last step, due to $2(1-\theta)<1$. To obtain from and , it remains to show that $$\|n\|_{L^3(\Omega)} \leq C\left(1+\int_{\Omega}|{\nabla}\sqrt n|^2dx\right).$$ Indeed, thanks to the continuous three dimensional embedding $H^1(\Omega)\hookrightarrow L^6(\Omega)$, we have $$\|n\|_{L^3(\Omega)} = \|\sqrt n\|_{L^6(\Omega)}^2 \leq C\left(\int_{\Omega}|{\nabla}\sqrt n|^2dx + \int_{\Omega}|\sqrt n|^2dx\right) \leq C\left(1 + \int_{\Omega}|{\nabla}\sqrt n|^2dx\right)$$ thanks $\|n\|_{L^1(\Omega)} \leq C$ from Lemma \[L1Linf\].
\[a-priori\] The following [*a priori*]{} estimtates hold uniformly in ${\varepsilon}\geq 0$ and $m\in \mathbb N$, $$\begin{aligned}
&\sup_{t\in (0,T)}\left(\int_{\Omega}n(t)\log n(t) dx + \|{\nabla}\sqrt{c}(t)\|_{L^2(\Omega)}^2 + \|u(t)\|_{L^2(\Omega)}^2\right) \leq C_T,\\
&\int_0^T\int_{\Omega}|{\nabla}\sqrt{n}|^2dxdt + \int_0^T\int_{\Omega}|\nabla u|^2dxdt \leq C_T,\\
&\int_0^T\int_{\Omega}c|\nabla^2 \log c|^2dxdt + \int_0^T\int_{\Omega}\frac{|\nabla c|^4}{c^3}dxdt \leq C_T,\\
&\int_0^T\int_{\Omega}|{\nabla}\sqrt{c}|^2n dxdt \leq C_T,\\
&{\varepsilon}\int_0^T\int_{\Omega} n(n^2-1)\log n dxdt \leq C_T,
\end{aligned}$$ where $C_T$ is a constant depending continuously on $T>0$.
Define $$\label{def_F}
\mathcal F(n,c,u) = \int_{\Omega}n\log n dx + 2\int_{\Omega}|\nabla \sqrt{c}|^2dx + \int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}+ K\int_{\Omega}|u|^2dx$$ where $K$ is a large enough constant such that $$K\frac{C(\mu)}{2} \geq 16\|c\|_{L^\infty(\Omega)}$$ with $C(\mu)$ is in Lemma \[energy-NS\]. It follows from Lemmas \[lem:energy2\] and \[energy-NS\] that $$\label{k1}
\begin{aligned}
\mathcal F(n,c,u)(t) + C\int_s^t\left[\int_{\Omega}\Bigl(|{\nabla}\sqrt{n}|^2 + c|{\nabla}^2\log c|^2 + \frac{|{\nabla}c|^4}{c^3} + |{\nabla}\sqrt{c}|^2n\Bigr)dx + \|u\|_{H^1(\Omega)}^2\right]dr\\
+ {\varepsilon}\int_s^t\int_{\Omega}n(n^2-1)\log n dxdr\\
\leq \mathcal F(n,c,u)(s) + C(t-s) + C\delta\int_s^t\int_{\Omega}|{\nabla}\sqrt n|^2dxdr + C\int_s^t\int_{\Omega}|{\nabla}\sqrt c|^2dxdr
\end{aligned}$$ for any $\delta > 0$. In particular, by choosing $\delta$ small enough, it follows that $$\mathcal F(n,c,u)(t) \leq \mathcal F(n,c,u)(s) + C(t-s) + C\int_s^t\mathcal F(n,c,u)(r)dr$$ for all $0<s<t<T$, and thus for all $t\in (0,T)$, $$\mathcal F(n,c,u)(t) \leq C_T.$$ From this and we obtain the desired bounds in Lemma \[a-priori\]. for all $t\in (0,T)$.
Global existence of solutions {#proofs}
=============================
In one or two dimensions
------------------------
According to Proposition \[a-prop-local-solution-hom\], the system - admits a unique classical solution on $(0,T)$ for all $T<T_{\max}$ for the maximal time $T_{\max}\in(0,\infty]$. We can reformulate the blow up criterion from to Proposition \[a-prop-local-solution-hom\] in spatial dimension two to $$\label{c-formula-explosion-condition}
\|n(t)\|_{L^{3}(\Omega)}
+\|{\nabla}n(t)\|_{L^{3}(\Omega)}
+\|{\nabla}c(t)\|_{L^{4}(\Omega)} + \|A^\alpha u(t)\|_{L^2_\sigma(\Omega)}\rightarrow \infty \ \text{as } t\nearrow T_{\max}$$ if $T_{\max}<\infty$ using the Sobolev embedding $W^{1,3}(\Omega)\subset L^\infty(\Omega)$. In order to prevent blow up, the first step is to bound $n$ in $L^p$ for all $1\leq p <\infty$. Considering the time derivative of the $L^p$ norm of $n$ yields the following result, which is based on a lemma from [@Win12]. It’s worth to remark that this trick only works in one or two spacial dimensions.
[@Bra17 Lemma 3.7]\[e-lem-estimate-lp-grad-n-in-2D\] If $d\leq 2$ and $p>1$, then there exists a constant $C_p>0$ such that $$\begin{aligned}
\label{n_Lp}
\frac1p\frac{d}{dt} \int_\Omega n^pdx +\frac{p-1}2\int_\Omega|{\nabla}n|^2n^{p-2}dx\leq C_p\left(\int_\Omega|{\nabla}c|^4dx+1\right)\int_\Omega n^pdx
\end{aligned}$$ holds for all $t\in (0,T)$.
It is remarked that the proof of this Lemma does not use any information of the logistic source (as it was included in the model in [@Bra17]), and therefore it is applicable for .
From Lemma \[e-lem-estimate-lp-grad-n-in-2D\], it is crucial to get $$\label{crucial}
\int_0^T\int_{\Omega}|\nabla c|^4dxdt \leq C_T.$$ From the energy estimate in Lemma \[a-priori\] we have $\int_0^T\int_{\Omega}|\nabla \sqrt[4]{c}|^4 dxdt \leq C_T$. Also since $\|c(t)\|_{L^\infty(\Omega)}$ is bounded, thanks to Lemma \[L1Linf\], the desired inequality follows immediately. Applying Gronwall’s inequality to and taking into account, we obtain that $$\label{73}
\|n\|_{L^\infty(0,T;L^p(\Omega))} \leq C_{p,T}$$ for all $1\leq p < +\infty$.
The next step is to find a uniform bound for $\|A^\alpha u(t)\|_{L^2_\sigma(\Omega)}$, where $\frac{d}{4}<\alpha<1$. This can be done similarly as in [@Win12 Eq. (4.19), pages 339-340]. In [@Win12] it was shown that $\|A^\alpha u(t)\|_{L^2_\sigma(\Omega)}$ can be bounded if $\|n\|_{L^2(\Omega)}$ and $\|\nabla \varphi\|_{L^\infty(\Omega)}$ are bounded. Comparing to [@Win12], we only have assumed that $\nabla\varphi\in L^\rho(\Omega)$ for $\rho > 6$. However, we can apply his calculations for $\tilde n:= n|\nabla\varphi|$, which is uniformly bounded in $L^2(\Omega)$ thanks to Young’s inequality and , and $\nabla\varphi/|\nabla\varphi|\in L^\infty(\Omega)$. Using $\alpha>\frac d4$ yields that $u$ is uniformly bounded thanks to Sobolev embeddings.
Now we can proceed as in the proof of Lemmas 4.2 – 4.4 of [@Bra17] to obtain that $$c\in L^\infty((0,T);W^{1,10}(\Omega))\cap L^{10}((0,T);W^{2,10}(\Omega))$$ and $$n\in L^\infty(0,T;W^{1,8}(\Omega)).$$ These estimates are enough to see that the solution does not blow up and therefore $T_{\mathrm{max}}=\infty$.
In three dimensions
-------------------
In this section, we will again denote by $(n^{\varepsilon,m}, c^{\varepsilon,m}, u^{\varepsilon,m})$ the global classical solution to for each $\varepsilon>0$ and $\mathbb N \ni m < \infty$. The main task is to study the limit $\varepsilon\to 0$ and $m\to \infty$. We first have the following uniform estimates.
\[interpolation\] We have $$\label{n53}
\{n^{{\varepsilon},m}\} \quad \text{ is bounded in } \quad L^{\frac 53}(Q_T),$$ and $$\label{u103}
\{u^{{\varepsilon},m}\} \quad \text{ is bounded in } \quad L^{\frac{10}{3}}(Q_T)$$ uniformly in ${\varepsilon}>0$ and $m>0$.
From Lemma \[a-priori\] we have $$\{\sqrt{n^{{\varepsilon},m}}\} \quad \text{ is bounded in } \quad L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H^1(\Omega)).$$ Since $d = 3$, $H^1(\Omega)\hookrightarrow L^6(\Omega)$ continuously. Moreover, an interpolation inequality gives $$L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;L^6(\Omega)) \hookrightarrow L^{\frac{10}{3}}(Q_T)$$ continuously. Therefore $\{\sqrt{n^{{\varepsilon},m}}\}$ is bounded in $L^{\frac{10}{3}}(Q_T)$, which implies . The bound is proved similarly thanks to the fact that $\{u^{{\varepsilon},m}\}$ is bounded in $L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ which is followed from Lemma \[a-priori\].
\[limits\] As ${\varepsilon}\to 0$ and $m\to \infty$, up to a subsequence, we have the following convergences $$\label{conv-n}
n^{{\varepsilon},m} \rightarrow n \quad \text{ strongly in } L^{\frac 53-}(Q_T) \quad \text{ and weakly in } L^{\frac 54}(0,T;W^{1,\frac 54}(\Omega)),$$ $$\label{conv-c}
c^{{\varepsilon}, m} \rightarrow c \quad \text{ strongly in } L^{\infty-}(Q_T) \quad \text{ and weakly in } L^4(0,T;W^{1,4}(\Omega)),$$ $$\label{conv-u}
u^{{\varepsilon}, m} \rightarrow u \quad \text{ strongly in } L^{\frac{10}{3}-}(Q_T) \quad \text{ and weakly in } L^2(0,T;H^1(\Omega)).$$ Here we write $f^{{\varepsilon},m} \to f$ in $L^{p-}(Q_T)$ if $f^{{\varepsilon},m} \to f$ in $L^q(Q_T)$ for all $1\leq q < p$.
We will prove the convergences , and separately.
[*Convergence of $n^{{\varepsilon},m}$.*]{} From Lemma \[a-priori\], we have $$\label{est_n}
\|n^{{\varepsilon},m}\|_{L^{\frac 53}(Q_T)} + \int_0^T\int_{\Omega}\frac{|\nabla \sqrt{n^{{\varepsilon},m}}|^2}{n^{{\varepsilon},m}}dxdt \leq C_T.$$ By Hölder’s inequality we can estimate $$\begin{aligned}
\label{grad_n}
\int_0^T\int_{\Omega}|{\nabla}n^{{\varepsilon},m}|^{\frac 54}dxdt &= \int_0^T\int_{\Omega}\left(\frac{|{\nabla}n^{{\varepsilon},m}|^2}{n^{{\varepsilon},m}}\right)^{\frac 58}(n^{{\varepsilon},m})^{\frac 58}dxdt\\
&\leq \left(\int_0^T\int_{\Omega}\frac{|{\nabla}n^{{\varepsilon},m}|^2}{n^{{\varepsilon},m}}dxdt\right)^{\frac 58}\left(\int_0^T\int_{\Omega}(n^{{\varepsilon},m})^{\frac 53}dxdt\right)^{\frac 38} \leq C_T.\nonumber
\end{aligned}$$ By testing the equation of $n^{{\varepsilon},m}$ with a smooth test function $\psi\in C^\infty_0(\Omega\times[0,T))$ we have $$\label{weak-n}
\begin{aligned}
&-\int_0^T\int_{\Omega}n^{{\varepsilon},m} \psi_t dxdt - \int_{\Omega}n^{{\varepsilon},m}_0\psi(\cdot,0)dx\\
&= -\int_0^T\int_{\Omega}{\nabla}n^{{\varepsilon},m}{\nabla}\psi dxdt - \int_0^T\int_{\Omega}n^{{\varepsilon},m}{\nabla}c^{{\varepsilon},m} {\nabla}\psi dxdt\\
&+ \int_0^T\int_{\Omega}n^{{\varepsilon},m}u^{{\varepsilon},m}\cdot{\nabla}\psi dxdt+ {\varepsilon}\int_0^T\int_{\Omega}(n^{{\varepsilon},m} - (n^{{\varepsilon},m})^3)\psi dxdt.
\end{aligned}$$ From we can estimate $$\left|\int_0^T\int_{\Omega}\nabla n^{{\varepsilon},m}{\nabla}\psi dxdt\right| \leq C_T\|\psi\|_{L^{5}(Q_T)}.$$ Lemma \[a-priori\] gives $$\begin{aligned}
&\left|\int_0^T\int_{\Omega}n^{{\varepsilon},m}{\nabla}c^{{\varepsilon},m} {\nabla}\psi dxdt\right|\\
&\leq \int_0^T\int_{\Omega}|{\nabla}c^{{\varepsilon},m}|\sqrt{n^{{\varepsilon},m}} \sqrt{n^{{\varepsilon},m}} |{\nabla}\psi|dxdt\\
&\leq \left(\int_0^T\int_{\Omega}|{\nabla}c^{{\varepsilon},m}|^2n^{{\varepsilon},m} dxdt\right)^{\frac 12}\left(\int_0^T\int_{\Omega}|n^{{\varepsilon},m}|^{2}dxdt\right)^{\frac 12}\|{\nabla}\psi\|_{L^\infty(Q_T)}\\
&\leq C_T\|\psi\|_{L^\infty(0,T;W^{2,4}(\Omega))},
\end{aligned}$$ thanks to the estimates in three dimensions $\|\nabla \psi\|_{L^\infty(\Omega)}\leq C\|\nabla \psi\|_{W^{1,4}(\Omega)} \leq C\|\psi\|_{W^{2,4}(\Omega)}$. Using the same idea we estimate $$\begin{aligned}
\left|\int_0^T\int_{\Omega}n^{{\varepsilon},m}u^{{\varepsilon},m}\cdot {\nabla}\psi dxdt \right| &\leq \|\psi\|_{L^\infty(Q_T)}\int_0^T\int_{\Omega}|n^{{\varepsilon},m}||u^{{\varepsilon},m}|dxdt\\
&\leq C\|\psi\|_{L^\infty(0,T;W^{2,4}(\Omega))}\|u^{{\varepsilon},m}\|_{L^{\frac{10}{3}}(Q_T)}\|n^{{\varepsilon},m}\|_{L^{\frac{10}{7}}(Q_T)}\\
&\leq C\|\psi\|_{L^\infty(0,T;W^{2,4}(\Omega))},
\end{aligned}$$ thanks to and the fact that $\frac{10}{7} < \frac{5}{3}$. From it follows that $$\left|{\varepsilon}\int_0^T\int_{\Omega}(n^{{\varepsilon},m} - (n^{{\varepsilon},m})^3) \psi dxdt\right| \leq C_T\|\psi\|_{L^\infty(Q_T)}.$$ Combining these estimates we obtain $$\label{est_nt}
\{\partial_tn^{{\varepsilon},m}\} \quad \text{ is bounded } \quad \text{ in } \quad L^1(0,T;(W^{2,4}(\Omega))^*).$$
From , and it follows from Aubin-Lions lemma that $$n^{{\varepsilon},m} \to n \quad \text{ in } \quad L^{\frac 54}(Q_T)$$ as ${\varepsilon}\to 0$ and $m\to\infty$ (up to a subsequence). Moreover, since $\{n^{{\varepsilon},m}\}$ is bounded in $L^{\frac 53}(Q_T)$, thanks to Lemma \[interpolation\], we have in fact $$n^{{\varepsilon},m} \to n \quad \text{ in } \quad L^{\frac 53-}(Q_T).$$ This convergence and give the convergence for $n$ as in .
[*Convergence of $c^{{\varepsilon},m}$.*]{} From $\|c^{{\varepsilon},m}\|_{L^\infty(Q_T)} \leq C$ and $\int_0^T\int_{\Omega}\frac{|{\nabla}c^{{\varepsilon},m}|^4}{(c^{{\varepsilon},m})^3}dxdt \leq C_T$ we have $$\label{grad-c}
\int_0^T\int_{\Omega}|{\nabla}c^{{\varepsilon},m}|^4dxdt \leq C\int_0^T\int_{\Omega}\frac{|{\nabla}c^{{\varepsilon},m}|^4}{(c^{{\varepsilon},m})^3}dxdt \leq C_T.$$ By testing the second equation in with a smooth test function $\psi \in C^\infty_0(\Omega\times[0,T))$ we have $$\begin{aligned}
\label{weak-c}
&-\int_0^T\int_{\Omega}c^{{\varepsilon},m} \psi_t dxdt - \int_{\Omega}c^{{\varepsilon},m}_0\psi(\cdot,0)dx\nonumber\\
&= -\int_0^T\int_{\Omega}{\nabla}c^{{\varepsilon},m} {\nabla}\psi dxdt + \int_0^T\int_{\Gamma}g(x)(\gamma - c^{{\varepsilon},m})\psi {d\mathcal{H}_x^{d-1}}dt\nonumber\\
&+\int_0^T\int_{\Omega}c^{{\varepsilon},m}u^{{\varepsilon},m}{\nabla}\psi dxdt -\int_0^T\int_{\Omega}n^{{\varepsilon},m}c^{{\varepsilon},m}\psi dxdt.
\end{aligned}$$ We have the following estimates $$\left|\int_0^T\int_{\Omega}{\nabla}c^{{\varepsilon},m}{\nabla}\psi dxdt\right| \leq \|{\nabla}c^{{\varepsilon},m}\|_{L^4(Q_T)}\|{\nabla}\psi\|_{L^{\frac{4}{3}}(Q_T)} \leq C\|\psi\|_{L^{\frac 43}(0,T;W^{1,\frac 43}(\Omega))},$$ $$\begin{aligned}
\left|\int_0^T\int_{\Gamma}\kappa(x)(\gamma - c^{{\varepsilon},m})\psi {d\mathcal{H}_x^{d-1}}dt\right| &\leq C(1+\|c^{{\varepsilon},m}\|_{L^\infty((0,T)\times \Gamma)})\int_0^T\int_{\Gamma}|\psi|{d\mathcal{H}_x^{d-1}}dt\\
&\leq C_T\|\psi\|_{L^1(0,T;W^{1,1}(\Omega))},
\end{aligned}$$ $$\begin{aligned}
\left|\int_0^T\int_{\Omega}c^{{\varepsilon},m}u^{{\varepsilon},m}{\nabla}\psi dxdt\right| &\leq \|c^{{\varepsilon},m}\|_{L^\infty(Q_T)}\|u^{{\varepsilon},m}\|_{L^2(Q_T)}\|{\nabla}\psi\|_{L^2(Q_T)}\\
&\leq C_T\|\psi\|_{L^2(0,T;H^1(\Omega))}.
\end{aligned}$$ and $$\begin{aligned}
\left|\int_0^T\int_{\Omega}n^{{\varepsilon},m}c^{{\varepsilon},m}\psi dxdt\right| &\leq \|c^{{\varepsilon},m}\|_{L^\infty(Q_T)}\|n^{{\varepsilon},m}\|_{L^{\frac 53}(Q_T)}\|\psi\|_{L^{\frac 52}(Q_T)}\\
&\leq C\|\psi\|_{L^{\frac 52}(0,T;L^{\frac 52}(\Omega))}.
\end{aligned}$$ Therefore $$\left|\int_0^T\int_{\Omega}\partial_t c^{{\varepsilon},m}\psi dxdt\right| \leq C_T\|\psi\|_{L^{\frac 52}(0,T;W^{1,\frac 52}(\Omega))}.$$ thus $$\{\partial_t c^{{\varepsilon},m} \} \quad \text{ is bounded in } \quad L^{\frac 53}(0,T;(W^{1,\frac 52}(\Omega))^*).$$ Combining this with and the uniform bound of $c^{{\varepsilon},m}$, it follows from the Aubin-Lions lemma that $$c^{{\varepsilon},m} \to c \quad \text{ in } \quad L^4(Q_T)$$ as ${\varepsilon}\to 0$ and $m\rightarrow \infty$, and consequently thanks to the boundedness of $c^{{\varepsilon},m}$ in $L^\infty(Q_T)$.
[*Convergence of $u^{{\varepsilon},m}$.*]{} Testing the equation of $u^{{\varepsilon},m}$ in with $\psi \in C^\infty_0(\Omega\times [0,T))^3$ we get $$\begin{aligned}
\label{weak-u}
&-\int_0^T\int_{\Omega}u^{{\varepsilon},m}\cdot \psi_t dxdt - \int_{\Omega}u^{{\varepsilon},m}_0\cdot\psi(\cdot,0)dx\nonumber\\
&= \int_0^T\int_{\Omega}{\nabla}u^{{\varepsilon},m} \cdot {\nabla}\psi dxdt - \int_0^T\int_{\Omega}\mathcal P^mB(u^{{\varepsilon},m},u^{{\varepsilon},m})\psi dxdt- \int_0^T\int_{\Omega}\mathcal{P}^m(n^{{\varepsilon},m}{\nabla}\varphi)\psi dxdt.
\end{aligned}$$ We estimate the terms on the right hand side as following $$\begin{aligned}
\left|\int_0^T\int_{\Omega}{\nabla}u^{{\varepsilon},m} \cdot {\nabla}\psi dxdt \right| \leq \|{\nabla}u^{{\varepsilon},m}\|_{L^2(Q_T)}\|{\nabla}\psi\|_{L^2(Q_T)} \leq C_T\|\psi\|_{L^2(0,T;H^1(\Omega))},
\end{aligned}$$ $$\begin{aligned}
\left|\int_0^T\int_{\Omega}\mathcal P^mB(u^{{\varepsilon},m},u^{{\varepsilon},m})\psi dxdt\right| \leq \|u^{{\varepsilon},m}\|_{L^{\frac{10}{3}}(Q_T)}^2\|{\nabla}\psi\|_{L^{\frac 52}(Q_T)} \leq C_T\|\psi\|_{L^{\frac 52}(0,T;W^{1,\frac 52}(\Omega))},
\end{aligned}$$ and $$\begin{aligned}
\left|\int_0^T\int_{\Omega}\mathcal{P}^m(n^{{\varepsilon},m}{\nabla}\varphi)\psi dxdt\right| \leq \|\varphi\|_{W^{1,\rho}(\Omega)}T\|n^{{\varepsilon},m}\|_{L^{\frac 53}(Q_T)}\|\psi\|_{L^{\frac{5\rho}{2\rho - 5}}(Q_T)} \leq C_T\|\psi\|_{L^5(Q_T)},
\end{aligned}$$ where we used $\frac{5\rho}{2\rho - 5} < 5$ at the end since $\rho > 6$. Therefore $$\{\partial_t u^{{\varepsilon},m} \} \quad \text{ is bounded in } \quad L^{\frac 54}(0,T;(W^{1,5}(\Omega))^*).$$ Now the Aubin-Lions gives us the strong convergence $$u^{{\varepsilon},m} \to u \quad \text{ in } \quad L^2(Q_T)$$ as ${\varepsilon}\to 0$ and $m\to \infty$ (up to a subsequence). Finally, follows from the fact that $\{u^{{\varepsilon},m}\}$ is bounded in $L^{\frac{10}{3}}(Q_T)$ from Lemma \[interpolation\].
We need one more preparation which was proved in [@Bra17 Lemma 4.11].
[@Bra17 Lemma 4.11]\[conv\_logistic\] The sequence $\{{\varepsilon}(n^{{\varepsilon},m} - (n^{{\varepsilon},m})^3)\}$ is weakly precompact in $L^1(Q_T)$.
We are now ready to prove Theorem \[thm:main3D\].
It is sufficient to show that the limits function $(n,c,u)$ obtained in Lemma \[limits\] is a weak solution in the sense of Definition \[weak\_sol\]. In order to do that, we need to take care of the limits ${\varepsilon}\to 0$ and $m\to \infty$ in , and .
For the first term on the right hand side of , we write ${\nabla}n^{{\varepsilon},m} = 2{\nabla}\sqrt{n^{{\varepsilon},m}}\sqrt{n^{{\varepsilon},m}}$ and use ${\nabla}\sqrt{n^{{\varepsilon},m}} \rightharpoonup {\nabla}\sqrt{n}$ in $L^2(Q_T)$ and $\sqrt{n^{{\varepsilon},m}}\to \sqrt{n}$ in $L^2(Q_T)$ we get that ${\nabla}n^{{\varepsilon},m} \rightharpoonup {\nabla}n$ weakly in $L^2(Q_T)$, hence $$\int_0^T\int_{\Omega}{\nabla}n^{{\varepsilon},m}\cdot {\nabla}\psi dxdt \xrightarrow{{\varepsilon}\to 0, \; m\to\infty} \int_0^T\int_{\Omega}{\nabla}n\cdot {\nabla}\psi dxdt.$$ From ${\nabla}c^{{\varepsilon},m} \rightharpoonup {\nabla}c$ weakly in $L^4(Q_T)$ and $n^{{\varepsilon},m} \to n$ strongly in $L^{\frac 43}(Q_T)$ it follows $$\int_0^T\int_{\Omega}n^{{\varepsilon},m}{\nabla}c^{{\varepsilon},m}\cdot {\nabla}\psi dxdt \xrightarrow{{\varepsilon}\to 0, \; m\to\infty} \int_0^T\int_{\Omega}n{\nabla}c\cdot {\nabla}\psi dxdt.$$ From and we have $n^{{\varepsilon},m} \to n$ strongly in $L^{\frac{20}{13}}(Q_T)$ and $u^{{\varepsilon},m}\to u$ strongly in $L^{\frac{20}{7}}(Q_T)$, and thus $n^{{\varepsilon},m}u^{{\varepsilon},m} \to nu$ strongly in $L^1(Q_T)$ and consequently $$\int_0^T\int_{\Omega}n^{{\varepsilon},m}u^{{\varepsilon},m}\cdot{\nabla}\psi dxdt \xrightarrow{{\varepsilon}\to 0, \; m\to\infty} \int_0^T\int_{\Omega}nu\cdot{\nabla}\psi dxdt.$$ The convergence of the last term $$\int_0^T\int_{\Omega}{\varepsilon}(n^{{\varepsilon},m} - (n^{{\varepsilon},m})^3)\psi dxdt \xrightarrow{{\varepsilon}\to 0,\; m\to\infty} 0$$ follows from Lemma \[conv\_logistic\].
Similarly, the convergence of all terms in holds thanks to -- and the interpolation inequality $$\int_0^T\int_{\Gamma}|c^{{\varepsilon},m} - c|^2{d\mathcal{H}_x^{d-1}}dt \leq C\|c^{{\varepsilon},m} - c\|_{L^2(0,T;H^1(\Omega))}\|c^{{\varepsilon},m} - c\|_{L^2(0,T;L^2(\Omega)}.$$
All the terms in can be treated similarly thanks to -.
The uniform bound of the energy will be proved in Proposition \[bound\_energy\].
\[bound\_energy\] We have the following bound of the energy $$\sup_{t\in [0,\infty)}\left(\int_{\Omega}n(t)\log n(t)dx + \| {\nabla}\sqrt c(t)\|_{L^2(\Omega)}^2 + \|u(t)\|_{L^2(\Omega)}^2\right) \leq C.$$
From , we deduce by choosing $\delta$ small enough that $$\label{k2}
\mathcal F(n,c,u)(t) + C\int_s^t\mathcal E(n,c,u)(r)dr \leq \mathcal F(n,c,u)(s) + C(t-s) + C\int_s^t\int_{\Omega}|{\nabla}\sqrt c|^2dxdr$$ where $$\label{def_E}
\mathcal E(n,c,u) = \int_{\Omega}\left(|{\nabla}\sqrt{n}|^2 + c|{\nabla}^2\log c|^2 + \frac{|{\nabla}c|^4}{c^3} + |{\nabla}\sqrt{c}|^2n + \|u\|_{H^1(\Omega)}^2 \right)dx.$$ Looking at , it becomes clear that that last term on the right-hand side is troublesome as it prevents to obtain uniform bound in time of the energy while applying Gronwall’s lemma. To overcome this difficulty, we introduce an additional energy, namely $\int_{\Omega}s^\infty(c|\gamma)dx$, where we recall that $s^\infty$ is defined in . Remark that here we consider an extension of the surface function $\gamma: \Gamma \to \mathbb R$ into the (with a slight abuse of notation) $\gamma:\overline{\Omega}\to \infty$, with $\gamma \in H^1(\overline{\Omega})\cap L^\infty(\overline{\Omega})$. Moreover, thanks to , $\gamma(x) \geq \underline{\gamma} >0$ for all $x\in\overline{\Omega}$. We now show that $$\begin{aligned}
\label{k3}
\frac d{dt}\int_{\Omega}s^\infty(c|\gamma)dx+4\int_{\Omega}\left|\nabla\sqrt{c}\right|^2dx+\int_{\Gamma}\kappa s^\infty(\gamma|c) d\mathcal H^{d-1}_x\leq C.
\end{aligned}$$ Indeed, by direct computations, we have $$\begin{aligned}
\frac d{dt}\int_{\Omega}s^\infty(c|\gamma)dx+4\int_{\Omega}|{\nabla}\sqrt{c}|^2dx
&=
\int_{\Omega}\partial_tc\log\frac{c}\gamma dx +\int_{\Omega}\nabla c\cdot \nabla\log cdx
\\&= \int_{\Gamma}\kappa (\gamma- c) \log\frac{c}{\gamma}d\mathcal H^{d-1}_x-\int_{\Omega}cn \log\frac{c}{\gamma}dx.
\end{aligned}$$ We now observe the following two identities $$\begin{aligned}
(\gamma- c) \log\frac{c}{\gamma}&=
-c\log\frac{c}{\gamma}+c-\gamma
+\gamma\log\frac{c}{\gamma}+\gamma-c \\&=
-s^\infty(c|\gamma)- s^\infty(\gamma|c),
\end{aligned}$$ and $$\begin{aligned}
-cn \log\frac{c}{\gamma}
&=-n\left(c\log\frac{c}{\gamma}-c+\gamma\right)+n(\gamma-c)
=n(\gamma-c)-ns^\infty(c|\gamma).
\end{aligned}$$ Combing these calculations and using $s^\infty(x|y) \geq 0$ lead to $$\begin{aligned}
\frac d{dt}\int_{\Omega}s^\infty(c|\gamma)dx+4\int_{\Omega}|{\nabla}\sqrt c|^2 dx
+\int_{\Gamma}\kappa s^\infty(\gamma|c) d\mathcal H^{d-1}_x\\
\leq \int_{\Omega}n(\gamma-c)dx \leq \|\gamma\|_{L^\infty(\Omega)}\|n\|_{L^1(\Omega)} \leq C,
\end{aligned}$$ due to the non-negativity of $n$ and $c$, and $\|n\|_{L^1(\Omega)}\leq C$.
By multiplying by a large constant $L>0$, integrating the resultant on $(s,t)$, and adding the obtained inequality to , we get $$\label{k4}
\mathfrak X(n,c,u)(t) + C\int_s^t \mathfrak Z(n,c,u)(r)dr \leq \mathfrak X(n,c,u)(s) + C(t-s)$$ where $$\begin{aligned}
\label{X}
\mathfrak X(n,c,u)(t) = \mathcal F(n,u,c)(t) + L\int_{\Omega}s^{\infty}(c|\gamma)(t)dx
\end{aligned}$$ and $$\begin{aligned}
\mathfrak Z(n,c,u)(r) = \mathcal E(n,c,u)(r) + \int_{\Omega}|{\nabla}\sqrt c(r)|^2dx + \int_{\Gamma}\kappa s^\infty(\gamma|c){d\mathcal{H}_x^{d-1}}.
\end{aligned}$$ We will now prove for some constants $\lambda>0$ and $C>0$ that $$\label{eede}
\mathfrak Z(n,c,u) \geq \lambda \mathfrak X(n,c,u) - C.$$ From and , to obtain , it remains to show that $$\label{k5}
\int_{\Omega}|{\nabla}\sqrt n|^2dx \geq \lambda_1\int_{\Omega}n\log n dx - C$$ and $$\label{k6}
\int_{\Omega}|{\nabla}\sqrt{c}|^2dx \geq \lambda_2\int_{\Omega}s^{\infty}(c|\gamma) - C.$$ By the Logarithmic-Sobolev inequality we have, where $\overline{n} = \frac{1}{|\Omega|}\int_{\Omega}n(x)dx$, $$\begin{aligned}
\int_{\Omega}|{\nabla}\sqrt n|^2dx &= \frac 14\int_{\Omega}\frac{|{\nabla}\sqrt n|^2}{n}dx \geq \frac 14 C_{LSI}\int_{\Omega}n\log\frac{n}{\overline{n}}dx\\
&= \frac 14C_{LSI}\int_{\Omega}n\log n dx - \frac 14 C_{LSI}\log \overline{n}\|n\|_{L^1(\Omega)},
\end{aligned}$$ hence , thanks to $\|n\|_{L^1(\Omega)} \leq C$. Similarly $$\begin{aligned}
\int_{\Omega}|{\nabla}\sqrt c|^2dx &\geq \frac 14C_{LSI}\int_{\Omega}c\log c\,dx - \frac 14C_{LSI}\log \overline{c}\|c\|_{L^1(\Omega)}\\
&= \frac 14C_{LSI}\int_{\Omega}s^\infty(c|\gamma)dx + \frac 14 C_{LSI}\int_{\Omega}(c\log \gamma + c - \gamma)dx - \frac 14C_{LSI}\frac{1}{|\Omega|}\|c\|_{L^1(\Omega)}^2
\end{aligned}$$ hence , due to $\|c\|_{L^\infty(\Omega)} \leq C$, $\|\gamma\|_{L^\infty(\Omega)} \leq C$ and $\gamma(x) \geq \underline{\gamma} > 0$. We have proved and , and consequently . Using in , it follows that $$\mathfrak{X}(n,c,u)(t) + C\int_s^t \mathfrak{X}(n,c,u)(r)dr \leq \mathfrak{X}(n,c,u)(s) + C(t-s).$$ Thus for some suitable constant $K$ and $\Theta(r) = \mathfrak{X}(n,c,u)(r) - K$, $$\Theta(t) + C\int_s^t\Theta(r)dr
\leq \Theta(s).$$ Defining $$\Xi(s) = \int_s^t \Theta(r)dr,$$ we have $$\Xi'(s) = -\Theta(s) \leq \Theta(t) - C\Xi(s)$$ and consequently $$\left(e^{Cs}\Xi(s)\right)' + e^{Cs}\Theta(t)\leq 0.$$ Integrating this from $0$ to $t$, noting that $\Xi(t) = 0$, gives $$-\Xi(0) + \Theta(t)\frac{e^{Ct}-1}{C}\leq 0.$$ From this we have $$\Theta(t)\frac{e^{Ct}-1}{C} \leq \Xi(0) = \int_0^t\Theta(r)dr \leq \frac{\Theta(0) -\Theta(t)}{C}$$ and therefore $$\Theta(t) \leq e^{-Ct}\Theta(0).$$ Replacing $\Theta(t) = \mathfrak{X}(n,c,u)(t)-K$ we finally obtain $$\mathfrak{X}(n,c,u)(t) \leq K + e^{-Ct}(\mathfrak X(n_0,c_0,u_0) - K) \leq C$$ for all $t>0$, which proves our claim thanks to and .
[**Acknowledgement.**]{} The authors gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics (Bonn) through the Junior Trimester Kinetic Theory. This work is partially supported by NAWI Graz, the International Research Training Group IGDK 1754 and by the Austrian Science Fund (FWF) project F 65.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In regions where stars form, variations in density and temperature can cause gas to freeze-out onto dust grains forming ice mantles, which influences the chemical composition of a cloud. The aim of this paper is to understand in detail the depletion (and desorption) of CO on (from) interstellar dust grains. Experimental simulations were performed under two different (astrophysically relevant) conditions. In parallel, Kinetic Monte Carlo simulations were used to mimic the experimental conditions. In our experiments, CO molecules accrete onto water ice at temperatures below 27 K, with a deposition rate that does not depend on the substrate temperature. During the warm-up phase, the desorption processes do exhibit subtle differences indicating the presence of weakly bound CO molecules, therefore highlighting a low diffusion efficiency. IR measurements following the ice thickness during the TPD confirm that diffusion occurs at temperatures close to the desorption. Applied to astrophysical conditions, in a pre-stellar core, the binding energies of CO molecules, ranging between 300 K and 850 K, depend on the conditions at which CO has been deposited. Because of this wide range of binding energies, the depletion of CO as a function of AV is much less important than initially thought. The weakly bound molecules, easily released into the gas phase through evaporation, change the balance between accretion and desorption, which result in a larger abundance of CO at high extinctions. In addition, weakly bound CO molecules are also be more mobile, and this could increase the reactivity within interstellar ices.'
author:
- 'S. Cazaux, R. Mart[í]{}n-Dom[é]{}nech, Y. J. Chen, G. M. Mu[ñ]{}oz Caro, C. Gonz[á]{}lez D[í]{}az'
title: 'CO depletion: a microscopic perspective'
---
Introduction
============
In the last decades, observing facilities have significantly increased in sensitivity allowing to study in detail the chemical composition of many places of our Universe. Molecules and atoms are powerful indicators of the gas characteristics of a medium and are used to derive detailed properties of astrophysical objects. [In particular,]{} observations of star forming environments have rapidly been confronted with the impossibility to explain the abundances of some species with gas-phase reactions only. In regions where stars [form]{}, about 1% of the mass is constituted by small dust particles ranging in size from few tens of Å to a few micrometer. However small and inconspicuous these dust grains seem, they interact with the gas phase and can dramatically alter its composition. In the first phases of star formation, the [molecular]{} clouds present some overly dense regions, called pre-stellar cores, which are the precursors of the stars. To reproduce the observations [of dense clouds]{}, about 90% of CO molecules should leave the gas phase, on average along the line of sight, and over 99% of them must deplete in the core nucleus (@caselli1999). This is due to CO freeze-out onto dust particles, which then form thick icy mantles (e.g. @ossenkopf1994; @pontoppidan2008). When the protostar forms and heats its surroundings, a rich molecular chemistry is triggered driven by thermal desorption of [the ice mantles]{} (@cazaux2003). This chemically rich phase is called hot core/hot corino (for high/low mass stars) and characterised by an abundant organic inventory (water and organics such as H$_2$CO and CH$_3$OH (@schoier2002), complex O- and N-bearing molecules such as formic acid and acetaldehyde (@cazaux2003)). In order to explain these observations, the understanding of CO interaction with dust surfaces is unavoidable.
In this study we followed experimentally the formation of CO ices deposited at different [conditions]{}. For this purpose, we [performed]{} two experimental simulations focusing on two different accretion processes: 1) accretion onto a water ice substrate at a decreasing temperature, from 80 K to 8 K, and 2) accretion at a constant temperature of 14 K. After accretion, the temperature of the [substrate]{} was increased [in both cases]{}, and CO molecules evaporated and could be measured in the gas phase. This is called a temperature programmed desorption (TPD) experiment. The present study [aims to]{} understand whether TPD experiments can be sensitive to the different accretion processes. [In this work, we show that subtle deviation between TPDs are crucial to constrain the binding energies and the diffusion of CO molecules in the ice. Furthermore, combining quadrupole mass spectrometry, infrared spectroscopy and laser interference allows to follow simultaneously the solid and gaseous phases of the CO molecules during the TPD and further constrain the desorption and diffusion processes.]{} The experimental results are supported by theoretical calculations taking into account the microphysics occurring in ices. Our results are then exported to astrophysical conditions.
This paper is articulated as follow. In section 2 the two experiments performed in this study are described as well as their results. In section 3, the theoretical model and assumptions are described and used to reproduce the experimental results. In section 4 the model is extended to pre-stellar cores in order to reproduce the CO depletion observed in these objects.
Experiments
===========
Deposition [followed by]{} warming up of CO ices has been studied experimentally in two different setups, [under two different conditions]{} in order to address whether the deposition conditions influence the [subsequent thermal]{} desorption of the CO ices. The two experimental setups are described in the following sections, as well as the experimental results.
{width="55.00000%"}
Experimental setup: the InterStellar Astrochemistry Chamber
------------------------------------------------------------
[The experimental simulations corresponding to the accretion of CO molecules at a decreasing temperature onto a previously deposited water ice substrate were]{} performed using the InterStellar Astrochemistry Chamber (ISAC) at the Centro de Astrobiología. [The aim of these experiments was to determine the maximum temperature at which CO molecules were able to accrete onto the water ice and form an ice mantle.]{} ISAC is an ultra-high vacuum (UHV) chamber with pressure typically in the range P = 2.5-4.0 $\times$ 10$^{-11}$ mbar, which corresponds to dense cloud interiors. A [schematic representation]{} of this setup is shown in figure \[setup2\]. The chemical components used in the experiments were H$_2$O (vapor, triply distilled), and CO (gas), that were introduced into the chamber from an independent gas-line system through a capillary tube, condensing onto a KBr substrate and forming an ice analog. [A closed-cycle helium cryostat and a tunable heater, combined with]{} a silicon diode temperature sensor and a LakeShore Model 331 temperature controller were used [to control the sample temperature]{}, reaching a sensitivity of about 0.1 K. The evolution of the solid sample was monitored by in situ Fourier transform infrared (FTIR) spectroscopy in transmittance (model Bruker Vertex 70, equipped with a deuterated triglycine sulfate detector, or DTGS), with a spectral resolution of 2 cm$^{-1}$ (@munozcaro2010). Column densities of each species in the ice were calculated from the IR spectra using the formula $$N=\frac{1}{A}\int_{\rm{band}} \tau_{\nu} \ d\nu$$ where $N$ is the column density in molecules cm$^2$, $\tau_{\nu}$ the optical depth of the absorption band, and $A$ the band strength in cm molecule$^{-1}$, where we adopt a value of $A$(CO)=1.1 $\times$ 10$^{-17}$ cm molecule$^{-1}$ (@jiang1975) and [A(H$_2$O)=2.0 $\times$ 10$^{-16}$ cm molecule$^{-1}$ (@hagen1981)]{}. [A total of 35 ML (1 ML = 10$^{15}$ molecules cm$^{-2}$) of amorphous solid water (ASW) were first deposited onto the KBr substrate at 80 K with an accretion rate of 6 ML/min.]{} Then, CO gas was admitted in the chamber [at a constant pressure]{}, and the temperature of the [substrate was gradually decreased from 80 K to 8 K at a constant rate of 0.5 K/min]{}. Once the CO ice was deposited on top of the water ice, the substrate was warmed up from 8 K at a rate of 0.5 K/min, leading to the desorption of the CO molecules that were detected by a quadrupole mass spectrometer (QMS).
Experimental results
--------------------
Figure \[acc\] shows the CO [in the gas phase as measured by]{} the QMS [while]{} the temperature of the substrate was cooled down from 80 to 8 K with a rate of 0.5 K/min. [ The drastic decrease observed at 26.5 K is due to accretion of CO molecules on the substrate. This is confirmed by IR spectroscopy, as shown in Fig. \[acc2\], left panel. The CO IR band at $\sim$2139 cm$^{-1}$ was not observed at temperatures higher than 26.5 K. This means that only 1 ML(at most) of CO (which is the sensitivity of our FTIR spectrometer) could have been accreted on top of the ASW before that temperature was reached. However, once the temperature decreases below 26.5 K, the solid CO IR feature is observed, increasing its intensity at a constant rate (see Fig. \[acc2\], right panel), which corresponds to an accretion rate of 1.5 ML/min. This feature does not present any shoulder at $\sim$2152 cm$^{-1}$, typical of CO molecules interacting with dangling OH bonds (see, e.g., @collings2003b; @martindomenech2014). Therefore, CO diffusion into the ASW structure does not take place in our experiment, at least to a significant extent. This is due to the lower porosity of the water ice deposited at 80 K compared to that deposited at lower temperatures (see, e.g., @bossa2012). This shoulder at $\sim$2152 cm$^{-1}$ may not be observed either in astronomical spectra (see, e.g., @cuppen2011).]{} [Once the substrate reached a temperature of 8 K, and the temperature increased at a rate of 0.5 K/min (TPD). The desorbing CO molecules were detected by the QMS (see Fig.\[tpd2\]). While an important desorption peak can be seen near 30 K, corresponding to the desorption of the bulk of the CO molecules, an extended shoulder ranges from 30 K to $\sim$60 K, corresponding to the desorption in the sub-monolayer regime (@noble2011).]{}
[This TPD behavior is similar to what was previously reported for CO ices accreted at a fixed temperature under different conditions (see, e.g., @collings2004;@martindomenech2014). Therefore, at first glance, the deposition conditions do not seem to affect the subsequent thermal desorption process. ]{}
{width="55.00000%"}
{width="45.00000%"} {width="45.00000%"}
{width="55.00000%"}
Experimental setup: the Interstellar Photoprocess System
--------------------------------------------------------
Additional experimental simulations corresponding to the CO ice accretion at a constant temperature of 14 K, and subsequent TPD, were performed using the Interstellar Photoprocess System (IPS) described in Chen et al. (2014). [The aim of these experiments was to confirm that a CO ice accreted at different conditions from that used in the experimental simulations described in the previous section presented similar TPD curves. In addition, as mentioned in Section 1, laser interference and infrared spectroscopy used in these series of experiments in order to get a better understanding of the desorption process.]{} A schematic representation of IPS is shown in figure \[ips\]. IPS is an ultra-high-vacuum (UHV) chamber with a base pressure of 1.3 $\times$ 10$^{-10}$ mbar. [The substrate for interstellar solid analogs (usually a KBr window)]{} is located at the sample holder, placed on the tip of a cold finger from a closed-cycle helium cryostat (CTI-M350), which reaches temperatures as low as 14 K. Two silicon diodes are used to monitor the temperature of both the substrate and the cold finger, with an accuracy of 0.1 K. [As in the ISAC setup, the species]{} were allowed to enter the UHV chamber through a capillary tube, condensing onto a KBr substrate [and forming the ice analogs]{}. A FTIR spectrometer (model ABB FTLA-2000-104) equipped with a mercury-cadmium-telluride (MCT, more sensitive than a DTGS) detector monitors the solid sample. [In this case,]{} the ice sample position is at 45 $\deg$ from the IR beam (see Fig. \[ips\]) and, therefore, the thickness experienced by the IR beam (effective ice thickness) is larger than the actual ice thickness by a factor of $\frac{\sqrt{2}}{2}$. This is taken into account during the experimental simulations. A QMS covering the range of 1 - 200 amu with 0.5 amu resolution, provides monitoring of the introduced gas during the deposition, and measures the presence of desorbing molecules in the gas phase during the warm-up phase.
{width="8cm"}
[In addition, a laser reflective interference system (not included in the ISAC setup) counts with a He-Ne laser ($\lambda$ = 632.8 nm), and a power meter (Newport M835) calibrated for measuring the reflected intensity of the laser at 632.8 nm (power $<$ 5 mW). The He-Ne laser reaches the solid sample with an incident angle of 2 degrees with respect to the normal, and the reflective light is subsequently directed by a mirror to the power meter (see Fig. \[ips\]). The reflected intensity of the laser oscillates between constructive and destructive interference, leading to a sinusoidal pattern. This interference pattern was monitored during both a blank experiment (warming up of a KBr substrate alone) and the TPD of a CO ice deposited directly onto the bare KBr substrate at 14 K. During the blank experiment, the sinusoidal interference pattern responded to the (constructive and destructive) interference between the light reflected from the front KBr surface and the rear KBr surface. When the KBr window is held at a specific temperature, its thickness is constant and does not cause any variation of light interference, but this changes during the warm up, leading to a variation in the sinusoidal pattern. On the other hand, during the TPD of the deposited CO ice, the pattern responded to the interference between the light reflected from the ice surface and the KBr window, and changed as the CO molecules desorbed from the ice.]{}
Results
-------
[The evolution of the CO ice was studied by using both laser interference to follow the CO solid sample (Fig. \[ir\], and Fig.\[tpdqms\] left panel) and the QMS to follow the TPD and therefore the CO in the gas (Fig. \[tpdqms\], right panel). In this experiment, the sample temperature was increased from 14 K to 70 K at an identical rate of 1 K/min. ]{}
The results corresponding to [the blank experiment (warming up of a KBr substrate alone) and the TPD of a CO ice deposited onto a bare KBr substrate at a constant temperature of 14 K using laser interference to study the subsequent desorption of the CO ice are shown in Fig. \[ir\]. The interference pattern as a function of the temperature during TPD of the CO ice is shown in red, and compared to the blank experiment in black. The variation of the sinusoidal pattern during the TPD experiment is due to the thickness variation of both the KBr substrate and the CO ice. Therefore, the laser interference pattern cannot be used to estimate the thickness variation of the CO ice alone because it is difficult to determine the contributed weights from the ice sample and the substrate to this effect. However, the differences in the variation of this sinusoidal pattern during the TPD of the CO ice with respect to the blank experiment allow us to determine if some CO is retained on the KBr surface, and if its thickness considerably changes. Between 13 K and 28 K, the measurements indicate that no important changes of the ice thickness are observed for temperatures prior to CO desorption (region I), since the variation of the sinusoidal pattern is similar to that of the blank experiment. Around 30K, a fast decrease and a subsequent increase of the laser intensity (red line) is observed due to the sublimation of CO molecules and the subsequent thickness variation. For temperatures higher than 32 K, in region II, the phase of the red line (CO ice) is delayed compared to the black line (blank experiment). This implies that some CO molecules are still present on the surface. In region III, above 54 K, the phases of the red line (CO ice) and black line (KBr substrate) become the same, which suggests that all CO molecules have desorbed. The decrease of the CO ice column density due to thermal desorption was simultaneously monitored from integration of the IR band at $\sim$2139 cm$^{-1}$ during the TPD experiment, and is reported in the left panel of Fig. \[tpdqms\]. The IR measurements support the idea that approximately 1 monolayer of CO remains on the surface at temperatures above $\sim$30 K (region II in Fig. \[ir\]) and desorbs at T$\le$60 K.]{}
The TPD measurements are reported in the right panel of Fig. \[tpdqms\]. CO deposition was studied on three different types of surfaces, namely: the aforementioned bare KBr substrate (this experiment is represented in black in Fig. \[tpdqms\]), water ice previously deposited at 80 K with a rate of 0.1 ML/s (blue), and methanol ice deposited at the same temperature with a rate of 0.07 ML/s (red). Results are similar in the three cases, and also to those presented in the previous section, thus confirming that the deposition conditions do not greatly affect, at first glance, the subsequent thermal desorption process. CO ices stay adsorbed on the substrate until $\sim$ 30 K. At this temperature, the CO molecules that constitute the bulk of the ice desorb into the gas phase and are detected by the QMS, leading to the desorption peak in the right panel of Fig. \[tpdqms\]. From $\sim$30 K to $\sim$50 K, the TPD measurements from the QMS show a slow decrease of the signal. This, along with the IR and laser interference measurements (only shown for the TPD of CO ice deposited onto the KBr substrate in Fig. \[ir\]), suggests that around 1 ML of CO is still present on the different surfaces until $\sim$54 K, as expected from previous experiments (see, e.g., @noble2011).
The slow decrease in intensity of the TPD signal inferred from the QMS data collected during the three experiments suggests that [these]{} CO molecules are strongly bound to the different surfaces, with binding energies ranging between $\sim$1000 K and 1600 K. The binding energies of the molecules can be directly estimated from the maximum desorption temperature as described in [@luna2017], where E$_{\rm{bin}}$ (in K)=30.9 $\times$ T$_{\rm{des}}$ (in K).
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Accretion of CO: summary of experimental results
------------------------------------------------
The two experiments differed in the CO ice deposition temperature (decreasing temperature from 80 K to 8 K [versus]{} fixed at 14 K), as well as in the resulting deposition rates (1.5 ML/min [versus]{} 7.3 ML/min, respectively). Deposition of CO with decreasing surface temperature in the ISAC setup shows that CO accretes on the surface [at temperatures below]{} 26.5 K. At this temperature, the drop in the CO ion current, due to the important accretion on the substrate, reflects the multilayer regime, where the temperature is low enough so that CO can be bound to CO ice. This suggests that CO molecules deposited at 26.5 K are able to find binding sites on the CO ice with binding energies of the order of 800 K. [The accretion rate was found to remain constant as the substrate temperature decreased, suggesting that the sticking coefficient of the CO molecules remains constant with temperature, as reported in @dawson1965. ]{} In addition, the two experiments show that deposition at the two different conditions mentioned above result in similar TPD curves ([TPD curves are also very similar when the CO ice is accreted on different surfaces, as shown in Fig. \[tpdqms\]]{}), in which CO desorbs at $\sim$30 K in the multilayer regime, while the sub-monolayer regime desorbs between 30 K and 60 K (see also @noble2011). [The QMS measurements were confirmed by IR spectroscopy and reflective laser interference of the solid sample in the case of experiment 2 performed with the IPS setup. This seems to indicate that no structural changes in the CO ice during warming up could be detected during our experiments, which is in agreement with previous work (see, e.g., @munozcaro2016). However, a phase change from amorphous to crystalline CO ice is known to take place prior to thermal desorption (@kouchi1990). This indicates that the transition from amorphous CO as deposited at low temperatures (typically between 8 and 20 K) to crystalline CO must occur at temperatures close to the peak of desorption in the TPD curve see section 2.6]{}
[A desorption peak at around 50 K attributed to monolayer desorption has been reported in some previous experimental studies (see, e.g., @collings2004) but was not observed in the TPD curves shown in the previous sections. When the TPD is performed with a heater located close to the sample, a temperature gradient may be created from the sample to other parts of the cryostat. In this scenario, desorption of CO molecules adsorbed on surfaces different from the substrate can take place when the temperature of the substrate is higher than the temperature of this surface, leading to a desorption peak in the TPD curve that is actually an artifact. This is not the case of the present work.]{}
[While the TPD for CO deposition at fixed T=14 K or at 80 K downwards show similar features, their representation in logarithm scale shows a small bump near 20 K. This bump is most prominent for CO deposited at T=14 K (red) than at 80 K downwards (black). This indicates that CO molecules with lower binding energies are present on the CO ice in both cases, but also that deposition at 14 K provides more of such weakly bound CO molecules compared to deposition from 80 K downwards. ]{}
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Amorphous versus crystalline CO ice.
------------------------------------
[As mentioned in the previous section, the transition between amorphous to crystalline CO ice occurs at temperatures close to the peak of desorption in the TPD curve. We performed a series of TPD measurements for CO deposited at temperatures from 8 K to 27 K. For each experiment of CO deposited at fixed temperature, the temperature is then decreased to 8 K and the TPD is then performed until 50 K with a rate of 1 K/min. Figure \[amorph\_crys\] shows the different TPD peaks for the different deposition temperatures. The first obvious result is that TPD for low temperatures deposition is composed of 2 peaks, one located at 27 K and the other at 28 K. For the lowest deposition temperatures, at 8 and 20 K, the peak at 27 K is more intense, while for deposition at 23 and 24 K, the peak at 28 K is dominating. For deposition at 25K, although the beginning of the TPD is the same as for lower temperatures, only one peak is seen. This is also the case for deposition at 25.5 K and 26 K, which also show an increasing displacement at the beginning of the TPD. The change in the intensity of the peaks indicates that the phase change occurs above 24K (and is already evident at 25K), but the stronger shifts at 26 and 27 K indicate that there is still a fraction of amorphous ice mixed with the crystalline phase. Our study is in agreement with previous studies showing the phase change of CO from amorphous to crystalline being close to the desorption temperature (@kouchi1990).]{}
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Simulations
===========
We used a step-by-step Monte Carlo simulation to follow the formation of CO ices through deposition and [subsequent]{} evaporation in the gas phase. Our model is described in [@cazaux2016]. CO molecules originating from the gas phase arrive at a random time and location [to the substrate]{}, and follow a random path within the ice. The arrival time depends on the rate at which gas species collide with the surface (section 3.1). The molecules arriving on the surface can be bound to the substrate and to other CO molecules through van der Waals interactions. The binding energy of each CO molecule depends on its number of neighbours, as described in section 3.2. [Our theoretical approach to estimate binding energies is similar than the one from [@cuppen2007] and [@garrod2013] but here we had to determine how the binding energy of a single CO molecule increases with the number of CO neighbours around in order to reproduce the experimental results.]{} Depending on its binding energy, the CO molecules diffuse on the surface/in the CO ices. The diffusion is described in section 3.3. During warming-up, the CO molecules can evaporate from the substrate/ices (section 3.4). The number of molecules evaporating as a function of the temperature, corresponds to the QMS measurements.
Accretion
---------
In our model, we defined the surface as a grid with a size of 20$\times$20 sites. CO molecules from the gas-phase arrive on the grid and can be bound to the substrate (that we choose as water substrate to mimic the experiments) and/or to adsorbed CO molecules through van der Waals interactions. The accretion rate (in s$^{-1}$) depends on the density of the species, their velocity, and the cross section of the surface, and can be written as: $$R_{\rm{acc}} = n_{\rm{CO}} v_{\rm{CO}} \sigma \rm{S},$$ $v_{\rm{CO}}=\sqrt{8 k T_{\rm{gas}}/(\pi m_{\rm{CO}})} \sim 2.75 \times 10^4 \sqrt{\frac{T_{\rm{gas}}}{100}}$
cm s$^{-1}$ is the thermal velocity, S the sticking coefficient that we consider to be unity in this study. The cross section of the surface, $\sigma$, directly scales with the size of the grid we use for the simulations, which is 20$\times$20 sites in our calculations. Since the distance between two sites is 3 Å, the density of sites is what is typically assumed, i.e. $\sim$(3 Å)$^{-2}$$\sim$10$^{15}$cm$^{-2}$. The cross section scales with the size of the grid considered in our calculations, as $\sigma$ $\sim$ (3$\times$ 10$^{-8}$$ \times$ 20)$^2$cm$^2$=3.6 10$^{-13}$ cm$^2$ . The deposition rate is therefore: $R_{\rm{acc}} = 1.7 \ 10^{-8}$ n$_{\rm{CO}}$ s$^{-1}$, for T$_{\rm{gas}}$=300 K. In order to mimic experimental conditions with deposition rates of 7.8 ML/min $\sim$ 52 molecules/s and 1.5 ML/min $\sim$ 10 molecules/s, we set the density of CO molecules in the gas in cm$^{-3}$ as being n$_{\rm{CO}}$= 3 $\times$ 10$^{9}$ cm$^{-3}$ and n$_{\rm{CO}}$=6 $\times$ 10$^{8}$ cm$^{-3}$ cm$^{-3}$ respectively (1 ML corresponds to 400 molecules on a 20 $\times$ 20 grid).
Building CO ices.
-----------------
The desorption of a CO molecule on a water ice surface is observed between 30-50 K which corresponds to binding energies ranging between 900 and 1500 (@he2016,@martin2014, @noble2011). To account for the fact that the water substrate is not homogeneous, we describe the initial surface of water ice with a random distribution of binding energies centered around 1200 K with a dispersion of 180 K as shown in [@noble2011]. [In this sense, by using such a distribution, we are mimicking a non-flat and smooth substrate, as shown in many previous studies.]{} If the molecules are not bound to the surface, but start to pile up, the binding energies of CO molecules, due to CO-CO interactions, increase with the number of surrounding neighbours. The lowest interaction between two CO neighbour molecules is around 16 meV (185 K; @karssemeijer2014). In a multilayer regime, the binding energy of a CO molecule is about 830 K (@luna2014, @noble2011, @munozcaro2010, @pontoppidan2006, @Acharyya2007, @collings2003a). The binding energy as function of coverage of CO on ASW water ice has been reported by [@karssemeijer2014]. At low coverages, the binding energy of CO on the surface is $\sim$ 125 meV (1450 K), while for high coverages the surface becomes covered by CO molecules and the binding becomes $\sim$ 75 meV (870 K; TPD desorption temperature of 29 K, table 3 from [@martin2014]).
In order to estimate the binding energy of CO molecules as function of the number of neighbours, we use a simple approximation that is shown in Fig. \[cluster\]. The points show the interaction of a CO molecule with one single CO molecule, which is about 185 K and becomes of 860 K in the multilayer regime, which corresponds to 1 (CO adsorbed one CO molecule) to 3 direct neighbours (CO embedded on a top layer with 1 neighbour underneath and 2 neighbours around). By using a fit through these points we mimic a saturation for a high number of neighbours and can calculate the binding energy of a CO molecule as function of its number of neighbours nn: $$E_{\rm{CO}}=-3360*(nn+1)^{-2}+1040$$
In our calculations, we compute the binding energy by considering an effective number of neighbours nn, that scales with the distance between the neighbour CO$_{\rm{n}}$ and the CO molecule. In that sense, one neighbour would contribute as +1/(r$_{CO-CO_{\rm{n}}})^6$ depending on the CO-CO$_{\rm{n}}$ distance, to account for the fact that van der Waals interactions depend on distance as 1/r$^6$. For direct neighbours, this distance is 1, while for neighbours on the sides this distance is $\sqrt 2$ and for the neighbours located in a corner the distance is $\sqrt 3$.
\[h\] {width="53.00000%"}
[In the theoretical model from [@cuppen2007], the binding energy of individual molecules is the sum of the binding with their neighbours. [@garrod2013] performed off lattice KMC method to compute the reactivity and porosity of ices, also considering that the binding energy of one species is the sum of the pair-wise interaction potentials with its neighbours. While this method is more sophisticated than the present method since it allows to determine the distance of the species explicitly, we here consider the distance between CO molecules to be equal, and concentrate on defining the binding energies as function of neighbours. However, because our method does not compute the distance between CO molecules, the diffusion could be different because (1) the distances are not identical (2) using a Lennard-Jones potential, such as in [@garrod2013] instead of a square barrier, would change the diffusion efficiency. This could have an impact on the number of holes/micropores present in the ices deposited at very low temperatures.]{}
Diffusion
---------
A recent study on the diffusion of CO on hexagonal water ice surface shows that diffusion barriers are of the order of 50 meV (@karssemeijer2014), which represent only $\sim$ 30$\%$ of the binding energy. We define the diffusion rates depending on the number of neighbours interacting with the CO molecules, that we call $nn$ in the above section. For a position (i,j,k) of a CO molecule in the grid, we calculate the associated binding energy Ei and identify the possible sites where the molecule can diffuse to as i$\pm$1; j$\pm$1; k$\pm$1. The final binding energy Ef is calculated as function of the neighbours present around this site. The diffusion rate, from an initial site with an energy Ei to a final site with an energy Ef, is illustrated in Fig.\[barrier\]. The barrier to go from Ei to Ef is defined as follows if Ei$\le$Ef (Fig.\[barrier\]; left panel): $$\rm{Ed= \alpha \times min(Ei,Ef)}, \hspace{1cm} if \ \rm{Ei<Ef}$$ If Ei$>$Ef, on the other hand, the barrier becomes (Fig.\[barrier\]; right panel) $$\rm{Ed= \alpha \times min(Ei,Ef) +\Delta E}, \hspace{1cm} if \ \rm{Ei>Ef}$$ with $\Delta$E = max(Ei,Ef)-min(Ei,Ef). By defining the barriers in such a manner, we do take into account microscopic reversability in this study (@cuppen2013). The barriers to move from one place to another should be identical to the reverse barrier. The diffusion barriers scale with the binding energies with a parameter $\alpha$. [While this parameter is found to be of 30$\%$ for CO on water ice from [@karssemeijer2014] which is of the same order of the water-on-water diffusion derived experimentally (@collings2003b), recent studies also point out to very small values for the CO diffusion barriers (@lauck2015). However, the CO-on-CO diffusion has not been determined, but studies highlight the large differences between bulk and surface diffusion (@ghesquiere2015). The diffusion parameter $\alpha$ sets the temperature at which CO molecules can re-arrange and diffuse in the ices to form more dense ices. In this study, we reproduce experimental results in order to constrain the diffusion parameter.]{}
\[h\] {width="50.00000%"}
The diffusion rate, in s$^{-1}$, for a CO molecule can be written as: $$R_{\rm{diff}} = 4 \times \sqrt{\frac{\rm{Ei-Es}}{\rm{Ef-Es}}}\times \nu \exp\left({-\frac{E_{\rm{act}}}{\rm{T}}}\right),$$ where $\nu$ is the vibrational frequency of a CO molecule in its site (that we consider as 10$^{12}$ s$^{-1}$), T is the temperature of the substrate (water ice or CO ice) and Es is the energy of the saddle point, which is Es=(1-$\alpha$)$\times$ min(Ei,Ef). This formula differs from typical thermal hopping because the energy of the initial and final sites are not identical (@cazaux2004).
Evaporation
-----------
The CO molecules present on the surface can return into the gas phase because they evaporate. This evaporation rate depends on the binding energy of the species with the surface/ice. As mentioned previously, the binding energy of a CO molecule depends on its number of neighbours, or wether the molecule is directly bound to the surface. The binding energy Ei of the CO molecule sets the evaporation rate as: $$R_{\rm{evap}}(X)=\nu \exp\left({-\frac{Ei}{T}}\right),$$ where $\nu$ is the oscillation factor of the CO molecule on the surface, which is typically $\nu$=10$^{12}$ s$^{-1}$, and T the temperature of the substrate.
Theoretical results
-------------------
### Deposition
We have performed simulations for the two different deposition rates used experimentally (1.5 ML/min for CO deposited from 80 K with decreasing temperatures of 0.5 K/min down to 8 K, and 7.8 ML/min for CO deposited at 14 K).
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[Our first goal is to constrain the diffusion parameter $\alpha$ (ratio between diffusion barrier and binding energy) from the constant accretion measured in experiment 1. In this first experiment, the CO molecules are deposited [on a water ice surface]{} as the temperature of the substrate is decreased from 80 K to 10 K. In our calculations, the diffusion coefficient $\alpha$ is set to 0.9 and 0.7. Our results presented in the left panel of Fig. \[stick\] show that in order to reproduce the experimental accretion rate, the diffusion doesn’t need to be efficient, since the experimental results are reproduced with $\alpha$=0.9 (low diffusion, see Sect. 3.3). This is because of the very high flow of CO arriving on the surface, needed to reproduce the experimental accretion rate. If the flux was 6 times lower (corresponding to a density of 10$^8$ cm$^{-3}$), then the accretion would be much lower for $\alpha$=0.9. This is because for lower flows, there is a competition between diffusion (which allow to find higher binding sites with more neighbours) and evaporation. Therefore, the first experiment does not allow to constrain the diffusion parameter.\
We then tried to constrain the diffusion parameter $\alpha$, from the second experiment. For this purpose we computed the evolution of the thickness with increasing temperature, as in the second experiment. Our results, considering a diffusion parameter $\alpha$=0.7 and 0.9, are shown in figure \[stick\], right panel. The thickness of the CO ice depends on the diffusion of CO molecules within the ices. If the diffusion barrier is of 90$\%$ of the binding energy ($\alpha$=0.9), then the re-organisation of the CO molecules occurs just before desorption, and this cannot be seen in the change of thickness of the ices. However, for a diffusion barrier of 70$\%$ of the binding energy ($\alpha$=0.7), re-organisation of CO molecules in the ices imply a decrease of the thickness at around $\sim$22 K (which corresponds to $\sim$0.7$\times$ E$_{\rm{bin}}$/30.9 where E$_{\rm{bin}} \sim$900 K is the energy of CO in the bulk), which is not seen in the experiments. In order to observe no thickness decrease between 14 and 27 K, as shown experimentally, the diffusion parameter should be either $\alpha \ge$ 0.9, so that the thickness decrease occurs around 27 K, or $\alpha$ $<$ 0.5 (T$\sim$0.5$\times$900 /30.9$\sim$ 14.5 K) so that the thickness decrease occurs below 14 K. Therefore, we conclude that CO diffusion within CO ices has to be either lower than 50$\%$ of the binding energy or higher than 90$\%$ of the binding energy. ]{}
[In the experiments, the ices obtained after deposition were heated to higher temperatures and the desorption was measured. To mimic the warming up, we performed simulations with the ices deposited from 80 K to 8 K. To simulate the warming up of the ices, we used conditions similar to the ones met in the first experiment so that the temperature of the substrate was heated at a rate of 0.5 K/min until 80 K. The TPDs obtained are shown in Fig. \[tpd\]. We computed the TPD for two different $\alpha$ of 0.9 and 0.7. Our simulations can reproduce the multilayer peak located around 30 K as well as the monolayer contribution, extending after 40 K. Also, the small desorption bump occurring at $\sim$20 K in the experiment can be reproduce if the diffusion is inefficient ($\alpha$=0.9). If the diffusion is higher, then the CO molecules do re-organise instead of desorbing. Our model therefore confirms the presence of weakly bound CO molecules in the TPD, that can be explained only if the diffusion of CO in the ice is inefficient. Note that the amount of CO desorbing around $/sim$20 K is small so we used a larger grid of 40$\times$40 to make these computations.]{}
Our simulations show that even if ices present different structures and different binding energies, the TPD measurements are very similar. However, the subtle differences seen in the TPD, especially at low temperatures, do highlight the different structure of the ice and the presence of weakly bound CO molecules.
![Simulated TPD from Monte Carlo simulations with two different values for $\alpha$. []{data-label="tpd"}](COtpd.pdf){width="45.00000%"}
In our simulations, we use a grid of 20 $\times$ 20 sites, and each CO molecule is represented by a box as shown in figures \[COads\] left panel, for the first experiment and \[COads\], right panel, for the second experiment. The different colours show the binding energy of each of the CO molecules. These binding energies range from 300 K (a CO molecule with one direct neighbour and neighbours on the sides) to $\sim$1700 K (a CO molecule on the deeper sites of the water substrate).
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To mimic the first experiment with our simulations, CO molecules are admitted on the surface, and we follow the build up of the CO ice, which takes place layer by layer as the temperature goes down. [The first accreted monolayer, formed at temperatures higher than 26.5 K, has no empty sites, since CO molecules can diffuse on the surface to find the binding sites with binding energies high enough so that they can stay on the surface. That is, CO molecules first populate the strongest binding sites. As the temperature decreases further, the weakest binding sites on the water surface can also be filled. This makes the first CO layer (directly bound to the water surface) completely filled.]{} The drop in gas phase CO at 26.5 K implies that the temperature is low enough to allow the adsorption of CO [molecules]{} on the first layer of CO. For these temperatures, the multilayer regime is reached and CO from the gas phase disappears while CO [ice]{} can be seen on the surface, as shown in Fig. \[acc\]. [The next layers of CO are built on a similar manner on top of previously deposited CO (one layer without holes) at 26.5 K. These layers are also very well organised without any holes. As a result, the first layers of ice (deposited at higher temperatures) show almost no holes compared to the ice grown at low temperatures (see below). As the temperature decreases further, a temperature range is reached at which the molecules do not diffuse anymore. These next layers show more holes and weaker binding CO molecules. The top layer (deposited at 8 K), present many holes and weakly bound molecules. This CO ice is a hybrid ice made of a compact CO ice where no holes can be seen at the bottom layers, and weakly bound molecules and many holes becoming present as the number of layers increase.]{}
The ice [structure]{} after deposition in the second experiment is shown in Fig. \[COads\], right panel. In the second experiment, CO molecules are deposited at 14 K on different types of surfaces. In our simulations, we concentrate on the adsorption of CO on water ice. The resulting CO ice shows many empty spaces, and weakly bound CO molecules. This is due to the fact that the deposition temperature is low enough for the CO molecules to be weakly bound, [and]{} not to evaporate. The CO molecules arriving on the surface with low binding energies ($\sim$ 200 K) can go to a site where the binding energy is high enough to settle in that site.
Therefore, experiments and simulations show that ices deposited at 80 K downwards, and at fixed temperature of 14 K have different structures. We conclude that the deposition temperature sets the binding energy of the CO molecules in the ice.
Astrophysical applications
==========================
In starless cores, an important CO depletion has been measured (@bergin2002). Evidence of depletions by factor of 4-15 ([75-94$\%$ of CO missing from the gas phase]{}) in many of these cores has been found (@bacman2002). However, by observing CO isotopologues, a [higher]{} CO depletion of [up to]{} 100 and 1000 in the center of the B68 and L1544 [dense clouds]{}, respectively, [has been]{} measured ($\ge$99% of CO missing from the gas phase; @caselli1999 [@bergin2002]).
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The most common explanation is that CO is frozen on dust grains at high densities and low temperatures. However, in order to match the observed CO spectra, [@keto2010] found that the desorption rate due to cosmic ray strikes, ([@Hasegawa1993]), should be increased by a factor of 30. At this rate, desorption and depletion have equal time-scales at a density of about 10$^4$ cm$^{-3}$. There are other [non-thermal]{} processes in addition to direct cosmic ray strikes that cause desorption and could increase the gas-phase abundance of CO. These processes are photodesorption with UV photons (@oberg2007 [@oberg2009a; @munozcaro2010; @fayolle2011; @munozcaro2016]); formation of [H$_{2}$]{}(@takahashi2000), non canonical explosions (@rawlings2013); direct cosmic ray sputtering (@dartois2015) or cosmic ray induced explosive chemical desorption (@shen2004); chemical desorption (@dulieu2013) and impulsive spot heating on grains (@ivlev2015).
In order to understand which processes allow to keep [the required]{} fraction of the depleted CO in the gas phase, we first compute the binding energies of the CO molecules arriving on the surface with our Monte Carlo simulations. We calculate the binding energies of each CO molecule as they arrive on the CO ices in the dense core conditions (at T$_{\rm{dust}}$=6 K and n$_{\rm{H}}$=10$^{6}$). We here therefore consider only CO in multilayer [regime]{} (not directly bound to the surface, [but to other CO molecules]{}). [We consider a diffusion parameter $\alpha$ of 0.9 as derived in the experiments. The resulting structure of the CO ice and binding energies of CO molecules in the ices, after moving and being relocalised on the surface is shown in Fig. \[binding\]. The CO molecules in the ices are weakly bound and many holes are present. The CO ices deposited in conditions present in pre-stellar cores show many holes and a wide range of binding energies. ]{}
In order to address how [this range of]{} CO binding energies influence the freeze out of CO molecules from the gas phase, we used a time dependent gas-grain model to follow the abundances of species in the gas, as well as in the ices [in a pre-stellar core]{}. We used a three-phase chemical model that combines gas-phase chemistry with surface and bulk chemistry. The grain surface chemistry model (surface + bulk) takes into account the different binding energies of the species on bare or icy surfaces and includes evaporation, reactions, photodissociation, and thermal and photodesorption processes, which transform surface species either into other surface species or into gas-phase species, as in [@cazaux2016]. Note that in this study, we only consider CO freeze out on the dust surface but do not allow surface reactions with CO. [In addition,]{} our model does not take into account the diffusion of species from bulk to surface and from surface to bulk. In this sense, when the coverage has reached one layer, the accreting species become bulk species with higher energies and lower diffusion rates (because the diffusion depends on the binding energy). The gas-phase chemical model is adopted from the KIDA database (@wakelam2012), while the surface chemistry model is described in [@cazaux2016]. The input parameters to mimic the temperature and density profile of a pre-stellar core are taken from [@keto2010] and shown in Fig. \[DT\].
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The C$^+$, C, CO, O and H$_2$O abundances in the gas phase as a function of extinction are reported in Fig. \[gas\] for 10$^4$ years, left panel. Our results show that if we consider a binding energy of CO, typical of CO on CO ices $\sim$ 830 K, then the CO depletion at extinctions of A$_{\rm{V}}$ $\sim$ 20 can reach 5 orders of magnitude [(0.001% of CO in the gas phase, that is, 99.999% of CO frozen-out in the ice and missing from the gas phase)]{}. However, as the binding energy of weakly bound CO is considered, such as 400 K (green), 350 K (blue) and 300 K (pink), as shown in Fig. \[gas\] right panel, the depletion is strongly decreased and CO in the gas phase reaches 0.1$\%$, 10$\%$ and [100]{}$\%$ respectively.
Therefore considering the weakly bound CO molecules can completely change the depletion of CO at high A$_{\rm{V}}$. As the cloud evolves and reaches 10$^5$ years, C$^+$, C, CO, O and H$_2$O abundances also evolve as shown in Fig. \[gas2\], left panel. For typical binding energies of CO with CO ($\sim$ 830 K), the depletion is more pronounced from extinctions of A$_{\rm{V}}$ $\sim$2. When the weakly bound CO molecules are also considered, as shown in Fig. \[gas2\] right panel, differences from the depletion of CO can be already seen for binding energies of 400 K (green). This is due to the fact that the abundances of CO are set by the accretion versus evaporation processes. In this case, the CO bound at low energies will be earlier in equilibrium, which implies that the abundances of CO in the gas phase for bindings of CO of 300 and 350 K do not change between 10$^4$ and 10$^5$ years. However, for CO molecules more strongly bound to the surface, the equilibrium is not reached yet at 400 K and 830 K and depletion is still increasing.
[While we showed that considering low binding energies for CO would change the depletion of CO in dense cores, we could not compute with a rate equation model the depletion of CO due to the distribution of binding energies on the last layer of the ices. To perform such calculations, one should use a more detailed model considering layers to mimic the icy mantle covering dust grains (@taquet2012). However, our simulations indicate that in the conditions met in dense cores, the CO depletion could be described by higher CO binding energies at low A$_V$ (between 0 and 4) and lower binding energies after A$_{\rm{V}}$=4. From figures \[gas\] and \[gas2\] right panels, the CO depletion would therefore range between the red and green curves (CO binding between 830 and 400 K) below A$_{\rm{V}}$=4 and would range between the green and blue curves (CO binding between 400 and 350 K) for higher A$_{\rm{V}}$. ]{}
Summary and conclusions
=======================
We showed experimentally that CO ices [can accrete at temperatures below 26.5 K, with an accretion rate that does not depend on the substrate temperature. In addition, CO ices deposited under different conditions (decreasing temperatures from $\sim$ 80 K until 8 K versus constant low temperature of 14 K) show TPD spectra with small differences. These differences highlight the two different structure of the CO ices and the presence of weakly bound CO molecules desorbing around $\sim$20K. Using Monte Carlo simulations, we can reproduce such differences and the weakly bound CO molecules at the condition that the mobility is very inefficient. We show that the diffusion barrier should be 90$\%$ of the binding energies to reproduce such experimental results. ]{} [In this work we show that the deposition conditions (flow and temperature) set the binding energy. During the warming up of the CO ices, the CO molecules re-organise in the most stable configuration, which result in almost identical TPD spectra. However, while these TPDs appear similar, the subtle differences highlight the differences of the structure of the CO ices. Re-organisation occurs only at temperatures close to desorption. The different structure becomes clear when CO is deposited at 25.5-27 K. At these temperatures closes to the maximum in the desorption during TPD experiments, the CO is deposited mainly as crystalline $\alpha$-CO. The proportion of amorphous-to-crystalline ice decreases as the deposition temperature approaches the temperature of maximum desorption.]{}
In environments where stars [form]{}, the temperature can be so low that weakly bound molecules do not re-organise and stay weakly bound to dust grains. This has an impact on the gas phase composition of the environment, but also on the chemistry occurring on [the surface of dust grains]{}, which could be more efficient as weakly bound species are more mobile. In pre-stellar cores, CO molecules are seen to be depleted as the medium becomes denser and cooler. While this is attributed to the freeze out of CO molecules from the gas phase, actual models overestimate the freezing of CO on dust. In this work we show that considering weekly and strongly bound molecules in the CO ices changes the CO depletion in pre-stellar cores and allow a less severe depletion.
S. C. is supported by the Netherlands Organization for Scientific Research (NWO; VIDI project 639.042.017) and by the European Research Council (ERC; project PALs 320620). This work was supported by the MOST grants MOST 103-2112-M-008-025- MY3 (Y.J.C.) and by the Spanish MINECO under projects AYA2011-29375, AYA2014-60585-P and AYA2015-71975-REDT (R. M.-D. and G. M. M. C.). We acknowledge Dr. Wing-Fai Thi for useful feedback on the manuscript. We would like to thank the anonymous referee for his/her valuable comments which helped to improve the manuscript.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Provable safety is one of the most critical challenges in automated driving. The behavior of numerous traffic participants in a scene cannot be predicted reliably due to complex interdependencies and the indiscriminate behavior of humans. Additionally, we face high uncertainties and only incomplete environment knowledge. Recent approaches minimize risk with probabilistic and machine learning methods – even under occlusions. These generate comfortable behavior with good traffic flow, but cannot guarantee safety of their maneuvers. Therefore, we contribute a safety verification method for trajectories under occlusions. The field-of-view of the ego vehicle and a map are used to identify critical sensing field edges, each representing a potentially hidden obstacle. The state of occluded obstacles is unknown, but can be over-approximated by intervals over all possible states. Then set-based methods are extended to provide occupancy predictions for obstacles with state intervals. The proposed method can verify the safety of given trajectories (e.g. if they ensure collision-free fail-safe maneuver options) w.r.t. arbitrary safe-state formulations. The potential for provably safe trajectory planning is shown in three evaluative scenarios.
*Index Terms*— ADAS, automated vehicles, formal verification, reachability analysis, risk assessment, occlusions, field-of-view.
author:
-
bibliography:
- 'root.bib'
title: ' Tackling Occlusions & Limited Sensor Range with Set-based Safety Verification '
---
Introduction
============
Over the last decades research effort about ADAS and fully automated vehicles has increased drastically in academia and the commercial sector. The latter presenting a worrying push to early market introduction as at least two major challenges, safety and scalability, have not been solved yet [@mobileye_formal_2017].
Scalability is an issue, because most state-of-the-art approaches require sensors and processing power that is not, nor will likely be available for prices that allow mass sales in the near future [@mobileye_formal_2017].
Provable safety on the other hand is a must with regard to accountability, user acceptance and in consequence as a prerequisite for legal permission. As extensively explained in [@mobileye_formal_2017], provable safety is of special interest when applying machine learning approaches as they often lack formal validation methods. Therefore, they introduce the notion of blame, define a concept of Responsibility Sensitive Safety (RSS) and explain how to develop automated vehicles that provably fulfill RSS. Another promising and meanwhile complementary approach to safety verification is the computation of reachable sets of obstacles (reachable states limited by physics and traffic law) [@althoff_set-based_2016]. These are used to prove if the ego trajectory allows a fail-safe maneuver in the next planning step and, therefore, is safe in itself. They formally introduce the method for sets of initial obstacle states, but de facto developed only over-approximations for exactly known initial obstacle states.
In reality, we don’t know exact obstacle states, because we encounter a variety of uncertainties and incomplete environment knowledge. The former arise in all stages of an ADAS pipeline, from measurement noise and sensor limitations, over processing steps as localization, tracking, prediction and planning (due to modeling errors or unexpected situations), up to the imprecise realization of planned trajectories. Additionally, the environment knowledge is incomplete because of limited perception range and due to occlusions from static and dynamic obstacles alike. A typical example is illustrated in \[fig:eye\_catcher\].
![Dangerous intersection with occlusions. Yellow: The obstacle is occluded by a container, such that it and its reachable set is not known to the ego vehicle. Red: We can over-approximate the reachable set of possibly hidden vehicles and adapt the ego trajectory (transparent blue) to still guarantee safety. Imagery © 2018 Google, Map data © 2018 GeoBasis-DE/BKG. []{data-label="fig:eye_catcher"}](figures/eye_catcher){width="\columnwidth"}
Related Work {#sec:related_work}
============
A lot of research has been done to approach the problem of risk assessment. A well-arranged survey is given in [@lefevre_survey_2014].
Most of the methods presented try to develop behavior models and then check for collisions under the assumption of those models or detect deviations from these models. Either way such risk assessment approaches assume that all possible maneuvers in a given scenario can be modeled or that unexpected situations can be detected reliably. Additionally, many do not consider the incompleteness of an environment model at all.
Recently focus on occlusion-aware[^1] risk assessment and behavior generation has increased [@hoermann_entering_2017; @lee_collision_2017; @chung_safe_2009; @brechtel_probabilistic_2014; @miller_isaac_team_2008; @sadou_occlusions_2004; @bouraine_passively_2014; @zhan_non-conservatively_2016]. These range from simple visibility modeling that improves the tracking of previously detected obstacles [@miller_isaac_team_2008], to sophisticated multi-layered environment models [@hoermann_entering_2017].
Some of them explicitly consider uncertainties [@zhan_non-conservatively_2016; @hoermann_entering_2017; @brechtel_probabilistic_2014], while others use visibility analysis to define velocity constraints [@chung_safe_2009; @lee_collision_2017]. Only two publications among those prove at least passive motion safety[^2] [@bouraine_passively_2014] or prove collision freedom at discrete time steps of their trajectories[^3] [@zhan_non-conservatively_2016].
A promising approach is given by [@hoermann_entering_2017]. They model and predict the environment with three grid map layers: Object-based, object-free and unobservable environment. During planning, they treat a cell as occupied as soon as one of the layers is predicted as occupied. Still, these methods minimize the risk of collisions at most [@brechtel_probabilistic_2014; @chung_safe_2009; @hoermann_entering_2017; @lee_collision_2017] and give no or too weak [@bouraine_passively_2014; @zhan_non-conservatively_2016] safety guarantees. Some of the earliest approaches show even fundamental problems like ignoring not yet observed obstacles [@miller_isaac_team_2008], analyzing occluded areas without prediction [@sadou_occlusions_2004] or only investigating occlusions from static obstacles [@chung_safe_2009].
Concluding, all these methods lack verification of higher levels of safety[^4] considering limited environment knowledge.
Vehicles with such risk assessment strategies will try to follow their intended behavior as long as no noteworthy risk is detected. But this could still lead to situations where a collision is inevitable.
We want to promote an inverse methodology. An automated vehicle should only follow its intended trajectory as long as it can prove that it is safe. Let us ensure safety first and then increase comfort and traffic flow, not the other way around. Such a development approach will lead to more conservative behavior in the first years, but on the other hand help in earlier legal permission and to gain users confidence.
Thus, we tackle the safety verification problem in scenarios with occlusions by over-approximating all possible states instead of engineering discrete maneuvers or maneuver classes.
Hence, our safety concept does not rely on e.g. the performance of an intention estimation module. It depends only on the reliability of an ego localization[^5], the detection of obstacles, of occluded areas and on a map[^6].
Therefore, in this work we characterize potential risk from occlusions and limited sensing (\[sec:main\_risk\_from\_occlusion\]), enhance the reachable set over-approximations introduced by [@althoff_set-based_2016] to serve well for these critical perception field bounds (\[sec:main\_reachable\_sets\_for\_intervals\]) and show its potential for safe trajectory planning (\[sec:main\_planning\]) in several simulative scenarios (\[sec:results\]).
Our main contribution, addressed in \[sec:main\_risk\_from\_occlusion,sec:main\_reachable\_sets\_for\_intervals\], is twofold.
1. We formalize the potential risk from occlusions and limited sensing by over-approximating all possible states of unobservable obstacles with state intervals.
2. We derive reachable set over-approximations for obstacles with such initial state intervals.
The approach is not only applicable for fully automated vehicles as in our evaluation, but also for level 1–4 ADAS [@committee_taxonomy_2014], e.g. collision warning systems. Additionally, our modeling enables an easy integration of uncertainties from measurements and can be used with any safety definition.
Risk from Occlusions and Limited Sensing {#sec:main_risk_from_occlusion}
========================================
As a necessary preparation for the following chapters, we first characterize the potential risk that results from occlusions and limited sensor range.
We model the environment mostly as in [@althoff_set-based_2016], meaning that the road geometry and topology is given as a lanelet map [@bender_lanelets_2014], a lane consists of consecutive lanelets, the ego vehicle and other (visible) obstacles have rectangular shape and the over-approximations of predicted occupancies are modeled with polygons. The main difference in modeling is that we explicitly represent critical sensing field borders.
To do so we need a representation of the sensing field of the ego vehicle. This can be provided by accurate sensor models or explicitly mapped from sensor data. The mapping is straight forward for 3D range sensors, e.g. all range measurements (occupied or free) can be modeled as rays with specified beam angle and used directly as geometrical shapes or transferred into a visibility layer of a grid map at the cost of discretization errors.
In our simulative evaluation we use a simple visibility model, assuming a $\ang{360}$ range sensor with $\SI{50}{\meter}$ viewing range mounted on top of the vehicle center using a direct geometrical representation, see the light blue filled area in \[fig:critical\_edges\].
![Critical sensing field edges. The one-way driving direction of each lane is visualized as an arrow. Relevant edges are red, irrelevant green. Edge 1 is not relevant as it is on a lane that does never cross or lead to lanelets we travel. Edge 2 is not relevant as its travel direction leads outside of the sensing field. Edge 3 is relevant as its lane leads to and crosses the ego lane. Edge 4 and 5 are not relevant as their risk is already covered by edge 3.[]{data-label="fig:critical_edges"}](figures/critical_edges){width="\columnwidth"}
The borders of the sensing field can then be extracted and intersected with all lanelets. Each of these intersections generates at least one border segment and can be classified into relevant and irrelevant sections. For computational efficiency we assume the border to be modeled as a polygon, thus the border sections consist of line segments.
The classification of relevant sections is not self-evident, but can be reduced to false positives only, which are not critical. In general only sections that could hide obstacles with right of way need to be examined. Falsely considering non relevant sections as relevant comes at an additional performance cost, but does not affect the safety verification result. That is because the occupancy of corresponding potential obstacles does not intersect with the current or any of the coming ego lanelets. Employed on the lanelet representation, relevant sections can be classified as non relevant without false-positives in these incomplete cases:
- Border segments that do not intersect any lanelet which can lead to or cross any of the lanelets the ego vehicle is currently or will be traveling.
- Segments on lanelets without right of way.
- Segments in the ego lane that are behind the ego vehicle.[^7]
- Edges that lead outside of the sensing field[^8].
- For multiple segments on the same side[^9] of the same lane only the foremost segment\[ft:intersection\] is relevant.
See \[fig:critical\_edges\] for a descriptive example of critical sensing field edges.
Having identified the critical sensing field edges, the key question arises: What is the potential risk that results from these? The illustrative answer is that each occluded area might contain at least one obstacle with unknown state. A naive approach of simply spawning countless virtual obstacles with randomly chosen possible states and predict their occupancy is not an option as a stochastic approach will not enable us to verify planned trajectories with reasonable computational cost. But even though the state is unknown, we can over-approximate possible states as bounded sets, such that other states are either physically impossible or would not lead to an accident of our blame (as for tremendous speeding of the obstacle for example). Therefore, we define one virtual obstacle for each critical edge $e$ with the following state set, described as intervals in orientation $\psi_e(0)$ and velocity $v_e(0)$ and a line segment in the initial position $\bm{s}_e(0)$ defined by the two edge vertices $\bm{s}_1$ and $\bm{s}_2$. We use the following notation:
\[eq:x\_edge\] $$\begin{aligned}
\bm{s}_e(0) &\in \left[ \begin{pmatrix} s_{1,x} \\ s_{1,y} \end{pmatrix}, \begin{pmatrix} s_{2,x} \\ s_{2,y} \end{pmatrix} \right]\\
\psi_e(0) &\in [\psi_{\min}, \psi_{\max}] \\
v_e(0) &\in [v_{\min}, v_{\max}]
\end{aligned}$$
In the following we will enhance the reachable set approximations introduced by [@althoff_set-based_2016] and released as source code [@koschi_spot_2017] for such initial state sets.
Reachable Sets for Intervals of Initial State {#sec:main_reachable_sets_for_intervals}
=============================================
In this section we derive reachable set over-approximations for obstacles with initial state intervals \[eq:x\_edge\]. These are based on the $M_1$ and $M_2$ over-approximations from [@althoff_set-based_2016]. $M_1$ describes the physically reachable area based on Kamm’s circle, limiting the possible absolute acceleration and prohibiting driving backwards as an assumption. To incorporate a lane-following property $M_2$ describes the longitudinally reachable area with maximum velocity and maximum engine power. The maximum velocity should be set to a realistic value which represents expectable speeding (e.g. $\SI{110}{\percent}$ of the speed limit), the maximum engine power can be set to infinity.
The following subsections expand the $M_1$ and $M_2$ formulas step by step to intervals of initial state. We keep the original notation wherever feasible, but redefine some variables to keep a clean notation, free of avoidable indices and accents.
Acceleration-Based Occupancy $M_1$ {#subsec:main_reachable_sets_m1}
----------------------------------
For simplicity, we assume the initial obstacle state to be represented in local coordinates, w.l.o.g.:
\[eq:x\_local\] $$\begin{aligned}
\bm{s}(0) &\in \left[ \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} \overline{s_x} \\ \overline{s_y} \end{pmatrix} \right]\\
\psi_e(0) &\in [-\psi_{\max}, \psi_{\max}]\\
v(0) &\in [\underline{v}, \overline{v}]
\end{aligned}$$
We use a formulation of Kamm’s circle with center $\bm{c}(t)$ and radius $r(t)$ and the function $\bm{b}(t)$ bounding that circle over time from [@althoff_set-based_2016]:
\[eq:kamms\_circle\] $$\begin{aligned}
\bm{c}(t) &= \begin{pmatrix} s_x(0) \\ s_y(0) \end{pmatrix} + \begin{pmatrix} v_x(0) \\ v_y(0) \end{pmatrix} t\\
r(t) &= \tfrac{1}{2}a_{\max}t^2\\
b_x(t) &= v_0t-\frac{a^2_{\max}t^3}{2v_0}\\
b_y(t) &= \sqrt{\tfrac{1}{4}a^2_{\max}t^4-\left(\frac{a^2_{\max}t^3}{2v_0}\right)^2}
\end{aligned}$$
Please see \[fig:occ\_vel\] for a graphical representation.
### Interval of Initial Velocities {#subsec:main_reachable_sets_m1_vel}
With an interval in the initial velocity $v_0 \in [\underline{v}, \overline{v}]$, but known orientation $\psi(0)=0$ and $\bm{s}(0)=(0,0)^T$, we can define
$$\begin{aligned}
\label{eq:c_interval}
\underline{c}(t) &= c(t, \underline{v}), \quad
\overline{c}(t) = c(t, \overline{v})\end{aligned}$$
and $b_x, b_y$ likewise. The occupancy of the obstacle for a time period of $\tau_k = [t_k, t_{k+1}]$ can then be over-approximated by the polygon spanned by the points $\bm{q}_1, \dots, \bm{q}_6$:
\[eq:occ\_vel\] $$\begin{aligned}
\bm{q}_1 &= \left(\underline{c_x}(t_k)-r(t_k),\ r(t_k)\right)^T\\
\bm{q}_2 &= \left(\underline{b_x}(t_{k+1}),\ r(t_{k+1})\right)^T\\
\bm{q}_3 &= \left(\overline{c_x}(t_{k+1})+r(t_{k+1}),\ r(t_{k+1})\right)^T\\
\bm{q}_4 &= \left(\overline{c_x}(t_{k+1})+r(t_{k+1}),\ -r(t_{k+1})\right)^T\\
\bm{q}_5 &= \left(\underline{b_x}(t_{k+1}),\ -r(t_{k+1})\right)^T\\
\bm{q}_6 &= \left(\underline{c_x}(t_k)-r(t_k),\ -r(t_k)\right)^T
\end{aligned}$$
as visualized in \[fig:occ\_vel\].
[0.4]{} {width="\columnwidth"}
[0.3]{} {width="\columnwidth"}
[0.25]{} {width="\columnwidth"}
The left part is equivalent to the original $\mathcal{O}_1(\tau_k,\underline{v})$, but vertices $\bm{q}_3$, $\bm{q}_4$ are computed using $\overline{v}$. This occupancy encloses all $v_i \in [\underline{v}, \overline{v}]$ as each circle $C_{k+1}(v_i)$ has the same radius $r(t_{k+1})$, and center $c_y=0, {c_x}(t_{k+1}) \in [\underline{c_x}(t_{k+1}), \overline{c_x}(t_{k+1})]$. Therefore, it is enclosed by the polygon $P(\bm{q}_1, \bm{q}_2, \bm{q}_3, \bm{q}_4, \bm{q}_5, \bm{q}_6)$ spanned from $\underline{C_{k}}$, $\underline{C_{k+1}}$ and $\overline{C_{k+1}}$. Consequently, this polygon encloses all $C_{t}(v_i)$ with $t \in [t_k, t_{k+1}]$, proving that this polygon is an over-approximation of all reachable states for an obstacle with interval velocities.
### Interval of Initial Orientations {#subsec:main_reachable_sets_m1_phi}
With additionally an initial orientation interval of $\psi(0) \in [-\psi_{\max},\psi_{\max}]$ the whole occupancy $P(\bm{q}_1, \bm{q}_2, \bm{q}_3, \bm{q}_4, \bm{q}_5, \bm{q}_6)$ rotates around the origin depending on the real initial orientation of the obstacle. We can again over-approximate this set. The borders of the set can be derived by rotating $\bm{q}_1, \bm{q}_2, \bm{q}_3$ counterclockwise to $\overline{\bm{q}_1}, \overline{\bm{q}_2}, \overline{\bm{q}_3}$ and $\bm{q}_4, \bm{q}_5, \bm{q}_6$ clockwise to $\underline{\bm{q}_4}, \underline{\bm{q}_5}, \underline{\bm{q}_6}$ by $\psi_{\max}$ and over-approximating the circle given through the rotation of the furthest longitudinal point $\bm{p}_{long} = (\overline{c_x}(k+1) + r(t_{k+1}),\ 0)^T$ with
\[eq:eq\_circle\_approx\] $$\begin{aligned}
\bm{w}_0 &= \left(\frac{\overline{c_x}(t_{k+1})+r(t_{k+1})}{\cos\frac{\theta}{2}},\ 0\right)^T,
\quad \theta=\frac{\psi_{\max}}{n}\\
\overline{\bm{w}_j} &= R_{\theta_j}\bm{w}_0\\
\underline{\bm{w}_j} &= -R_{\theta_j}\bm{w}_0
\end{aligned}$$
for integers $j \in [1, n]$. A reasonable approximation of the circle over $\psi_{\max}$ of around $\ang{45}$ can already be achieved with $n=3$.
\[fig:occ\_vel\_xi\] illustrates the construction. The prove that each polygon with rotation $\psi_i \in [-\psi_{\max},\psi_{\max}]$ is enclosed by the polygon
$$\begin{aligned}
\label{eq:polygon_vel_xi}
P(\overline{\bm{q}_1}, \overline{\bm{q}_2}, \overline{\bm{q}_3},
\overline{\bm{w}_n}, \dots, \overline{\bm{w}_1}, \bm{w}_0, \underline{\bm{w}_1}, \dots, \underline{\bm{w}_n},
\underline{\bm{q}_4}, \underline{\bm{q}_5}, \underline{\bm{q}_6})\end{aligned}$$
follows trivially from its construction.
### Interval of Initial Positions {#subsec:main_reachable_sets_m1_pos}
The transfer to position intervals, meaning linear interpolations between both line segment vertices, can be realized by computing the occupancy $\underline{P}$ for $\underline{\bm{s}}(0)=(0, 0)^T$ as described in the previous subsection, creating a duplicate $\overline{P}$ that has been translated by $\overline{\bm{s}}(0)=(s_x, s_y)^T$ and computing the convex hull over both occupancies $\mathcal{O}_1(\tau_k)=\text{Conv}(\underline{P}, \overline{P})$.
The occupancy for each $\bm{s}_i \in \left[\left(\begin{smallmatrix}0\\0\end{smallmatrix}\right), \left(\begin{smallmatrix}s_x\\s_y\end{smallmatrix}\right)\right]$ (each possible position on the line segment) is enclosed by the convex hull $\mathcal{O}_1(\tau_k)$ due to the linearity of translations and line segments. Thus, $\mathcal{O}_1(\tau_k)$ is provably an over-approximation of the reachable set of an obstacle with an unknown but bounded initial state, modeled as \[eq:x\_local\].
The resulting over-approximation is visualized for realistic example parameters in \[fig:occ\_vel\_xi\_pos\].
Lane-Following Occupancy $M_2$ {#subsec:main_reachable_sets_m2}
------------------------------
The adaption of the original $M_2$ formulation for interval initial states follows quickly. We use the same computation of shortest paths in lanes as [@althoff_set-based_2016], but choose the start- and end-point wisely from our interval. To do so, we first sort both vertices of the line segment $\bm{s}_e(0)$ w.r.t. the longitudinal path coordinates of the corresponding lane or lanelet and name them $\underline{\bm{s}}(0)$, $\overline{\bm{s}}(0)$ such that $\underline{\bm{s}}(0) < \overline{\bm{s}}(0)$ without loss of generality. The start border of the lane occupancy is then given by $b_{\bm{s}tart}(\underline{\bm{s}_x}(0), \underline{\bm{s}_y}(0), \underline{v}(0))$. The maximal traveled distance $\xi_f(t)$ can be obtained with the limited maximum speed and engine power model as in [@althoff_online_2014]:
$$\begin{aligned}
\label{eq:acceleration_model}
a_{c2,long} =
\begin{cases}
a_{\max}\frac{v_S}{v}, & v_S<v<v_{\max} \land u_2>0,\\
a_{\max}, & (0<v \leq v_S \lor (v > v_S \land u_2 \leq 0)\\
0, & v \leq 0 \lor (v \geq v_{\max} \land u_2 \geq 0)
\end{cases}\end{aligned}$$
This enables us to compute the end border using the inflection point segmentation algorithm as $b_{end}(\overline{\bm{s}_x}(0), \overline{\bm{s}_y}(0), \overline{v}(0))$.
Finally, the occupancy prediction is obtained by computing and intersecting the reachable set approximations $M_1$ and $M_2$ for each critical sensing field edge in each prediction step period $\tau$ over the whole prediction horizon $[0, t_f]$. A trajectory planner can use this prediction to verify if its intended trajectory is safe or not.
Note that in case of known initial state, where the intervals collapse to precise values $\bm{s}_e(0)=[\bm{s}_0, \bm{s}_0]$, $\psi(0)=[\psi_0, \psi_0]$, $v(0)=[v_0, v_0]$, our formulation results in the same reachable set over-approximation as with the original method in [@althoff_set-based_2016].
Trajectory Planning with Reachable Sets {#sec:main_planning}
=======================================
The strategy for trajectory planning has been profoundly motivated and extensively explained in [@althoff_set-based_2016]. Yet we want to briefly summarize the method and explain our proof-of-concept implementation to give a better understanding of the following evaluation.
The planning approach assumes that we start in a safe state with a reliable planning frequency of $\frac{1}{{\scriptscriptstyle\Delta}t}$. The safety verification relies on induction. Starting from a safe state with a verified fail-safe trajectory at hands the planner will search for a desirable trajectory that will guarantee a fail-safe maneuver choice in the next planning step too. If it fails to find one, it switches to the fail-safe trajectory that has been found and verified in the previous planning step. Thus, the vehicle will always follow a trajectory whose safety has been proven.
To do so a reference trajectory will be planned in each planning step $t$, based on the current environment model. This might also contain reasonably good predictions of all obstacles, though it is not a prerequisite. But the better the prediction the higher the probability for a comfortable ride. As unexpected situations become rare events, the planner will rarely be forced to switch to the fail-safe maneuver. In order to generate safe trajectories a potential trajectory is computed. It consists of an intended part, the first part of the reference trajectory over $[t, t + {{\scriptstyle\Delta}t}]$, and a following fail-safe part over $[t + {{\scriptstyle\Delta}t}, t_\text{f}]$ towards a safe state. If we can prove that the potential trajectory does not intersect with any of the reachable set over-approximations of other obstacles, it is verified as safe. If the verification fails, the intended trajectory will be iteratively adapted until a verifiably safe trajectory, in the worst case the fail-safe trajectory of the previous planning step $t - {{\scriptstyle\Delta}t}$, is found.
Our proof-of-concept planner initializes the reference trajectory according to the intelligent driver model [@intelligent_driver_model] along the centerline of the ego lane. If the verification fails the intended acceleration is gradually changed towards the fail-safe trajectory until the potential trajectory can be verified as safe.
The planning and prediction horizon $t_\text{f}$ is defined by the time needed to reach a safe state. This raises two questions: What is a safe state? And in case we need to decide for a fail-safe maneuver, do we want to reach the safe state fast or comfortably?
The safe state itself depends on the intended maneuver. As long as we don’t want to cross or intersect with other lanes a full stop can be seen as safe state in urban areas. But as soon as we merge into or cross the traffic of another lane we need to ensure a safe cut-in time. Hence, a full stop is not a safe state in these cases. For a further discussion on safe states please refer to e.g. [@mobileye_formal_2017].
Choosing the desired trajectory towards a safe state is a trade-off between the comfort during a fail-safe maneuver and the probability of having to switch to one. Comfortable trajectories i.a. have low jerk and acceleration such that they need a long time to reach the safe state. This durations then defines the required prediction horizon for safety verification. With longer prediction horizons prediction occupancies are bigger, such that it becomes more likely that replanning is necessary to find a trajectory that succeeds in verification. As a consequence, if the prediction horizon is too long due to too comfortable fail-safe trajectories, the probability of too conservative behavior (from a users’ or traffic participants’ perspective) will be high. In our evaluation we used a compromise desired fail-safe deceleration of $\SI{4}{{\tfrac{\meter}{\second^2}}}$.
Results and Evaluation {#sec:results}
======================
We evaluated our trajectory validation method on three critical urban scenarios with occlusion from static or dynamic obstacles and show its usability for safe trajectory planning. All scenarios represent or are inspired by real intersections in the city of Karlsruhe and a small town Fürstenfeldbruck. They are based on the *commonroad* scenarios *DEU\_Ffb1\_1\_T1* and *DEU\_Ffb2\_1\_T1* [@althoff_commonroad_2017]. Our modified versions have been contributed to the *commonroad* benchmark as scenarios *DEU\_Ffb1\_4*, *DEU\_Ffb1\_5\_T1* and *DEU\_Ffb2\_3\_T1*.
We assume perfect perception from a $\ang{360}$ range sensor with $\SI{50}{\meter}$ viewing range mounted on top of the vehicle center. The desired ego velocity is set to $\SI{9}{{\tfrac{\meter}{\second}}}$ with a maximum comfortable acceleration of $\SI{2}{{\tfrac{\meter}{\second^2}}}$ and maximum possible acceleration of $\SI{8}{{\tfrac{\meter}{\second^2}}}$, a realistic value for dry asphalt [@wallman_friction_2001]. For an easier understanding of the effects of occupancies, we do not adapt the desired velocity based on curvature. The prediction and planning horizon is defined by a desired fail-safe deceleration of $\SI{4}{{\tfrac{\meter}{\second^2}}}$, a compromise between comfortable fail-safe trajectories and keeping the desired driving speed. Other occupancy prediction parameters are $t_\text{f} = \SI{9}{{\tfrac{\meter}{\second}}}/\SI{4}{{\tfrac{\meter}{\second^2}}} = \SI{2,25}{\second}$, ${{\scriptstyle\Delta}t}=\SI{0.1}{\second}$, $a_{\max}=\SI{10}{{\tfrac{\meter}{\second^2}}}$, $\underline{v}=\SI{0}{{\tfrac{\meter}{\second}}}$, $\overline{v}=\num{1.1} \cdot v_{\text{lim}}$, $\psi_{\max}=\ang{22.5}$, $n=3$.
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Merging on a T junction, static occlusion {#subsec:results_T_junction}
-----------------------------------------
The first scenario *DEU\_Ffb2\_3\_T1* is a T-junction with a major road leading from east to west with a speed limit of $v_{\text{lim}}=\SI{14}{{\tfrac{\meter}{\second}}}$ and a minor road from the north. One edge of this intersection is occupied by a container, which we doubled in size to dramatize the occlusion effect.
The ego vehicle drives from the north and wants to merge the major road to the west. The container occludes the eastern arm of the intersection, such that this arm will only be visible well enough around $\SI{2}{\meter}$ before merging into the lane.
The first row in \[fig:evaluation\] shows the behavior without incorporating occupancies, only based on occupancy predictions of visible obstacles. As to expect, the ego vehicle does not reduce its velocity in the first $\SI{2.1}{\second}$, then detects the obstacle and has to decelerate at maximum rate, as it does not find a safe trajectory anymore. But the emergency braking comes too late and a collision is unavoidable if the other vehicle does not react or lacks enough reaction time, e.g. with a speeding of $\SI{10}{\percent}$ $(\SI{15.4}{{\tfrac{\meter}{\second}}})$ or with bad friction, e.g. because of a wet road.
The effectiveness of an occlusion aware occupancy prediction with our provided method is shown in the second row of \[fig:evaluation\]. The ego vehicle reduces its velocity as soon as the potential trajectory cannot be verified as safe, because it intersects the occupancy of a potentially occluded obstacle. As in fact there is an obstacle appearing behind the occlusion, the ego vehicle safely comes to a full stop to give way. Without an obstacle it could safely continue merging after it has decelerated to around $\SI{2.4}{{\tfrac{\meter}{\second}}}$ at the point to see far enough to the east.
Crossing an X junction, static occlusion {#subsec:results_X_junction_cross}
----------------------------------------
The second scenario, shown in row 3 of \[fig:evaluation\] is an intersection in the residential area of Fürstenfeldbruck with a speeding limit of $\SI{40}{{\tfrac{\kilo\meter}{\hour}}}$. We model it as an uncontrolled intersection, meaning without traffic lights or stop lines and that the priority-to-the-right rule applies, with $v_{\text{lim}}=\SI{11}{{\tfrac{\meter}{\second}}}$.
The ego vehicle comes from the south and wants to cross the intersection. Similarly to the previous scenario, a static obstacle, in this case a residential building, occludes the easterly lanes.
The simulation without dynamic obstacles shows that the ego vehicle has to slow down significantly, in order to guarantee safety, but can finally pass the intersection. This is truly an appropriate behavior for such a dangerous setting as such intersections usually feature at least stop lines to force drivers to slow down and have a second look. The real intersection in Fürstenfeldbruck actually is even regulated by traffic lights.
Turning at an X junction, dynamic occlusion {#subsec:results_X_junction_turn}
-------------------------------------------
In the last scenario we use the road geometry from the second Fürstenfeldbruck intersection, but additionally incorporate a dynamic obstacle that passes from west to east.
The ego vehicle wants to cross the intersection to the west, while the dynamic obstacle occludes its view of the easterly lane. Again the question arises at what time and position it is safe to cross the intersection.
As can be seen in the last row of \[fig:evaluation\], the ego vehicle slows down more than without the dynamic obstacle due to the additional occlusion. Specifically this is the result of assessing every critical sensing field edge with the same prior, i.e. the same initial state intervals in velocity and orientation. It is apparent from the simulation sequence that the east lane is already partly visible before the dynamic obstacle occludes that area. As a consequence one could derive a better prior for this area based on those observations and continue driving earlier. We will briefly discuss this possible type of performance improvement in \[sec:conclusion\].
Another observation in all scenarios is that the fail-safe trajectory has been activated even without dynamic obstacles. However, a sophisticated planning method should slow down the vehicle earlier in order to optimize its approaching time and velocity such that the switch to a fail-safe maneuver will rarely be necessary.
Despite those inefficiencies the evaluation shows the need for occlusion-awareness in safety verification and thus highlights the value of our occlusion-aware occupancy prediction.
Conclusions and Future Work {#sec:conclusion}
===========================
We motivate this contribution with the purpose of ensuring safety before comfort or traffic efficiency, especially when facing incomplete environment knowledge. We therefore characterize the risk from unperceived space as potentially hidden obstacles with an initial state that is unknown, but can be over-approximated with intervals. We enhance the reachable set approach introduced in [@althoff_set-based_2016] to predict occupancy over-approximations of such obstacles. To show the usefulness and potential of our approach, we implement a proof-of-concept planner which uses the occupancy prediction to plan provably safe trajectories. The performance is shown in three intersection scenarios with occlusions from static and dynamic obstacles. All collisions can be prevented while still moving through traffic fast enough.
Having a method to guarantee safety under occlusions w.r.t. an arbitrary safe-state, further work to improve comfort and efficiency can be done.
Comfort and traffic flow will increase with better prediction, because scenarios with low conflict probability can be approached and passed quicker. The fail-safe maneuver will still be guaranteed, but only a less comfortable one. On the other hand the vehicle could decelerate earlier when approaching intersections with high conflict probability.
Similarly, also clever fail-safe velocity profiles can lead to a good compromise between comfortable, but not too conservative behavior.
Tracking of occluded areas would allow to reduce the initial state intervals of hidden obstacles by reasoning. Hence, the predicted occupancies will be significantly reduced in dynamic scenarios, while still guaranteeing safety.
Finally, we will incorporate the proposed method in our prototype vehicle in the coming months and analyze the performance under real world conditions.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Daimler AG for the fruitful collaboration and the support for this work.
[^1]: We will use “occlusion” for any area outside of our field of view. This includes areas occluded by obstacles, but also areas outside the sensor range.
[^2]: “If a collision takes place, the robot will be at rest.” [@bouraine_passively_2014]
[^3]: Collisions between discrete time steps will not be detected here.
[^4]: Meaning higher than passive or even passive friendly safety [@macek_towards_2009], but at least RSS [@mobileye_formal_2017].
[^5]: w.r.t. the map.
[^6]: Featuring the road topology, geometry and traffic rules.
[^7]: This can be motivated by the concept of blame [@mobileye_formal_2017] or similar, as the blame of an accident is on the rear car as long as the ego did not cut in the others’ lane with an unsafe longitudinal distance.
[^8]: w.r.t. the driving direction.
[^9]: \[ft:intersection\]w.r.t. intersection.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Recent observations of near-infrared and X-ray flares from Sagittarius A$%
^{\ast }$, which is believed to be a supermassive black hole at the Galactic center, show that the source exhibits about 20-minute periodic variability. Here we provide arguments based on a quantitative analysis that supermassive objects at galactic centers may be bubbles of dark matter axions rather than black holes. An oscillating axion bubble can explain periodic variability of Sagittarius A$^{\ast }$ and yields the axion mass about $0.6$ meV which fits in the open axion mass window. The bubble scenario with no other free parameters explains lack of supermassive “black holes" with mass $M<10^{6}$M$_{\odot }$. Low-mass bubbles decay fast and as a result are very rare. We also found that the mass of an axion bubble can not exceed $1.5\times 10^{9}$M$_{\odot }$, in agreement with the upper limit on the supermassive “black hole" mass obtained from observations. Our finding, if confirmed, suggests that Einstein general relativity is invalid for strong gravity and the gravitational field for the bubble effectively becomes repulsive at large potential. Imaging a shadow of the “black hole" at the Galactic center with VLBI in the next decade can distinguish between the black hole and the oscillating axion bubble scenarios. In the case of axion bubble, a steady shadow will not be observed. Instead, the shadow will appear and disappear periodically with a period of about $20$ min.
author:
- 'Anatoly A. Svidzinsky'
title: Oscillating axion bubbles as alternative to supermassive black holes at galactic centers
---
Introduction
============
Originally introduced to explain why the strong interaction, in contrast to weak interactions, does not violate CP symmetry [@Pecc77], hypothetical axions have since become one of the leading particle candidates for the cold dark matter in the Universe [@Brad03]. The axion appears as a pseudo Nambu-Goldstone boson of a spontaneously broken Peccei-Quinn symmetry [Pecc77]{}, whose scale $f$ determines the axion mass $m$,$$m\approx \frac{m_{\pi }f_{\pi }}{2f}=0.62\text{ eV}\cdot \frac{10^{7}\text{
GeV}}{f} \label{f1}$$and suppresses the coupling to Standard Model particles, $\propto 1/f$. Here $m_{\pi }=135$ MeV is the neutral pion mass and $f_{\pi }=93$ MeV its decay constant [@Raff02; @Brad03]. Astrophysical and cosmological arguments constrain the axion mass $m$ to be in the range of $10^{-6}-3\times 10^{-3}$ eV [@Brad03]. Axions in this mass range could provide much or all of the cold dark matter in the Universe. Properties of stars impose the upper limit on the axion mass via constraints on axion interaction with photons, leptons and nucleons. However, such interactions are model-dependent. Observations of the cosmological large-scale structure constrain the axion-pion coupling which yield weaker upper mass limit $m<1.05$ eV [@Hann05].
Interaction of axions with QCD instantons generates the axion mass and periodic interaction potential [@Kim87] $$V(\varphi )=m^{2}f^{2}[1-\cos (\varphi /f)], \label{p01}$$where $\varphi $ is a real scalar axion field.
Here we argue that oscillating axion bubbles, rather then supermassive black holes, could be present at galactic centers. Recent observations of near-infrared and X-ray flares from Sagittarius A$^{\ast }$, which is believed to be a $3.6\times 10^{6}$M$_{\odot }$ black hole at the Galactic center, show that the source exhibits about 20-minute periodic variability [@Genz03; @Gill06; @Bela06]. An oscillating axion bubble can explain periodic variability of Sagittarius A\* and yields the axion mass about $0.6$ meV. Fig. \[u0\] explains the mechanism of flare variability.
Moreover, the bubble scenario explains observed lack of supermassive black holes" with mass $M<10^{6}$M$_{\odot }$. As we discuss later in this paper the bubble life-time $t\propto M^{9/2}$, it becomes less then the age of the Universe for $M\lesssim 5\times 10^{6}$M$%
_{\odot }$. The bubble at our Galactic center would decay within about $%
5\times 10^{9}$ yrs. If, however, $M<10^{6}$M$_{\odot }$ the decay time becomes very short, $t\lesssim 10^{7}$ yrs, and as a result such objects are very rare.
=0.45=0.41
Finally, the axion bubbles with no free parameters (if we fix $m=0.6$ meV based on Sagittarius A\* flare variability) explain the upper limit ($%
1.5\times 10^{9}$M$_{\odot }$) on the supermassive black hole" mass found in recent analysis of the measured mass distribution [Wu02]{}.
In recent years, the evidence for the existence of an ultra-compact concentration of dark mass associated with the radio source Sagittarius A\* in the Galactic Center has become very strong. However, an unambiguous proof that this object is a black hole is still lacking. A defining characteristic of a black hole is the event horizon. To a distant observer, the event horizon casts a relatively large shadowwith an apparent diameter of about $10$ gravitational radii due to bending of light. The predicted size ($\sim $30 micro-arcseconds) of this shadow for Sagittarius A\* approaches the resolution of current radio-interferometers. Hence, there exists a realistic expectation of imaging the shadow of a black hole with very long-baseline interferometry within the next decade [Falc00,Falc01,Shen05,Huan07]{}. Such imaging will allow us to distinguish between the black hole and the oscillating axion bubble scenario which we propose in this paper. If the axion bubble, rather then a black hole, is present at the Galactic center, the steady shadow will not be observed. Instead, the shadow will appear and disappear periodically with a period of about $20$ $\min $.
Axion bubbles
=============
We introduce dimensionless coordinates and define the unit of distance, time and $\varphi $ as $$r_{0}=\frac{\hbar }{mc},\quad t_{0}=\frac{\hbar }{mc^{2}},\quad \varphi _{0}=%
\frac{1}{\sqrt{4\pi G}},$$where $c$ is the speed of light and $G$ is the gravitational constant. For the moment we omit gravity. Further we use natural units for which $\hbar
=c=1$. Energy of the axion field in units of $m_{\text{pl}}^{2}/m$ is given by $$E=\int d\mathbf{r}\left[ \frac{1}{2}\left( \frac{\partial \varphi }{\partial
t}\right) ^{2}+\frac{1}{2}(\nabla \varphi )^{2}+V(\varphi )\right] ,
\label{ef}$$where $$V(\varphi )=\frac{1}{\alpha ^{2}}[1-\cos (\alpha \varphi )],\quad \alpha =%
\frac{1}{\sqrt{4\pi G}f}=\frac{m_{\text{pl}}}{\sqrt{4\pi }f}$$is the dimensionless potential and the coupling parameter respectively, $m_{%
\text{pl}}=\sqrt{\hbar c/G}=1.2\times 10^{19}$ GeV is the Planck mass.
The interaction potential $V$ has degenerate minima $V=0$ at $\varphi =2\pi
n/\alpha $, where $n$ is an integer number. As a consequence, equation for the axion field $\varphi $ has bubble-like solutions. The bubble surface is an interface between two degenerate vacuum states with $\varphi =2\pi
n/\alpha $ ($r<R$) and $\varphi =0$ ($r>R$). In this paper we consider spherical bubbles with surface width much smaller then its radius $R$ and $%
n=1$. Energy density of the axion field is nonzero only at the bubble surface. Energy of a static thin-wall bubble is $E=4\pi \sigma R^{2}$, where $\sigma $ is the surface energy per unit area (which equals to the surface tension for the domain wall we consider [@Ipse84]) determined by an integral over one potential period [@Svid04; @Svid04b] $$\sigma =\frac{1}{4\pi }\int \sqrt{2V}d\varphi =\frac{2}{\pi \alpha ^{2}}.
\label{y0}$$Surface tension $\sigma $ depends only on the axion interaction strength. The later, however, slightly depends on the axion model which can change Eq. (\[y0\]) by a factor of the order of one [@Huan85].
Under the influence of surface tension an initially static bubble collapses to its center which yields reduction of the surface energy. However, if we include gravity this gives an additional energy contribution. Such a contribution could substantially alter the bubble evolution and, in particular, prevent collapse as we discuss in the next section.
One should mention that decay of axions into photons is suppressed in the bubble. Such a decay is not allowed by energy conservation. The bubble surface is approximately a one dimensional kink. In the kink’s reference frame the kink is static and the distribution of the axion field is obtained by minimization of the energy functional (\[ef\]) subject to the boundary conditions that far from the kink we have fixed vacuum states ($\varphi =0$ from one side and $\varphi =2\pi /\alpha $ from the opposite side). The optimized field distribution determines the kink’s energy. Any small change in $\varphi $ would increase the total energy of the axion field. Axions in the kink cannot decay into photons because annihilation of the axion would change the distribution of the axion field in the kink and, hence, increase the kink’s energy. In such a process both the axion field and photon acquire energy which violates energy conservation.
The alternative theory of gravity vs Einstein general relativity
================================================================
So far Einstein general relativity has successfully passed all available tests. However such tests have inspected the theory only at weak gravitational field [@Will06]. One should note that observations of binary pulsars yet have not provided a test of general relativity at strong gravity. Rather, such observations tested Einstein equations in the post-Newtonian approximation and the strong equivalence principle [Will06]{}.
Are Einstein equations valid for strong gravity? The answer to this question remains unknown and only appropriate observational tests can shed light on it. It is well known that in Einstein general relativity the gravitational field disobeys the principle of superposition. This is the consequence of the postulate that the space-time metric is determined by the Einstein equations $$R_{ik}-\frac{1}{2}g_{ik}R=8\pi T_{ik}, \label{ee}$$with the particular choice of the matter energy-momentum tensor $T_{ik}$ proposed by Einstein. However the Einstein theory can be modified by modifying the energy-momentum tensor. This yields a possibility to satisfy the superposition principle by a proper choice of $T_{ik}$.
In Appendix A we derive a space-time metric produced by a static mass distribution based only on the superposition principle. The answer is given by the Yilmaz exponential metric. Then we show that the Einstein equations yield the exponential metric if $T_{ik}$ is taken as an electrostatic energy-momentum tensor.
In the weak field limit the exponential metric is equivalent to those obtained in Einstein theory and, hence, the exponential metric agrees with the four classic tests of general relativity. However in the opposite limit of strong gravity the exponential metric is dramatically different. Since the superposition principle is satisfied the exponential metric predicts no black holes, but rather stable compact objects with no event horizon and very large, but finite, gravitational redshift (dark red holes"). This suggests that gravitational field for those objects effectively becomes repulsive at large gravitational potential.
Here we analyze properties of compact objects at galactic centers and show that they are in favour of the exponential metric. Our conclusion is based on a quantitative analysis which is independent of the particular choice of the time-dependent theory of gravity [@pop]. This is possible because the main part of the bubble dynamics we use for the quantitative comparison occurs in the well-tested limit of Newtonian gravity. Only a small part of the trajectory near the lower radius turning point (where gravity effectively becomes repulsive) is beyond Newtonian description. As a result, e.g., the period of bubble oscillation can be accurately obtained using Newtonian gravity, independent of which theory of gravity yields the repulsive force at small radius.
Bubbles in exponential metric
-----------------------------
In Appendix A we derive a metric for a static mass distribution assuming that the gravitational field obeys the principle of superposition for any field strength. The answer is given by an exponential isotropic line element of the class proposed by Yilmaz [@Yilm58; @Yilm77]
$$ds^{2}=-e^{2\phi }dt^{2}+e^{-2\phi }(dx^{2}+dy^{2}+dz^{2}), \label{ymet}$$
where for a spherically symmetric thin-wall bubble of radius $R$ (units are $%
c=G=1$) $$\phi (r)=\left\{
\begin{array}{c}
-M/r,\quad r\eqslantgtr R \\
-M/R,\quad r<R,%
\end{array}%
\right. \label{ymet1}$$and $M$ is the bubble mass. The exponential metric does not have the singularity of the Schwarzschild solution at finite radius, and therefore replaces the concept of black holes with that of dark red holes".
In the reference frame of a distant observer the energy of a static bubble is given by [@Clap73] $$U(R)=4\pi \sigma R^{2}\exp \left( \frac{M}{R}\right) , \label{y1}$$where $\sigma $ is the intrinsic surface energy density (as it would appear to an observer located at the bubble surface) given by Eq. (\[y0\]), $M$ is the dimensionless bubble mass in units of $m_{\text{pl}}^{2}/m$ and $R$ is the dimensionless bubble radius in units of $r_{0}$. If the radius of a spherical bubble changes with time then the total energy is$$E=U(R)+E_{k}, \label{y2}$$where $E_{k}\geq 0$ is the kinetic energy. The total energy $E$ is a constant of motion, then based on the equivalence principle we obtain $M=E=$const. Evolution of the bubble radius is similar to a one dimensional motion of a particle in the effective potential $U(R)$. We plot $U(R)$ in Fig. [u1]{}. The effective potential has a shape of a well and depends on the total energy (mass). At $R\gg M$ one can omit gravity and $U(R)\simeq 4\pi \sigma
R^{2}$ is just a surface energy (tension) which tends to contract the bubble. At $R\ll M$ gravity produces large repulsive effective potential which forces the bubble to expand. As a result the bubble radius $R(t)$ oscillates between two turning points determined by $U(R)=M$.
Fig. \[tp\] shows numerical solution of the equation for turning points $%
M=4\pi \sigma R^{2}\exp \left( M/R\right) $. For $4\pi \sigma
M<4/e^{2}=0.541 $ equation has two solutions; radius of such bubbles oscillates with time between turning points $R_{1}$ and $R_{0}$. If $M=M_{%
\text{max}}=1/(e^{2}\pi \sigma )$ the bubble is static with $R=M/2$. Such static bubble possesses maximum possible mass. Bubbles with $M>M_{\text{max}%
} $ do not exist. As we show below, for the Galactic center bubble $M\lll M_{%
\text{max}}$.
=0.45=0.35
=0.45=0.35
In a general case to describe $R(t)$ quantitatively we need to use dynamic equations. These equations depend on a particular choice of the time-dependent theory of gravity. In this paper, however, we do not need them which makes our results quite general. The point is that if the Galactic center object is an axion bubble the main part of its periodic oscillation occurs in the limit $R(t)\gg M$. In this region we can omit gravity and use the well-tested special theory of relativity that yields the following mass-radius equation for a relativistic bubble $$M=\frac{4\pi \sigma R^{2}}{\sqrt{1-(dR/dt)^{2}}}$$which has a simple solution$$R(t)=R_{0}\text{cn}\left( \frac{\sqrt{2}t}{R_{0}},\frac{1}{\sqrt{2}}\right) ,
\label{y3}$$where $R_{0}=\sqrt{M/4\pi \sigma }$ is the maximum bubble radius and cn$%
(x,k) $ is Jacobian elliptic cosine. For small $R(t)$ the solution (\[y3\]) is not applicable. In this region the bubble shrinking slows down and after reaching the inner turning point $R_{1}$ the bubble starts to expand. Assuming that Eq. (\[y3\]) is accurate for the main part of the motion we obtain that the period of bubble oscillation is $T\simeq 2.622R_{0}$. Taking into account that $M=4\pi \sigma R_{0}^{2}=8R_{0}^{2}/\alpha ^{2}$, we get in dimension units $T\simeq 0.262\sqrt{M/m}\hbar /fc^{2}$. Then using Eq. (\[f1\]) we find$$T\simeq 0.523\frac{\hbar }{c^{2}}\frac{\sqrt{Mm}}{m_{\pi }f_{\pi }}=\sqrt{%
\frac{M}{10^{6}M_{\odot }}\frac{m}{10^{-3}\text{eV}}}\times 15.27\text{ min}.
\label{y4}$$
If $M=3.6\times 10^{6}$M$_{\odot }$ and $T=22.2$ min [@Bela06] then Eq. (\[y4\]) yields the axion mass $m\simeq 0.6$ meV ($f\simeq 10^{10}$ GeV). One should mention, however, that due to time dilation the period of flare variability depends on the distance between the flare source and the bubble center (see Fig. \[u0\]) and, thus, could differ from Eq. (\[y4\]) by a factor of the order of one. This yields an inaccuracy in the axion mass determination in the same factor.
Next we discuss the bubble life time. The bubble decay occurs by means of axion emission. Due to spherical symmetry there is no radiation of gravitation waves. Bubble surface, the interface between different vacuum states, is a soliton (or a kink) that is studied in many areas of nonlinear physics. One dimension solitons, contrary to 2D or 3D, are stable and preserve their shape under reflection from a boundary. Because a thin-wall bubble surface can be treated as a 1D soliton this insures its very long life time. However, due to finite bubble radius the 1D treatment is only approximate. Deviation of the problem from 1D leads to slow decay of the soliton by emission of particles (axions).
We estimate the bubble decay rate as the time of energy loss by the bubble with the radius $R(t)$ oscillating between the outer $R_{0}$ and inner $%
R_{1} $ turning points. Energy loss by the bubble surface becomes substantial only when $R(t)\lesssim R_{0}^{2/3}$ (see Appendix B below). In our case $R_{1}\gg R_{0}^{2/3}$ and, therefore, the region of intensive energy dissipation is not accessible. As a result, the energy emission is negligible yielding long-lived bubbles. In Appendix B we estimate the bubble life-time. The answer is given by Eq. (\[s9\]) which in dimension units reads $$t\sim \frac{R_{0}}{c}\left( \frac{R_{1}}{R_{0}}\right) ^{4}\left( \frac{R_{1}%
}{r_{0}}\right) ^{2}. \label{y5}$$If $m=0.6$ meV then the bubble surface has thickness $r_{0}=\hbar /mc=0.3$ mm. For a bubble with mass $M=3.6\times 10^{6}$M$_{\odot }$ the maximum radius is $R_{0}=483R_{\odot }$, while the gravitational radius $%
R_{g}=16.1R_{\odot }$. One can find the inner turning point $R_{1}$ from the equation $R_{0}^{2}=R_{1}^{2}\exp (M/R_{1})$ which yields $R_{1}=1.1R_{\odot
}$. Using Eq. (\[y5\]) we then obtain the bubble life time $t\sim 5\times
10^{9}$ yrs. This can explain observed lack of supermassive black holes" with $M<10^{6}$M$_{\odot }$. Axion bubbles with such masses decay fast with life time $t\propto M^{9/2}$.
For bubbles with $M\ll M_{\max }$ one can obtain $R_{1}$ in terms of the bubble mass $M$ analytically$$R_{1}=\frac{M}{2\ln \left[ \frac{2R_{0}}{M}\ln \left( \frac{2R_{0}}{M}%
\right) \right] },\quad R_{0}=\alpha \sqrt{\frac{M}{8}}.$$Substitute this into Eq. (\[y5\]) yields the following expression for the bubble life-time
$$t\sim \frac{\pi ^{3/2}\hbar m_{\pi }^{3}f_{\pi }^{3}\sqrt{m}M^{9/2}}{\sqrt{8}%
c^{2}m_{\text{pl}}^{12}\ln ^{6}[x\ln (x)]}=$$$$=\left( \frac{M}{10^{6}M_{\odot }}\right) ^{9/2}\sqrt{\frac{m}{10^{-3}\text{%
eV}}}\times \frac{4.73\times 10^{11}}{\ln ^{6}[x\ln (x)]}\text{ years,}
\label{ylt}$$where$$x=\frac{m_{\text{pl}}^{2}\sqrt{m}}{\sqrt{2\pi }m_{\pi }f_{\pi }\sqrt{M}}=140%
\sqrt{\frac{m}{10^{-3}\text{eV}}\frac{10^{6}M_{\odot }}{M}}.$$Eq. (\[ylt\]) shows that the bubble life-time $t\propto M^{9/2}$ and it becomes less then the age of the Universe for $M\lesssim 5\times 10^{6}$M$%
_{\odot }$. For $M<10^{6}$M$_{\odot }$ the decay time becomes very short, $%
t\lesssim 10^{7}$ yrs, this is why we do not observe supermassive black holes" with such masses.
Recent analysis of the mass distribution for the compact objects at galactic centers shows existence of an upper limit for the supermassive black hole“ mass [@Wu02]:$$M_{\max }=1.2_{-0.4}^{+2.6}\times 10^{9}M_{\odot }. \label{ymass}$$Next we calculate the maximum possible mass of an axion bubble. In dimension units it is given by$$M_{\max }=\frac{0.0215m_{\text{pl}}^{4}m}{m_{\pi }^{2}f_{\pi }^{2}}.
\label{y6}$$For $m=0.6$ meV Eq. (\[y6\]) yields $M_{\max }=1.5\times 10^{9}$M$_{\odot
} $. This value agrees with the upper limit on the supermassive black hole” mass (\[ymass\]) measured for galactic nuclei. Radius of the static bubble with $M_{\max }$ is $R=1673R_{\odot }$.
Bubble in Einstein general relativity
-------------------------------------
In Einstein general relativity for a spherically symmetric bubble the metric can be written in the form
$$ds^{2}=-hdt^{2}+gdr^{2}+r^{2}d\Omega ^{2},$$
where $g$, the radial metric, and $h$, the lapse, are functions of $t$ and $%
r $ with $r$ being the circumferential radius. For a static thin-wall bubble of radius $R$ the Einstein equations yield $$h(r)=\left\{
\begin{array}{c}
1-2M/r,\quad r\eqslantgtr R \\
1-2M/R,\quad r<R,%
\end{array}%
\right.$$$$g(r)=\left\{
\begin{array}{c}
\frac{1}{1-2M/r},\quad r>R \\
1,\quad r<R.%
\end{array}%
\right.$$Note that $g(r)$ undergoes a jump at the bubble surface, while exponential metric (\[ymet\]), (\[ymet1\]) is continuos. Energy of the thin-wall bubble with radius $R(t)$ is given by [@Blau87; @Auri91] $$E=\frac{4\pi \sigma R^{2}}{\sqrt{1-(dR/d\tau )^{2}}}-8\pi ^{2}\sigma
^{2}R^{3},$$where $\tau $ is the interior coordinate time ($d\tau ^{2}=hdt^{2}$). The corresponding effective potential$$U(R)=4\pi \sigma R^{2}-8\pi ^{2}\sigma ^{2}R^{3}$$is pictured in Fig. \[ue\].
In Einstein theory small" bubbles shrink toward the gravitation radius $R_{g}=2M$ and at $t\gg R_{g}/c$ behave as black holes, while large bubbles expand infinitely. This is dramatically different from bubble evolution in the exponential metric which we discussed in the previous section. At the same time, the exponential and the isotropic form of the Schwarzschild metric (\[a2\]) are the same to second order in the gravitational potential $\phi $ in the temporal part and to first order in the spatial part (see Appendix A). This is sufficient to insure that they both give identical results in the four classic weak-field tests of general relativity.
=0.45=0.35
Discussion
==========
If axion bubbles, rather then supermassive black holes, are located at galactic centers then what is the mechanism of their nucleation? Dark matter axions, if they exist, form halos around galaxies. The halo of axions is in a quantum degenerate non-equilibrium regime. Evolution of the axion halo is governed by the self-gravity and axion interaction $V(\varphi )$. The interaction $V(\varphi )$ becomes important only for dense axion clumps. Dynamics of a dilute galactic halo is determined by self-gravity. Seidel and Suen have studied evolution of a massive, self-gravitating real scalar field in Newtonian limit (omitting self-interaction $V(\varphi )$) [@Seid94]. They have shown that independent of the initial conditions a scalar field configuration collapses to form a compact object by ejecting part of the scalar field, carrying out the excess kinetic energy. The cooling occurs due to nonlinear effects of the self-gravitation of the field. Characteristic cooling time is a free falling time to the center due to self gravity $%
t\simeq 2R_{\text{halo}}^{3/2}/\sqrt{2GM_{\text{halo}}}$, where $R_{\text{%
halo}}$ and $M_{\text{halo}}$ is an initial radius and mass of the axion halo in a galaxy. For $R_{\text{halo}}=60$ kpc and $M_{\text{halo}}=10^{12}$M$_{\odot }$ we obtain the characteristic cooling time $t\sim 10^{8}$ yrs.
Thus, within about $10^{8}$ yrs the gravitational cooling mechanism yields formation of compact axion clumps in a galactic halo. Evolution of such clumps is then governed by self-interaction $V(\varphi )$ which leads to bubble formation. This is shown by three-dimensional numerical simulation of the evolution of inhomogeneities in the axion field due to the self-interaction $V(\varphi )$ [@Kolb94]. Such a simulation (which omits gravity) has indeed demonstrated formation of bubble-like structures (see Fig. 5a in Ref. [@Kolb94]). The mass of the nucleated bubbles is much smaller then the mass of the halo they are born in. However for typical halos many bubbles are born with masses much greater then $10^{6}$M$_{\odot
} $ and hence they are long-lived objects.
In Einstein general relativity axion bubbles under the influence of surface tension collapse fast into black holes. However, the Einstein theory yet to be tested in the limit of strong gravitational field. There is a possibility that at strong field the gravity is not described by Einstein general relativity, and rather by an alternative theory which also passes all available tests. In this paper we consider axion bubbles in a very general approach avoiding a particular choice of the alternative time-dependent theory of gravity. Our results are valid for any metric theory which in based on the principle of superposition. This principle yields the exponential metric in the static limit, as shown in Appendix A. One should mention that if the space-time geometry is described by the exponential metric then compact supermassive objects at galactic centers can not be made of baryonic matter. Maximum mass of a compact (neutron star like) baryonic object in such a metric can not exceed about $12$M$_{\odot }$ [@Robe99]. Hence, dark matter of non baryonic origin is the only alternative for their composition.
We found that in the exponential metric the axion bubbles with $M>10^{6}$M$%
_{\odot }$ are very long lived. Instead of collapsing into a black hole the bubble radius oscillates between two turning points determined by the net mass. Such oscillating bubbles, rather then supermassive black holes, could be present at galactic centers. Our result can account for periodic variability observed in near-infrared and X-ray flares from Sagittarius A\* [@Genz03; @Gill06; @Bela06] and yields the axion mass about $0.6$ meV.
Moreover, the bubble scenario with no free parameters (if we fix $m=0.6$ meV based on Sagittarius A\* flare variability) explains lack of supermassive black holes“ with $M<10^{6}$M$_{\odot }$. We find that if $%
M<10^{6}$M$_{\odot }$ the bubble life time becomes very short, $t\lesssim
10^{7}$ yrs, and as a result such objects are very rare. We also found that for the exponential metric the bubble mass can not exceed $M_{\max
}=1.5\times 10^{9}$M$_{\odot }$. This, again with no free parameters, explains the upper limit on the supermassive black hole” mass measured for galactic centers [@Wu02].
For axion with mass $m=0.6$ meV the axion-photon coupling constant is $%
g_{a\gamma }\sim 10^{-13}$GeV$^{-1}$ [@Raff07]. Recently it was argued that the solar corona $X-$ray emission can be explained by solar axions of the Kaluza-Klein type (that is by axions propagating into extra dimensions) which are gravitationally trapped by the Sun and decay near the solar surface [@Dile03; @Ziou04]. The estimated value of $g_{a\gamma }$ from the analysis of solar corona $X-$rays is similar to our finding; this is an interesting coincidence.
Observation of the Galactic center with very long-baseline interferometry within the next few years will be capable to test theories of gravitation in the strong field limit. Such an observation will allow us to distinguish between the black hole (predicted by Einstein general relativity) and the oscillating axion bubble scenario which we propose in this paper. If future observations indeed discover periodic appearance of the shadow from the Galactic center object this will also be a strong evidence for the axion nature of dark matter and will lead to an accurate measurement of the axion mass.
I am very grateful to E. Sezgin and N. Suntzeff for useful remarks.
Derivation of the static exponential metric from the principle of superposition
===============================================================================
Let us consider a point mass $M$ located at $r=0$. Static gravitational field produced by the point mass possesses spherical symmetry. Without loss of generality one can look for the metric in an isotropic form $$ds^{2}=-h(r)dt^{2}+g(r)(dr^{2}+r^{2}d\theta ^{2}+r^{2}\sin ^{2}\theta
d\varphi ^{2}). \label{a1}$$Einstein equations yield well known Schwarzschild solution [@Land95]$$h(r)=\left( \frac{1-M/2r}{1+M/2r}\right) ^{2},\quad g(r)=\left( 1+\frac{M}{2r%
}\right) ^{4}. \label{a2}$$
Here we derive $h(r)$ and $g(r)$ in a different way. First we note that a static gravitational field of any strength has a potential [@Land95]$$\phi (r)=\ln \sqrt{h(r)} \label{a3}$$and the gravitational force acting on a test rest particle with mass $m$ is$$\mathbf{f}=-m\nabla \phi \quad \left( f_{\alpha }=-m\frac{\partial \phi }{%
\partial x^{\alpha }}\right) . \label{a3f}$$ In Minkowski space-time the potential $\phi $ satisfies the Poisson equation $\Delta \phi =4\pi M\delta (\mathbf{r})$. Writing the Laplacian and the delta-function in curvilinear coordinates with metric $g_{ik}$ the Poisson equation yields
$$\frac{1}{\sqrt{-|g_{ik}|}}\frac{\partial }{\partial x^{i}}\left( \sqrt{%
-|g_{ik}|}g^{ik}\frac{\partial \phi }{\partial x^{k}}\right) =\frac{4\pi }{%
\sqrt{-|g_{ik}|}}M\delta (\mathbf{r}), \label{a4}$$
where $|g_{ik}|$ is determinant of the metric tensor and $g^{ik}$ is the tensor reciprocal to $g_{ik}$, that is $g_{ik}g^{kl}=\delta _{i}^{l}$. For the metric (\[a1\]) Eq. (\[a4\]) reduces to$$\frac{\partial }{\partial r}\left( r^{2}\sqrt{h(r)g(r)}\frac{\partial \phi
(r)}{\partial r}\right) =4\pi M\delta (r). \label{a5}$$Eq. (\[a5\]) (with $\phi $ from Eq. (\[a3\])) describes a relation between the functions $h$ and $g$ which the metric must satisfy. We note that Eq. (\[a5\]) is consistent with Einstein general relativity because the Schwarzschild solution (\[a2\]) obeys Eq. (\[a5\]).
To find the functions $h$ and $g$ we need an additional constraint. Here we postulate that the force of gravity must obey the principle of superposition at any strength of the gravitational field. This is the only difference from Einstein general relativity we introduce. Such way of thoughts makes an appealing connection with the quantum theory.
The principle of superposition is formulated in coordinate systems in which we measure space coordinates by ideal rods unaffected by gravity (e.g. of atomic constitution). This assures that the coordinate system is independent of the position and the value of masses. Let us consider masses $M_{1}$ and $%
M_{2}$ located at coordinates $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ in the mentioned above coordinate system. The superposition principle means that the gravitational force on any test particle due to masses $M_{1}$ and $%
M_{2} $ equals to the vector sum of the force due to the mass $M_{1}$ located at $\mathbf{r}_{1}$ if there is no mass $M_{2}$ and the force due to the mass $M_{2}$ located at $\mathbf{r}_{2}$ if there is no mass $M_{1}$.
Please note that we formulate the superposition principle for the covariant force vector $f_{\alpha }$ as in Eq. (\[a3f\]). This makes a proper connection with the Newtonian limit in which the measurable gravitational force is a covariant vector (under coordinate transformation it transforms like derivatives of a scalar). The covariant force $f_{\alpha }$ on a test particle is defined as [@Land95] $f_{\alpha }=md^{2}x_{\alpha }/ds^{2}$, where $x_{\alpha }$ is the coordinate of the test particle with mass $m$ and $ds$ is the interval. If at the moment when the force is measured the particle has zero velocity (that is when Eq. (\[a3f\]) applies) then $%
ds=d\tau $, where $\tau $ is the proper time and, hence, $f_{\alpha
}=md^{2}x_{\alpha }/d\tau ^{2}$. This equation shows that the definition of the covariant force is unaffected by gravity (if the coordinates $x_{\alpha
} $ are unaffected) and therefore $f_{\alpha }$ is the relevant vector to formulate the superposition principle. On the other hand, the contravariant vector $f^{\alpha }=g^{\alpha \beta }f_{\beta }$ contains metric in its definition and hence cannot be used in the superposition principle.
It follows from Eqs. (\[a3f\]) and (\[a5\]) that the force of gravity satisfies the principle of superposition if and only if $$h(r)g(r)=1, \label{a19}$$which yields for arbitrary field strength$$\frac{\partial }{\partial r}\left( r^{2}\frac{\partial \phi (r)}{\partial r}%
\right) =4\pi M\delta (r).$$Hence $\phi (r)=-M/r$ and $$h(r)=\exp (-2M/r),\quad g(r)=\exp (2M/r). \label{a20}$$Eq. (\[a20\]) is known as Yilmaz exponential metric [@Yilm58].
For small $M/r$ both the Schwarzschild (\[a2\]) and the exponential ([a20]{}) metrics yield the same expansion$$h(r)=1-\frac{2M}{r}+\frac{2M^{2}}{r^{2}}+\ldots ,\quad g(r)=1+\frac{2M}{r}%
+\ldots \label{a21}$$and hence both metrics pass the four classic weak-field tests of general relativity. Accuracy of current tests is yet far from ability to check the next terms in the expansion (\[a21\]) where the two metrics start to deviate from each other [@Will06].
The principle of superposition allows us to find the metric in the case of $%
N $ point masses $M_{1}$, ..., $M_{N}$ located at $\mathbf{r}_{1}$, ... $%
\mathbf{r}_{N}$. To apply the principle of superposition we must choose the coordinate system in which we measure space coordinates by ideal rods unaffected by gravity. Since the speed of light measured by such rods and clocks (e.g. of atomic constitution) is independent of direction in a gravitational field the metric in such coordinates is isotropic, that is $$ds^{2}=-h(\mathbf{r})dt^{2}+g(\mathbf{r})(dx^{2}+dy^{2}+dz^{2}).
\label{a21a}$$For the metric (\[a21a\]) Eq. (\[a4\]) yields$$\frac{\partial }{\partial x}\left( \sqrt{hg}\frac{\partial \phi }{\partial x}%
\right) +\frac{\partial }{\partial y}\left( \sqrt{hg}\frac{\partial \phi }{%
\partial y}\right) +\frac{\partial }{\partial z}\left( \sqrt{hg}\frac{%
\partial \phi }{\partial z}\right) =$$$$=4\pi \lbrack M_{1}\delta (\mathbf{r}-\mathbf{r}_{1})+\ldots +M_{N}\delta (%
\mathbf{r}-\mathbf{r}_{N})]. \label{a21b}$$The principle of superposition for the potential $\phi $ (and hence for the force of gravity $\mathbf{f}$) is satisfied provided $hg=1$. Then Eqs. ([a21b]{}) and (\[a3\]) give the following answer for $N-$body space-time geometry in isotropic Cartesian coordinates $$ds^{2}=-e^{2\phi }dt^{2}+e^{-2\phi }(dx^{2}+dy^{2}+dz^{2}), \label{a22}$$where $\phi $ is the $N-$body potential$$\phi (\mathbf{r})=-\frac{M_{1}}{|\mathbf{r}-\mathbf{r}_{1}|}-...-\frac{M_{N}%
}{|\mathbf{r}-\mathbf{r}_{N}|}. \label{a23}$$Thus, finding the space-time geometry reduces to a simple electrostatic" problem.
Next we discuss how to obtain the exponential metric from Einstein equations.
Derivation of exponential metric from Einstein equations
--------------------------------------------------------
Einstein equations read$$R_{ik}-\frac{1}{2}g_{ik}R=8\pi T_{ik}, \label{a24}$$where $R_{ik}$ is the Ricci tensor, $R$ is the scalar space curvature and $%
T_{ik}$ is the energy-momentum tensor of matter. Let us consider $N$ fixed point masses $M_{1}$, ..., $M_{N}$ located at $\mathbf{r}_{1}$, ... $\mathbf{%
r}_{N}$. For such a system the energy-momentum tensor in Einstein general relativity is$$T_{0}^{0}=\sum_{j=1}^{N}M_{j}\delta (\mathbf{r}-\mathbf{r}_{j}),\quad \text{%
all other }T_{i}^{k}=0. \label{a25}$$Solution of Eq. (\[a24\]) with this $T_{ik}$ yields black holes and no superposition principle.
To obtain exponential metric we must use another energy-momentum tensor. Let us write $T_{ik}$ by analogy with electrostatic. For $N$ fixed point electric charges $q_{1}$, ..., $q_{N}$ located at $\mathbf{r}_{1}$, ... $%
\mathbf{r}_{N}$ the energy-momentum tensor is given by [@Land95] (in curvilinear coordinates)$$T_{ik}=\frac{1}{4\pi }\left( \frac{\partial \phi }{\partial x^{i}}\frac{%
\partial \phi }{\partial x^{k}}-\frac{1}{2}g_{ik}(\nabla \phi )^{2}\right) ,
\label{a26}$$where $(\nabla \phi )^{2}=E_{\alpha }E^{\alpha }=g^{\alpha \beta }(\partial
\phi /\partial x^{\alpha })(\partial \phi /\partial x^{\beta })$ and $\phi $ is the electric potential satisfying the Poisson equation$$\frac{\partial }{\partial x^{i}}\left( \sqrt{-|g_{ik}|}g^{ik}\frac{\partial
\phi }{\partial x^{k}}\right) =4\pi \sum_{j=1}^{N}q_{j}\delta (\mathbf{r}-%
\mathbf{r}_{j}). \label{a27}$$We assume that for the system of $N$ fixed point masses $M_{1}$, ..., $M_{N}$ the energy-momentum tensor $T_{ik}$ is given by Eqs. (\[a26\]) and ([a27]{}) with the change $q_{j}\rightarrow M_{j}$. Substituting this tensor into Einstein equations (\[a24\]) we obtain the solution for the metric given by formulas (\[a22\]) and (\[a23\]).
One should mention that we can find the proper energy-momentum tensor ([a26]{}) simply by plugging the exponential metric (\[a22\]) into the left hand side of Einstein equations (\[a24\]). Then the electrostatic" energy-momentum tensor (\[a26\]) is obtained automatically.
The result discussed here is valid for a static gravitational field. How to generalize it for time-dependent fields is beyond the scope of the present paper.
Energy emission from a shrinking bubble
=======================================
Here we calculate energy loss by a shrinking spherically symmetric bubble caused by emission of scalar particles (axions). For an order of magnitude estimate one can omit the effect of gravity. Then the evolution of the scalar field $%
\varphi (t,r)$ is described by sine-Gordon equation $$\ddot{\varphi}-\varphi ^{\prime \prime }+\frac{1}{\alpha }\sin (\alpha
\varphi )=2\varphi ^{\prime }/r, \label{s1}$$where $r$ is the radial coordinate. Without right-hand side, Eq. (\[s1\]) has an exact, so-called kink, solution $$\varphi _{0}=\frac{4}{\alpha }\arctan \left\{ \exp \left[ \pm \frac{%
(r-vt-R_{0})}{\sqrt{1-v^{2}}}\right] \right\} , \label{s2}$$where $R_{0}\gg 1$ is the initial bubble radius. The solution describes a kink (space region where $\varphi $ changes from $2\pi /\alpha $ to $0$) propagating with constant velocity $v$; the kink’s size is $l\sim \sqrt{%
1-v^{2}}$.
If $l\ll R(t)$, where $R(t)$ is the bubble radius, r.h.s. of (\[s1\]) may be treated as a small perturbation. Eq. (\[s1\]) possesses approximate solution in the form of the kink (\[s2\]) with parameters slowly changing in time under the action of the perturbation. In particular, the kink shrinks due to its surface tension so that the bubble radius and the velocity evolve as [@Malo87a] $$R(t)=R_0cn(\sqrt{2}t/R_0,1/\sqrt{2})\text{, \ }v(t)=\sqrt{1-R^4(t)/R_0^4}%
\text{,}$$ where $cn$ stands for the elliptic cosine with the modulus $1/\sqrt{2}$. Such a process is accompanied by emission of scalar particles which yields the energy loss. We estimate the energy loss following the original work of Malomed [@Malo87a; @Malo87b]. In terms of the inverse scattering technique, the spectral density of the emitted energy $E_e(t,q)$ is
$$\frac{dE_{e}}{dq}=\frac{4}{\pi \alpha ^{2}}|B(t,q)|^{2}, \label{s3}$$
where $q$ is the radiation wavenumber and the perturbation-induced evolution equation for the complex amplitude $B(t,q)$ is given by [Malo87a,Malo87b]{} $$\frac{dB}{dt}=-\frac{i}{2(\lambda ^{2}+\gamma ^{2})}\int_{-\infty }^{\infty
}dr\left( \lambda ^{2}-\gamma ^{2}-2i\lambda \gamma \tanh \left[ \frac{r-vt}{%
\sqrt{1-v^{2}}}\right] \right) \exp \left( i\sqrt{1+q^{2}}t-iqr\right)
\partial _{r}\varphi _{0}, \label{s4}$$where $\lambda =\sqrt{1+q^{2}}-q$ and $\gamma =(1+v)/2\sqrt{1-v^{2}}$. Calculating the integral in (\[s4\]) yields $$\frac{dB}{dt}=\frac{i\pi \left[ \lambda ^{2}(1-v)(1-\sqrt{1+v})-v/2\right] }{%
(1+v)/4+\lambda ^{2}(1-v)}\frac{\exp \left( i\sqrt{1+q^{2}}t-iqvt\right) }{%
\cosh \left[ \pi q\sqrt{1-v^{2}}/2\right] }. \label{s5}$$If $v$ slowly varies with time one can take $v\approx const$ in Eq. (\[s5\]), then after integration we obtain $$B(t,q)=\frac{\pi \left[ \lambda ^{2}(1-v)(1-\sqrt{1+v})-v/2\right] }{\left[
(1+v)/4+\lambda ^{2}(1-v)\right] (\sqrt{1+q^{2}}-qv)}\frac{\exp \left( i%
\sqrt{1+q^{2}}t-iqvt\right) -1}{\cosh \left[ \pi q\sqrt{1-v^{2}}/2\right] }.$$Therefore $$\frac{dE_{e}}{dq}=\frac{16\pi \left[ \lambda ^{2}(1-v)(1-\sqrt{1+v})-v/2%
\right] ^{2}}{\alpha ^{2}\left[ (1+v)/4+\lambda ^{2}(1-v)\right] ^{2}(\sqrt{%
1+q^{2}}-qv)^{2}}\frac{\sin ^{2}\left[ \left( \sqrt{1+q^{2}}-qv\right) t/2%
\right] }{\cosh ^{2}\left[ \pi q\sqrt{1-v^{2}}/2\right] }. \label{s6}$$Integration of (\[s6\]) over $dq$ gives the emitted energy as a function of time $E_{e}(t)=\int_{-\infty }^{\infty }dq(dE_{e}/dq)$. In Eq. (\[s6\]) sine is a fast oscillating function, so we substitute $\sin
^{2}(x)\rightarrow 1/2$. The radiation power increases when the kink’s velocity $v$ approaches the speed of light $c=1$. Assuming $1-v\ll 1$, integration of Eq. (\[s6\]) yields $$E_{e}(t)\approx \frac{2.51}{\alpha ^{2}(1-v(t))^{3/2}}\approx \frac{7.10}{%
\alpha ^{2}}\left( \frac{R_{0}}{R(t)}\right) ^{6} \label{s7}$$The emitted energy becomes comparable with the initial bubble energy $%
E_{0}=4\pi \sigma R_{0}^{2}=8R_{0}^{2}/\alpha ^{2}$ when the bubble radius reaches the value $R_{\ast }\approx R_{0}^{2/3}$. This value agrees with those obtained in [@Widr89].
If the bubble shrinks from the outer turning point $R_{0}$ to the inner turning point $R_{1}\gg R_{\ast }$ the radiated energy is $$E_{e}\sim \frac{7.10}{\alpha ^{2}}\left( \frac{R_{0}}{R_{1}}\right) ^{6}%
\text{.} \label{s8}$$To emit all its energy the bubble must oscillate between $R_{1}$ and $R_{0}$ about $E_{0}/E_{e}$ cycles. As a result, the bubble life-time is $$t\sim R_{0}\frac{E_{0}}{E_{e}}\approx \frac{R_{1}^{6}}{R_{0}^{3}}.
\label{s9}$$
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R=R_{s}e^{\phi }$ which yields a factor $e^{-2\phi }$ if we express $U_{s}$ in terms of $R$. A redshift factor $e^{\phi }$ reduces the energy for the distant observer to $U=4\pi \sigma R^{2}e^{-\phi }$.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this note we study the image of ${{\mathbb P}}^2$ in $\text{Gr}(2, {{\mathbb C}}^{4})$ given by tangent bundle of ${{\mathbb P}}^2. $ We show that there is component $\mathcal{H}$ of the Hibert scheme of surfaces in $\text{Gr}(2, {{\mathbb C}}^{4})$ with no point of it corresponds to a smooth surface.'
address:
- 'Laboratoire de Mathématiques de Lens EA 2462 Faculté des Sciences Jean Perrin Rue Jean Souvraz, SP18 F-62307 LENS Cedex France'
- 'Institute of Mathematical Sciences C.I.T. campus, Taramani, chennai 600113,India'
author:
- 'A. El Mazouni'
- 'D.S. Nagaraj'
title: |
Tangent bundle of ${{\mathbb P}}^2$ and morphism from ${{\mathbb P}}^2$ to $\text{Gr}(2,
{{\mathbb C}}^{4})$
---
Projective plane; Tangent bundle; Morphisms; Grassmannian.
Introduction
============
Let ${{\mathbb P}}^2 $ denote the projective plane over the field of complex numbers ${{\mathbb C}}$ and $\text{Gr}(2, {{\mathbb C}}^{4})$ Grassman variety of two dimensional quotients of the vector space ${{\mathbb C}}^4.$
The aim of this paper is to study the image of ${{\mathbb P}}^2$ by non constant morphisms ${{\mathbb P}}^2 \to \text{Gr}(2, {{\mathbb C}}^{4})$ obtained by tangent bundle $T_{{{\mathbb P}}^2}$ of ${{\mathbb P}}^2.$ The bundle $T_{{{\mathbb P}}^2}$ is generated by sections and hence it is generated by four($=\text{rank}(T_{{{\mathbb P}}^2})
+\text{dim}({{\mathbb P}}^2)$) independent global sections. Any set $S$ of four independent generating sections of $T_{{{\mathbb P}}^2}$ defines a morphism $$\phi_S: {{\mathbb P}}^2 \to \text{Gr}(2, {{\mathbb C}}^{4}),$$ such that the $\phi_S^*(Q) = T_{{{\mathbb P}}^2},$ where $Q$ is the universal rank two quotient bundle on $\text{Gr}(2, {{\mathbb C}}^{4}):$ $$\mathcal{O}_{\text{Gr}(2, {{\mathbb C}}^{4})}^4 \to Q \to 0.$$
According to a result of Tango [@Ta], if $ \phi: {{\mathbb P}}^2 \to Gr(2, {{\mathbb C}}^{4})$ is an imbedding then the pair of Chern classes $(c_1(\phi^*(Q)), c_2(\phi^*(Q)))$ is equal to one of the following pairs: $(H,0), (H,H^2), (2H,H^2)$ or $(2H,3H^2),$ where $H$ is the ample generator of $\text{H}^2({{\mathbb P}}^2, \mathbb{Z}).$
Since $c_1(T_{{{\mathbb P}}^2}) = 3H$ and $c_2(T_{{{\mathbb P}}^2}) = 3H^2,$ the morphism $\phi_S$ defined by a set $S$ of four independent generating sections of $T_{{{\mathbb P}}^2}$ is not an imbedding. Thus, it is natural to ask, does there exists a set $S$ of four independent generating sections of $T_{{{\mathbb P}}^2}$ for which $\phi_S$ is generically injective? In this direction we have the following (Theorem \[thm1\]):
For general choice of an ordered set $S$ of four independent generating sections of $T_{{{\mathbb P}}^2}$ the morphism $$\phi_S: {{\mathbb P}}^2 \to {\rm Gr}(2, {{\mathbb C}}^{4})$$
is generically injective.
We also, show that in fact one can find an ordered set of generators $S$ of $T_{{{\mathbb P}}^2}$ the morphism is an immersion i.e., the morphism induces an injection on all the tangent spaces.
As by product of our result we obtain the following (Theorem \[hilb\]):
There is an irreducible component $\mathcal{H}$ of the Hilbert scheme of surfaces in ${\rm Gr}(2, {{\mathbb C}}^{4})$ no point which corresponds to a smooth surface.
The Tangent bundle of ${{\mathbb P}}^2.$
========================================
The tangent bundle of ${{\mathbb P}}^2$ fits in an exact sequence called the “Euler sequence”: $$\label{eq1}
0 \to \mathcal{O}_{{{\mathbb P}}^2} \to \mathcal{O}_{{{\mathbb P}}^2}(1)^{3} \to
T_{{{\mathbb P}}^n} \to 0.$$ This exact sequence together with the fact $\text{H}^1(\mathcal{O}_{{{\mathbb P}}^n})= 0,$ implies that $\text{dim}\text{H}^0(T_{{{\mathbb P}}^2}) = 8,$ where $\text{H}^i$ denotes the $i$ th sheaf cohomology group. Since the rank two bundle $T_{{{\mathbb P}}^2}$ on ${{\mathbb P}}^2$ is ample and generated by sections, a minimal generating set of independent sections has cardinality four. Any set $S$ of four independent generators of $T_{{{\mathbb P}}^2}$ gives to an exact sequence: $$0\to E_S \to \mathcal{O}_{{{\mathbb P}}^2}^4 \to T_{{{\mathbb P}}^2} \to 0.$$ This in turn corresponds to a morphism $\phi_S :{{\mathbb P}}^2 \to \text{Gr}(2,{{\mathbb C}}^{4}),$ where $\phi_S(x) = \{ \mathbb{C}^4 = \mathcal{O}_{{{\mathbb P}}^2}^4|_x \to
T_{{{\mathbb P}}^2}|_x \to 0\}.$
The main result
===============
Note that in the “Euler sequence” (\[eq1\]) the injective map $$0 \to \mathcal{O}_{{{\mathbb P}}^2} \to \mathcal{O}_{{{\mathbb P}}^2}(1)^{3}$$ is given by the section $v=(X,Y,Z),$ where $X,Y,Z$ is the standard basis of $\text{H}^0(\mathcal{O}_{{{\mathbb P}}^n}(1)).$ For a section $v_i$ of $\mathcal{O}_{{{\mathbb P}}^n}(1)^3$ we denote $w_i$ the image section of $T_{{{\mathbb P}}^2}$ under the surjection $$\mathcal{O}_{{{\mathbb P}}^2}(1)^{3} \to T_{{{\mathbb P}}^n} \to 0$$ in (\[eq1\]). Let $\tilde{S} = (v_1,v_2,v_3,v_4)$ be an ordered set of four linearly independent sections of $\mathcal{O}_{{{\mathbb P}}^2}(1)^{3}$ and $S = (w_1,w_2,w_3,w_4)$ be the corresponding ordered set of sections of $T_{{{\mathbb P}}^2}.$ Clearly,the set $S$ is generating set of independent sections of $T_{{{\mathbb P}}^2}$ if and only if $\tilde{S}\cup \{v\}$ is a generating set of independent sections of $\mathcal{O}_{{{\mathbb P}}^2}(1)^{3}.$
\[lem1\] Let $v_1=(X,0,0),v_2=(0,Y,0),v_3=(Y,Z,X),v_4=(Z, X, Y)$ be four sections of $\mathcal{O}_{{{\mathbb P}}^2}(1)^3$ and $\tilde{S}=(v_i|1\leq i \leq 4).$ Then the ordered set $\tilde{S}$ with $ \{v=(X,Y,Z) \}$ is a generating set of independent sections of $\mathcal{O}_{{{\mathbb P}}^2}(1)^{3}.$ Hence the corresponding ordered set of sections $S =(w_1,w_2,w_3,w_4)$ generate $T_{{{\mathbb P}}^2},$ where $w_i$ is the image of $v_i$ under the map given in exact sequence (\[eq1\]).
[**Proof:**]{} Clearly, $v$ generate a subspace of $\mathcal{O}_{{{\mathbb P}}^2}(1)^{3}$ dimension one at every point of ${{\mathbb P}}^2.$ Hence $w_i,w_j$ is not independent at a point $p \in {{\mathbb P}}^2$ if and only if the section $v_{ij}=v_i\wedge v_j \wedge v$ of $\mathcal{O}_{{{\mathbb P}}^2}(3)$ vanishes at $p.$ Thus, if the six independent sections $\{v_i\wedge v_j \wedge v| 1\leq i<j \leq 4\}$ has no common zero implies $\tilde{S}$ with $\{v=(X,Y,Z) \}$ is a generating set of independent sections of $\mathcal{O}_{{{\mathbb P}}^2}(1)^3.$ Note that $v_{12}=XYZ,v_{13}=X(Z^2-XY),
v_{14}=X(XZ-Y^2), v_{23}=Y(X^2-YZ),v_{24}=Y(YX-Z^2),
v_{34}=3XYZ-(X^3+Y^3+Z^3).$ It is easy to see that the set $\{v_{12}, v_{13},v_{14},v_{23},v_{24},v_{34}\}$ of sections of $\mathcal{O}_{{{\mathbb P}}^2}(3)$ has no common zero in ${{\mathbb P}}^2$ and hence the ordered set $\tilde{S}$ with $\{v=(X,Y,Z) \}$ is a generating set of independent sections of $\mathcal{O}_{{{\mathbb P}}^2}(1)^{3}.$ Hence the corresponding ordered set of sections $S =(w_1,w_2,w_3,w_4)$ generate $T_{{{\mathbb P}}^2}.$ $\hfill{\Box}$
\[lem2\] Let $f: {{\mathbb P}}^2 \to {{\mathbb P}}^n$ be a non constant morphism and $f^*(\mathcal{O}_{{{\mathbb P}}^n}(1))= \mathcal{O}_{{{\mathbb P}}^2}(m).$ Assume that there exists a linear subspace $W$ of codimension two such that $W\cap f({{\mathbb P}}^2)$ consists of exactly $m^2$ points. Then the morphism $f$ is generically injective.
[**Proof:**]{} Note as the morphism $f$ is non constant $f^*(\mathcal{O}_{{{\mathbb P}}^n}(1))=\mathcal{O}_{{{\mathbb P}}^2}(m)$ with $m>0$ and hence is ample. This means $f$ is finite map. Set $r= \text{deg}(f),$ the number of elements $f^{-1}(f(x))$ for a general $x\in {{\mathbb P}}^2.$ If $d$ to be the degree of $f({{\mathbb P}}^2)$ in ${{\mathbb P}}^n$ then it is easy to see that $m^2 =d.r.$ On the other hand the assumption, $W\cap f({{\mathbb P}}^2)$ consists of exactly $m^2$ points, implies $d\geq m^2.$ Thus we must have $d=m^2$ and $r=\text{deg}(f)=1.$ Thus $f$ is generically injective. $\hfill{\Box}$
\[lem3\] With the notations of Lemma(\[lem1\]), the surjection of vector bundles on ${{\mathbb P}}^2$ $$\mathcal{O}_{{{\mathbb P}}^2}^4 \to T_{{{\mathbb P}}^2}$$ given by $S$ defines a generically injective morphism $$\phi_S : {{\mathbb P}}^2 \to {\rm Gr}(2, {{\mathbb C}}^{4}).$$
[**Proof:**]{} Let $p: \text{Gr}(2, {{\mathbb C}}^{4}) \to {{\mathbb P}}^5$ be the Pluker imbedding given by the determinant of the universal quotient bundle. Then $p\circ \phi_S $ is given by $$\begin{array}{l}
(x;y;z) \mapsto \\
(xyz;x(z^2-xy);x(xz-y^2));y(x^2-yz);y(xy-z^2);\\
3xyz-(x^3+y^3+z^3)).
\end{array}$$ To prove the map $\phi_S$ is generically injective it is enough to prove the map $p\circ \phi_S $ is so. Set $(Z_0,\ldots, Z_5)$ as the homogeneous coordinates of ${{\mathbb P}}^5$ and $W$ be the codimension two subspace of ${{\mathbb P}}^5$ defined by $Z_0= 0 = Z_5.$ Then $W\cap p\circ \phi_S({{\mathbb P}}^2)$ is equal to $$\{(0,-\omega^i, 1,0,0,0);(0,0,0,-\omega^i,
1,0);(0,\omega^i, 1,\omega^i,1,0)|1\leq i\leq 3 \},$$ where $\omega$ is a primitive cube root of unity. Note that $$(p\circ \phi_S)^*(\mathcal{O}_{{{\mathbb P}}^5}(1)) = \mathcal{O}_{{{\mathbb P}}^2}(3).$$ Hence,the required result follows from Lemma(\[lem2\]). $\hfill{\Box}$
We show (see Lemma \[lem5\]) that $p\circ \phi_S$ is an immersion. i.e., the induced linear map on the tangent space at every point of ${{\mathbb P}}^2$ is injective and one to one except finitely many points.
Next we recall the follwing \[See, Lemma(3.13)[@ALN]\]:
\[lem4\] Let $X$ and $Y$ be two irreducible projective varieties. Let $T$ be an irreducible quasi-projective variety and $t_0 \in T$ be a point. Let $$F : X \times T \to Y$$ be a morphism. Assume that $F_t :=F|_{X\times t} : X \to Y$ is finite for all $t\in T$ and $F_{t_0}$ is a birational onto its image. Then there is an open subvariety $U$ of $T$ such that $t_0 \in U$ and for $t \in U$ the morphism $F_t$ is birational onto its image.
[*Proof:*]{} For the sake of completeness we reproduce the proof here. Consider the morphism $G=F\times Id_T : X \times T \to Y \times T.$ Then the assumption $F_t$ is finite implies the morphism $G$ is finite and proper. Hence ${\mathcal G}= G_{*}({{\mathcal O}}_{X\times T})$ is coherent sheaf of ${{\mathcal O}}_{Y\times T}$ modules. Let $Z \subset Y\times T$ be the subvariety on which the sheaf $G_{*}({{\mathcal O}}_{X\times S})$ is supported. Then clearly the map $p: Z \to T,$ restriction of the natural projection, is surjective. The section $1 \in {{\mathcal O}}_{X\times T}$ gives an inclusion of $ {{{\mathcal O}}_{Z}}$ in ${\mathcal G}.$ Let ${\mathcal F} = {\mathcal G}/{{{\mathcal O}}_{Z}}.$ Let $Z_1\subset Y\times T $ be the subvariety on which the sheaf ${\mathcal F} $ supported. Let $q : Z_1 \to T$ be the natural projection and let $U = \{ t \in T| {\rm dim}{q^{-1}(t)} < {\rm dim}(X) \} $ then we see that by semi continuity \[See, page 95, Exercise (3.22) [@Ha]\], $U$ is an open subset and is non-empty as $t_0 \in U.$ For $t \in U$ the morphism $F_t$ is an isomorphism on $X\times t - G^{-1}(q^{-1}(t).$ Since $G$ is finite $G^{-1}(q^{-1}(t)$ is proper closed subset of $X\times t$ and hence the morphism $F_t$ is birational onto its image. This proves the Lemma. $\hfill{\Box}$
\[thm1\] For a generic choice of an ordered set $S$ of four independent generating sections of $T_{{{\mathbb P}}^2}$ the morphism $$\phi_S: {{\mathbb P}}^2 \to {\rm Gr}(2, {{\mathbb C}}^{4})$$ is generically injective.
[*Proof:*]{} It is easy to see that the ordered set of four sections $S$ generating $T_{{{\mathbb P}}^2}$ is an irreducible quasi projective variety. In fact it is an open subvariety of the affine space $V^4,$ where $V=\text{H}^0(T_{{{\mathbb P}}^2}). $ The theorem at once follows from Lemma(\[lem4\]), if we show the existence of one $S$ for which $\phi_S$ is generically injective. But the existence of one such $S$ follows from Lemma(\[lem3\]). $\hfill{\Box}$
An example
==========
The result of the previous section can be used give an example of a component of a Hilbert Scheme of $Gr(2,{{\mathbb C}}^4)$ with out any point corresponding to a smooth surface.
\[lem5\] The morphism $p\circ \phi_S: \mathbb{P}^2 \to \mathbb{P}^5$ of Lemma(\[lem2\]) is an immersion i.e., the induced linear map on the tangent space at every point of ${{\mathbb P}}^2$ is injective. Moreover, $p\circ \phi_S$ one to one except $$S_1=\{ (1;0;0),(0;1;0), (0;0;1) \} \mapsto (0;0;0;0;0;1)$$ and $$S_2=\{ (1;1;1),(\omega;\omega^2;1),(\omega^2;\omega;1)\} \mapsto (1;0;0;0;0;0),$$ where $\omega$ is a primitive cube root of unity.
[*Proof:*]{} Let $X, Y, Z$ be the homogeneous coordinates functions on $\mathbb{P}^2$ and $Z_0, Z_1,Z_2,Z_3,Z_4,Z_5 $ be the homogeneous coordinates functions on $\mathbb{P}^5.$ Clearly under the morphism $p\circ \phi_S: \mathbb{P}^2 \to \mathbb{P}^5$ the set $S_1$ maps to $(0;0;0;0;0;1)$ and the set $S_2$ maps to $(1;0;0;0;0;0).$ Note that the lines $X=0, Y=0,$ and $Z= 0$ mapped to nodal cubics $Z_0=Z_1=Z_2=Z_3^3+Z_4^3-Z_3Z_4Z_5 =0,$ $Z_0=Z_3=Z_4=Z_1^3+Z_2^3+Z_1Z_2Z_5 =0,$ and $Z_0=Z_1-Z_4=Z_2+Z_3=Z_1^3+Z_2^3-Z_1Z_2Z_5 =0$ respectively. Thus we can conclude that the morphism $p\circ \phi_S$ is an immersion on these three lines. On the complement of these lines the morphism $p\circ \phi_S$ can be described as $$(x,y) \mapsto (1/y-x,x/y-y,x-y/x,y-1/x,3-x^2/y-y^2/x-1/xy)$$ from $\mathbb{C}^2-\{xy=0\} \to \mathbb{C}^4.$ If $(x,y)$ and $(x_1,y_1)$ maps to the same point then we get the following equations: $$\label{eq2}
1/y-x = 1/{y_1}-x_1$$ $$\label{eq3}
x/y-y = {x_1}/{y_1}-y_1$$$$\label{eq4}
y/x-x= {y_1}/{x_1}-x_1$$$$\label{eq5}
1/x-y = 1/x_1-y_1.$$ The equations \[eq2\] and \[eq5\] gives us $$\frac{xy-1}{y} = \frac{x_1y_1-1}{y_1}; \,\, \frac{xy-1}{x} = \frac{x_1y_1-1}{x_1} .$$ Since $xy\neq 0$ and $x_1y_1\neq 0$ we see that either $xy-1 \neq 0 $ and $(x,y) = (x_1,y_1)$ or $xy-1 = 0 =x_1y_1-1 .$ Hence we get $(p\circ \phi_S)$ is one to one out side the set $ \{(1,1),(\omega,\omega^2),(\omega^2,\omega)\}$ and this is mapped to $(0,0,0,0,0). $ The assertion about the immersion of the given morphism $$\mathbb{C}^2-\{xy=0\} \to \mathbb{C}^4$$ can be checked by looking at the two by two minors of the below jacobian matrix of the morphism: $$\left( \begin{matrix}
-1 & 1/y & 1+y/{x^2} & 1/{x^2} & -2x/y+y^2/{x^2}+ 1/{x^2y}\\
-1/{y^2} & -x/{y^2}-1 & -1/x & 1 & {x^2}/{y^2}-2y/x+ 1/{xy^2}
\end{matrix}
\right)$$ $\hfill{\Box}$
\[hilb\] Let $\mathcal{H}$ be the irreducible component of the Hilbert scheme of ${\rm Gr}(2, {{\mathbb C}}^{4})$ containing the point corresponding to the image surface of the morphism $$\phi_S : {{\mathbb P}}^2 \to {\rm Gr}(2, {{\mathbb C}}^{4})$$ of \[lem3\]. Then no point of $\mathcal{H}$ corresponds to a smooth surface.
[**Proof:**]{} Let $ p:{\rm Gr}(2, {{\mathbb C}}^{4}) \to {{\mathbb P}}^5$ be the Pluker imbedding. Since $(p\circ \phi_S)^*(\mathcal{O}_{{{\mathbb P}}^5}(1)) = \mathcal{O}_{{{\mathbb P}}^2}(3)$ and by Lemma \[lem5\] the morphism $ p\circ \phi_S $ is an imbedding outside finite set of points. Moreover, general hyperplane section of $(p\circ \phi_S)({{\mathbb P}}^2)$ in ${{\mathbb P}}^5$ is smooth curve of genus one. If a point of the irreducible $\mathcal{H}$ of corresponds to a smooth surface $Y$ then it has to have the same cohomology class as that of $(p\circ \phi_S)({{\mathbb P}}^2)$ namely $(3,6) \in \rm{H}^4({\rm Gr}(2, {{\mathbb C}}^{4}), {{\mathbb Z}}).$ Also, the general hyperplane section of $p(Y)$ has to be a smooth curve of genus one. But according to the classification of smooth surfaces of type $(3,6)$ in ${\rm Gr}(2, {{\mathbb C}}^{4})$ (see, [@GM Theorem 4.2]) implies that there are no smooth surface of type $(3,6)$ with hyperplane section a smooth curve of genus one. This contradiction proves that no point of $\mathcal{H}$ corresponds to a smooth surface. $\hfill{\Box}$
The component $\mathcal{H}$ of the the Hilbert Scheme in Theorem \[hilb\] is reduced irreducible of dimension 23. In fact computing the normal sheaf associated to the morphism $\phi_S$ of Lemma \[lem3\] and counting the dimension of space of all such morphisms we see that $\mathcal{H}$ is a reduced irreducible of dimension 23.
[**Acknowledgment**]{} This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). The second author thanks university D’artois, Lens and the University of Lille. We thank Laytimi Fatima for help during the work.
[111]{}
El Mazouni, A.; Laytimi, F.; Nagaraj, D. S. *Morphisms from ${{\mathbb P}}^2$ to $Gr(2,{{\mathbb C}}^4).$* J. Ramanujan Math. Soc. 26 (2011), no. 3, 321–332.
Gross, Mark.; *Surfaces of bidegree $(3,n)$ in $\text{Gr}(1,\mathbb{P}^3).$* Math. Z. 212 (1993), no. 1, 73–106.
R. Hartshorne: Algebraic Geometry. Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977.
Hiroshi Tango: *On (n-1)-dimensional projective spaces contained in the Grassmann variety $Gr(n,1).$* J. Math. Kyoto Univ. 14-3 (1974) 415-460.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: |
The double electron capture half-lives of $^{156}$Dy, $^{162}$Er and $^{168}$Yb are evaluated using the pseudo SU(3) model, which describes ground and excited bands as well as their B(E2) and B(M1) transition strengths in remarkable agreement with experiment. The best candidate for experimental detection is the decay $^{156}$Dy $\rightarrow ^{156}$Gd, with $\tau^{1/2} (0^+_{gs} \rightarrow 0^+_{gs}) = 2.74 \times 10^{22}$ yrs and $\tau^{1/2} (0^+_{gs} \rightarrow 0^+_1) = 8.31 \times 10^{24}$ yrs.
\
PACS numbers: 23.40.-s, 21.60.Fw, 27.70.+q\
Keywords: double electron capture, $^{156}$Dy, $^{162}$Er and $^{168}$Yb, pseudo SU(3) model.
author:
- |
Victoria E. Cerón$^a$ and Jorge G. Hirsch$^{b}$.\
[*$^a$ Departamento de Física, Centro de Investigación y Estudios Avanzados del IPN,*]{}\
[*Ap. Postal 14-740, 07000 México D.F., México*]{}\
[*$^b$ Instituto de Ciencias Nucleares, UNAM,*]{}\
[*Circuito Exterior C.U., Ap. Postal 70-543, 04510 México D. F., México* ]{}\
title: 'Double electron capture in $^{156}$Dy, $^{162}$Er and $^{168}$Yb'
---
The double beta decay and its relationship with weak-interactions and neutrino physics have been intensively studied in recent years. Measuring $\beta^-\beta^-\,_{2\nu}$ lifetimes of the order $10^{19}-10^{22}$ yrs and establishing the limit $\langle m_{\nu_e} \rangle \leq 1$eV for the Majorana mass of the neutrino are major experimental achievements [@Suh98].
The double positron emitting decay ($\beta^+\beta^+\,_{2\nu}$) and the accompanying electron capture (EC) processes have been more elusive. Many candidates have low natural abundances or Q-values [@Boe92]. While the relevance of $\beta^+\beta^+\,_{0\nu}$ as a lepton violating process has been known for many years [@Ver83], only recently nuclear matrix elements were calculated in the context of the pn-QRPA [@Sta91]. Searches for these decays, both with and without neutrino emission, have been performed in $^{54}$Fe, $^{78}$Kr, $^{92}$Mo, $^{106}$Cd and $^{130,132}$Ba [@Sae94].
Double electron capture processes (ECEC$_{2\nu}$) have larger Q-values, but the decay to the ground state of the final nuclei ($0^+_{gs}
\rightarrow 0^+_{gs}$) has only X-rays emitted, making its detection difficult [@Suh98]. The ECEC$_{2\nu}$ decay to an excited state in the final nuclei ($0^+_{gs} \rightarrow 0^+_{1}$) could be detected through the two characteristic gamma rays emitted by the final nuclei [@Bar94]
In the present letter we study the ECEC$_{2\nu}$ of $^{156}$Dy, $^{162}$Er and $^{168}$Yb using the pseudo SU(3) model. The motivation for this study is twofold: i) The first two nuclei have been mentioned as possible candidates for experimental detection, with half-lives calculated using rough estimates for the nuclear matrix elements [@Bar94]. ii) The QRPA has undesirable features which makes its predictions unreliable [@Vog98], while shell model calculations provide more reliable nuclear matrix elements for light and medium mass nuclei [@Vog98; @Nak96]. The pseudo SU(3) shell model describes many features of heavy deformed nuclei, as ground and excited bands, B(E2) and B(M1) transition strengths in remarkable agreement with experiment using a symmetry truncated shell model theory with Hamiltonians which includes single particle energies, quadrupole-quadrupole and pairing interactions. The same formalism has proved its effectiveness for even-even [@Beu98] and odd-even [@Var99] nuclei.
As mentioned above the $\beta^+\beta^+\,_{2\nu}$ decay can occur in three different ways:
1. The $\beta^+\beta^+\,_{2\nu}$ proper $$(A, Z+2) \longrightarrow (A, Z) + 2e^{+} + 2\nu$$ is easy to detect through the annihilation of the two positrons, but strongly suppressed by the low Q-value and the Coulomb repulsion between the positrons and the atomic nuclei.
2. The $\beta^+EC\,_{2\nu}$, which captures one bound electron $e^-_b$ $$e^{-}_b +(A, Z+2) \longrightarrow (A, Z) + e^{+} + 2\nu$$ shares with first one the hindering factors.
3. The ECEC$_{2\nu}$ double electron capture $$2e^{-}_b +(A, Z+2) \longrightarrow (A, Z) + 2\nu$$ has the largest Q-values, no Coulomb suppression but is very difficult to detect, because only two X rays are emitted together with the neutrinos.
The double electron capture decay to excited states in the final nuclei $$\begin{aligned}
(A,Z+2) + 2e^{-}_b \longrightarrow &(A, Z)^{*} + 2\nu
~~~~~~~~~~~~~~~ \\
& ~~~~^{\mid}\!\!\!\longrightarrow (A, Z) + 2 \gamma\end{aligned}$$ has been proposed as a good candidate to be measured [@Ver83; @Bar94]. The two gammas are far easier to detect than the X rays. A sensitivity close to $\sim 10^{22}$yr has been estimated for this type of experiments [@Bar94].
The decay scheme for $^{156}$Dy is shown in Fig. 1. It can proceed directly to the ground state of $^{156}$Gd, or to two excited $0^+$ states, which decay mainly to the first $2^+$ state emitting 960.55 keV or 1079.24 keV gammas, followed by a gamma with energy 88.97 keV from the decay of the $2^+$ state [@nndc]. The two gammas in coincidence are the signature of the double electron capture process.
In the pseudo $SU(3)$ shell model coupling scheme [@Rat73] normal parity orbitals $(\eta ,l,j)$ are identified with orbitals of a harmonic oscillator of one quanta less $\tilde \eta = \eta-1$. This set of orbitals with $\tilde j
= j = \tilde l + \tilde s$, pseudo spin $\tilde s =1/2$ and pseudo orbital angular momentum $\tilde l$ define the so called pseudo space. The orbitals with $j = \tilde l \pm 1/2$ are nearly degenerate. For configurations of identical particles occupying a single j orbital of abnormal parity a convenient characterization of states is made by means of the seniority coupling scheme.
The many particle states of $n_\alpha$ nucleons in a given shell $\eta_\alpha$, $\alpha = \nu $ or $\pi$, can be defined by the totally antisymmetric irreducible representations $\{ 1^{n^N_\alpha}\} $ and $\{1^{n^A_\alpha}\}$ of unitary groups. The dimensions of the normal $(N)$ parity space is $\Omega^N_\alpha = (\tilde\eta_\alpha + 1) (\tilde\eta_\alpha +2)$ and that of the unique $(A)$ space is $\Omega^A_\alpha =
2\eta_\alpha +4$ with the constraint $n_\alpha = n^A_\alpha + n^N_\alpha$. Proton and neutron states are coupled to angular momentum $J^N$ and $J^A$ in both the normal and unique parity sectors, respectively. The wave function of the many-particle state with angular momentum $J$ and projection $M$ is expressed as a direct product of the normal and unique parity ones, as:
$$|J M > = \sum\limits_{J^N J^A} [|J^N> \otimes |J^A>]^J_M$$
We are interested in describing the low-lying energy states. In the normal subspace only pseudo spin zero configurations are taken into account. Additionally in the abnormal parity space only seniority zero configurations are taken into account. This simplification implies that $J^A_\pi = J^A_\nu = 0$. This is a very strong assumption quite useful in order to simplify the calculations. It is possible to describe the intruder states using an SU(3) formalism in the [*real*]{} space including S = 0 and 1 states [@Var98]. Calculations performed in this scheme involve a more sophisticated formalism which we are just starting to explore.
The double beta decay, when described in the pseudo SU(3) scheme, is strongly dependent on the occupation numbers for protons and neutrons in the normal and abnormal parity states $n^N_\pi, n^N_\nu, n^A_\pi,
n^A_\nu$ [@Cas94]. These numbers are determined filling the Nilsson levels from below, as discussed in [@Cas94]. In particular the $\beta^+ \beta^+$ decay is allowed only if they fulfil the following relationships
$$\begin{array}{l}
n^A_{\pi ,f} = n^A_{\pi ,i} - 2~~,
\hspace{1cm}n^A_{\nu ,f} = n^A_{\nu ,i}~~, \\
n^N_{\pi ,f} = n^N_{\pi ,i}~~ ,
\hspace{1.7cm} n^N_{\nu ,f} = n^N_{\nu ,i} + 2 ~~.\label{num}
\end{array}$$
If these relations are not satisfied the $\beta^+ \beta^+$ decay becomes forbidden. This is the first selection rule the pseudo SU(3) formalism imposes on the double beta decay, similar to that found for $\beta^-\beta^-$ processes [@Cas94].
In $^{156}$Gd there is one dominant component in the ground state wave function[@Beu98]. Assuming a small deformation to satisfy Eq. (\[num\]) the ground state of this nuclei can be described as
$$\begin{aligned}
|^{156}\hbox{Gd}, 0^+\rangle \approx&|&\{{2^{5}}\}_{\pi} \,(10,4)_{\pi}
,\,\{{2^{4}}\}_{\nu}
\,(18,4)_{\nu};\,(28,8)1 \, L=M=0>_{N} \nonumber \\
&& \nonumber \\
&|&(h_{11/2})_{\pi}^{6} \, J_{\pi}^{A} = 0, \;(i_{13/2})_{\nu}^{4} \,
J_{\nu}^{A} = 0 >_{A} . \label{sdg}\end{aligned}$$
In a similar way, the ground state of $^{156}$Dy will be dominated by
$$\begin{aligned}
|^{156}\hbox{Dy}, 0^+\rangle \approx &|& \{{2^{5}}\}_{\pi} \,(10,4)_{\pi}
;\,\{{2^{3}}\}_{\nu} \,(18,0)_{\nu};\,(28,4)1 \, L=M=0>_{N} \nonumber \\
&& \nonumber \\
&|&(h_{11/2 })_{\pi}^{6} \, J_{\pi}^{A} = 0,\;(i_{13/2})_{\nu}^{2} \,
J_{\nu}^{A} = 0 >_{A} .\label{sdy}\end{aligned}$$
The inverse half life of the two neutrino mode of the $ECEC_{2\nu}$-decay can be expressed in the form [@Doi85]
$$\left[\tau^{1/2}_{2\nu}(0^+ \rightarrow 0^+_\sigma)\right]^{-1} =
G_{2\nu}(\sigma) \ | \ M_{2\nu}(\sigma) \ |^2 \ \ .$$
where $\sigma$= g.s., 1 or 2 labels the final state $0^+_\sigma$, $G_{2\nu}(\sigma)$ are kinematical factors which depend on $E_{\sigma} = {1 \over 2} [{\it Q}_{\beta \beta}-
E(0^+_\sigma )]$ which is half of the total energy released when the electron binding energies are neglected. The nuclear matrix element is evaluated using the pseudo SU(3) formalism [@Hir95]. For the $^{156}$Gd $\rightarrow$ $^{156}$Dy case it can be written as
$$\begin{array}{ll}
M_{2\nu}^{GT}(\sigma)~ =~a~b(n^A_\pi) ~{\cal
E}_{\sigma}^{-1}\\
\hspace{1cm}\sum\limits_{(\lambda_0 \mu_0 ) K_0}
<(\tilde\eta 0)1 \tilde l,(\tilde\eta 0)1 \tilde l \| (\lambda_0 \mu_0
) 0 0>_1 \sum\limits_\rho
<(18,0)1~0,(\lambda_0 \mu_0 )K_0 J\|(18,4 )
1 J>_\rho\\
\hspace{1cm}\sum\limits_{\rho'}
\left[\begin{array}{cccc} (10,4) &(0,0) &(10,4) &1\\
(18,0) &(\lambda_0 \mu_0 ) &(18,4) &\rho' \\
(28,4) &(\lambda_0 \mu_0 ) &(\lambda \mu )_\sigma &\rho \\
1 &1 &1 \end{array} \right]
<(18,4)\mid\mid\mid [a^\dagger_{\tilde \eta 0),{1\over 2}}
a^\dagger_{\tilde \eta 0),{1\over 2}}]^{(\lambda_0 \mu_0 )}
\mid\mid\mid (18,0)>_{\rho'} \label{mgt}
\end{array}$$
In the above formula $<..,..\|,,>$ denotes the SU(3) Clebsch-Gordan coefficients [@Dra73], the symbol $[...]$ represents $9-\lambda\mu$ recoupling coefficients [@Mil78], $<..\mid\mid\mid ..\mid\mid\mid ..>$ are the triple reduced matrix elements [@Hir95c] and the following notation was introduced:
$$\begin{array}{l}
a = {{4\eta}\over{(2\eta + 1)\sqrt{2\eta - 1}}},
\hspace{4cm}
b(n^A_\pi ) = [n^A_\pi(\eta +2 -
n^A_\pi/2)]^{1/2},\\ ~\\
{\cal E}_{\sigma} =
E_{\sigma} -\hbar \omega k_\pi 2 j_\pi + \Delta_C
\hspace{.4cm}\hbox{ with } \hspace{.4cm}
\Delta_C ={ 0.70 \over A^{1/3}} [2 Z + 1 - 0.76 ( (Z+1)^{4/3} -Z^{4/3} )]
\end{array}$$
The SU(3) tensorial components $(\lambda_0 ,\mu_0)$ of the normal part of the double Gamow-Teller operator must be able to couple the proton and total irreps (18,0) and (28,4) associated with the ground state of $^{156}$Gd to the corresponding irreps (18,4) and $(\lambda
\mu)_\sigma $ = (28,8), (30,4) and (32,0), which characterize the ground and excited rotational bands in $^{156}$Gd. If these irreps cannot be connected by $(\lambda_0 ,\mu_0 )$ the $\beta^+ \beta^+$ decays to the $0^+$ states are forbidden. This is a second selection rule imposed by the model to the $\beta \beta$ decay.
----------------------------------------------------------------------------------------------------------------------------------------
$G_{2\nu}$(yr$^{-1}$) $| \ $\tau^{1/2}$(yr)
M_{2\nu}(\sigma) \ |$
----------------------------------- --------------------------------- ----------------------- ------------------ -----------------------
$^{156}$Dy $\rightarrow ^{156}$Gd $0^{+} \rightarrow 0^{+}(g.s.)$ 9.79$\times 10^{-21}$ 0.061 2.74$\times 10^{22}$
$0^{+} \rightarrow 0^{+}(1)$ 1.65$\times 10^{-22}$ 0.027 8.31$\times 10^{24}$
$0^{+} \rightarrow 0^{+}(2)$ 7.58$\times 10^{-23}$ 0.035 1.08 $\times 10^{25}$
$^{162}$Er $\rightarrow ^{162}$Dy $0^{+} \rightarrow 0^{+}(g.s.)$ 8.06$\times 10^{-21}$ 0.066 2.85$\times10^{22}$
$0^{+} \rightarrow 0^{+}(1)$ 1.60$\times 10^{-24}$ 0.013 3.70$\times10^{27}$
$^{168}$Yb $\rightarrow ^{168}$Er $0^{+} \rightarrow 0^{+}(g.s.)$ 2.47$\times 10^{-21}$ 0.045 2.00$\times 10^{23}$
$0^{+} \rightarrow 0^{+}(1)$ 5.18$\times 10^{-28}$ 0.0006 5.36$
\times 10^{33}$
----------------------------------------------------------------------------------------------------------------------------------------
Table 1: Half-lives for the $ECEC_{2\nu}$ decay to the ground and excited states of the final nuclei.\
In Table 1 we present the $ECEC_{2\nu}$ decay of $^{156}$Dy, $^{162}$Er and $^{168}$Yb to the ground and excited states of $^{156}$Gd, $^{162}$Dy and $^{168}$Er respectively. The kinematical factors $G_{2\nu}(\sigma)$ were evaluated following the prescriptions given in [@Doi88]. When the energy released in the decay to an excited stated ($2 E_\sigma$) is small the available phase space $G_{2\nu}$ is strongly reduced, and the half life could be very large. It is the case in the double electron capture decay to the first excited $0^+$ state in $^{168}$Er. Details of the calculation of the nuclear matrix elements, as well as the energy spectra of the nuclei involved will be presented elsewhere [@Cer99]. The nuclear matrix elements associated with the decay to the ground state of the final nuclei $| M_{2\nu}(g.s.) \ |$ have values close to 0.05 - 0.06, a factor of 5 smaller than the assumption of Barabash [@Bar94], and are similar for the three nuclei studied. The nuclear matrix elements to excited states $| M_{2\nu}(1,2) \ |$ show a wide spread, being close to those of the ground state for $^{156}$Dy, suppressed by a factor 5 for $^{162}$Er and by a factor 80 for $^{168}$Yb. While in general it is confirmed that deformed nuclei have smaller nuclear matrix elements than spherical [@Cas94], $^{156}$Dy appears to be the best candidate of this group for experimental detection, with a half-life around $10^{24}$ years for the double electron capture to the first excited $0^+$ state.
In summary, we have used the pseudo SU(3) shell model to investigate the double electron capture decays in three heavy deformed nuclei, and found that experiments with sensitivities around $10^{24}$ years could detect the decay of $^{156}$Dy to the first excited state in $^{156}$Gd, while detecting the doubled electron capture decay in $^{162}$Er and $^{168}$Yb would be very difficult.
The authors thanks Esteban Roulet and Roelof Bijker for the critical reading of the manuscript. Partial economical support by Conacyt, México, is acknowledged.
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Figure Captions:
Energy levels relevant for the $\beta^+\beta^+$ decay of $^{156}$Dy.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of s near a in a Hamiltonian system with continuous symmetries. In particular we prove that under appropriate hypotheses there exist s near relative equilibria even when these relative equilibria are singular points of the corresponding moment map, i.e. when the reduced spaces are singular.
[**Sur les oscillations normales relatives**]{}
[R[ÉSUMÉ]{}.]{} On généralise le théorème d’existence de Weinstein-Moser des oscillations normales non-linéaires voisines d’un équilibre dans un système hamiltonien, à un théorème d’existence des orbites périodiques relatives voisines d’un équilibre relatif dans un système hamiltonien à symétrie continue. Entre autres on démontre, moyennant quelques hypothèses, qu’il existe des orbites périodiques relatives près des équilibres relatifs même lorsque ceux-ci sont des points singuliers de l’application de moment, c’est-à-dire lorsque les espaces réduits sont singuliers.
address: 'Department of Mathematics, University of Illinois, Urbana IL 61801, USA'
author:
- 'E. Lerman'
- 'T. F. Tokieda'
date: '28/10/98'
title: On relative normal modes
---
[Version fran[ç]{}aise abrégée]{}.
Dans cette note on généralise le théorème de Weinstein-Moser [@W1; @Ms; @W2; @MnRS; @Ba] sur les oscillations normales non-linéaires voisines d’un équilibre dans un système hamiltonien, à un théorème sur les orbites périodiques relatives voisines d’un équilibre relatif dans un système hamiltonien symétrique.
Soit $M$ une variété symplectique munie d’une opération hamiltonienne d’un groupe de Lie connexe $G$ et donc d’une application de moment $\Phi: M \to {{\mathfrak g}}^{\ast}$. Une orbite du champs hamiltonien $X_h$ d’un hamiltonien $G$-invariant $h\in C^{\infty}(M)^G$ est un [*équilibre relatif*]{} \[resp. une [*orbite périodique relative*]{}\] si son image dans $M/G$ est un point \[resp. un lacet\].
Les améliorations ultérieures du théorème de Weinstein-Moser se prêtent toutes à des généralisations par notre méthode; néanmoins, afin de ne pas alourdir l’exposé on traitera seulement la version originale:
\[Theorem1\] [\[W1\]]{} Soit $h$ un hamiltonien sur un vectoriel symplectique $V$ avec $dh(0)$ nul et $d^2h(0)$ défini positif. Alors pour tout $\varepsilon > 0$ petit, $h\inv(h(0) + \varepsilon)$ porte au moins $\frac{1}{2}\dim V$ orbites périodiques de $X_h$.
Soit $x\in M$ un point d’un équilibre relatif pour un hamiltonien $G$-invariant. Existe-t-il des orbites périodiques relatives voisines? A cela nulle difficulté si $x$ est un point régulier de $\Phi$, ou même dans le cas singulier si la dimension de la strate de son image $\bar{x}$ dans l’espace réduit est strictement positive, car le problème se ramène alors au théorème de Weinstein sur cette strate, qui est stable par rapport à la dynamique de l’espace réduit stratifié [@AMM; @SL]. Mais que dire si la strate de $\bar{x}$ est un point?
On verra qu’en l’occurrence il existe encore des familles d’orbites périodiques relatives voisines de l’équilibre relatif, pourvu qu’un certain ‘rempla[ç]{}ant’ du hessien soit défini en tant que forme quadratique et que le groupe d’isotropie de $\Phi(x)$ soit un tore. On a en effet un théorème de correspondance que voici:
\[Theorem3\] Soit $M$ une variété symplectique munie d’une opération hamiltonienne d’un groupe de Lie connexe $G$ et d’une application de moment $\Phi: M \to {{\mathfrak g}}^{\ast}$, et soit $h\in C^{\infty}(M)^G$ un hamiltonien $G$-invariant.
Supposons que $x\in M$ appartienne à un équilibre relatif pour $h$ et que le groupe d’isotropie de $\Phi(x)$ soit un tore. Alors il existe un vectoriel symplectique $V$ muni d’une opération hamiltonienne du groupe d’isotropie $G_x$ de $x$ et un hamiltonien $G_x$-invariant $h_V\in C^{\infty}(V)^{G_x}$ tels que
1. l’origine $0\in V$ est un équilibre pour $h_V$;
2. le hessien $d^2h_V(0)$ de $h_V$ peut être calculé à partir de $h$;
3. toute orbite périodique $G_x$-relative pour $h_V$ dans $V$ donne lieu à une famille d’orbites périodiques $G$-relatives pour $h$ dans $M$.
Ici $V$ désigne le sous-espace symplectique maximal de $\ker d\Phi(x)$ (dit [*tranche symplectique*]{} en $x$) et $d^2h_V(0)$ est la forme quadratique induite sur $V$ par $d^2(\augh)(x)|_{\ker d\Phi(x)}$, où $\xi$ est une [*vitesse*]{} de $x$ que l’on trouve dans $$x \text{ appartient \`a un \'equilibre relatif pour } h \
\Longleftrightarrow\ \exists \xi\in {{\mathfrak g}}, \quad d(\augh)(x) = 0.$$ La forme normale locale de Guillemin-Sternberg et de Marle [@GS; @Mr] permet de construire une fonction $h_V$ sur $V$ dont le hessien est $d^2h_V(0)$. Cette notion de hessien intervient dans l’étude de la stabilité et des bifurcations des équilibres relatifs aux points singuliers de l’application de moment [@LS; @Ortega; @OR].
Le Théorème 1, joint par exemple au théorème de Weinstein, conduit au résultat escompté:
\[Corollary\] Soient $M$, $G$, $\Phi : M \to {{\mathfrak g}}^{\ast}$, $h\in C^{\infty}(M)^G$ comme dans le Théorème 1. Supposons que $x\in M$ appartienne à un équilibre relatif et que le groupe d’isotropie de $\Phi(x)$ soit un tore; appelons $V$ la tranche symplectique en $x$. S’il existe une vitesse $\xi$ de $x$ telle que $d^2(\augh)(x)|_V$ est défini positif, alors pour tout $\varepsilon > 0$ petit, $\{ y \in M \mid (\augh)(y) = (\augh)(x) + \varepsilon \}$ porte des familles d’orbites périodiques relatives.
On peut sans doute se passer de l’hypothèse selon laquelle le groupe d’isotropie de $\Phi(x)$ est un tore, mais la démonstration ne promet guère d’être si simple. Signalons que dans le cas où $G$ est compact, l’hypothèse est satisfaite de fa[ç]{}on [*générique*]{}.
[*Esquisse de la démonstration.*]{}
D’abord, on se ramème au cas où $G$ est un tore en passant à un sous-système hamiltonien dont le groupe de symétrie est le groupe d’isotropie de $\Phi(x)$, grâce à un théorème de Guillemin-Sternberg (voir [@GLS], Corollaire 2.3.6). Ensuite, on plonge, par un symplectomorphisme équivariant, un voisinage de $G\cdot x$ dans $(T^{\ast}L\times V)/\Gamma$ ($L$ est le tore complémentaire de la composante connexe $K$ de $1$ dans $G_x$, et $\Gamma = G_x/K$); en réduisant par $L$, on réduit ce dernier espace à $V$ et $h$ à $h_V$. Enfin, on vérifie que les orbites périodiques relatives pour $h_V$ se relèvent à des orbites périodiques relatives pour $h$: les unes et les autres correspondent aux orbites périodiques du système obtenu soit en réduisant $V$ par $\Gamma\times K$, soit en réduisant $(T^{\ast}L\times V)/\Gamma$ par $L\times K$.
Comme un exemple d’application, on remarque que le Corollaire assure l’existence des orbites périodiques relatives d’une paire de corps rigides à symétrie axiale liés par un potentiel qui dépend de l’angle entre les corps.
------------------------------------------------------------------------
Introduction.
=============
In this paper we generalize the Weinstein-Moser theorem ([@W1; @Ms; @W2; @MnRS; @Ba] and references therein) on the nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the s near a in a symmetric Hamiltonian system.
Let $M$ be a symplectic manifold, with a Hamiltonian action of a connected Lie group $G$ and a moment map $\Phi: M \to
{{\mathfrak g}}^{\ast}$. Recall that an orbit of the Hamiltonian vector field $X_h$ of a $G$-invariant Hamiltonian $h\in C^{\infty}(M)^G$ is a [**]{} if its image in the orbit space $M/G$ is a point, and that an orbit of $X_h$ is a [**]{} if its image in $M/G$ is a closed curve.
Later improvements of the Weinstein-Moser theorem lend themselves to generalizations by our method; to streamline the presentation, however, we shall treat only the original version:
\[Theorem1\] [\[W1\]]{} Let $h$ be a Hamiltonian on a symplectic vector space $V$ such that its differential at the origin $dh(0)$ is zero and its Hessian at the origin $d^2h(0)$ is positive definite. Then for every small $\varepsilon > 0$, the energy level $h\inv(h(0) + \varepsilon)$ carries at least $\frac{1}{2}\dim V$ periodic orbits of the Hamiltonian vector field of $h$.
Now suppose $x\in M$ lies on a for a $G$-invariant Hamiltonian $h$. If $x$ is a [*regular*]{} point of the moment map $\Phi$, then the reduced space at $\mu = \Phi(x)$ is smooth near the image $\bar{x}$ of $x$. Provided appropriate conditions hold on the Hessian of the reduced Hamiltonian, Weinstein’s theorem applied to the reduced system guarantees at least $\frac{1}{2} \dim (\text{reduced
space})$ periodic orbits, and hence as many families of s near the . (If $x$ lies on a , then the orbit through $g\cdot x$ is relative periodic also for every $g\in G$.) On the other hand, if $x$ is a [*singular*]{} point of $\Phi$, then the reduced space at $\mu$ is a stratified space, and the reduced dynamics preserves the stratification [@AMM; @SL]. Unless the stratum through $\bar{x}$ is an isolated point, we have again $\frac{1}{2}\dim (\text{stratum})$ families of s, provided appropriate conditions hold on the Hessian of the restriction of the reduced Hamiltonian to the stratum. But what if the stratum through $\bar{x}$ is an isolated point? It is difficult to make sense of Hessians on singular spaces.
We shall show that in this case also there are families of s near the provided a certain substitute for the Hessian is definite and the isotropy group $G_{\mu}$ of $\mu$ is a torus. The proof amounts to a computation in ‘good coordinates’ that allows us to reduce our problem to Weinstein’s theorem. We proceed via the following ‘correspondence theorem’:
\[Theorem2\] Let $M$ be a symplectic manifold, with a Hamiltonian action of a connected Lie group $G$ and a moment map $\Phi: M \to
{{\mathfrak g}}^{\ast}$, and let $h\in C^{\infty}(M)^G$ be a $G$-invariant Hamiltonian.
Suppose $x\in M$ lies on a for $h$ and the isotropy group of $\Phi(x)$ is a torus. Then there exist a symplectic vector space $V$ with a Hamiltonian action of the isotropy group $G_x$ of $x$ and a $G_x$-invariant Hamiltonian $h_V\in
C^{\infty}(V)^{G_x}$ such that
1. the origin $0\in V$ is an equilibrium for $h_V$;
2. the Hessian $d^2h_V(0)$ of $h_V$ can be computed in terms of $h$;
3. every $G_x$- for $h_V$ on $V$ sufficiently close to the origin gives rise to a family of $G$-s for $h$ on $M$.
The meaning of the vector space $V$ and of the quadratic form $d^2h_V(0)$ is as follows. Note that $$x\in M \text{ lies on a \re\ for } h \ \Longleftrightarrow\
d(\augh)(x) = 0 \; \text{ for some } \xi\in {{\mathfrak g}}.$$ The vector $\xi$ is called a [*velocity*]{} of $x$. Velocity is not unique: any two velocities of $x$ differ by a vector in the isotropy Lie algebra ${{\mathfrak g}}_x$ of $x$.
The function $\augh$ is constant along the orbit $G_{\mu}\cdot x$, where as above $G_{\mu}$ is the isotropy group of $\mu = \Phi (x)$. The form $d^2(\augh)(x)|_{\ker d\Phi(x)}$ is therefore always degenerate and descends to a well-defined form on the vector space $V := \ker d\Phi (x) / T_x
(G_{\mu}\cdot x)$; alternatively we can think of $V$ as the maximal symplectic subspace of $\ker d\Phi (x)$. $V$ is called the [*symplectic slice*]{} at $x$. It follows from the local normal form theorem of Guillemin-Sternberg and Marle [@GS; @Mr] that there exists a $G_x$-invariant symplectic submanifold $\Sigma$ passing through $x$ such that $T_x \Sigma = V$. Thus, locally $\Sigma \simeq V$ as symplectic manifolds, with $x$ corresponding to the origin in $V$. The function $h_V$ in Theorem \[Theorem2\] is the restriction $(\augh)|_\Sigma =(\augh)|_V $. Since Hessians are functorial, $d^2 h_V (0) = d^2(\augh)|_V(0) = d^2(\augh)(x) |_V.$ This notion of Hessian has been used in [@LS; @Ortega; @OR] to study the stability and bifurcation of relative equilibria at singular points of the moment map.
As an example of applications of Theorem \[Theorem2\], we combine it with Weinstein’s theorem to obtain:
\[Corollary\] Let $M$, $G$, $\Phi :M \to {{\mathfrak g}}^* $, $h\in C^\infty (M)^G$ be as in Theorem \[Theorem2\]. Suppose $x\in M$ lies on a for $h$ and the isotropy group of $\Phi(x)$ is a torus; call $V$ the symplectic slice at $x$. If there is a velocity $\xi$ of $x$ for which $d^2(\augh)(x)|_V$ is positive definite, then for every small $\varepsilon > 0$, the level set $\{ y \in M \mid (\augh)(y) = (\augh)(x) + \varepsilon\}$ carries families of s.
We expect that the assumption on the isotropy group of $\Phi(x)$ being a torus can be dropped, but the proof is unlikely to be quite so simple. In case $G$ is compact, the assumption is satisfied [*generically*]{}.
Proof of Theorem \[Theorem2\].
===============================
Let $M$, $G$, $\Phi: M \to {{\mathfrak g}}^{\ast}$, $h\in
C^{\infty}(M)^G$ be as in the statement of Theorem \[Theorem2\]. We are supposing that $x\in M$ lies on a for $h$ and that the isotropy group $G_{\mu}$ of $\mu = {\Phi(x)}$ is a torus.
First, we show that it suffices to consider the case when the whole $G$ is a torus. Since $G_{\mu}$ is compact, we can produce an $Ad^\dagger (G_{\mu})$-invariant inner product on ${{\mathfrak g}}^*$ and take the corresponding $G_{\mu}$-equivariant splitting ${{\mathfrak g}}= {{\mathfrak g}}_{\mu} \oplus {{\mathfrak h}}$. Then a small enough $G_{\mu}$-invariant neighborhood $B$ of $\mu$ in the affine plane $\mu + {{\mathfrak h}}^\circ $ is transverse to the moment map (here ${{\mathfrak h}}^\circ $ denotes the annihilator of ${{\mathfrak h}}$ in ${{\mathfrak g}}$). Hence $R := \Phi \inv (B)$ is a $G_{\mu}$-invariant submanifold of $M$ containing $x$. In fact, by the symplectic cross-section theorem of Guillemin and Sternberg (cf. [@GLS], Corollary 2.3.6), $R$ is a [*symplectic*]{} submanifold of $M$ and the action of $G_{\mu}$ on $R$ is Hamiltonian; the moment map for this action is the restriction of $\Phi$ to $R$ followed by the natural projection ${{\mathfrak g}}^* \to
{{\mathfrak g}}_{\mu}^*$. Since ${{\mathfrak g}}^* \to {{\mathfrak g}}_{\mu}^*$ restricted to $\mu + {{\mathfrak h}}^\circ$ is an isomorphism, the moment map for the action of $G_{\mu}$ on $R$ is $\Phi|_R$ up to a linear isomorphism. It follows that $$\ker d\Phi(y) = \ker d(\Phi|_R)(y)$$ for every $y\in R$ (cf. [@LS], Lemma 2.5). Moreover, because $h$ is $G$-invariant, the flow of $X_h$ preserves the fibers of the moment map, and so the flow preserves $R$. It follows that $$(X_h)|_R = X_{(h|_R)}$$ (cf. [@L], p.218). Thus, we have found a Hamiltonian sub-system $(R, G_{\mu}, \Phi|_R, h|_R)$ for which the symmetry group $G_{\mu}$ is a torus. Passing to this sub-system, we may and shall assume without loss of generality that $G$ is a torus and $G = G_{\mu}$.
Second, we construct the Hamiltonian $h_V$. The connected component $K$ of $1$ in the isotropy group $G_x$ of $x$ is a torus and has a complementary torus $L$, so that $G = L\times K$. The finite abelian group $\Gamma = G_x/K$ may be regarded as a subgroup of $L$. Then $\Gamma$ acts symplectically (by the lifted action) on $T^*L = L\times {{\mathfrak l}}^*$ and on the symplectic slice $V$ at $x$. Hence $\Gamma$ acts diagonally on $T^*L \times V$. Note that $G$ too acts on $T^*L \times V$: $L$ acts on its own cotangent bundle and $K$ acts on $V$. Hence $G$ acts in a Hamiltonian fashion on $(T^*L \times V)/\Gamma$. The orbit of $[1,0,0] \in (T^*L
\times V)/\Gamma$ is isotropic and is isomorphic to $G/(\Gamma\times K)
\simeq G\cdot x$. By the equivariant isotropic embedding theorem, there exist $G$-invariant neighborhoods $U_0$ of $[1,0,0]$ in $(T^*L \times V)/\Gamma$ and $U$ of $x$ in $M$ and an equivariant symplectomorphism $\sigma : U_0\to U$ such that $\sigma
([1,0,0]) = x$ and the derivative $d \sigma ([1,0,0]) $ sends $V\subset T_{[1,0,0]} (T^*L \times V)/\Gamma $ to $V\subset \ker
d\Phi (x) \subset T_x M$ by the identity map. Define $\th = \sigma ^* (h - \langle\Phi | \xi\rangle )$. Because $G$ is assumed to be abelian, $\xi$ is trivially in the center of ${{\mathfrak g}}$, and so $\th$ is $G$-invariant. At this juncture it is convenient to pretend that $U_0 = (T^*L \times
V)/\Gamma$. By the invariance of $\th \in C^\infty ((T^*L \times
V)/\Gamma)^{L\times K}$, $\th ([l, \lambda, v]) = \th ([1, \lambda, v])$ for all $(l,\lambda, v) \in L\times {{\mathfrak l}}^* \times V$. Now define $h_V (v) = \th ([1, 0, v])$. This $h_V \in C^\infty (V)^{\Gamma\times K}$ is the desired Hamiltonian: we have $$dh_V (0) = d\th([1, 0, 0])|_V = 0$$ and $$d^2 h_V (0) = d^2\th ([1, 0, 0])|_V
= d^2\left(\sigma^*(\augh)\right)([1, 0, 0])|_V
= \sigma^*\left( d^2(\augh)(x)|_V\right).$$
Third and last, we prove that s for $h_V$ give rise to s for $h$. Consider the fully reduced system $(P, h_P)$ obtained by reducing the system $(V, h_V)$ by the action of $\Gamma\times K$. We can arrive at $(P, h_P)$ also by reducing the system $( (T^*L\times V)/\Gamma, \th\, )$ by the action of $G = L\times K$. Thus, s for $h_V$ correspond to periodic orbits for $h_P$, which in turn give rise to s for $\th$, or equivalently to s for $\augh$. But s for $\augh$ are s for $h$. Indeed, writing $\varphi_f^t$ for the Hamiltonian flow of $f$, we have $\varphi_{\augh}^t =
\varphi_{-\langle\Phi | \xi \rangle}^t \circ \varphi_h^t$ by the $G$-invariance of $h$. Therefore, $\varphi_{\augh}^t(x) = g\cdot x$ for some $g\in G$ if and only if $\varphi_h^t(x) = g^{\prime}\cdot x$ with $g^{\prime} = \exp(-t\xi)\, g\in G$. This concludes the proof of Theorem \[Theorem2\].
Concluding remarks
==================
I. Corollary proves the existence of s for a pair of axially symmetric rigid bodies subject to a coupling potential which depends on the angle between the bodies. We expect the theorem to prove the existence of s for the motion of Riemann ellipsoids.\
II. In Theorem \[Theorem2\], we compute $d^2h_V(0)$ from $d^2(\augh)(x)|_{\ker d\Phi(x)}$ by ‘killing’ the direction of degeneracy. This computation needs only the [*existence*]{} of the ‘Darboux coordinates’ $\sigma : (T^*L \times V)/\Gamma \supset U_0 \to
M$ (see the proof above), but by computing without the explicit form of $\sigma$, we lose some information. Alternatively we can compute the Taylor expansion of $\sigma$ and translate the higher-order information on the jet of the original Hamiltonian $h$ into the information on the jet of the model Hamiltonian $h_V$.
[XXXX]{}
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V. Guillemin, E. Lerman, and S. Sternberg, [*Symplectic fibrations and multiplicity diagrams*]{}, Cambridge University Press, Cambridge, 1996.
V. Guillemin and S. Sternberg, A normal form for the moment map, in: [*Differential Geometric Methods in Mathematical Physics*]{} (ed. S. Sternberg), Reidel, Dordrecht, 1984.
E. Lerman, On the centralizer of invariant functions on a Hamiltonian $G$-space, [*J. Diff. Geom.*]{} [**30**]{} (1989) 805–815.
E. Lerman and S. F. Singer, Stability and persistence of relative equilibria at singular values of the moment map, [*Nonlinearity*]{} [**11**]{} (1998) 1–13.
C.-M. Marle, Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique, [*Rend. del Seminario Matematico*]{} [**43**]{} (1985) 227–251.
J. A. Montaldi, R. M. Roberts, and I. N. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, [*Phil. Trans. R. Soc. Lond.*]{} A [**325**]{} (1988) 237–293.
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Measurements in Liquid Argon Time Projection Chamber (LArTPC) neutrino detectors, such as the MicroBooNE detector at Fermilab [@Acciarri2017], feature large, high fidelity event images. Deep learning techniques have been extremely successful in classification tasks of photographs, but their application to LArTPC event images is challenging, due to the large size of the events. Events in these detectors are typically two orders of magnitude larger than images found in classical challenges, like recognition of handwritten digits contained in the MNIST database or object recognition in the ImageNet database. Ideally, training would occur on many instances of the entire event data, instead of many instances of cropped regions of interest from the event data. However, such efforts lead to extremely long training cycles, which slow down the exploration of new network architectures and hyperparameter scans to improve the classification performance. We present studies of scaling a LArTPC classification problem on multiple architectures, spanning multiple nodes. The studies are carried out on simulated events in the MicroBooNE detector. We emphasize that it is beyond the scope of this study to optimize networks or extract the physics from any results here. Institutional computing at Pacific Northwest National Laboratory and the *SummitDev* machine at Oak Ridge National Laboratory’s Leadership Computing Facility have been used. To our knowledge, this is the first use of state-of-the-art Convolutional Neural Networks for particle physics and their attendant compute techniques onto the DOE Leadership Class Facilities. We expect benefits to accrue particularly to the Deep Underground Neutrino Experiment (DUNE) LArTPC program, the flagship US High Energy Physics (HEP) program for the coming decades.'
address: 'Pacific Northwest National Laboratory, Richland, WA, USA'
author:
- 'A Hagen, E Church, J Strube, K Bhattacharya, and V Amatya'
bibliography:
- 'acat\_2019.bib'
title: Scaling the training of particle classification on simulated MicroBooNE events to multiple GPUs
---
Introduction\[sec:introduction\]
================================
Use of convolutional networks to analyze time projection chamber data is often performed on cropped data because of large image sizes. Training and inference on uncropped TPC data is desired to minimize physics information loss before training. The high fidelity and large size of the image data requires scaling of computing resources past the 1s to 10s and 100s of GPUs.
The MicroBooNE Detector and data format used
--------------------------------------------
This work formats its simulated data with inspiration from the MicroBooNE experiment [@Acciarri2017]. MicroBooNE is a $170\;\mathrm{tonne}$ liquid argon time projection chamber (LArTPC) with the express interest of analyzing neutrino physics. Readout of MicroBooNE consists of 2 induction planes with $2400\;\mathrm{wires}$ each and 1 collection plane with $3156\;\mathrm{wires}$. Readout occurs every $4.8\;\mathrm{ms}$ (which is $2.2\times$ the TPC drift time) for $9600\;\mathrm{digitizations}$.
This work presents classification on simulated single particle events in a format similar to that of MicroBooNE [^1]. GEANT4 Monte Carlo particle transport code was used to simulate interactions with the wires in a LArTPC. The seed particles were single particles of type $\mu^{+}$, $e^{+}$, $e^{-}$, $K^{+}$, $\pi^{+}$, and $\gamma$. The interactions with the collection plane were then tallied across $3200$ time ticks and padded with zeros to increase the time dimension to $3600$. The collection plane dimension was also padded with zeros to reach size $3600$. The final simulated data was a $3600\times 3600$ “image”, with examples for each particle shown in Figure \[fig:events\].
Previous work showed not only decreased training time for simple convolution networks on 2 to 14 GPUs (on PNNL’s Institutional Computing), but improved performance, achieving lower losses with 14 GPUs versus with only 1 [@Bhattacharya2018]. This work extends upon that scaling study, moving from 1 to 10s of GPUs to the 100s.
Methodology
===========
A simplified convolutional neural network (CNN) model was developed for testing[^2]. As shown in figure \[fig:network-architecture\], the network architecture used two subsequent blocks consisting of convolutional layers with kernel size 5, padding of 2, and exponential linear unit (ELU) activation; followed by max pool layers of pooling size 5. Then, two more subsequent blocks of convolutional layers with kernel size 4, padding of two, and ELU activation followed by max pooling layers of pool size 4. These four blocks have increasing numbers of convolutional filters with the scheme of 1, 10, 64, 128, 256. After these four blocks, a linear layer with 20736 inputs, 32 outputs and ELU activation was appended. A linear layer with 32 inputs, 5 outputs, and softmax activation, one output for each of the particle types ($e^{+}$ and $e^{-}$ were grouped together), finished the network. For a dense data representation, this network architecture was implemented using `Torch`’s `Conv2d` and `MaxPool2d` layers and is hereafter called `JishNet`. For a sparse data representation, discussed further in section \[sec:sparse\], this network used `SparseConvNet`’s `SubmanifoldConvolution` and `MaxPooling` layers and is hereafter referred to as `SCNet`. It should be noted that the padding behavior of the convolutions in `JishNet` and `SCNet` differ subtly.
Resources
---------
Many industrial applications of convolutional networks can apply 10s to 100s of GPUs to speed training and enable fast iteration in network design [@Mikami2018; @Sergeev2018].
### Hardware
This work presents one of the first known instances of scaling a physics classificaton problem to this scale on Oak Ridge National Laboratory Leadership Computing Facility’s [*SummitDev*]{} computer. [*SummitDev*]{}, which is a test stand for the eventual *Summit* computer, was used in this study. The [*SummitDev*]{} machine boasts 50 nodes, each with 2 22-core IBM Power8NVL CPUs and 4 NVIDIA P100 GPUs. The interconnections between nodes are Mellanox EDR 100G InifiBand; interconnections between GPU to host node are NVLink. It should also be noted that there is a 4 hour time limit to all [*SummitDev*]{} jobs.
### Software
There is an embarassment of options for the development of neural networks and data parallel training of these neural networks. While previous scaling studies used Pacific Northwest National Laboratory’s [`MaTEx`]{} [@Bhattacharya2018; @Amatya2018] code for data parallel training, this work was based on the use of [`Torch`]{} [@Paszke2017] for neural network definition and training, and [`Horovod`]{} [@Sergeev2018] for data parallelization.
Due to the large size of images and their large number, compression was needed to store the dense representation of the images in memory when loading[^3]. Compression using `Blosc` [@Alted2019] when loading images from file into CPU memory. At training time, a minibatch of these images were uncompressed from memory and transferred to GPU memory. Due to the large size of each datum, only 7 images alongsize the `JishNet` model could fit into a single NVIDIA P100’s memory. Thus, all training was performed with a minibatch size of $7$. After training of a single minibatch on all GPUs allocated for each training instance, [`Horovod`]{}’s `allreduce` function was used to transfer and allocate all gradients to the head node. This has the effect of generating a batch size of $7 \times N_{GPU}$. The loss and accuracy local to each GPU was written to disk before this `allreduce` operation.
The use of many GPUs significantly affects the training dynamics of the CNN in this training example. A prolonged study was performed examining the training dynamics of this task using stochastic gradient descent (SGD) optimizer and warmup and scaling suggested in Goyal [@Goyal2017]. The SGD optimization study ultimately underperformed, so other options were explored. Distributed Adam optimization was tested and ultimately successful; thereafter, all results used Adam optimization [@Kingma2014]. The training dynamics were highly sensitive to parameters of Adam optimizer, final parameters were $lr=0.001$, $\beta = \left[0.9, 0.999\right]$, $\epsilon = 1\times 10^{-8}$, and $decay = 0.01$. Figures \[fig:jishnet\_loss\] and \[fig:scnet\_loss\] illustrate expected training behavior and verify the use of the Adam optimizer.
Sparse\[sec:sparse\]
--------------------
Most of the data passed to the dense convolutional network is zeros, as no events happen in large parts of the TPC. Sparse convolutional networks are a more efficient network architecture for LArTPC event classificaton. After conversion from a dense format to a sparse data format, training using `SubmanifoldConvolution` layers from Facebook’s `SparseConvNet` [@Graham2017] codebase and the modified CNN `SCNet` was performed.
Results
=======
Results show increasing the number of GPUs for both dense (`JishNet`) and sparse (`SCNet`) CNNs was successful in decreasing the training time on large images in large datasets.
Dense
-----
[0.45]{}
[0.45]{}
Scaling to multiple GPUs shows an approximately linear speedup with respect to the number of GPUs, as shown in Figure \[fig:jishnet\_speedup\]. Figure \[fig:jishnet\_loss\] shows the loss versus training wall time with increasing GPUs. With more GPUs, a lower loss can be achieved in the allotted 4 hours, and the time to a loss (TTL) of $1.2$ decreases with the number of GPUs used (shown in inset).
In Figure \[fig:jishnet\_loss\], there is a marked difference in the time at which the first loss was reported with different number of GPUs. This time, the data loading time (DLT), is affected by how many images must be loaded into CPU before beginning training. Each training example was performed on 15,000 images, so each CPU had to load and compress $\nicefrac{15,000}{N_{GPU}}$ into memory. Optimization of this compression routine is left for future work. It is hypothesized that the TTL speedup is sublinear ($\propto N_{GPU}^{0.7)}$) because of this DLT effect.
Sparse
------
[0.45]{}
[0.45]{}
`SCNet` shows interesting behavior during training compared to `JishNet`. Figure \[fig:scnet\_speedup\] again shows an approximately linear speedup when trained on increasing numbers of GPUs, however the highest speed is $\sim 2.5\;\nicefrac{\mathrm{epochs}}{\mathrm{min}}$, which is $60\;\mathrm{\%}$ faster than `SCNet`. However, the TTL, shown in Figures \[fig:scnet\_loss\] is $4\times$ lower than that in Figure \[fig:jishnet\_loss\].
Figure \[fig:scnet\_loss\] shows comparable loss to `JishNet` after the full training set, however there are several interesting characteristics. Compared to Figure \[fig:jishnet\_loss\], there is very little evident DLT in Figure \[fig:scnet\_loss\]. Despite this, the TTL speedup is only $\propto N_{GPU}^{0.6}$.
It should be noted that the 7 image GPU memory limitation did not apply for `SCNet`, due to the small size of each datum, yet we kept the minibatch size at 7, nevertheless. This study did not explore varying the minibatch size (and in extension batch size), but the researchers hypothesize that increased speedup could be realized using more optimal batch sizes during training of `SCNet`.
Discussion and Conclusions
==========================
[0.45]{}
[0.45]{}
Training both dense and sparse convolutional networks on 10s to 100s of GPUs proved successful in this study. The wall time to a training loss decreased linearly with increased amounts of GPU resources, and allowed training on large images for physics related classificatoin studies. The final accuracy metrics for both of these networks are shown in Tables \[tab:cm\_dense\] and \[tab:cm\_sparse\]. The final validation accuracy for `JishNet` was $78.8\% \pm 3.8\%$, and for `SCNet` was $76.8\% \pm 3.3\%$. Particle labeling confusion is observed among species for which physics intuition suggests it is not unexpected. This study also elucidated several other important aspects related to training CNNs with large image sizes. These aspects included the sensitivity of training to the optimizer parameters and seeming inappropriateness of suggestions in the literature for training with SGD. We also uncovered the importance of minibatch size and tradeoff between GPU memory resources and optimal training dynamics.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors gratefully acknowledge the MicroBooNE collaboration for permission to work on simulated LArTPC data to focus on compute resources and performance scaling. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. Thus, the Oak Ridge Leadership Computing Facility and its staff are also gratefully acknowledged for use of the [*SummitDev*]{} computer. The authors also gratefully acknowledge Pacific Northwest National Laboratory’s Institutional Computing for use of its resources.
References {#references .unnumbered}
==========
[^1]: Permission from MicroBooNE collaboration was granted to focus on compute resources and performance studies
[^2]: It is stressed that this model was generated to provide model results while focusing on studying the computatonal resources. The CNN itself could be improved in numerous ways, but that is past the scope of this work.
[^3]: No compression was needed to store the sparse representation of these events.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The effect of spin of particles in the propagation of plasma waves is studied using a semi-classical kinetic theory for a magnetized plasma. We focus in the simple damping effects for the electrostatic wave modes besides Landau damping. Without taking into account more quantum effects than spin contribution to Vlasov’s equation, we show that spin produces a new damping or instability which is proportional to the zeroth order magnetization of the system. This correction depends on the electromagnetic part of the wave which is coupled with the spin vector.'
author:
- 'Pablo S. Moya'
- 'Felipe A. Asenjo'
title: The effect of spin magnetization in the damping of electron plasma oscillations
---
One of the most important physical results in the propagation of plasma waves which cannot be deduced by a fluid description is Landau damping [@landau]. This effect predicts that an electron plasma wave, in an collisionless plasma, suffers damping owing to the wave-particle interaction. Damping mechanism will depend on the velocity distribution of the particles which are often Maxwellian. The mathematical procedures to obtain Landau damping are very standard in kinetic theory. Thus, the formalism has been extended for other kind of electron interactions, for example thermal ion Landau damping effects [@goldston], and Landau damped electron waves by photons [@bing] or neutrinos [@losilva].
On the other hand, recently there have been a huge interest in the field of plasma physics for the dynamics of the spin of electrons, and how its quantum nature affects the different known modes of propagations [@shukla; @misra; @shukla2] or other properties [@gosh; @eliason]. These treatments are useful in, for example, astrophysical systems [@baring] or high-energy lasers [@kremp]. Often these effects are important in high density, low temperature or strong magnetic fields conditions, but it has been shown that for some systems at high temperaure the spin dynamics play a crucial role [@manfredi].
In a previous work [@faz], we present a first approach to calculate a correction to Landau damping due to spin. In this letter, we complete the analysis done in the previous work finding a full solution for the damping produced by the spin to electrostatic modes. We will show that this new damping is proportional to the spin magnetization of the plasma, and it depends on the electromagnetic part of the wave.
To obtain the correction to the Landau damping produced by the spin of the plasma constituents, we use the semi-classical kinetic theory constructed in Ref. [@brodin] which start from the Pauli Hamiltonian. Here, the dynamics of the spin of particles is included in a Vlasov equation for a generalized distribution function. Other quantum corrections, as Bohm potential, spin-spin interaction force and high order force terms in the spin evolution equation are neglected. Thus, Vlasov equation, for particles with velocity ${\bf v}$, spin vector ${\bf s}$ and with distribution function $f=f({\bf r}, {\bf v}, {\bf s}, t)$ is $$\begin{aligned}
\frac{\partial f}{\partial t}&+&{\bf v}\cdot\nabla f+\left[\frac{q}{m}\left({\bf E}+{\bf
v}\times{\bf B}\right)+\frac{2\mu_e}{m\hbar}\nabla\left({\bf s}\cdot{\bf
B}\right)\right]\cdot\nabla_v f\nonumber\\
&&\qquad\qquad\qquad\qquad+\frac{2\mu_e}{\hbar}\left({\bf s}\times{\bf B}\right)\cdot\nabla_s f=0\, ,
\label{vlasov}\end{aligned}$$ where $q=-e$ and $m$ are the electron charge and mass respectively, $\mu_e=-g
e\hbar /(4m)$ is the electron magnetic moment and $g\approx 2.002319$ is the electron spin factor. The Fermi-Dirac equilibrium distribution function $f_0({\bf v}, {\bf s})$ is [@brodin] $$f_0({\bf v}, {\bf s})=\frac{B_0 \mu_e{\tilde f}_0({\bf v})}{4\pi k_B
T\sinh (B_0 \mu_e/k_B T)}\exp\left(\frac{2\mu_e {\bf s}\cdot{\bf B}_0}{\hbar
k_B T}\right)\, ,
\label{distribu0}$$ where $T$ is the temperature, $k_B$ is the Boltzmann constant, ${\bf
B}_0$ is a background magnetic field ($B_0=|{\bf B}_0|$) and ${\tilde f}_0$ is the classical Maxwellian distribution function $${\tilde f}_0({\bf v})= \left(\frac{m}{2\pi k_B T}\right)^{3/2}\exp\left(-\frac{m v^2}{2 k_B T}\right)\, ,
\label{distrMaxwe}$$ with $v=|{\bf v}|$. The distribution function is normalized as $\int f_0 d{\bf v} d{\bf s}=1$, where the integration is made over the three degree of freedom in velocity space and the two degree of freedom in spin space.
Now, we use the kinetic formalism and a similar analysis of Ref. [@faz] to derive the dispersion relation for electron plasma oscillations in a magnetized plasma which interacts with the spin of the particles. The electric and magnetic fields will be perturbed in the form ${\bf E}={\bf E}_1$ and ${\bf B}={\bf B}_0+{\bf B}_1$ respectively. The terms with subscript $0$ are the zeroth order equilibrium quantities, and the terms with subscript $1$ are the first order perturbed quantitites. The distribution function is perturbed as $f({\bf r}, {\bf v},
{\bf s}, t)=f_0({\bf v}, {\bf s})+{\hat f}_1({\bf r}, {\bf v}, {\bf
s}, t)$ and we choose ${\bf B}_0=B_0\hat z$. The perturbed distribution ${\hat f}_1$ will have the form ${\hat f}_1({\bf
r}, {\bf v}, {\bf s}, t)=f_1({\bf v}, {\bf s})\exp(i{\bf k}\cdot{\bf
r}-i\omega t)$, with similar assumption for other perturbed quantities. Here, ${\bf k}$ and $\omega$ are the wavenumber and the frequency of the wave.
Linearizing Eq. , with the velocity ${\bf v}$ and spin ${\bf s}$ as independet variables and following Ref. [@faz], we can find the perturbed distribution function as $$\begin{aligned}
f_1&=&\frac{-i}{\omega-{\bf k}\cdot{\bf v}}\left(\frac{q}{m}{\bf E}_1\cdot\nabla_v f_0\right.\nonumber\\
&&\left.+\frac{2\mu_e}{m\hbar}\nabla\left({\bf s}\cdot{\bf B}_1\right)\cdot\nabla_v
f_0+\frac{2\mu_e}{\hbar}\left({\bf s}\times{\bf B}_1\right)\cdot\nabla_s f_0\right)\,
,\nonumber\\
&&
\label{f1}\end{aligned}$$ because ${\bf v}\times{\bf B}_1\cdot\nabla_v f_0=0$.
From here, we are going to focus in the study of spin correction to the Landau damping for electrostatic modes. Let us concentrate the charge density $\rho$ of the plasma which is given by $\rho=qn_0\int f_1 d{\bf
v}d{\bf s}$, where $n_0$ is the equilibrium density. Using the Maxwell equation ${\bf k}\times{\bf E}_1=\omega{\bf B}_1$, the perturbed distribution function , and defining the quantities ${\bf
E}_\perp={\bf k}\times{\bf E}_1/k$ and $E_\parallel={\bf k}\cdot{\bf E}_1/k$ with $k=|{\bf k}|$, the charge density becomes in $$\begin{aligned}
\rho&=&-iqn_0\int d{\bf s}\int_{-\infty}^\infty \frac{d{\bf v}}{\omega-{\bf k}\cdot{\bf v}}\left(\frac{q}{mk}E_\parallel\left({\bf k}\cdot\nabla_v f_0\right)\right.\nonumber\\
&&\left.+\frac{i2\mu_e k}{m\hbar\omega}\left({\bf s}\cdot{\bf E}_\perp\right){\bf k}\cdot\nabla_v
f_0+\frac{2\mu_ek}{\hbar\omega}\left({\bf s}\times{\bf E}_\perp\right)\cdot\nabla_s f_0\right)\,
.\nonumber\\
&&
\label{f12}\end{aligned}$$
This charge density must be used in the Poisson’s equation $i k E_\parallel=4\pi
\rho$ to obtain the dispersion relation for electrostatic modes. In this way, the dispersion relation is
$$%\begin{eqnarray}
1=\frac{\omega_p^2}{k^2}\int d{\bf s}\int_{-\infty}^\infty \frac{d{\bf v}}{{\bf k}\cdot{\bf
v}-\omega}\left(1+\frac{2i\mu_e k^2}{q\hbar\omega}\left(\frac{{\bf s}\cdot{\bf
E}_\perp}{E_\parallel}\right)\right){\bf k}\cdot\nabla_v f_0%\nonumber\\
+\frac{\omega_p^2}{k^2}\int d{\bf s}\int_{-\infty}^\infty \frac{d{\bf v}}{{\bf k}\cdot{\bf
v}-\omega}\left(\frac{2\mu_e k^2 m}{q\hbar\omega}\right)\left(\frac{{\bf s}\times{\bf
E}_\perp}{E_\parallel}\right)\cdot\nabla_s f_0\, ,
\label{disperrelation}
%\end{eqnarray}$$
where $\omega_p^2=4\pi e^2 n_0/m$ is the square of the plasma frequency. When the spin contribution is neglected ($\mu_e=0$), we reobtain the classical dispersion relation for electrostatic modes.
To solve the dispersion relation , we need to evaluate the two integral involving the spin contribution. The integration in the two degree of freedom in spin space is done in spherical coordinates such that $d{\bf s}\equiv d\Omega_s=d(\cos\theta_s)d\phi_s$ where the subindex $s$ is for spin coordinates. In the same sense, the spin vector will be ${\bf s}=-\hbar/2\hat s=-\hbar/2\left(\sin\theta_s\cos\phi_s\hat
x+\sin\theta_s\sin\phi_s\hat y+\cos\theta_s\hat z\right)$, and $\nabla_s f_0=2\mu_e
B_0\sin\theta_s f_0/(\hbar k_B T)\hat \theta$. The choice on the spin orientation is to minimize the magnetic moment energy, which is consistent with paramagnetism [@brodin2]. On the other hand, the above integrations in velocity and spin space can be simplified introducing the one-dimensional distribution $$F_0(u,{\bf s})\equiv \int f_0 \ \delta\left(u-\frac{{\bf k}\cdot{\bf v}}{k}\right)d{\bf v}\, .
\label{F0}$$
We can use to rewrite the dispersion relation as
$$%\begin{eqnarray}
1=\frac{\omega_p^2}{k^2}\int d{\Omega_s}\int_{-\infty}^\infty \frac{d
u}{u-\omega/k}\left(1-\frac{i\mu_e k^2}{q\omega}\left(\frac{{\hat s}\cdot{\bf
E}_\perp}{E_\parallel}\right)\right)\frac{\partial F_0}{\partial u}%\nonumber\\
-\frac{2\omega_p^2\mu_e^2 m B_0}{q\hbar\omega k_B T}\int d{\Omega_s}\int_{-\infty}^\infty \frac{d u \sin\theta_s F_0}{u-\omega/k}\frac{\left({\hat s}\times{\bf E}_\perp\right)\cdot\hat\theta}{E_\parallel}\, .
\label{disperrelation2}
%\end{eqnarray}$$
The integrals in Eq. must be evaluated as a contour integral considering the singularity at $u_\phi\equiv\omega/k$. This is the origin of classical Landau damping. We consider the case of large phase velocity $u_\phi$ and weak damping, where the pole lies near the real $u$ axis. In this case, $F_0$ and $\partial{F}_0/\partial u$ are both small near $u_\phi$. Neglecting the thermal correction to the real part of the frequency, the first two integrals are given by [@chen] $$\int d{\Omega_s} \int_{-\infty}^\infty \frac{du}{u-\omega/k}\frac{\partial F_0}{\partial u}\simeq \frac{k^2}{\omega^2}+i\pi\left.\frac{\partial{\tilde F}_0}{\partial u}\right|_{u=u_\phi}\, ,
\label{integral1}$$ $$\begin{gathered}
\int d{\Omega_s} \int_{-\infty}^\infty \frac{du}{u-\omega/k}\frac{{\hat s}\cdot{\bf
E}_\perp}{E_\parallel}\frac{\partial F_0}{\partial u}\simeq\\
-\chi\eta\left(\alpha\right)\left(\frac{k^2}{\omega^2}+i\pi\left.\frac{\partial{\tilde F}_0}{\partial u}\right|_{u=u_\phi}\right)\, ,
\label{integral2}\end{gathered}$$ where $\chi=\hat z\cdot{\bf E}_\perp/E_\parallel$, $\alpha=\mu_e B_0/k_BT$ and $\eta(x)=\coth(x)-1/x$ is the Langevin function. ${\tilde F}_0$ comes from and it is defined as $${\tilde F}_0(u)\equiv \int {\tilde f}_0 \ \delta\left(u-\frac{{\bf k}\cdot{\bf v}}{k}\right)d{\bf v}\, .
\label{F02}$$
The third integral in dispersion relation vanish due the only relevant spin contribution is anti parallel to the background magnetic field. Then, the dispersion relation becomes $$1=\left(\frac{\omega_p^2}{\omega^2}+\frac{i\pi\omega_p^2}{k^2}\left.\frac{\partial{\tilde
F}_0}{\partial u}\right|_{u_\phi}\right)\left(1+\frac{i\mu_e
k^2}{q\omega}\chi\eta(\alpha)\right)\,.
\label{disperrelation3}$$
We seek a frequency which has a real and an imaginary part given by $\omega=\omega_r+i\omega_i$ such that $\omega_i \ll \omega_r $. Using this in Eq., and solving for the real and imaginary parts, we can obtain the frequency for the electrostatic modes $$\omega=\omega_r\left(1+\frac{i\pi \omega_r^2}{2 k^2} \left.\frac{\partial{\tilde F}_0}{\partial u}\right|_{u_\phi}\right)+\frac{i k^2 M_0 \chi}{2 n_0 q}\, ,
\label{disp4}$$ where, neglecting terms of order $(\partial\tilde F / \partial u)|^2_{u_\phi} $, the real part of the frequency is given by
$$\label{disp5}
\omega_r=\omega_p\left(1 - \frac{M_0\chi\pi\omega_p}{2 q n_0} \left.\frac{\partial{\tilde F}_0}{\partial u}\right|_{u_\phi}\right)\,,$$
and $M_0=n_0\mu_e\eta(\alpha)$ is value of the spin magnetization of the system. This is because the distribution $f_0$ of Eq. , gives the zeroth order magnetization of the system ${\bf M}_0=(2\mu_e n_0/\hbar)\int{\bf s} f_0d{\bf v}{d\bf s}=M_0\hat z$ [@marklund; @brodin]. Thus, using Eqs. and the imaginary part of the frequency is $$\label{disp6}
\omega_i=\frac{\pi \omega_p^3}{2 k^2}\left.\frac{\partial{\tilde F}_0}{\partial u}\right|_{u_\phi}+\frac{k^2 M_0 \chi}{2 n_0 q}\, ,$$
From we note that there is a correction in the imaginary part of the frequency which appears due to the magnetization of the plasma due to spin of their constituents. This correction depends on the electromagnetic part of the waves, which is coupled with spin vector of each particle. As we discuss in our previous work [@faz], the interaction of spin with the perturbed magnetic field in a magnetized plasma is the responsible of this contribution to the damping of an electrostatic mode. Moreover, in the present analysis we show that the energy trasfer between waves and particles depends on the shape of the equilibrium function (classical Landau damping) and also on the spin magnetization of the plasma.
The spin damping correction of Eq. depends on the ratio of the magnetization and the charge of each particle, i.e, $M_0/n_0q = \hbar g\eta(\alpha)/4m$. It is expected that spin effects will be important at low temperatures, high densities and huge magnetic fields, and due to the dependences of $\eta$ on the $\alpha$ parameter we can see that, in fact, the spin correction to the Landau damping is higher for high values of density and background magnetic field, and for low temperatures. However, this spin damping is proportional to $\hbar$ and the main effect is the classical Landau damping.
Besides, we can see that the correction to the damping is proportional to $\chi$, which depends on the transversal part of the electromagnetic wave. If $\chi>0$ in the case of an electron plasma, $\eta(\alpha)<0$ and the correction is a damping. Also, for the same electron plasma, if $\chi<0$ the correction is an instability. When the wave has no electromagnetic transverse component $|{\bf
E_{\perp}}|=0$, then $\chi=0$ and there are no spin correction to the damping. The exact value of this coefficient should be obtained solving the dynamical Maxwell equations with the current density ${\bf j}=qn_0\int {\bf v} f_1
d{\bf v}d{\bf s}$. The complete implications of the value of $\chi$ is being studied.
On the other hand, from Eq. we note that there is a correction in the frequency of plasma oscillations. This corrections depends on the magnetization of the plasma and also on the shape of the distribution function. As spin corrections are proportional to $\hbar$ and the derivative of the distribution function is small, as in the case of the imaginary part, the correction is small and the frequency of electron waves is near $\omega_p$ as expected.
In conclusion, we have shown that the incorporation of spin to kinetic theory in a magnetized plasma produces corrections to the classical Landau damping for electrostatic waves that depends on the magnetization of the system, and is due to the coupling between the spin vector and the electromagnetic part of the wave. In other words, in addition to the Landau mechanism to transfer energy from waves to particles, the inclusion of spin allows the energy transfer through the quantum interaction between spin and magnetic fields. However, the corrections are of $\hbar$ order and, in the case of electron plasma and mawellian distribution functions, the electrostatic wave will show an evolution similiar to its classical dynamics when the spin is not included. In addition we shown that spin contribution also introduces a correction to the frequency of plasma oscillations which is also proportional to $\hbar$ and it is due to the magnetization of the plasma.
All of these results show the importance of the kinetic theory of plasmas. The same formalism can be used to introduce other quantum contributions to classical plasma physics and derive new corrections and effects for strongly coupled plasmas, as well as plasmas in presence of large magnetic fields when quantum effects are relevant.
Pablo S. Moya is grateful to CONICyT D-21070397 Doctoral Fellowship.
[17]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Theorem 1.2 in their paper arXiv:1904.00999v1 \[math.AP\] 30 Mar 2019 “Reconstruction of unknown cavity by single measurement” is not valid.
AMS: 35R30
KEY WORDS: No response test, enclosure method, probe method
author:
- 'Masaru IKEHATA[^1]'
title: 'Remarks on Lin-Nakamura-Wang’s paper'
---
R[[**R**]{}]{} \#1[**]{} \#1[(\#1)]{}
A counter example
=================
In [@LNW] they state[^2] if $\overline D\not\subset\overline G$, then $I(G)=\infty$. However, in this note we give a simple example that $\displaystyle\overline D\not\subset\overline G$, however $I(G)=0$.
Let $\Omega=\{x\in\Bbb R^2\vert\,\vert x\vert<R\}$ with $R>1$ and $D=\{x\in\Bbb R^2\,\vert\,\vert x\vert<1\}$. Let $u$ solve $$\left\{\displaystyle
\begin{array}{ll}
\displaystyle
\Delta u=0 & \text{in $\Omega\setminus\overline D$,}
\\
\\
\displaystyle
\frac{\partial u}{\partial\nu}=0 & \text{on $\partial D$,}\\
\\
\displaystyle
u(R\cos\theta,R\sin\theta)=\left(R+\frac{1}{R}\right)\cos\theta, & \text{$\theta\in\,[0,\,2\pi[$}.
\end{array}
\right.
\tag {1.0}$$ Note that the solution has the explict form $$\displaystyle
u(r\cos\theta,r\sin\theta)=\left(r+\frac{1}{r}\right)\cos\theta.$$ The key point of this note is the following trivial fact: $u$ has an extension to the domain $\tilde{\Omega}=\{x\in\Bbb R^2\vert\,0<\vert x\vert<R\}=\Omega\setminus\{0\}$ as a solution of the Laplace equation.
Let $0<\delta<1$ and choose $G=\{x\in\Bbb R^2\,\vert\,\vert x\vert<1-\delta\}$. We have $\overline G\subset D$ and thus $\overline D\not\subset\overline G$.
Given $\epsilon>0$ let $g\in H^{1/2}(\partial\Omega)$ be an arbitrary function such that the solution $z_g$ of $$\left\{
\begin{array}{ll}
\displaystyle
\Delta z_g=0 & \text{in $\Omega$,}\\
\\
\displaystyle
z_g=g & \text{on $\partial\Omega$}
\end{array}
\right.$$ satisfies $$\displaystyle
\Vert z_g\Vert_{H^1(G)}<\epsilon.
\tag {1.1}$$
By Lemma 2.1 in [@LNW] we have $$\displaystyle
\int_{\partial\Omega}\partial_{\nu}w\cdot g\,ds
=-\int_{\partial D} u\cdot\partial_{\nu}z_g\,ds,
\tag {1.2}$$ where $w=u-v$ and $v$ solves $$\left\{
\begin{array}{ll}
\displaystyle
\Delta v=0 & \text{in $\Omega$,}\\
\\
\displaystyle
v=u & \text{on $\partial\Omega$.}
\end{array}
\right.$$ Let $\tilde{u}$ denote the [*harmonic extension*]{} of $u$ into $\tilde{\Omega}$, that is $$\displaystyle
\tilde{u}(r\cos\theta,r\sin\theta)=\left(r+\frac{1}{r}\right)\cos\theta.$$ Let $C=\{x\in\Bbb R^2\,\vert\,\vert x\vert=1-\delta'\}$ with $\delta<\delta'<1$. We have $C\subset G$.
Write $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
-\int_{\partial D} u\cdot\partial_{\nu}z_g\,ds
\\
\\
\displaystyle
=\int_{\partial D}\left(\partial_{\nu}u\cdot z_g-u\cdot\partial_{\nu}z_g\right)\,ds
\\
\\
\displaystyle
=\int_{\partial D}\left(\partial_{\nu}\tilde{u}\cdot z_g-\tilde{u}\cdot\partial_{\nu}z_g\right)\,ds.
\end{array}$$ Since $\tilde{u}$ and $z_g$ are harmonic in $1-\delta'<\vert x\vert<1$, one has the expression $$\displaystyle
\int_{\partial D}\left(\partial_{\nu}\tilde{u}\cdot z_g-\tilde{u}\cdot\partial_{\nu}z_g\right)\,ds
=\int_{C}\left(\partial_{\nu}\tilde{u}\cdot z_g-\tilde{u}\cdot\partial_{\nu}z_g\right)\,ds.$$ Thus (1.2) becomes $$\displaystyle
\int_{\partial\Omega}\partial_{\nu}w\cdot g\,ds
=\int_{C}\left(\partial_{\nu}\tilde{u}\cdot z_g-\tilde{u}\cdot\partial_{\nu}z_g\right)\,ds.$$ It is easy to see that this right-hand side has the bound $O(\Vert z_g\Vert_{H^1(G)})$. Thus the condition (1.1) yields $$\displaystyle
\left\vert\int_{\partial\Omega}\partial_{\nu}w\cdot g\,ds
\right\vert
\le C\epsilon,$$ where $C$ is independent of $g$. Hence $I_{\epsilon}(G)\le C\epsilon$ and $I(G)=\lim_{\epsilon\downarrow 0}I_{\epsilon}(G)=0$.
Looking at the example in Section 1 a little more
=================================================
Let $u$ be the solution of (1.0) and $\tilde{u}$ its harmonic extension to $\tilde{\Omega}$. In this section $G$ denotes an arbitrary open subset of $\Omega$ such that $\overline G\subset\Omega$ and $\Omega\setminus\overline G$ is connected. In this section we prove
\(a) If $(0,0)\in G$, then $I(G)=0$.
\(b) If $(0,0)\not\in\overline G$, then, for all $\epsilon$ $I_{\epsilon}(G)=\infty$.
[*Proof.*]{} First we prove (a). In this case one can find a cirecle $S$ centered at $(0,0)$ such that $S\subset G$. At this time, the following equation is obtained as in the previous section: $$\displaystyle
\int_{\partial\Omega}\partial_{\nu}w\cdot g\,ds
=\int_{S}\left(\partial_{\nu}\tilde{u}\cdot z_g-\tilde{u}\cdot\partial_{\nu}z_g\right)\,ds.$$ Note that $z_g$ is the same as before. Thus this together with (1.2) yield $I_{\epsilon}(G)\le C\epsilon$ with a positive constant $C$ independent of $g$. And hence $I(G)=\lim_{\epsilon\downarrow 0}I_{\epsilon}(G)=0$.
Next we prove (b). For this we claim the identity: $$\displaystyle
\int_{\partial\Omega}\partial_{\nu}w\cdot g\,ds
=-2\pi\nabla z_g(0,0)\cdot\mbox{\boldmath $e$}_1,
\tag {2.1}$$ where $\mbox{\boldmath $e$}_1=(1,0)^T$.
First of all admit equation (2.1) and move on. Consider the case $(0,0)\not\in\overline G$. One can find an open disc $B$ centered at $(0,0)$ and radius $t_0$ such that $\overline B\subset\Omega\setminus\overline G$. Let $B_t=\{x\in\Bbb R^2\,\vert\,\vert x\vert<t\}$ with $0<t<t_0$. Since the function $$\displaystyle
E_t(x)=\log \vert x-t\mbox{\boldmath $e$}_1\vert$$ is harmonic in a neighbourhood of $\overline{G}\cup \overline{B_{t/2}}$, the Runge approximation property yields: there exists a sequence $\{g_j\}$ such that $$\displaystyle
\lim_{j\rightarrow\infty}\Vert z_{g_j}-E_t\Vert_{H^1(G\cup B_{t/2})}=0.
\tag {2.2}$$ Then an interior regulerity estimate yields $z_{g_j}$ together with its all derivatives converges to $E_t$ and the corresponding derivatives compact uniformly in $B_{t/2}$. Thus (2.1) yields $$\displaystyle
\lim_{j\rightarrow\infty}\int_{\partial\Omega}\partial_{\nu}w\cdot g_j\,ds
=\frac{2\pi}{t}.
\tag {2.3}$$ Note also that we have $$\displaystyle
\lim_{j\rightarrow\infty}\Vert z_{g_j}\Vert_{H^1(G)}=\Vert E_t\Vert_{H^1(G)}.$$
Given $\epsilon>0$ define $$\displaystyle
\tilde{g}_j=\frac{\epsilon}{2\Vert E_t\Vert_{H^1(G)}}g_j.$$ Since the map $g\mapsto z_g$ is linear, we have $$\displaystyle
\Vert z_{\tilde{g}_j}\Vert_{H^1(G)}=\frac{\epsilon}{2\Vert E_t\Vert_{H^1(G)}}\Vert z_{g_j}\Vert_{H^1(G)}<\epsilon$$ for all $j>>1$.
And (2.3) gives $$\displaystyle
\lim_{j\rightarrow\infty}\int_{\partial\Omega}\partial_{\nu}w\cdot \tilde{g}_j\,ds
=\frac{2\pi}{t}\cdot\frac{\epsilon}{2\Vert E_t\Vert_{H^1(G)}}
\tag {2.4}$$ Since $\overline B\cap\overline G=\emptyset$, Lebesgue’s dominated convergence theorem gives $\lim_{t\downarrow 0}\Vert E_t\Vert_{H^1(G)}=\Vert E_0\Vert_{H^2(G)}<\infty$. Thus the right-hand side on (2.4) blows up as $t\downarrow 0$. This yields $I_{\epsilon}(G)=\infty$.
$\Box$
[**Remarks.**]{}
\(i) The case $(0,0)\in\partial G$ seems delicate (at the present time).
\(ii) This type of sequence satisfying (2.2) has been used in the [*probe method*]{} [@IProbe] which aims at reconstructing unknown discontinuities such as cavities, inclusions and cracks. However, the probe method employs the Dirichlet-to-Neumann map, i.e., infinitely many pairs of the Cauchy data of the governing equation. Instead in the proof of (b) a single pair of Cauchy data is [*fixed*]{} and sequences $z_{g_j}$ produced by [*infinitely many*]{} $g_j$ are used as test functions.
\(iii) The choices of $\{g_j\}$ in two cases (a) and (b) are different. Since we do not know the position of $\{(0,0)\}$ in advance, we have the question: what is the [*good choice*]{} of $\{g_j\}$ [*common*]{} to two cases. This is also a problem about the no response test.
Proof of (2.1)
--------------
Same as before, we have, for all circles $S_{\eta}$ centered at $(0,0)$ with radius $\eta\in\,]0,\,1[$ $$\displaystyle
\int_{\partial\Omega}\partial_{\nu}w\cdot g\,ds
=\int_{S_{\eta}}\left(\partial_{\nu}\tilde{u}\cdot z_g-\tilde{u}\cdot\partial_{\nu}z_g\right)\,ds.$$ We compute the limt of this right-hand side as $\eta\downarrow 0$.
First we have $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_{S_{\eta}}\partial_{\nu}\tilde{u}\cdot z_g\,ds
\\
\\
\displaystyle
=\left(1-\frac{1}{\eta^2}\right)\eta
\int_0^{2\pi}
\cos\theta\cdot z_g(\eta\cos\theta,\eta\sin\theta)d\theta\\
\\
\displaystyle
=-\left(1-\frac{1}{\eta^2}\right)\eta\int_0^{2\pi}
\sin\theta\cdot \frac{d}{d\theta}\left\{z_g(\eta\cos\theta,\eta\sin\theta)\right\}d\theta\\
\\
\displaystyle
=-\left(1-\frac{1}{\eta^2}\right)\eta^2
\int_0^{2\pi}
\sin\theta\cdot \nabla z_g(\eta\cos\theta,\eta\sin\theta)
\cdot(-\sin\theta,\cos\theta)^T\,d\theta
\\
\\
\displaystyle
\rightarrow
\int_0^{2\pi}
\sin\theta\cdot \nabla z_g(0,0)
\cdot(-\sin\theta,\cos\theta)^T\,d\theta
\\
\\
\displaystyle
=-\pi\nabla z_g(0,0)\cdot\mbox{\boldmath $e$}_1.
\end{array}$$ Second we have $$\begin{array}{l}
\displaystyle
\,\,\,\,\,\,
\int_{S_{\eta}}\tilde{u}\cdot\partial_{\nu}z_g\,ds
\\
\\
\displaystyle
=(\eta^2+1)\int_0^{2\pi}\cos\theta\cdot\nabla z_g(\eta\cos\theta,\eta\sin\theta)\cdot
(\cos\theta,\sin\theta)^T\,d\theta
\\
\\
\displaystyle
\rightarrow
\pi\nabla z_g(0,0)\cdot\mbox{\boldmath $e$}_1.
\end{array}$$ This completes the proof.
One can not apply Fatou’ s lemma
================================
The key point of their argument on page 5 is the definiteness of the signature of $\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)$ for $x\in N_{y_0}\cap\partial D$ and $y\rightarrow y_0$ along the axis of the cylinder $N_{y_0}$. Here we give an example of $D$ that does not ensure this property.
Let $D$ be a bounded domain and in $x_3<0$. We assume that $y_0=(0,0,0)\in\partial D$ and $N_{y_0}\cap\partial D$ is [*flat*]{} and included in the plane $x_3=0$. Thus $\nu_x=\nu_{y_0}=\mbox{\boldmath $e$}_3$.
Let $E(x)=\frac{1}{\vert x\vert}$. We have $$\displaystyle
\partial_3E(x)=-\frac{x_3}{\vert x\vert^3},$$ and $$\displaystyle
\partial_3^2E(x)
=\frac{1}{\vert x\vert^5}
(3x_3^2-\vert x\vert^2).$$
Since $\mbox{\boldmath $a$}=\nu_{y_0}=\mbox{\boldmath $e$}_3$, we have, for all $x\in N_{y_0}\cap\partial D$ and $y=(0,0,y_3)$ with $0<y_3<<1$ $$\displaystyle
\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)=-\partial_3^2E(x-y)$$ and thus $$\displaystyle
\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)=
-\frac{1}{\vert x-y\vert^5}
(2y_3^2-x_1^2-x_2^2).$$ Therefore we have
\(i) if $x_1^2+x_2^2<2y_3^2$, then $\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)<0$;
\(ii) if $x_1^2+x_2^2>2y_3^2$, then $\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)>0$.
Thus as $y_3\downarrow 0$ the sign of the function $\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)$ of $x\in N_{y_0}\cap\partial D$ can not have a definite sign.
This implies, one can not apply Fatou’s lemma as done (3.4) in this simplest case.
Another reason of invalidness of (3.5) on page 5: A heuristic explanation
=========================================================================
Even general case one can not obtain (3.5). Its heuristic explanation is the following.
Since $\mbox{\boldmath $a$}=\nu_{y_0}$, if $x\in N_{y_0}\cap \partial D$ we expect $$\displaystyle
\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)
\sim
-\partial_{\nu_{x_0}}^2E(x-y).$$ However, $E$ satisfies the Laplace equation we have $$\displaystyle
\partial_{\nu_{x_0}}^2E(x-y)
=-(\partial_{x_1}^2+\partial_{x_2}^2)E(x-y),$$ where $x_1$ and $x_2$ are [*tangential directions*]{} at $y_0$. Thus we can expect $$\displaystyle
\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)
\sim
(\partial_{x_1}^2+\partial_{x_2}^2)E(x-y).$$ Then the integral $$\displaystyle
\int_{N_{y_0}\cap\partial D}u(x)\cdot\partial_{\nu_x}F_{\mbox{\boldmath $a$}}(x,y)ds(x)$$ may become $$\displaystyle
\sim
\int_{N_{y_0}\cap\partial D}u(x)\cdot
(\partial_{x_1}^2+\partial_{x_2}^2)E(x-y)ds(x).$$ Then applying integration by parts to this right-hand, one can reduce the singularity of integrand twice and gets an integral and additional terms which are bounded as $y\rightarrow y_0$.
Some comments on references
===========================
In [@I1] (1999!) using a single set of the Cauchy data, we have already given the reconstruction formula of the convex hull of unknown polygonal cavity $D$ and done its numerical testing in [@IO]. The method developed in this paper is called the [*enclosure method*]{} and based on the asymptotic behaviour of the integral with respect to a large parameter $\tau$ $$\displaystyle
\int_{\partial\Omega}\partial_{\nu}w\,g\,ds,$$ where $g=e^{\tau x\cdot(\omega+i\omega^{\perp})}$ with two unit vectors $\omega$ and $\omega^{\perp}$ perpendicular each other. Note that in this case $z_g(x)=e^{\tau x\cdot(\omega+i\omega^{\perp})}$.
Besides, in the case when $\Omega$ is an ellipse, even though the homogeneous background is [*unknown*]{}, the enclosure method works and yields a reconstruction formula of the convex hull of the union of the polygonal cavity and the focal points of $\Omega$ by using a single flux corresponding to a [*band-limited*]{} surface potential [@I3].
These informations are missed in [@LNW].
Extendability
=============
The point is the extendability of the potential $u$ from $\Omega\setminus\overline D$ across $\partial D$ into $D$, for example, if $\partial D$ is a real analytic surface, then by applying the Cauchy-Kovalevskaya theorem one has such an extension locally. In this case, we can prove that, by doing the procedure above locally around $y_0\in\partial D\setminus\overline G$ on page 5 in [@LNW], (3.5) in [@LNW] is not valid. The enclosure method in [@I1] catches a corner where one can not have an extention of the potential (due to Friedman-Isakov’s extension argument [@FI] under the condition $\text{diam}\,D<\text{dist}\,(D,\partial\Omega)$).
So at least we have to find an argument that employs explicitly the impossibility of applying the Cauchy-Kovalevskaya theorem on $\partial D$.
Conclusion
==========
The problem is not simple and still unsolved! I guess the complete version of the no response test with a single measurement tells us the limt of the extension of the soultion (continuation as a solution of the governing equation). Proposition 2.1 is an evidence of this belief.
$$\quad$$
[**Acknowledgments**]{}
The author was partially supported by Grant-in-Aid for Scientific Research (C)(No. 17K05331) and (B)(No. 18H01126) of Japan Society for the Promotion of Science.
$$\quad$$
[99]{}
Friedman, A. and Isakov, V., On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math. J., [**38**]{}(1989), 563-579.
Ikehata, M., Reconstruction of the shape of the inclusion by boundary measurements, Comm. PDE., [**23**]{}(1998), 1459-1474.
Ikehata, M., Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, [**15**]{}(1999), 1231-1241.
Ikehata, M., A remark on the enclosure method for a body with an unknown homogeneous background conductivity, CUBO A Mathematical Journal, [**10**]{}(2008), No.2, 31-45.
Ikehata, M. and Ohe, T., A numerical method for finding the convex hull of polygonal cavities using the enclosure method, Inverse Problems, [**18**]{}(2002), 111-124.
Lin, Y-H., Nakamura, G. and Wang, H., Reconstruction of unknown cavity by single measurement, arXiv:1904.00999v1 \[math.AP\] 30 Mar 2019.
e-mail address
[email protected]
[^1]: Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashihiroshima 739-8527, JAPAN
[^2]: Please refer to their paper [@LNW] for the symbols used in this note without explanation.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The COSINE-100 dark matter search experiment is an array of NaI(Tl) crystal detectors located in the Yangyang Underground Laboratory (Y2L). To understand measured backgrounds in the NaI(Tl) crystals we have performed Monte Carlo simulations using the Geant4 toolkit and developed background models for each crystal that consider contributions from both internal and external sources, including cosmogenic nuclides. The background models are based on comparisons of measurement data with Monte Carlo simulations that are guided by a campaign of material assays and are used to evaluate backgrounds and identify their sources. The average background level for the six crystals (70 kg total mass) that are studied is 3.5 counts/day/keV/kg in the (2–6) keV energy interval. The dominant contributors in this energy region are found to be $^{210}$Pb and $^3$H.'
author:
- 'P. Adhikari'
- 'G. Adhikari'
- 'E. Barbosa de Souza'
- 'N. Carlin'
- 'S. Choi'
- 'W.Q. Choi'
- 'M. Djamal'
- 'A.C. Ezeribe'
- 'C. Ha'
- 'I.S. Hahn'
- 'A.J.F. Hubbard'
- 'E.J. Jeon'
- 'J.H. Jo'
- 'H.W. Joo'
- 'W.G. Kang'
- 'M. Kauer'
- 'W.S. Kang'
- 'B.H. Kim'
- 'H. Kim'
- 'H.J. Kim'
- 'K.W. Kim'
- 'M.C. Kim'
- 'N.Y. Kim'
- 'S.K. Kim'
- 'Y.D. Kim'
- 'Y.H. Kim'
- 'V.A. Kudryavtsev'
- 'H.S. Lee'
- 'J. Lee'
- 'J.Y. Lee'
- 'M.H. Lee'
- 'D.S. Leonard'
- 'W.A. Lynch'
- 'R.H. Maruyama'
- 'F. Mouton'
- 'S.L. Olsen'
- 'H.K. Park'
- 'H.S. Park'
- 'J.S. Park'
- 'K.S. Park'
- 'W. Pettus'
- 'H. Prihtiadi'
- 'S. Ra'
- 'C. Rott'
- 'A. Scarff'
- 'N.J.C. Spooner'
- 'W.G. Thompson'
- 'L. Yang'
- 'S.H. Yong'
date: 'Received: date / Accepted: date'
title: 'Background model for the NaI(Tl) crystals in COSINE-100'
---
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Introduction {#intro}
============
COSINE-100 is a dark matter search experiment consisting of a 106 kg array of eight ultra-pure NaI(Tl) crystals [@kims-nai2014; @kims-nai2015]. Its primary goal is to test DAMA/LIBRA’s assertion of an observation of annual modulation signal [@bernabei08; @bernabei10; @bernabei13; @bernabei18]. The experiment has been operating at the Yangyang Underground Laboratory (Y2L) since September 2016 [@cosinedet17]. COSINE-100 is one of several NaI(Tl)-based dark matter searches in operation (DM-Ice17 [@dmice17], ANAIS [@amare14; @amare16]) or under development (DM-Ice [@dmice14], Kam-LAND-PICO [@kamland-pico], SABRE [@sabre17], COSINUS [@cosinus]). Previously, the KIMS-CsI experiment put a limit on interaction rates in CsI crystals [@hslee07; @sckim12] that precluded the interpretation of the DAMA modulation signal as being due to WIMPs scattering from I or Tl nuclei, considering the different quenching factors of iodine and thallium for NaI(Tl) and CsI(Tl).
The COSINE-100 crystal array is immersed in a tank of liquid scintillator (LS) that tags backgrounds that originate from outside the LS as well as decays of $^{40}$K nuclides inside the crystals. To determine the sources of the backgrounds, we have performed Monte Carlo simulations using the Geant4 toolkit (V.4.9.6.p02) [@geant4] and built a background model for the eight detectors by iteratively fitting their contributions to the measured energy spectra; two crystals are excluded in this paper due to their low light yields, which result in a background spectrum without characteristic peaks of isotopes by the worse energy resolution, and relatively higher background contamination in the low energy region.
The paper is structured as follows: the COSINE-100 experimental setup is described in Sect. \[sec:2\]. Section \[sec:3\] describes the background modeling, with the simulation method described in Sect. \[sec:3.2\], sources of the background internal and external to the crystal and of cosmogenic origin in Sects. \[sec:3.2\] – \[sec:3.4\]. Section \[sec:4\] describes the comparison and fit to the data, and Sect. \[sec:5\] provides discussions of the background developed by the fits. Finally Sect. \[conc\] provides conclusions.
COSINE-100 setup and simulation geometry {#sec:2}
========================================
--------------------------------------------------------- --------------------------------------------------------
{width="40.00000%"} {width="47.00000%"}
\(a) Front view \(b) Side view
--------------------------------------------------------- --------------------------------------------------------
The experimental setup is described in detail in Ref. [@cosinedet17]. The simplified geometry used for the simulations is shown in Fig. \[detector-setup\]. Eight NaI(Tl) crystals, arranged in two layers, are located in the middle of a four-layer shielding structure. From outside inward, the four shielding layers are plastic scintillator panels, a lead-brick castle, a copper box, and a scintillating liquid. The eight NaI(Tl) crystal assemblies and their support table are immersed in the scintillating liquid that serves both as an active veto and a passive shield.
The eight NaI(Tl) crystals were grown out of batches of powder provided by Alpha Spectra [@as-inc] with successive improvements. The first attempts, which produced an order of magnitude reduction in $^{40}$K, were AS-B and AS-C. This was followed by WIMPScint-II (AS-WSII) which reduced the $^{210}$Pb contamination, and WIMPScint-III (AS-WSIII) which resulted in another factor of two reduction of $^{40}$K. The results are summarized in Table \[measured-activity\]. The final crystals are cylindrically shaped and hermetically encased in OFE copper tubes with wall thickness of 1.5 mm and quartz windows (12.0 mm thick) at each end. Each crystal’s lateral surfaces were wrapped in roughly 10 layers of 250 $\mu$m-thick PTFE reflective sheets. The quartz windows are optically coupled to each end of the crystal via 1.5 mm thick optical pads. These, in turn, are optically coupled to 3-inch Hamamatsu R12669SEL photomultiplier tubes (PMTs) with a thin layer of high viscosity optical gel. The PMTs are protected from the liquid scintillator by a housing made of copper and PTFE.
The following components of the detector have been included in the simulation: PTFE reflective sheets, copper tubes, quartz windows, optical gel, PMT housing, and PMTs.
Background modeling {#sec:3}
===================
Simulation method {#sec:3.1}
-----------------
The Physics list classes of G4EmLivermorePhysics for low energy electromagnetic process and G4Radioactive-Decay for radioactive decay process were used [@geant4:lowEM; @geant4:lowEMpackage; @geant4:2016]. The $^{238}$U and $^{232}$Th decay chains were treated as broken at the long-lived parts of the chain. The $^{238}$U chain was broken into five distinct groups and the $^{232}$Th chain was broken into three groups. The details are reported in Ref. [@kims-nai-bkg17].
Each simulated event record includes all energy deposited in the crystals within an event window of 10 $\mu$s from the time a decay is generated, to account for the conditions in the data acquisition system (DAQ) of the experimental setup [@cosinedet17]. Sometimes decays with relatively short half-lives, such as $^{212}$Po decay (with a half-life of 300 ns) and the subsequent daughter decays will appear in the 10 $\mu$s time window, resulting in pileup events. They are treated as a single event in the simulation.
The simulated spectrum was convolved with an energy dependent energy resolution function developed during a calibration run. Calibration points were measured using $\gamma$–ray sources: 59.5 keV($^{241}$Am), 1173.2 keV and 1332.5 keV ($^{60}$Co). Internal background peaks at 3.2 and 1460.8 keV from $^{40}$K, 67.3 keV from $^{125}$I, and 609.3 keV from $^{214}$Bi were used to calibrate the measured spectra; peaks at 3.2 keV, 59.5 keV, and 67.3 keV were used for the low energy calibration below 70 keV.
Internal backgrounds in the NaI(Tl) crystals {#sec:3.2}
--------------------------------------------
----------- ------ -------------------------- ---------- ----------------- -------------- ----------- -------------
Crystal Mass Size (inches) Powder $\alpha$ Rate $^{40}$K $^{238}$U $^{228}$Th
(kg) (diameter$\times$length) (mBq/kg) (ppb) (ppt) (ppt)
Crystal-1 8.3 $5.0\times7.0$ AS-B $3.20\pm0.08$ $34.7\pm4.7$ <0.02 $1.3\pm0.4$
Crystal-2 9.2 $4.2\times11.0$ AS-C $2.06\pm0.06$ $60.6\pm4.7$ <0.12 <0.6
Crystal-3 9.2 $4.2\times11.0$ AS-WSII $0.76\pm0.02$ $34.3\pm3.1$ <0.04 $0.4\pm0.2$
Crystal-4 18.0 $5.0\times15.3$ AS-WSII $0.74\pm0.02 $ $33.3\pm3.5$ <0.3
Crystal-5 18.3 $5.0\times15.5$ AS-C $2.06\pm0.05 $ $82.3\pm5.5$ $2.4\pm0.3$
Crystal-6 12.5 $4.8\times11.8$ AS-WSIII $1.52\pm0.04 $ $16.8\pm2.5$ <0.02 $0.6\pm0.2$
Crystal-7 12.5 $4.8\times11.8$ AS-WSIII $1.54\pm0.04 $ $18.7\pm2.8$ <0.6
Crystal-8 18.3 $5.0\times15.5$ AS-C $2.05\pm0.05 $ $54.3\pm3.8$ <1.4
----------- ------ -------------------------- ---------- ----------------- -------------- ----------- -------------
After the insertion of the crystals into the shield and prior to filling the liquid scintillator container, their background levels were measured to verify that they were free of any additional contamination. Overall, the eight crystals have acceptable $^{238}$U and $^{232}$Th contaminations as shown in Table \[measured-activity\] [@cosinedet17]. Secular equilibrium in the chains is assumed for the interpretation of $^{238}$U and $^{232}$Th related radioactivity measurements, with the exception of $^{210}$Pb.
In order to estimate the background contributions from $^{238}$U, $^{232}$Th, $^{40}$K, and $^{210}$Pb, we simulated background spectra from the internal radioactive contaminants and normalized them by their measured activities in Table \[measured-activity\]. In the normalization we assumed a chain equilibrium and, thus, all related activities within the chains are equal to the $^{238}$U, $^{232}$Th, and $^{40}$K activities multiplied by the branching ratios for decay of the daughter isotopes. We also added the background simulation of internal $^{210}$Pb by considering the measured $\alpha$ rate. The resultant background contributions, except for those from $^{40}$K and $^{210}$Pb, were negligible in all eight crystals.
The $^{40}$K contribution is reduced by the LS veto detector. To measure the reduction efficiency of the $^{40}$K generated 3.2 keV emission background provided by tagging the accompanying 1460.8 keV $\gamma$-ray in one of the other NaI(Tl) crystals or the LS, and to compare this to the efficiency provided by the other crystals alone, we generated $^{40}$K decays at random locations inside a NaI(Tl) crystal for the cases with and without the LS veto. From these simulations, we determined that the Crystal-6 tagging efficiency by other crystals without LS is 31.7$\pm$0.1$\,\%$ and by the LS only is 64.9$\pm$0.2$\,\%$. The total combined efficiency is 81.7$\pm$0.3$\%$. The efficiency is measured in the crystal energy range between 2 and 6keV by requiring the LS energy deposit be larger than 20keV. Efficiencies vary depending on the crystal location in the detector. For example, Crystal-1 (at the corner of the 4$\times$2 array) shows higher coverage by the LS (75%) than neighboring crystals (17%), but the combined efficiency is similar to that of Crystal-6 (82%). The tagging efficiency of the 1460.8keV $\gamma$-ray in the LS-only case is lower because the range of the $\gamma$-ray in the NaI(Tl) crystal is shorter than in the LS. Therefore, more $\gamma$-rays are stopped in the other crystals than in the LS. These estimated efficiencies are in agreement with measurements [@cosinedet17]. Accordingly, the $^{40}$K background level is reduced by as much as 80% by requiring single-hit crystal events with no signal in the LS.
The $^{210}$Pb contribution is estimated by modeling the background from bulk $^{210}$Pb and surface $^{210}$Pb as discussed in Sect. \[sec:4\].
External background sources {#sec:3.3}
---------------------------
--------------------- --------------- ---------------- --------------
External source U($^{214}$Bi) Th($^{228}$Ac) ($^{40}$K)
PMT [@kims-nai2014] 25 $\pm$ 5 12 $\pm$ 5 58 $\pm$ 5
(R12669SEL$^{b}$)
Quartz window $<$1.8 $<$7.5 $<$20
PTFE reflector $<$0.5 $<$1.0 $<$6.4
Cable ties $<$4.2 $<$3.5 149 $\pm$ 32
LS $<$2.7 $<$3.3 7 $\pm$ 4
--------------------- --------------- ---------------- --------------
: Radioactivity levels in detector components inside the shielding. (a) The radioactivities were measured with a HPGe detector at Y2L; upper limits are quoted with 90% C.L. The PMTs are measured in units of mBq/PMT and the other external sources are measured in units of mBq/kg (b) SEL means “selected for high quantum efficiency”. []{data-label="photomultipliertubes"}
The external $\gamma$ background from the radioactive isotopes in the surrounding rocks is shielded by the 20 cm-thick lead castle and the 3 cm-thick copper box. By using the full shielding structure with $N_{2}$ gas flowing into the inside of the copper shield to avoid backgrounds from $^{222}$Rn in the air at Y2L (measured to be $1.20\pm 0.49$ pCi/L [@kims-radon2011]), we reduced the environmental background by a factor of 10,000 based on the measurements of a high-purity Ge (HPGe) detector, thus ensuring that those contributions are negligibly small.
Despite all the efforts to block backgrounds due to external sources, some backgrounds from radioactive contaminations in detector components inside the shielding are still expected, including from the PMTs, grease, copper case, bolts, cables, acrylic supports, liquid scintillator, copper box, and steel that supports the lead block housing. We simulated background spectra from those external sources to test their effects and compared the shapes of contributions to the crystals’ energy spectra. We found that all the spectra from these external sources are similar in shape and, thus, could be represented by a spectrum that is obtained by simulating $^{238}$U, $^{232}$Th, and $^{40}$K, distributed randomly in the volume outside the eight crystals. Because the PMTs are the main contributer to the external background we used two kinds of spectra for the external background modeling; one is the spectrum from the PMTs and another is the spectrum from the other external sources that is treated as a parameter floating in the fit. The radioactivity levels of the PMTs and PMT surrounding parts were measured underground with a HPGe detector and the results are listed in Table \[photomultipliertubes\]. We used the measured activities from the PMTs to constrain the data fitting and treated background contributions from the PMTs in nine groups as broken at the long-lived parts of the chain.
Treatment of cosmogenic radionuclides {#sec:3.4}
-------------------------------------
[c|c|c]{}\
Cosmogenic & Half-life & Production rate\
isotopes & (days) & at sea level [@walter-thesis]\
& & (counts/kg/day)\
$^{125}$I & 59.4 & 221\
$^{121}$Te & 19.17 & 93\
$^{121m}$Te & 164.2 & 93\
$^{123m}$Te & 119.2 & 52\
$^{125m}$Te & 57.4 & 74\
$^{127m}$Te & 106.1 & 93\
$^{113}$Sn & 115.1 & 9.0\
$^{109}$Cd & 461.4 & 4.8\
$^{3}$H & 4500 & 26\
$^{22}$Na & 951 & 66\
[c|c|c]{}\
Crystal & Exposure & Radioactivity\
& time (see text) & cooling time at Y2L\
& (years) & (years)\
Crystal-1 & 2 & 3\
Crystal-2 & 0.75 & 2.75\
Crystal-3 & & 1.2\
Crystal-4 & 1.7 & 0.5\
Crystal-6 & 0.3 & 0.6\
Crystal-7 & 0.3 & 0.6\
Although the eight NaI(Tl) crystals had underground radioactivity cooling times that ranged from several months to three years, there are still background contributions due to the long-lived cosmogenic isotopes that were activated by cosmic rays while they were on the surface.
To consider these backgrounds, we first checked the list of cosmogenic radioactive isotopes that are produced in NaI(Tl), as reported in Ref. [@walter-thesis; @cosmogenic-amre15; @cosmogenic-villar18; @cosmogenic-amre18]. In Table \[cosmogenic\] (a), we list the contributing cosmogenic isotopes with their half lives; short-lived isotopes, for which half lives are less than a year, are $^{125}$I, $^{121}$Te, $^{121m}$Te, $^{123m}$Te, $^{125m}$Te, $^{127m}$Te, and $^{113}$Sn and long-lived isotopes are $^{109}$Cd, $^{3}$H, and $^{22}$Na. The radioactivity cooling time at Y2L for each crystal at the time data-taking for COSINE-100 started, is listed in Table \[cosmogenic\] (b). The short-lived isotopes are not expected to contribute to either Crystal-1 or Crystal-2 because their cooling times are long enough to reduce these activities to a negligible level.
However, we expect some backgrounds from the sho-rt-lived isotopes in other crystals because their production rates at sea level, as listed in Table \[cosmogenic\] (a), are high and their cooling times are less than or equal to a year. In addition, there are long-lived $^{109}$Cd, $^{3}$H, and $^{22}$Na nuclides that are potentially hazardous background sources; ${\it e.g.}$, the beta-decay spectrum of tritium has an endpoint energy of 18 keV. We thus need to understand their background contributions in the low energy region, especially in the (2–6) keV WIMP signal region of interest (ROI). Because it is impossible to compute the initial activities of those isotopes from the production rates in each crystal at Y2L without knowing the cosmic ray exposure conditions: time, location, altitude, etc. [@walter-thesis], we investigated the correlation of characteristic peaks produced by $\gamma$/X-rays from the decay of cosmogenic isotopes.
- $^{109}$Cd decays by electron capture to the isomeric state of $^{109}$Ag depositing in the crystal the binding energy of the Ag K-shell electrons (25.5 keV), that will be accompanied by the 88 keV $\gamma$ ray from the isomer transition of $^{109}$Ag having a mean time of 57.4 seconds. By using the timing information of two adjacent events that have each 25.5 keV and 88 keV, we measured the background contribution of $^{109}$Cd in Crystal-4 and found it to be 0.10$\pm$0.01 mBq/kg.\
- $^{22}$Na decays via positron emission (90%) and electron capture (10%), followed by 1274.6 keV $\gamma$-ray emission with a mean lifetime of 3.8 yr. The electron capture decay produces 0.9 keV emissions. Therefore, $\sim$10% of the $^{22}$Na decay will produce 0.9 keV X-rays and 1274.6 keV $\gamma$ rays simultaneously. Meanwhile, the positron will be converted to two 511 keV annihilation $\gamma$ rays.
However, it is generally difficult to measure long-lived cosmogenics’ activities, such as those for $^{3}$H, directly from the data due to their long half-lives. Therefore, we simulated background spectra from cosmogenic isotopes listed in Table \[cosmogenic\] (a) and used their shapes in the data fitting, while floating their unknown fractions. The details of their treatment in the background model for each NaI(Tl) crystal are discussed in section \[sec:4\].
Conclusion {#conc}
==========
We have studied, using the Geant4 toolkit, the background of the NaI(Tl) crystal detectors that are being used in the COSINE-100 dark matter search experiment. The crystals have different exposure histories and underground radioactivity cooling times.
In the background modeling the overall energy spectrum summed over all simulations is well matched to the data not only for single-hit events but also for multi-ple-hit events. Crystal-1 and Crystal-2 that had cooling times as long as three years at the Y2L are dominated by $^{210}$Pb and $^{3}$H for energies below 20 keV. The background contribution of $^{3}$H in Crystal-2 is smaller than that in Crystal-1 due to its shorter surface exposure time. Crystal-6 and 7 show clear contributions from $^{125}$I due to their short cooling times underground, as expected. Crystal-3 had an additional treatment for a repair that increased the background near 10 keV that is well modeled by surface $^{210}$Pb on the PTFE wrapping foil. Crystal-4 was exposed to surface cosmic rays for two years and only had a six month-long underground cooling time. As a result, this crystal has significant background contributions from both short-lived and long-lived cosmogenic isotopes.
Background contributions from external sources and internal $^{40}$K are reduced to the level of 0.03 dru and about 0.1 dru in the energy range of 2–6 keV, respectively, by the LS veto detector that surrounds the crystals.
The average background rate in the (2-6) keV energy range for the six crystals (with a total mass of 70 kg) studied here is 3.5 counts/day/keV/kg. The dominant contributions in this energy range are from $^{210}$Pb and $^3$H.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the Korea Hydro and Nuclear Power (KHNP) Company for providing underground laboratory space at Yangyang. This work is supported by: the Institute for Basic Science (IBS) under project code IBS-R016-A1, Republic of Korea; UIUC campus research board, the Alfred P. Sloan Foundation Fellowship, NSF Grants no. PHY-1151795, PHY-1457995, DGE-1122492 and DGE-1256259, WIPAC, the Wisconsin Alumni Research Foundation, Yale University and DOE/NNSA Grant no. DE-FC52-08NA28752, United States; STFC Grant ST/N000277/1 and ST/K001337/1, United Kingdom; and CNPq and Grant no. 2017/02952-0 FAPESP, Brazil.
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Introduction
============
Taking motivation from the work of Wang, Banica, Bichon and others (see [@free; @wang; @ban1; @ban2; @bichon; @univ1] and references therein), we have embarked on a programme to formulate and study various types of ‘quantum isometry groups’ in the setting of (possibly noncommutative) Riemannian geometry. It began with our formulation of quantum isometry group based on a ‘Laplacian’ in [@goswami], and then followed up by a formulation of ‘quantum group of orientation preserving isometries’ in [@qorient] (see also [@qdisc; @q_sphere; @jyotish; @jyotish_1] for many explicit computations). The basic idea in all these papers is the following: first get an operator theoretic characterisation of an isometric (or orientation preserving and isometric) group action on a Riemannian manifold, then give an analogous definition of (compact) quantum group action, and finally try to see whether the category of the compact quantum groups having such action admits a universal object. However, the transition from group to quantum group action creates a crucial problem, which stems from the fact that unlike the classical group actions implemented by some unitary representation on a Hilbert space which always preserve the usual trace, a quantum group action may not do so. This problem shows up even in the context of finite dimensional algebras like $M_n$, and we do not in general get a universal object in the category of quantum groups mentioned before. To get rid of this problem one has to fix a suitable functional (to be interpreted as a choice of ‘volume form’) on the underlying algebra, and then look at the subcategory of the (isometric and orientation preserving) isometric quantum group actions which also preserve this given functional. It has been shown in [@qorient] that this subcategory always has a universal object, which was called there the quantum group of orientation and volume preserving isometries.
The aim of this paper is to provide an alternative to the choice of a volume form. We prove here that if the manifold (possibly noncommutative, i.e.given by a spectral triple) has a real structure, then one can get a universal object in the natural subcategory of compact quantum groups whose action, besides being ‘orientation-preserving’ in the sense of [@qorient], preserves also the real structure in a suitable sense. The idea of the proof is very similar to that of [@qorient], and we mainly sketch in the present article the arguments which are different from those of [@qorient], but avoid repetition of those which are more or less the same. The main idea is to construct a canonical compact quantum group, which is a free product of countably infinitely many copies of the universal quantum groups of the form $A_u(Q)$ (notation as in [@wang]), such that any quantum group in the category under consideration can be identified with a quantum subgroup of this free product. In [@qorient], the volume preserving property was used precisely at this step: namely to show that given any eigenvalue $\lambda$ of $D$ there is a canonical quantum group $A_u(Q_\lambda)$, say, such that the restriction of the action of any quantum group in the above-mentioned category must factor through the canonical representation of $A_u(Q_\lambda)$ on the eigenspace corresponding to $\lambda$. The present work relies on the crucial observation that a canonical choice (but different from those in [@qorient]) of $A_u(Q_\lambda)$ can also be made using the assumption of preservation of the real structure instead of the volume form for an orientation preserving isometric quantum group action.
Preliminaries
=============
We shall mostly use the notation and terminologies of [@qorient], some of which we briefly recall here again. We begin by recalling the definition of compact quantum groups and their actions from [@woro; @woro1; @vandaelenotes]. A compact quantum group (to be abbreviated as CQG from now on) is given by a pair $({\cal S}, \Delta)$, where ${\cal S}$ is a unital $C^*$-algebra equipped with a unital $\ast$-homomorphism $\Delta : {\cal S} \rightarrow {\cal S} \otimes {\cal S}$ (where $\otimes$ denotes the injective tensor product of $C^*$-algebras) satisfying
$(ai)$ $(\Delta \otimes {\rm id}) \circ \Delta=({\rm id} \otimes \Delta) \circ \Delta$ (co-associativity), and
$(aii)$ each of the linear spans of $ \Delta({\cal S})({\cal S} \otimes 1)$ and $\Delta({\cal S})(1 \otimes {\cal S})$ is norm-dense in ${\cal S} \otimes {\cal S}$.
We say that the compact quantum group $({\cal S},\Delta)$ (co)-acts on a unital $C^*$-algebra ${\cal B}$, if there is a unital $\ast$-homomorphism (called an action) $\alpha : {\cal B} \rightarrow {\cal B} \otimes {\cal S}$ satisfying the following
$(bi)$ $(\alpha \otimes {\rm id}) \circ \alpha=({\rm id} \otimes \Delta) \circ \alpha$, and
$(bii)$ the linear span of $\alpha({\cal B})(1 \otimes {\cal S})$ is norm-dense in ${\cal B} \otimes {\cal S}$.
A unitary (co)representation of a compact quantum group $ ({\cal S},\! \Delta ) $ on a Hilbert space $ {\cal H} $ is a map $ U $ from $ {\cal H} $ to the Hilbert ${\cal S}$-module $ {\cal H} \otimes {\cal S} $ such that the element $ \widetilde{U} \in {\cal M} ( {\cal K} ( {\cal H} ) \otimes {\cal S} ) $ given by $\widetilde{U}( \xi \otimes b)=U(\xi)(1 \otimes b)$ ($\xi \in {\cal H}, b \in {\cal S})$) is a unitary satisfying $({\rm id} \otimes \Delta ) \widetilde{U} = {\widetilde{U}}_{12} {\widetilde{U}}_{13},$ where for an operator $X \in {\cal B}({\cal H}_1 \otimes {\cal H}_2)$ we have denoted by $X_{12}$ and $X_{13}$ the operators $X \otimes I_{{\cal H}_2} \in {\cal B}({\cal H}_1 \otimes {\cal H}_2 \otimes {\cal H}_2)$, and $\Sigma_{23} X_{12} \Sigma_{23}$ respectively ($\Sigma_{23}$ being the unitary on ${\cal H}_1 \otimes {\cal H}_2 \otimes {\cal H}_2$ which flips the two copies of ${\cal H}_2$).
Given a unitary representation $U$ we shall denote by $\alpha_U$ the $\ast$-homomorphism $\alpha_U(X)=\widetilde{U}(X \otimes 1){\widetilde{U}}^*$ for $X \in {\cal B}({\cal H})$. For a not necessarily bounded, densely defined (in the weak operator topology) linear functional $\tau$ on ${\cal B}({\cal H})$, we say that $\alpha_U$ preserves $\tau$ if $\alpha_U$ maps a suitable (weakly) dense $\ast$-subalgebra (say ${\cal D}$) in the domain of $\tau$ into ${\cal D} \otimes_{\rm alg} {\cal S}$ and $( \tau \otimes {\rm id}) (\alpha_U(a))=\tau(a)1_{\cal S}$ for all $a \in {\cal D}$. When $\tau$ is bounded and normal, this is equivalent to $(\tau \otimes {\rm id}) (\alpha_U(a))=\tau(a) 1_{\cal S}$ for all $a \in {\cal B}({\cal H})$.
We say that a (possibly unbounded) operator $T$ on ${\cal H}$ commutes with $U$ if $T \otimes I$ (with the natural domain) commutes with $\widetilde{U}$. Sometimes such an operator will be called $U$-equivariant.
Let us now recall the concept of universal quantum groups as in [@univ1; @free] and references therein. We shall use most of the terminologies of [@free], e.g. Woronowicz $C^*$-subalgebra, Woronowicz $C^*$-ideal etc, however with the exception that we shall call the Woronowicz $C^*$-algebras just compact quantum groups, and not use the term compact quantum groups for the dual objects as done in [@free]. For an $n \times n$ positive invertible matrix $Q=((Q_{ij}))$, let $A_u(Q)$ be the compact quantum group defined and studied in [@wang; @univ1], which is the universal $C^{*}$-algebra generated by $ \{ u^{Q}_{kj}, \ k,j=1,\dots ,n \}$ such that $u:=((u_{kj} \equiv u^{Q}_{kj} ))$ satisfies $$\label{wangalg} u u^*=I_n =u^{*}u, \qquad u^{\prime} Q \overline{u} Q^{-1}=I_n=Q{\overline{u}} Q^{-1} u^{\prime}.$$ Here $u^{\prime} =(( u_{ji} ))$ and $\overline{u}=(( u_{ij}^{*} ))$, and also note that we have made the identification of an $n \times n$ matrix $B$ with its trivial ampliation $B \otimes 1$ in $M_n({\mathbb C}) \otimes {\cal A}$ for any $C^*$-algebra ${\cal A}$. The coproduct, say $\tilde{\Delta}$, is given by, $\tilde{\Delta}(u_{ij})=\sum_k u_{ik} \otimes u_{kj}.$ It may be noted that $A_u(Q)$ is the universal object in the category of compact quantum groups generated by the coefficients of a unitary representation $v$ on ${\mathbb C}^n$ such that the adjoint action ${\rm Ad}_v$ on $M_n({\mathbb C})$ preserves the functional $M_n \ni x \mapsto {\rm Tr({Q}^{\prime} x)}$ (see [@wangergodic]), where we refer the reader to [@univ1] for a detailed discussion on the structure and classification of such quantum groups.
Given a $C^*$-algebra ${\cal S}$ we shall denote by $\tilde{J}_{\cal S}$ the antilinear map $a \mapsto a^*$. For any faithful state (which exists whenever ${\cal S}$ is separable) this map can be viewed as a closable unbounded antilinear map on the GNS space of the state, and the corresponding closed extension will be denoted by the same notation.
We now give a definition of the real structure along the lines of [@dab_real] and [@landi_real], which is a suitable modification of Connes’ original definition (see [@connes]) to accommodate the examples coming from quantum groups and quantum homogeneous spaces.
An odd spectral triple with a real structure is given by a spectral triple $({\cal A}^\infty{,} {\cal H}{,} D)$ along with a (possibly unbounded, invertible) closed anti-linear operator $\tilde{J}$ on ${\cal H}$ such that ${\cal D}:={\rm Dom}(D) \subseteq {\rm Dom}(\tilde{J}) $, $\tilde{J} {\cal D} \subseteq {\cal D}$, $\tilde{J}$ commutes with $D$ on ${\cal D}$, and the antilinear isometry $J$ obtained from the polar decomposition of $\tilde{J}$ satisfies the usual conditions for a real structure in the sense of [@landi_real], for a suitable sign-convention given by $(\epsilon, \epsilon^\prime)
\in \{ \pm 1 \} \times \{ \pm 1 \}$ as described in [@varilly page 30], i.e. $J^2=\epsilon I$, $JD=\epsilon^\prime DJ$, and for all $x,y \in {\cal A}^\infty$, the commutators $[x, JyJ^{-1}]$ and $[JxJ^{-1},[D,y]]$ are compact operators.
If the spectral triple is even, a real structure with the sign-convention given by a triplet $(\epsilon, \epsilon^\prime, \epsilon^{\prime \prime})$ as in [@varilly page 30] is similar to a real structure in the odd case (with the sign-convention $(\epsilon, \epsilon^\prime)$), but with the additional requirement that $J \gamma = \epsilon^{\prime \prime} \gamma J$.
We now recall from [@qorient] the definition of quantum family of orientation preserving isometries and then appropriately adapt it to the framework of real structure.
\[def\_q\_fam\]
A quantum family of orientation preserving isometries for the spectral triple $({{\cal A}^\infty}, {\cal H}, D)$ is given by a pair $({\cal S}, U)$ where ${\cal S}$ is a separable unital $C^*$-algebra and $U$ is an ${\mathbb C}$-linear map from ${\cal H}$ to the Hilbert module ${\cal H} \otimes {\cal S}$ such that the ${\cal S}$-linear map $\widetilde{U}$ given by $\widetilde{U}( \xi \otimes b)=U(\xi) (1 \otimes b)$ $(\xi \in {\cal H}$, $b \in {\cal S}$) extends to a unitary element of $ {\cal M}({\cal K}({\cal H}) \otimes {\cal S})$ satisfying the following
$(i)$ $\tilde{U}$ commutes with $D \otimes I$, and
$(ii)$ $({\rm id} \otimes \phi) \circ \alpha_U(a) \in ({{\cal A}^\infty})^{\prime \prime}$ $\forall a \in {\cal A}^\infty$ for every state $\phi$ on ${\cal S}$, where $\alpha_U(x):=\widetilde{U}( x \otimes 1) {\widetilde{U}}^* $ for $x \in {\cal B}({\cal H})$.
In case the $C^*$-algebra ${\cal S}$ has a coproduct $\Delta$ such that $({\cal S},\Delta)$ is a compact quantum group and $U$ is a unitary representation of $({\cal S}, \Delta)$ on ${\cal H}$, we say that $({\cal S}, \Delta)$ acts by orientation preserving isometries on the spectral triple.
Given a quantum family of orientation preserving isometries $({\cal S}, U)$ as above, note that, since $D$ has finite dimensional eigenspaces which are preserved by $U$, we have $U{\cal D}_0 \subseteq {\cal D}_0 \otimes_{\rm alg} {\cal S}$, where ${\cal D}_0$ denotes the linear span of eigenvectors of $D$.
Suppose that the (odd) spectral triple $({\cal A}^\infty, {\cal H}, D)$ is equipped with a real structure given by $\tilde{J}$. We say that a quantum family of orientation preserving isometries $({\cal S}, U)$ also preserves the real structure if the following holds on ${\cal D}_0$: $$\label{equiv_real}(\tilde{J} \otimes \tilde{J}_{\cal S}) \circ U=U \circ \tilde{J}.$$ In case the $C^*$-algebra ${\cal S}$ has a coproduct $\Delta$ such that $({\cal S},\Delta)$ is a compact quantum group and $U$ is a unitary representation of $({\cal S}, \Delta)$ on ${\cal H}$, we say that $({\cal S}, \Delta)$ acts by orientation and real structure preserving isometries on the spectral triple.
Similar definitions can be given in the even case, with the additional requirement being that $U$ commutes with $\gamma$.
Given a compact quantum group ${\cal Q}$ acting on ${\cal A}$, such that the action is implemented by a unitary representation $U$ of the quantum group on ${\cal H}$, it is easy to see that the notion of equivariance of the spectral triple with the real structure as proposed in [@dab_real] is equivalent to saying that $({\cal Q}, U)$ is a quantum group acting by orientation and real structure preserving isometries in our sense. We refer the reader to [@dab_real] for related discussions and examples of such equivariant real spectral triples.
As in [@qorient], we consider the category ${\bf Q} \equiv {\bf Q}(D)$ with the object-class consisting of all quantum families of orientation and real structure preserving isometries $({\cal S}, U)$ of the given spectral triple, and the set of morphisms ${\rm Mor}(({\cal S},U),({\cal S}^\prime,U^\prime))$ being the set of unital $\ast$-homomorphisms $\Phi : {\cal S} \rightarrow {\cal S}^\prime$ satisfying $({\rm id} \otimes \Phi) (U)=U^\prime$. We also consider another category ${\bf Q}^\prime \equiv {\bf Q}^\prime(D)$ whose objects are triplets $({\cal S}, \Delta, U)$, where $({\cal S},\Delta)$ is a compact quantum group acting by orientation and real structure preserving isometries on the given spectral triple, with $U$ being the corresponding unitary representation. The morphisms are the homomorphisms of compact quantum groups which are also morphisms of the underlying quantum families of orientation preserving isometries. The forgetful functor $F: {\bf Q}^\prime \rightarrow {\bf Q}$ is clearly faithful, and we can view $F({\bf Q}^\prime)$ as a subcategory of ${\bf Q}$. Our aim is to show that the above categories admit universal object, which we prove in the next section.
Main results and examples
=========================
Let us fix a spectral triple $({\cal A}^\infty, {\cal H}, D)$ which is of compact type along with a real structure given by $\tilde{J}$. We shall work with an odd spectral triple, but remark that all the arguments will go through almost verbatim, with some obvious and minor changes at places, in the even case. The sign-convention of the real structure is not explicitly mentioned, since it is not going to be needed anywhere, and we remark that our arguments are valid for any possible choice of the signs. The $C^*$-algebra generated by ${\cal A}^\infty$ in ${\cal B}({\cal H})$ will be denoted by ${\cal A}$. Let $\lambda_0=0, \lambda_1, \lambda_2, \ldots$ be the eigenvalues of $D$ with $V_i$ denoting the ($d_i$-dimensional, $d_i<\infty$) eigenspace for $\lambda_i$. Let $\{ e_{ij}, j=1,\dots , d_i \}$ be an orthonormal basis of $V_i$. Clearly, $\{ \tilde{J}(e_{ij}), \ i \geq 0, \ 1 \leq j \leq d_i \}$ is a linearly independent (but not necessarily orthogonal) set, and let $T_i$ denote the positive nonsingular matrix $\big( \langle \tilde{J}(e_{ij}), \tilde{J}(e_{ik}) \rangle \big)_{j,k=1}^{d_i}.$ Let us denote the CQG $A_u(T_i)$ by ${\cal U}_i$, with its canonical unitary representation $\beta_i$ on $V_i \cong {\mathbb C}^{d_i}$, given by $\beta_i(e_{ij})=\sum_k e_{ik} \otimes u^{T_i}_{kj}$. Let ${\cal U}$ be the free product of ${\cal U}_i$, $i=1,2,\dots $ and $\beta=\ast_i \beta_i$ be the corresponding free product representation of ${\cal U}$ on ${\cal H}$. We shall also consider the corresponding unitary element $\tilde{\beta}$ in ${\cal M}({\cal K}({\cal H}) \otimes {\cal U})$.
\[lem2\] Consider the real spectral triple $({\cal A}^\infty,{\cal H},D, \tilde{J})$ as before and let $({\cal S},U)$ be a quantum family of orientation and real structure preserving isometries of the given spectral triple. Moreover, assume that the map $U$ is faithful in the sense that there is no proper $C^*$-subalgebra ${\cal S}_1$ of ${\cal S}$ such that $\widetilde{U} \in {\cal M}({\cal K}({\cal H} ) \otimes {\cal S}_1)$. Then we can find a $\ast$-isomorphism $\phi : {\cal U}/
I \rightarrow {\cal S}$ between ${\cal S}$ and a quotient of ${\cal U}$ by a $C^*$-ideal $I$ of ${\cal U}$, such that $ U= ({\rm id}\otimes \phi) \circ ({\rm id} \otimes \Pi_I) \circ
\beta$, where $\Pi_I$ denotes the quotient map from ${\cal U}$ to ${\cal U}/I$.
If, furthermore, there is a compact quantum group structure on ${\cal S}$ given by a coproduct $\Delta$ such that $({\cal S},\Delta, U)$ is an object in ${\bf Q}^\prime(D)$, the ideal $I$ is a Woronowicz $C^*$-ideal and the $\ast$-isomorphism $\phi : {\cal U}/ I \rightarrow {\cal S}$ is a morphism of compact quantum groups.
We follow the line of arguments of a similar result in [@qorient], though with suitable modifications. It is clear that $U$ maps $V_i $ into $V_i \otimes
{\cal S}$ for each $i$. Let $v^{(i)}_{kj}$ ($j,k=1,\dots ,d_i$) be the elements of ${\cal S}$ such that $U(e_{ij})=\sum_k e_{ik} \otimes
v^{(i)}_{kj}$. Note that $v_i:=((v^{(i)}_{kj} ))$ is a unitary in $M_{d_i}({\mathbb C}) \otimes {\cal S}$. Moreover, the $\ast$-subalgebra generated by all $ \{ v^{(i)}_{kj}, \ i \geq 0, \ j,k=1 ,\dots ,d_i\}$ must be dense in ${\cal S}$ by the assumption of faithfulness.
Now, we shall make use of (\[equiv\_real\]). Fix any $i$ and let $\Lambda_i=(( \tau_{lm} ))$ be the matrix such that $\tilde{J}(e_{ij})=\sum_l \tau_{lj} e_{il}$. By assumption, $\Lambda_i$ is invertible, and it is clear that $\Lambda_i^* \Lambda_i=T_i$. Expanding both sides of $U(\tilde{J}e_{ij})=\sum_k \tilde{J}e_{ik} \otimes (v^{(i)}_{kj})^*$ we get $$\label{123}\sum_m e_{im} \otimes \left( \sum_l \tau_{lj} v^{(i)}_{ml} \right)=
\sum_m e_{im} \otimes \left( \sum_l \tau_{ml} (v^{(i)}_{lj})^* \right).$$ By comparing coefficients of $e_{im}$ in both sides of (\[123\]), we get $\sum_l \tau_{lj} v^{(i)}_{ml}=\sum_l \tau_{ml} (v^{(i)}_{lj})^*,$ that is, $v_i \Lambda_i=\Lambda_i \overline{v_i}$. It follows that $\overline{v_i}=\Lambda_i^{-1} v_i \Lambda_i, $ hence $\overline{v_i}$ is invertible, since $v_i$ is so. Moreover, taking the ${\cal S}$-valued inner product $\langle \cdot, \cdot \rangle_{\cal S}$ on both sides of $U(\tilde{J}e_{ij})=\sum_k \tilde{J}e_{ik} \otimes (v^{(i)}_{kj})^*$ we obtain $T_i=v_i^\prime T_i \overline{v_i}$. Thus, $T_i^{-1}v_i^\prime T_i$ must be the (both-sided) inverse of $\overline{v_i}$, from which we see that the relations (\[wangalg\]) are satisfied with $u$ replaced by $v_i$.
We get, by the universality of ${\cal U}_i$, a $\ast$-homomorphism from ${\cal U}_i$ to ${\cal S}$ sending $u^{(i)}_{kj} \equiv u_{kj}^{T_i} $ to $v^{(i)}_{kj}$, and by definition of the free product, this induces a $\ast$-homomorphism, say $\Pi$, from ${\cal U}$ onto ${\cal S}$, so that ${\cal U}/I
\cong {\cal S}$, where $I:={\rm Ker}(\Pi)$.
In case ${\cal S}$ has a coproduct $\Delta$ making it into a compact quantum group and $U$ is a quantum group representation, it is easy to see that the subalgebra of ${\cal S}$ generated by $\{ v^{(i)}_{kj},\ i \geq 0, \ j,k=1,\dots ,d_i \}$ is a Hopf algebra, with $\Delta(v^{(i)}_{kj})=\sum_l v^{(i)}_{kl} \otimes v^{(i)}_{lj}$. From this, it follows that $\Pi$ is Hopf-algebra morphism, hence $I$ is a Woronowicz $C^*$-ideal.
From the proof of the above result, it can be seen that the assumption of preserving the real structure implies that the ‘$R$-twisted volume form’ is preserved, where $R$ is given by $R|_{V_i}=T_i^\prime$. This connects the approach of the present article to that of [@qorient], and in some sense gives an explanation of how the proof of the above lemma works.
The rest of the arguments in [@qorient] goes through more or less verbatim and we have the following analogue of the main result of [@qorient]:
For any real $($odd or even$)$ spectral triple $({\cal A}^\infty, {\cal H}, D, \tilde{J})$, the category ${\bf Q}$ of quantum families of orientation and real structure preserving isometries has a universal $($initial$)$ object, say $(\widetilde{{\cal G}}, U_0)$. Moreover, $\widetilde{{\cal G}}$ has a coproduct $\Delta_0$ such that $(\widetilde{{\cal G}},\Delta_0)$ is a compact quantum group and $(\widetilde{{\cal G}},\Delta_0,U_0)$ is a universal object in the category ${\bf Q}^\prime$. The representation $U_0$ is faithful.
Let ${\cal G}$ denote the Woronowicz subalgebra of $\widetilde{{\cal G}}$ generated by elements of the form $\langle \xi \otimes 1, {\rm ad}_{U_0}(a)(\eta \otimes 1) \rangle $, where $\xi, \eta \in {\cal H}$, $a \in {\cal A}^\infty$, and where $\langle \cdot, \cdot\rangle$ denotes the $\widetilde{{\cal G}}$-valued inner product of the Hilbert module ${\cal H} \otimes \widetilde{{\cal G}}$. We shall call ${\cal G}$ the quantum group of orientation and real structure preserving isometries of the given spectral triple, and denote it by $ {QISO}^+ ( {\cal A}^{\infty}, {\cal H}, D, \tilde{J} ) $ or even simply as ${QISO}^{+}_{{\rm real}}(D)$. The quantum group $\widetilde{{\cal G}}$ is denoted by $\widetilde{QISO}^{+}_{{\rm real}}(D)$.
\[11\] It is clear from the definition that $QISO^+_{{\rm real}}\!(D)$ is a quantum subgroup of $QISO^+\!(D)\!$ whenever the later exists, since the former is the universal object in a subcategory of the category for which the latter is universal (if exists).
We conclude the article with two examples.
The standard spectral triple on the noncommutative two torus ${\cal A}_\theta$ (including the commutative case, i.e. $\theta=0$) has a canonical real structure. The Hilbert space ${\cal H}$ is in this case $L^2({\cal A}_\theta, \tau) \otimes {\mathbb C}^2$ (where $\tau$ is the canonical faithful trace on ${\cal A}_\theta$) and $D=\left( \begin{array}{cc} 0 & d_1+id_2\\ d_1-id_2 & 0 \end{array} \right), $ $J=\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right),$ where $d_1$, $d_1$ denote the canonical derivations (see [@connes]). This spectral triple (without taking into account the real structure) has been considered in Subsection 4.3 of [@qorient], where we have proved that the quantum group orientation preserving isometries exists and coincides with the classical group of such isometries, i.e. $C({\mathbb T}^2)$. Since it can easily be seen that this $C({\mathbb T}^2)$ action also preserves the real structure, it follows from Remark \[11\] that $QISO^+_{{\rm real}}(D)$ must be $C({\mathbb T}^2)$.
This is an example involving a quantum group action with nontrivial modularity. Consider the spectral triple on the Podles sphere $S^2_{\mu c}$ constructed in [@Dabrowski_et_al]. Note that in [@Dabrowski_et_al], a real structure has also been constructed and the spectral triple as well as the real structure are shown to be equivariant in the sense of [@dab_real] with respect to the canonical action of $SO_\mu(3)$. Thus, $QISO^+_{{\rm real}}(D)$ for this spectral triple has $SO_\mu(3)$ as a quantum subgroup, and then it follows from Theorem 3.39 of [@q_sphere] that $QISO^+_{{\rm real}}(D)$ must coincide with $SO_\mu(3)$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author acknowledges the support from Indian National Science Academy for the project ‘Noncommutative Geometry and Quantum Groups’ and UKIERI, British Council.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Embedding models for entities and relations are extremely useful for recovering missing facts in a knowledge base. Intuitively, a relation can be modeled by a matrix mapping entity vectors. However, relations reside on low dimension sub-manifolds in the parameter space of arbitrary matrices – for one reason, composition of two relations ${\bm{M}}_1,{\bm{M}}_2$ may match a third ${\bm{M}}_3$ (e.g. composition of relations `currency_of_country` and `country_of_film` usually matches `currency_of_film_budget`), which imposes compositional constraints to be satisfied by the parameters (i.e. ${\bm{M}}_1\cdot {\bm{M}}_2\approx {\bm{M}}_3$). In this paper we investigate a dimension reduction technique by training relations jointly with an autoencoder, which is expected to better capture compositional constraints. We achieve state-of-the-art on Knowledge Base Completion tasks with strongly improved Mean Rank, and show that joint training with an autoencoder leads to interpretable sparse codings of relations, helps discovering compositional constraints and benefits from compositional training. Our source code is released at [github.com/tianran/glimvec](github.com/tianran/glimvec).'
author:
- 'Ryo Takahashi\*^1^'
- 'Ran Tian\*^1^'
- |
Kentaro Inui^1,2^\
[**(\* equal contribution)**]{}\
^1^Tohoku University^2^ RIKEN, Japan\
[{ryo.t,tianran,inui}@ecei.tohoku.ac.jp]{}\
bibliography:
- 'acl2018.bib'
title: Interpretable and Compositional Relation Learning by Joint Training with an Autoencoder
---
Introduction
============
Broad-coverage knowledge bases (KBs) such as Freebase [@DBLP:conf/sigmod/BollackerEPST08] and DBPedia [@auer2007dbpedia] store a large amount of facts in the form of $\langle$, , $\rangle$ triples (e.g. $\langle$*The Matrix*, `country_of_film`, *Australia*$\rangle$), which could support a wide range of reasoning and question answering applications. The Knowledge Base Completion (KBC) task aims to predict the missing part of an incomplete triple, such as $\langle$*Finding Nemo*, `country_of_film`, ?$\rangle$, by reasoning from known facts stored in the KB.
![In joint training, relation parameters (e.g. ${\bm{M}}_1$) receive updates from both a *KB-learning objective*, trying to predict entities in the KB; and a *reconstruction objective* from an autoencoder, trying to recover relations from low dimension codings.[]{data-label="fig:model_arch"}](model_arch){width="0.9\columnwidth"}
As a most common approach [@DBLP:journals/tkde/WangMWG17], modeling entities and relations to operate in a low dimension vector space helps KBC, for three conceivable reasons. First, when dimension is low, entities modeled as vectors are forced to share parameters, so “similar” entities which participate in many relations in common get close to each other (e.g. *Australia* close to *US*). This could imply that an entity (e.g. *US*) “type matches” a relation such as `country_of_film`. Second, relations may share parameters as well, which could transfer facts from one relation to other similar relations, for example from $\langle$*x*, `award_winner`, *y*$\rangle$ to $\langle$*x*, `award_nominated`, *y*$\rangle$. Third, spatial positions might be used to implement *composition* of relations, as relations can be regarded as mappings from head to tail entities, and the composition of two maps can match a third (e.g. the composition of `currency_of_country` and `country_of_film` matches the relation `currency_of_film_budget`), which could be captured by modeling composition in a space.
However, modeling relations as mappings naturally requires more parameters – a general linear map between $d$-dimension vectors is represented by a matrix of $d^2$ parameters – which are less likely to be shared, impeding transfers of facts between similar relations. Thus, it is desired to reduce dimensionality of relations; furthermore, the existence of a composition of two relations (assumed to be modeled by matrices ${\bm{M}}_1,{\bm{M}}_2$) matching a third (${\bm{M}}_3$) also justifies dimension reduction, because it implies a *compositional constraint* ${\bm{M}}_1\cdot {\bm{M}}_2\approx {\bm{M}}_3$ that can be satisfied only by a lower dimension sub-manifold in the parameter space[^1].
Previous approaches reduce dimensionality of relations by imposing pre-designed hard constraints on the parameter space, such as constraining that relations are translations [@DBLP:conf/nips/BordesUGWY13] or diagonal matrices [@Yang2015], or assuming they are linear combinations of a small number of prototypes [@xie-EtAl:2017:Long]. However, pre-designed hard constraints do not seem to cope well with compositional constraints, because it is difficult to know *a priori* which two relations compose to which third relation, hence difficult to choose a pre-design; and compositional constraints are not always exact (e.g. the composition of `currency_of_country` and `headquarter_location` usually matches `business_operation_currency` but not always), so hard constraints are less suited.
In this paper, we investigate an alternative approach by training relation parameters jointly with an autoencoder (Figure \[fig:model\_arch\]). During training, the autoencoder tries to reconstruct relations from low dimension codings, with the reconstruction objective back-propagating to relation parameters as well. We show this novel technique promotes parameter sharing between different relations, and drives them toward low dimension manifolds (Sec.\[sec:analyzeautoenc\]). Besides, we expect the technique to cope better with compositional constraints, because it discovers low dimension manifolds posteriorly from data, and it does not impose any explicit hard constraints.
Yet, joint training with an autoencoder is not simple; one has to keep a subtle balance between gradients of the reconstruction and KB-learning objectives throughout the training process. We are not aware of any theoretical principles directly addressing this problem; but we found some important settings after extensive pre-experiments (Sec.\[sec:optimizationtricks\]). We evaluate our system using standard KBC datasets, achieving state-of-the-art on several of them (Sec.\[sec:mainresults\]), with strongly improved Mean Rank. We discuss detailed settings that lead to the performance (Sec.\[sec:trainingbase\]), and we show that joint training with an autoencoder indeed helps discovering compositional constraints (Sec.\[sec:compositionalconstraints\]) and benefits from compositional training (Sec.\[sec:gainscomptrain\]).
Base Model {#sec:basemodel}
==========
A knowledge base (KB) is a set ${\mathcal{T}}$ of triples of the form $\langle h, r, t\rangle$, where $h, t\in{\mathcal{E}}$ are entities and $r\in{\mathcal{R}}$ is a relation (e.g. $\langle$*The Matrix*, `country_of_film`, *Australia*$\rangle$). A relation $r$ has its inverse $r^{-1}\in{\mathcal{R}}$ so that for every $\langle h, r, t\rangle\in{\mathcal{T}}$, we regard $\langle t, r^{-1}, h \rangle$ as also in the KB. Under this assumption and given ${\mathcal{T}}$ as training data, we consider the Knowledge Base Completion (KBC) task that predicts candidates for a missing tail entity in an incomplete $\langle h, r, ?\rangle$ triple.
Most approaches tackle this problem by training a *score function* measuring the plausibility of triples being facts. The model we implement in this work represents entities $h,t$ as $d$-dimension vectors ${\bm{u}}_h,{\bm{v}}_t$ respectively, and relation $r$ as a $d\times d$ matrix ${\bm{M}}_r$. If ${\bm{u}}_h,{\bm{v}}_t$ are one-hot vectors with dimension $d=\lvert{\mathcal{E}}\rvert$ corresponding to each entity, one can take ${\bm{M}}_r$ as the adjacency matrix of entities joined by relation $r$, so the set of tail entities filling into $\langle h, r, ?\rangle$ is calculated by ${\bm{u}}_h^\top {\bm{M}}_{r}$ (with each nonzero entry corresponds to an answer). Thus, we have ${\bm{u}}_h^\top {\bm{M}}_{r}{\bm{v}}_t > 0$ if and only if $\langle h, r, t\rangle\in{\mathcal{T}}$. This motivates us to use ${\bm{u}}_h^\top {\bm{M}}_{r}{\bm{v}}_t$ as a natural parameter to model plausibility of $\langle h, r, t\rangle$, even in a low dimension space with $d\ll\lvert{\mathcal{E}}\rvert$. Thus, we define the score function as $$\label{eq:scrbilinear}
s(h,r,t):=\exp({\bm{u}}_h^\top{\bm{M}}_{r}{\bm{v}}_t)$$ for the basic model. This is similar to the bilinear model of @Nickel:2011:TMC:3104482.3104584, except that we distinguish ${\bm{u}}_h$ (the vector for head entities) from ${\bm{v}}_t$ (the vector for tail entities). It has also been proposed in @tian-okazaki-inui:2016:P16-1, but for modeling dependency trees rather than KBs.
More generally, we consider *composition* of relations $r_1/\ldots/r_l$ to model *path*s in a KB [@guu-miller-liang:2015:EMNLP], as defined by $r_1,\ldots,r_l$ participating in a sequence of facts such that the head entity of each fact coincides with the tail of its previous. For example, a sequence of two facts $\langle$*The Matrix*, `country_of_film`, *Australia*$\rangle$ and $\langle$*Australia*, `currency_of_country`, *Australian Dollar*$\rangle$ form a path of composition `country_of_film`/ `currency_of_country`, because the head of the second fact (i.e. *Australia*) coincides with the tail of the first. Using the previous $d=\lvert{\mathcal{E}}\rvert$ analogue, one can verify that composition of relations is represented by multiplication of adjacency matrices, so we accordingly define $$s(h,r_1/\ldots/r_l,t):=\exp({\bm{u}}_h^\top{\bm{M}}_{r_1}\cdots{\bm{M}}_{r_l}{\bm{v}}_t)$$ to measure the plausibility of a path. It is explored in @guu-miller-liang:2015:EMNLP to learn a score function not only for single facts but also for paths. This *compositional training* scheme is shown to bring valuable information about the structure of the KB and may help KBC. In this work, we conduct experiments both with and without compositional training.
In order to learn parameters ${\bm{u}}_h,{\bm{v}}_t,{\bm{M}}_r$ of the score function, we follow @tian-okazaki-inui:2016:P16-1 using a Noise Contrastive Estimation (NCE) [@DBLP:journals/jmlr/GutmannH12] objective. For each path (or triple) $\langle h,r_1/\ldots,t\rangle$ taken from the KB, we generate negative samples by replacing the tail entity $t$ with some random noise $t^{*}$. Then, we maximize $$\begin{gathered}
\mathcal{L}_1:=
\sum_{\text{path}}\ln\frac{s(h, r_1/\ldots, t)}{k+s(h, r_1/\ldots, t)}\\
+\sum_{\text{noise}}\ln\frac{k}{k+s(h, r_1/\ldots, t^{*})}\end{gathered}$$ as our *KB-learning objective*. Here, $k$ is the number of noises generated for each path. When the score function is regarded as probability, $\mathcal{L}_1$ represents the log-likelihood of “$\langle h,r_1/\ldots,t\rangle$ being actual path and $\langle h,r_1/\ldots,t^{*}\rangle$ being noise”. Maximizing $\mathcal{L}_1$ increases the scores of actual paths and decreases the scores of noises.
Joint Training with an Autoencoder {#sec:jointtraining}
==================================
Autoencoders learn efficient codings of high-dimensional data while trying to reconstruct the original data from the coding. By joint training relation matrices with an autoencoder, we also expect it to help reducing the dimensionality of the original data (i.e. relation matrices).
Formally, we define a *vectorization* ${\bm{m}}_r$ for each relation matrix ${\bm{M}}_r$, and use it as input to the autoencoder. ${\bm{m}}_r$ is defined as a reshape of ${\bm{M}}_r$ flattened into a $d^2$-dimension vector, and normalized such that $\lVert{\bm{m}}_r\rVert=\sqrt{d}$. We define $$\label{eq:coding}
{\bm{c}}_r:=\operatorname*{ReLU}({\bm{A}}{\bm{m}}_{r})$$ as the coding. Here ${\bm{A}}$ is a $c\times d^2$ matrix with $c\ll d^2$, and $\operatorname*{ReLU}$ is the Rectified Linear Unit function [@DBLP:conf/icml/NairH10]. We reconstruct the input from ${\bm{c}}_r$ by multiplying a $d^2\times c$ matrix ${\bm{B}}$. We want ${\bm{B}}{\bm{c}}_{r}$ to be more similar to ${\bm{m}}_r$ than other relations. For this purpose, we define a similarity $$\label{eq:reconscore}
g(r_1, r_2):=\exp(\frac{1}{\sqrt{dc}}{\bm{m}}_{r_1}^\top{\bm{B}}{\bm{c}}_{r_2}),$$ which measures the length of ${\bm{B}}{\bm{c}}_{r_2}$ projected to the direction of ${\bm{m}}_{r_1}$. In order to learn the parameters ${\bm{A}},{\bm{B}}$, we adopt the Noise Contrastive Estimation scheme as in Sec.\[sec:basemodel\], generate random noises $r^{*}$ for each relation $r$ and maximize $$\mathcal{L}_2:=
\sum_{r\in{\mathcal{R}}}\ln\frac{g(r, r)}{k+g(r, r)}
+\sum_{r^{*}\sim{\mathcal{R}}}\ln\frac{k}{k+g(r, r^{*})}$$ as our *reconstruction objective*. Maximizing $\mathcal{L}_2$ increases ${\bm{m}}_r$’s similarity with ${\bm{B}}{\bm{c}}_{r}$, and decreases it with ${\bm{B}}{\bm{c}}_{r^{*}}$.
During joint training, both $\mathcal{L}_1$ and $\mathcal{L}_2$ are simultaneously maximized, and the gradient $\nabla\mathcal{L}_2$ propagates to relation matrices as well. Since $\nabla\mathcal{L}_2$ depends on ${\bm{A}}$ and ${\bm{B}}$, and ${\bm{A}},{\bm{B}}$ interact with all relations, they promote indirect parameter sharing between different relation matrices. In Sec.\[sec:analyzeautoenc\], we further show that joint training drives relations toward a low dimension manifold.
Optimization Tricks {#sec:optimizationtricks}
===================
Joint training with an autoencoder is not simple. Relation matrices receive updates from both $\nabla\mathcal{L}_1$ and $\nabla\mathcal{L}_2$, but if they update $\nabla\mathcal{L}_1$ too much, the autoencoder has no effect; conversely, if they update $\nabla\mathcal{L}_2$ too often, all relation matrices crush into one cluster. Furthermore, an autoencoder should learn from genuine patterns of relation matrices that emerge from fitting the KB, but not the reverse – in which the autoencoder imposes arbitrary patterns to relation matrices according to random initialization. Therefore, it is not surprising that a naive optimization of $\mathcal{L}_1+\mathcal{L}_2$ does not work.
After extensive pre-experiments, we have found some crucial settings for successful training. The most important “magic” is the scaling factor $\frac{1}{\sqrt{dc}}$ in definition of the similarity function , perhaps being combined with other settings as we discuss below. We have tried different factors $1$, $\frac{1}{\sqrt{d}}$, $\frac{1}{\sqrt{c}}$ and $\frac{1}{dc}$ instead, with various combinations of $d$ and $c$; but the autoencoder failed to learn meaningful codings in other settings. When the scaling factor is too small (e.g. $\frac{1}{dc}$), all relations get almost the same coding; conversely if the factor is too large (e.g. $1$), all codings get very close to $0$.
The next important rule is to keep a balance between the updates coming from $\nabla\mathcal{L}_1$ and $\nabla\mathcal{L}_2$. We use Stochastic Gradient Descent (SGD) for optimization, and the common practice [@bottou2012stochastic] is to set the learning rate as $$\label{eq:commonpractice}
\alpha(\tau):=\frac{\eta}{1+\eta\lambda\tau}.$$ Here, $\eta,\lambda$ are hyper-parameters and $\tau$ is a counter of processed data points. In this work, in order to control the updates in detail to keep a balance, we modify to use a a step counter $\tau_r$ for each relation $r$, counting “number of updates” instead of data points[^2]. That is, whenever ${\bm{M}}_r$ gets a nonzero update from a gradient calculation, $\tau_r$ increases by $1$. Furthermore, we use different hyper-parameters for different “types of updates”, namely $\eta_1,\lambda_1$ for updates coming from $\nabla\mathcal{L}_1$, and $\eta_2,\lambda_2$ for updates coming from $\nabla\mathcal{L}_2$. Thus, let $\Delta_1$ be the partial gradient of $\nabla\mathcal{L}_1$, and $\Delta_2$ the partial gradient of $\nabla\mathcal{L}_2$, we update ${\bm{M}}_r$ by $\alpha_1(\tau_r)\Delta_1+\alpha_2(\tau_r)\Delta_2$ at each step, where $$\alpha_1(\tau_r):=\frac{\eta_1}{1+\eta_1\lambda_1\tau_r},\;\;
\alpha_2(\tau_r):=\frac{\eta_2}{1+\eta_2\lambda_2\tau_r}.$$
The rule for setting $\eta_1,\lambda_1$ and $\eta_2,\lambda_2$ is that, $\eta_2$ should be much smaller than $\eta_1$, because $\eta_1,\eta_2$ control the magnitude of learning rates at the early stage of training, with the autoencoder still largely random and $\Delta_2$ not making much sense; on the other hand, one has to choose $\lambda_1$ and $\lambda_2$ such that $\lVert\Delta_1\rVert/\lambda_1$ and $\lVert\Delta_2\rVert/\lambda_2$ are at the same scale, because the learning rates approach $1/(\lambda_1\tau_r)$ and $1/(\lambda_2\tau_r)$ respectively, as the training proceeds. In this way, the autoencoder will not impose random patterns to relation matrices according to its initialization at the early stage, and a balance is kept between $\alpha_1(\tau_r)\Delta_1$ and $\alpha_2(\tau_r)\Delta_2$ later.
But how to estimate $\lVert\Delta_1\rVert$ and $\lVert\Delta_2\rVert$? It seems that we can approximately calculate their scales from initialization. In this work, we use i.i.d. Gaussians of variance $1/d$ to initialize parameters, so the initial Euclidean norms are $\lVert{\bm{u}}_h\rVert\approx 1$, $\lVert{\bm{v}}_t\rVert\approx 1$, $\lVert{\bm{M}}_r\rVert\approx\sqrt{d}$, and $\lVert{\bm{B}}{\bm{A}}{\bm{m}}_r\rVert\approx\sqrt{dc}$. Thus, by calculating $\nabla\mathcal{L}_1$ and $\nabla\mathcal{L}_2$ using and , we have approximately $$\begin{gathered}
\lVert\Delta_1\rVert\approx\lVert{\bm{u}}_h{\bm{v}}_t^\top\rVert\approx 1,
\quad\text{and}\\
\lVert\Delta_2\rVert\approx\lVert\frac{1}{\sqrt{dc}}{\bm{B}}{\bm{c}}_r\rVert\approx
\frac{1}{\sqrt{dc}}\lVert{\bm{B}}{\bm{A}}{\bm{m}}_r\rVert\approx 1.\end{gathered}$$ It suggests that, because of the scaling factor $\frac{1}{\sqrt{dc}}$ in , we have $\lVert\Delta_1\rVert$ and $\lVert\Delta_2\rVert$ at the same scale, so we can set $\lambda_1=\lambda_2$. This might not be a mere coincidence.
Training the Base Model {#sec:trainingbase}
-----------------------
Besides the tricks for joint training, we also found settings that significantly improve the base model on KBC, as briefly discussed below. In Sec.\[sec:crucialsettings\], we will show performance gains by these settings using the FB15k-237 validation set.
#### Normalization
It is better to normalize relation matrices to $\lVert{\bm{M}}_r\rVert=\sqrt{d}$ during training. This might reduce fluctuations in entity vector updates.
#### Regularizer
It is better to minimize $\lVert {\bm{M}}_r^\top {\bm{M}}_r-\frac{1}{d}\operatorname*{tr}({\bm{M}}_r^\top {\bm{M}}_r)I\rVert$ during training. This regularizer drives ${\bm{M}}_r$ toward an orthogonal matrix [@tian-okazaki-inui:2016:P16-1] and might reduce fluctuations in entity vector updates. As a result, all relation matrices trained in this work are very close to orthogonal.
#### Initialization
Instead of pure Gaussian, it is better to initialize matrices as $(I+G)/2$, where $G$ is random. The identity matrix $I$ helps passing information from head to tail [@tian-okazaki-inui:2016:P16-1].
#### Negative Sampling
Instead of a unigram distribution, it is better to use a *uniform* distribution for generating noises. This is somehow counter-intuitive compared to training word embeddings.
Related Works
=============
KBs have a wide range of applications [@berant-EtAl:2013:EMNLP; @hixon-clark-hajishirzi:2015:NAACL-HLT; @DBLP:journals/pieee/Nickel0TG16] and KBC has inspired a huge amount of research [@DBLP:conf/nips/BordesUGWY13; @riedel-EtAl:2013:NAACL-HLT; @DBLP:conf/nips/SocherCMN13; @DBLP:conf/aaai/WangZFC14; @wang-EtAl:2014:EMNLP20145; @DBLP:conf/ijcai/xiao16; @nguyen-EtAl:2016:N16-1; @toutanova-EtAl:2016:P16-1; @das-EtAl:2017:EACLlong1; @hayashi-shimbo:2017:Short].
Among the previous works, TransE [@DBLP:conf/nips/BordesUGWY13] is the classic method which represents a relation as a translation of the entity vector space, and is partially inspired by @mikolov-yih-zweig:2013:NAACL-HLT’s vector arithmetic method of solving word analogy tasks. Although competitive in KBC, it is speculated that this method is well-suited for $1$-to-$1$ relations but might be too simple to represent $N$-to-$N$ relations accurately[@DBLP:journals/tkde/WangMWG17]. Thus, extensions such as TransR [@DBLP:conf/aaai/LinLSLZ15] and STransE [@nguyen-EtAl:2016:N16-1] are proposed to map entities into a relation-specific vector space before translation. The ITransF model [@xie-EtAl:2017:Long] further enhances this approach by imposing a hard constraint that the relation-specific maps should be linear combinations of a small number of prototypical matrices. Our work inherits the same motivation with ITransF in terms of promoting parameter-sharing among relations.
On the other hand, the base model used in this work originates from RESCAL [@Nickel:2011:TMC:3104482.3104584], in which relations are naturally represented as analogue to the adjacency matrices (Sec.\[sec:basemodel\]). Further developments include HolE [@DBLP:conf/aaai/NickelRP16] and ConvE [@dettmers2018conve] which improve this approach in terms of parameter-efficiency, by introducing low dimension factorizations of the matrices. We inherit the basic model of RESCAL but draw additional training techniques from @tian-okazaki-inui:2016:P16-1, and show that the base model already can achieve near state-of-the-art performance (Sec.\[sec:mainresults\],\[sec:crucialsettings\]). This sends a message similar to @kadlec-bajgar-kleindienst:2017:RepL4NLP, saying that training tricks might be as important as model designs.
Nevertheless, we emphasize the novelty of this work in that the previous models mostly achieve dimension reduction by imposing some pre-designed hard constraints [@DBLP:conf/nips/BordesUGWY13; @Yang2015; @DBLP:conf/icml/TrouillonWRGB16; @DBLP:conf/aaai/NickelRP16; @xie-EtAl:2017:Long; @dettmers2018conve], whereas the constraints themselves are not learned from data; in contrast, our approach by jointly training an autoencoder does not impose any explicit hard constraints, so it leads to more flexible modeling.
Moreover, we additionally focus on leveraging composition in KBC. Although this idea has been frequently explored before [@guu-miller-liang:2015:EMNLP; @neelakantan-roth-mccallum:2015:ACL-IJCNLP; @lin-EtAl:2015:EMNLP1], our discussion about the concept of compositional constraints and its connection to dimension reduction has not been addressed similarly in previous research. In experiments, we will show (Sec.\[sec:compositionalconstraints\],\[sec:gainscomptrain\]) that joint training with an autoencoder indeed helps finding compositional constraints and benefits from compositional training.
Autoencoders have been used solo for learning distributed representations of syntactic trees [@socher-EtAl:2011:EMNLP], words and images [@silberer-lapata:2014:P14-1], or semantic roles [@titov-khoddam:2015:NAACL-HLT]. It is also used for pretraining other deep neural networks [@Erhan:2010:WUP]. However, when combined with other models, the learning of autoencoders, or more generally *sparse codings* [@rubinstein2010dictionaries], is usually conveyed in an alternating manner, fixing one part of the model while optimizing the other, such as in @xie-EtAl:2017:Long. To our knowledge, joint training with an autoencoder is not widely used previously for reducing dimensionality.
Jointly training an autoencoder is not simple because it takes non-stationary inputs. In this work, we modified SGD so that it shares traits with some modern optimization algorithms such as Adagrad [@DBLP:journals/jmlr/DuchiHS11], in that they both set different learning rates for different parameters. While Adagrad sets them adaptively by keeping track of gradients for all parameters, our modification of SGD is more efficient and allows us to grasp a rough intuition about which parameter gets how much update. We believe our techniques and findings in joint training with an autoencoder could be helpful to reducing dimensionality and improving interpretability in other neural network architectures as well.
Experiments
===========
We evaluate on standard KBC datasets, including WN18 and FB15k [@DBLP:conf/nips/BordesUGWY13], WN18RR [@dettmers2018conve] and FB15k-237 [@toutanova-chen:2015:CVSC]. The statistical information of these datasets are shown in Table \[tab:datasets\].
Dataset
----------- -------- ------- --------- -------- --------
WN18 40,943 18 141,442 5,000 5,000
FB15k 14,951 1,345 483,142 50,000 59,071
WN18RR 40,943 11 86,835 3,034 3,134
FB15k-237 14,541 237 272,115 17,535 20,466
: Statistical information of the KBC datasets. $\lvert{\mathcal{E}}\rvert$ and $\lvert{\mathcal{R}}\rvert$ denote the number of entities and relation types, respectively; \#Train, \#Valid, and \#Test are the numbers of triples in the training, validation, and test sets, respectively.[]{data-label="tab:datasets"}
WN18 collects word relations from WordNet [@DBLP:journals/cacm/Miller95], and FB15k is taken from Freebase [@DBLP:conf/sigmod/BollackerEPST08]; both have filtered out low frequency entities. However, it is reported in @toutanova-chen:2015:CVSC that both WN18 and FB15k have information leaks because the inverses of some test triples appear in the training set. FB15k-237 and WN18RR fix this problem by deleting such triples from training and test data. In this work, we do evaluate on WN18 and FB15k, but our models are mainly tuned on FB15k-237.
For all datasets, we set the dimension $d=256$ and $c=16$, the SGD hyper-parameters $\eta_1=1/64$, $\eta_2=2^{-14}$ and $\lambda_1=\lambda_2=2^{-14}$. The training batch size is 32 and the triples in each batch share the same head entity. We compare the base model (<span style="font-variant:small-caps;">base</span>) to our joint training with an autoencoder model (<span style="font-variant:small-caps;">joint</span>), and the base model with compositional training (<span style="font-variant:small-caps;">base+comp</span>) to our joint model with compositional training (<span style="font-variant:small-caps;">joint+comp</span>). When compositional training is enabled (<span style="font-variant:small-caps;">base+comp</span>, <span style="font-variant:small-caps;">joint+comp</span>), we use random walk to sample paths of length $1+X$, where $X$ is drawn from a Poisson distribution of mean $\lambda=1.0$.
For any incomplete triple ${\langle h, r, ? \rangle}$ in KBC test, we calculate a score $s(h,r,e)$ from , for every entity $e\in{\mathcal{E}}$ such that ${\langle h, r, e \rangle}$ *does not appear in any of the training, validation, or test sets* [@DBLP:conf/nips/BordesUGWY13]. Then, the calculated scores together with $s(h,r,t)$ for the gold triple is converted to ranks, and the rank of the gold entity $t$ is used for evaluation. Evaluation metrics include Mean Rank (MR), Mean Reciprocal Rank (MRR), and Hits at 10 (H10). Lower MR, higher MRR, and higher H10 indicate better performance.
We consult MR and MRR on validation sets to determine training epochs; we stop training when both MR and MRR have stopped improving.
KBC Results {#sec:mainresults}
-----------
![Examples of relation codings learned from FB15k-237. Each row shows a 16 dimension vector encoding a relation. Vectors are normalized such that their entries sum to $1$.[]{data-label="fig:code-heatmap"}](code_heat){width="\columnwidth"}
-------------------------------------------------------------- ------------- -------------- ------------ -------------- -------------- -------------- -------------- ------------- -------------- --------------
(lr)[2-3]{} (lr)[4-5]{} (lr)[6-8]{} (l)[9-11]{} MR H10 MR H10 MR MRR H10 MR MRR H10
<span style="font-variant:small-caps;">joint</span> **277** **95.8** **53** **82.5** **4233** **.461**$^*$ **53.4** **212** .336 **52.3**$^*$
<span style="font-variant:small-caps;">base</span> 286 **95.8** **53** **82.5** 4371 .459 52.9 215 **.337**$^*$ **52.3**$^*$
<span style="font-variant:small-caps;">joint+comp</span> **191**$^*$ **94.8** **53** **69.7** **2268**$^*$ **.343** **54.8**$^*$ **197**$^*$ **.331** **51.6**
<span style="font-variant:small-caps;">base+comp</span> 195 **94.8** 54 69.4 2447 .310 54.1 203 .328 51.5
TransE [@DBLP:conf/nips/BordesUGWY13] 292 92.0 **66** 70.4 4311 .202 45.6 **278** .236 41.6
TransR [@DBLP:conf/aaai/LinLSLZ15] **281** 93.6 76 **74.4** **4222** .210 **47.1** 320 **.282** **45.9**
RESCAL [@Nickel:2011:TMC:3104482.3104584] 911 58.0 163 41.0 9689 .105 20.3 457 .178 31.9
HolE [@DBLP:conf/aaai/NickelRP16] 724 **94.3** 293 66.8 8096 **.376** 40.0 1172 .169 30.9
STransE [@nguyen-EtAl:2016:N16-1] 206 93.4 69 79.9 - - - - - -
ITransF [@xie-EtAl:2017:Long] **205** 94.2 65 81.0 - - - - - -
ComplEx [@DBLP:conf/icml/TrouillonWRGB16] - 94.7 - 84.0 **5261** .44 **51** 339 .247 42.8
Ensemble DistMult [@kadlec-bajgar-kleindienst:2017:RepL4NLP] 457 95.0 35.9 90.4 - - - - - -
IRN [@shen-EtAl:2017:RepL4NLP1] 249 95.3 38 **92.7$^*$** - - - - - -
ConvE [@dettmers2018conve] 504 95.5 64 87.3 5277 **.46** 48 **246** **.316** **49.1**
R-GCN+ [@DBLP:journals/corr/SchlichtkrullKB17] - **96.4**$^*$ - 84.2 - - - - .249 41.7
ProjE [@DBLP:conf/aaai/ShiW17] - - **34$^*$** 88.4 - - - - - -
-------------------------------------------------------------- ------------- -------------- ------------ -------------- -------------- -------------- -------------- ------------- -------------- --------------
The results are shown in Table \[tab:main-results\]. We found that joint training with an autoencoder mostly improves performance, and the improvement becomes more clear when compositional training is enabled (i.e., $\textsc{joint}\geq\textsc{base}$ and $\textsc{joint+comp}>\textsc{base+comp}$). This is convincing because generally, joint training contributes with its regularizing effects, and drastic improvements are less expected[^3]. When compositional training is enabled, the system usually achieves better MR, though not always improves in other measures. The performance gains are more obvious on the WN18RR and FB15k-237 datasets, possibly because WN18 and FB15k contain a lot of easy instances that can be solved by a simple rule [@dettmers2018conve].
Furthermore, the numbers demonstrated by our joint and base models are among the strongest in the literature. We have conducted re-experiments of several representative algorithms, and also compare with state-of-the-art published results. For re-experiments, we use @DBLP:conf/aaai/LinLSLZ15’s implementation[^4] of TransE [@DBLP:conf/nips/BordesUGWY13] and TransR, which represent relations as vector translations; and @DBLP:conf/aaai/NickelRP16’s implementation[^5] of RESCAL [@Nickel:2011:TMC:3104482.3104584] and HolE, where RESCAL is most similar to the <span style="font-variant:small-caps;">base</span> model and HolE is a more parameter-efficient variant. We experimented with the default settings, and found that our models outperform most of them.
Among the published results, STransE [@nguyen-EtAl:2016:N16-1] and ITransF [@xie-EtAl:2017:Long] are more complicated versions of TransR, achieving the previous highest MR on WN18 but are outperformed by our <span style="font-variant:small-caps;">joint+comp</span> model. ITransF is most similar to our <span style="font-variant:small-caps;">joint</span> model in that they both learn sparse codings for relations. On WN18RR and FB15k-237, @dettmers2018conve’s report of ComplEx [@DBLP:conf/icml/TrouillonWRGB16] and ConvE were previously the best results. Our models mostly outperform them. Other results include @kadlec-bajgar-kleindienst:2017:RepL4NLP’s simple but strong baseline and several recent models [@DBLP:journals/corr/SchlichtkrullKB17; @DBLP:conf/aaai/ShiW17; @shen-EtAl:2017:RepL4NLP1] which achieve best results on FB15k or WN18 in some measure. Our models have comparable results.
Intuition and Insight {#sec:analyzeautoenc}
---------------------
What does the autoencoder look like? How does joint training affect relation matrices? We address these questions by analyses showing that **(i)** the autoencoder learns sparse and interpretable codings of relations, **(ii)** the joint training drives relation matrices toward a low dimension manifold, and **(iii)** it helps discovering compositional constraints.
### Sparse Coding and Interpretability {#sec:interpretability .unnumbered}
Due to the $\operatorname*{ReLU}$ function in , our autoencoder learns sparse coding, with most relations having large code values at only two or three dimensions. This sparsity makes it easy to find patterns in the model that to some extent explain the semantics of relations. Figure \[fig:code-heatmap\] shows some examples.
In the first group of Figure \[fig:code-heatmap\], we show a small number of relations that are almost always assigned a near one-hot coding, regardless of initialization. These are high frequency relations joining two large categories (e.g. film and language), which probably constitute the skeleton of a KB.
In the second group, we found the $12$th dimension strongly correlates with `currency`; and in the third group, we found the $4$th dimension strongly correlates with `film`. As for the relation `currency_of_film_budget`, it has large code values at both dimensions. This kind of relation clustering also seems independent of initialization. Intuitively, it shows that the autoencoder may discover similarities between relations and promote indirect parameter sharing among them. Yet, as the autoencoder only reconstructs *approximations* of relation matrices but never constrain them to be exactly equal to the original, relation matrices with very similar codings may still differ considerably. For example, `producer_of_film` and `writer_of_film` have codings of cosine similarity 0.973, but their relation matrices only have[^6] a cosine similarity 0.338.
### Low dimension manifold {#low-dimension-manifold .unnumbered}
[0.48]{} ![By UMAP, relation matrices are embedded into a 2D plane. Colors show frequencies of relations; and lighter color means more frequent.[]{data-label="fig:umap"}](base "fig:"){width="\textwidth"}
[0.48]{} ![By UMAP, relation matrices are embedded into a 2D plane. Colors show frequencies of relations; and lighter color means more frequent.[]{data-label="fig:umap"}](joint "fig:"){width="\textwidth"}
[0.48]{} ![By UMAP, relation matrices are embedded into a 2D plane. Colors show frequencies of relations; and lighter color means more frequent.[]{data-label="fig:umap"}](base+comp "fig:"){width="\textwidth"}
[0.48]{} ![By UMAP, relation matrices are embedded into a 2D plane. Colors show frequencies of relations; and lighter color means more frequent.[]{data-label="fig:umap"}](joint+comp "fig:"){width="\textwidth"}
In order to visualize the relation matrices learned by our joint and base models, we use UMAP[^7] [@2018arXivUMAP] to embed ${\bm{M}}_r$ into a 2D plane[^8]. We use relation matrices trained on FB15k-237, and compare models trained by the same number of epochs. The results are shown in Figure \[fig:umap\].
We can see that Figure \[subfig:umap-base\] and Figure \[subfig:umap-base+comp\] are mostly similar, with high frequency relations scattered randomly around a low frequency cluster, suggesting that they come from various directions of a high dimension space, with frequent relations probably being pulled further by the training updates. On the other hand, in Figure \[subfig:umap-joint\] and Figure \[subfig:umap-joint+comp\] we found less frequent relations being clustered with frequent ones, and multiple traces of low dimension structures. It suggests that joint training with an autoencoder indeed drives relations toward a low dimension manifold. In addition, Figure \[subfig:umap-joint+comp\] shows different structures against Figure \[subfig:umap-joint\], which we conjecture could be related to compositional constraints discovered by compositional training.
### Compositional constraints {#sec:compositionalconstraints .unnumbered}
In order to directly evaluate a model’s ability to find compositional constraints, we extracted from FB15k-237 a list of $(r_1/r_2, r_3)$ pairs such that $r_1/r_2$ matches $r_3$. Formally, the list is constructed as below. For any relation $r$, we define a *content set* $C(r)$ as the set of $(h,t)$ pairs such that $\langle h,r,t\rangle$ is a fact in the KB. Similarly, we define $C(r_1/r_2)$ as the set of $(h,t)$ pairs such that $\langle h,r_1/r_2,t\rangle$ is a path. We regard $(r_1/r_2, r_3)$ as a compositional constraint if their content sets are similar; that is, if $\lvert C(r_1/r_2)\cap C(r_3) \rvert\geq 50$ and the Jaccard similarity between $C(r_1/r_2)$ and $C(r_3)$ is $\geq 0.4$. Then, after filtering out degenerated cases such as $r_1=r_3$ or $r_2=r_1^{-1}$, we obtained a list of 154 compositional constraints, e.g.\
(`currency_of_country`/`country_of_film`, `currency_of_film_budget`).
Model MR MRR
---------------------------------------------------------- ---------------- ---------------------
<span style="font-variant:small-caps;">joint+comp</span> **130$\pm$27** **.0481$\pm$.0090**
<span style="font-variant:small-caps;">base+comp</span> 150$\pm$3 .0280$\pm$.0010
<span style="font-variant:small-caps;">RandomM2</span> 181$\pm$19 .0356$\pm$.0100
: Performance at discovering compositional constraints extracted from FB15k-237[]{data-label="tab:compositional-constraints"}
For each compositional constraint $(r_1/r_2, r_3)$ in the list, we take the matrices ${\bm{M}}_1$, ${\bm{M}}_2$ and ${\bm{M}}_3$ corresponding to $r_1$, $r_2$ and $r_3$ respectively, and rank ${\bm{M}}_3$ according to its cosine similarity with ${\bm{M}}_1{\bm{M}}_2$, among all relation matrices. Then, we calculate MR and MRR for evaluation. We compare the <span style="font-variant:small-caps;">joint+comp</span> model to <span style="font-variant:small-caps;">base+comp</span>, as well as a randomized baseline where $M_2$ is selected randomly from the relation matrices in <span style="font-variant:small-caps;">joint+comp</span> instead (<span style="font-variant:small-caps;">RandomM2</span>). The results are shown in Table \[tab:compositional-constraints\]. We have evaluated 5 different random initializations for each model, trained by the same number of epochs, and we report the mean and standard deviation. We verify that <span style="font-variant:small-caps;">joint+comp</span> performs better than <span style="font-variant:small-caps;">base+comp</span>, indicating that joint training with an autoencoder indeed helps discovering compositional constraints. Furthermore, the random baseline <span style="font-variant:small-caps;">RandomM2</span> tests a hypothesis that joint training might be just clustering $M_3$ and $M_1$ here, to the extent that $M_3$ and $M_1$ are so close that even a random $M_2$ can give the correct answer; but as it turns out, <span style="font-variant:small-caps;">joint+comp</span> largely outperforms <span style="font-variant:small-caps;">RandomM2</span>, excluding this possibility. Thus, joint training performs better not simply because it clusters relation matrices; it learns compositions indeed.
Losses and Gains
----------------
In the KBC task, where are the losses and what are the gains of different settings? With additional evaluations, we show **(i)** some crucial settings for the base model, and **(ii)** joint training with an autoencoder benefits more from compositional training.
### Crucial settings for the base model {#sec:crucialsettings .unnumbered}
MR MRR H10
---------------------- --------- ---------- ----------
**214** **.338** **52.5**
no normalization 309 .326 49.9
no regularizer 400 .328 51.3
pure Gaussian 221 .336 52.1
unigram distribution 215 .324 50.6
: Ablation of the four settings of the base model as described in Sec.\[sec:trainingbase\][]{data-label="tab:crucial-settings"}
It is noteworthy that our base model already achieves strong results. This is due to several detailed but crucial settings as we discussed in Sec.\[sec:trainingbase\]; Table \[tab:crucial-settings\] shows their gains on the FB15k-237 validation data. The most dramatic improvement comes from the regularizer that drives matrices to orthogonal.
### Gains with compositional training {#sec:gainscomptrain .unnumbered}
One can force a model to focus more on (longer) compositions of relations, by sampling longer paths in compositional training. Since joint training with an autoencoder helps discovering compositional constraints, we expect it to be more helpful when the sampled paths are longer. In this work, path lengths are sampled from a Poisson distribution, we thus vary the mean $\lambda$ of the Poisson to control the strength of compositional training. The results on FB15k-237 are shown in Table \[tab:ablation-ae-comp\].
We can see that, as $\lambda$ gets larger, MR improves much but MRR slightly drops. It suggests that in FB15k-237, composition of relations might mainly help finding more appropriate candidates for a missing entity, rather than pinpointing a correct one. Yet, joint training improves base models even more as the paths get longer, especially in MR. It further supports our conjecture that joint training with an autoencoder may strongly interact with compositional training.
----------------------------------------------------- ----- -------- ----------- --------- -------- ----------- ---------
(lr)[3-5]{} (l)[6-8]{} MR MRR H10 MR MRR H10
<span style="font-variant:small-caps;">base</span> 0 209 .341 52.9 215 .337 52.3
<span style="font-variant:small-caps;">joint</span> 0 +1 -.001 -.2 **-3** -.001 0
<span style="font-variant:small-caps;">base</span> 0.5 204 .337 52.2 211 .332 51.7
<span style="font-variant:small-caps;">joint</span> 0.5 **-3** **+.002** **+.1** +1 **+.002** **+.2**
<span style="font-variant:small-caps;">base</span> 1.0 191 .334 52.0 203 .328 51.5
<span style="font-variant:small-caps;">joint</span> 1.0 **-5** **+.002** -.1 **-6** **+.003** **+.1**
----------------------------------------------------- ----- -------- ----------- --------- -------- ----------- ---------
: Evaluation of <span style="font-variant:small-caps;">base</span> and gains by <span style="font-variant:small-caps;">joint</span>, on FB15k-237 with different strengths of compositional training. Bold numbers are improvements.[]{data-label="tab:ablation-ae-comp"}
Conclusion
==========
We have investigated a dimension reduction technique which trains a KB embedding model jointly with an autoencoder. We have developed new training techniques and achieved state-of-the-art results on several KBC tasks with strong improvements in Mean Rank. Furthermore, we have shown that the autoencoder learns low dimension sparse codings that can be easily explained; the joint training technique drives high-dimensional data toward low dimension manifolds; and the reduction of dimensionality may interact strongly with composition, help discovering compositional constraints and benefit from compositional training. We believe these findings provide insightful understandings of KB embedding models and might be applied to other neural networks beyond the KBC task.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by JST CREST Grant Number JPMJCR1301, Japan. We thank Pontus Stenetorp, Makoto Miwa, and the anonymous reviewers for many helpful advices and comments.
Out-of-vocabulary Entities in KBC
=================================
Occasionally, a KBC test set may contain entities that never appear in the training data. Such out-of-vocabulary (OOV) entities pose a challenge to KBC systems; while some systems address this issue by explicitly learn an OOV entity vector [@dettmers2018conve], our approach is described below. For an incomplete triple ${\langle h, r, ? \rangle}$ in the test, if $h$ is OOV, we replace it with the most frequent entity that has ever appeared as a head of relation $r$ in the training data. If the gold tail entity is OOV, we use the zero vector for computing the score and the rank of the gold entity.
Usually, OOV entities are rare and negligible in evaluation; except for the WN18RR test data which contains about 6.7% triples with OOV entities. Here, we also report adjusted scores on WN18RR in the setting that all triples with OOV entities are removed from the test. The results are shown in Table \[tab:wn18rr-remove-oov\].
Model MR MRR H10
---------------------------------------------------------- ---------- ---------- ----------
<span style="font-variant:small-caps;">joint</span> **3317** **.493** **57.2**
<span style="font-variant:small-caps;">base</span> 3435 .492 56.7
<span style="font-variant:small-caps;">joint+comp</span> **1507** **.367** **58.7**
<span style="font-variant:small-caps;">base+comp</span> 1629 .332 58.0
: Adjusted scores on WN18RR.[]{data-label="tab:wn18rr-remove-oov"}
[^1]: It is noteworthy that similar compositional constraints apply to most modeling schemes of relations, not just matrices.
[^2]: Similarly, we set separate step counters for all head and tail entities, and the autoencoder as well.
[^3]: The source code and trained models are publicly released at <https://github.com/tianran/glimvec>, where we also show the mean performance and deviations of multiple random initializations, to give a more complete picture.
[^4]: <https://github.com/thunlp/KB2E>
[^5]: <https://github.com/mnick/holographic-embeddings>
[^6]: Cosine similarity 0.338 is still high for matrices, due to the high dimensionality of their parameter space.
[^7]: <https://github.com/lmcinnes/umap>
[^8]: UMAP is a recently proposed manifold learning algorithm based on the fuzzy topological structure. We also tried t-SNE [@maaten2008visualizing] but found UMAP more insightful.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For models in which dark matter annihilation is Sommerfeld-enhanced, the annihilation cross section increases at low relative velocities. Dwarf spheroidal galaxies (dSphs) have low characteristic dark matter particle velocities and are thus ideal candidates to study such models. In this paper we model the dark matter phase space of dSphs as isotropic and spherically-symmetric, and determine the $J$-factors for several of the most important targets for indirect dark matter searches. For Navarro-Frenk-White density profiles, we quantify the scatter in the $J$-factor arising from the astrophysical uncertainty in the dark matter potential. We show that, in Sommerfeld-enhanced models, the ordering of the most promising dSphs may be different relative to the standard case of velocity-independent cross sections. This result can have important implications for derived upper limits on the annihilation cross section, or on possible signals, from dSphs.'
author:
- 'Kimberly K. Boddy$^{1,}$[^1]'
- 'Jason Kumar$^{1,}$[^2]'
- 'Louis E. Strigari$^{2,}$[^3]'
- 'Mei-Yu Wang$^{2,}$[^4]'
title: 'Sommerfeld-Enhanced $J$-Factors For Dwarf Spheroidal Galaxies'
---
Introduction
============
A major strategy for the indirect detection of dark matter is the search for photons arising from dark matter annihilation in dwarf spheroidal galaxies (dSphs) [@Conrad:2015bsa]. The dark matter halo masses of dSphs are well constrained from stellar kinematics [@McConnachie:2012vd; @Walker:2012td; @Battaglia:2013wqa; @Strigari:2013iaa], and the systematic uncertainties associated with the expected backgrounds [@Winter:2016wmy] are relatively small compared with other astrophysical targets and channels used in indirect detection. Null detection results from Fermi-LAT place strong limits on the dark matter annihilation cross section for particles with mass $\lesssim 100$ GeV [@Geringer-Sameth:2014qqa; @Ackermann:2015zua], ruling out thermal relic dark matter for velocity-independent cross sections to some final states. Targeted ground-based observatories provide the most stringent limits for masses $\gtrsim 1$ TeV [@Ahnen:2016qkx].
In determining the flux of photons arising from dark matter annihilation, the main astrophysical dependence is encapsulated in the $J$-factor of the target. If the annihilation cross section $\sigma_A v$ is velocity-independent, the $J$-factor is simply an integral over the line-of-sight and over a given angular region of the square of the dark matter density profile of the target. With this assumption the $J$-factor is independent of the underlying particle physics such as the dark matter mass and cross section, and furthermore, it is independent of the particular phase space distribution of the dark matter. Many authors have determined the $J$-factors from the stellar kinematics of dSphs under the assumption of a velocity-independent annihilation cross section [@Essig:2010em; @Martinez:2013els; @Geringer-Sameth:2014yza; @Bonnivard:2015xpq; @Evans:2016xwx; @Sanders:2016eie].
However, from a theoretical perspective, the annihilation cross section may be velocity dependent; for example, theoretically well-studied models have $p$-wave suppressed dark matter annihilation ($\sigma_A v \propto v^2$). Additionally, it has been long appreciated that there are some dark matter models in which dark matter annihilation exhibits a [*Sommerfeld enhancement*]{} at low relative velocities [@ArkaniHamed:2008qn]. If the annihilation cross section is velocity-dependent, the photon flux arising from dark matter annihilation does in fact depend on the dark matter velocity distribution, and the astrophysical dependence cannot be entirely factorized from the particle physics [@Robertson:2009bh; @Ferrer:2013cla]. However, there has not yet been a systematic study of $J$-factors for velocity-dependent cross sections.
In this paper we determine the analog of the $J$-factor which is relevant for Sommerfeld-enhanced dark matter annihilation. We use a simple isotropic and spherically-symmetric model[^5] for the dark matter phase space distribution of the dSphs, which — under the assumption of a model for the gravitational potential — is constrained by the measured stellar velocity distributions. We show that this new astrophysics factor, denoted as $J_S$, depends on two parameters which are determined by the detailed particle physics of dark matter annihilation.
The results that we present have important practical applications for interpreting limits on (or establishing possible detections of) a dark matter annihilation signal from dSphs. For example, if we consider a single dSph target, the $J$-factor is the quantity which allows one to translate a statistical bound on (or an observed excess due to) the number of photons arising from dark matter annihilation into a corresponding limit on (or value for) the dark matter annihilation cross section. Similarly, a determination of the Sommerfeld-enhanced $J$-factor would allow one to translate observations of the photon flux into preferred and/or excluded values of the Sommerfeld-enhanced cross section.
In addition, we may consider the results from a combination of multiple dSph targets. If one observes multiple targets, the relative ordering of the $J$-factors also provides an important consistency check for the dark matter interpretation of any potentially observed excess. For a given dark matter annihilation cross section, the expected flux of photons arising from dark matter annihilation scales with the $J$-factor [@Fermi-LAT:2016uux]. Thus, if an excess is observed in one dwarf spheroidal target, one would also expect excesses in other targets with larger $J$-factors; a failure to see such excesses would draw into question the consistency of the dark matter interpretation of the photon signal. However, since the different dwarf spheroidal galaxies can have very different velocity dispersions, the relative ordering of the Sommerfeld-enhanced $J$-factors may be quite different from that of the ordinary $J$-factor. Thus, a pattern of excesses in the gamma-ray emissions of many dwarf spheroidal galaxies, which may appear inconsistent with velocity-independent dark matter annihilation, may still be consistent with Sommerfeld-enhanced dark matter annihilation.
The plan of this paper is as follows. In section \[sec:distribution\], we review the formalism for obtaining the dark matter velocity distribution from stellar observations. In section \[sec:sommerfeld\], we review the theoretical considerations underlying the Sommerfeld enhancement of dark matter annihilation and derive an expression for the Sommerfeld-enhanced $J$-factor, $J_S$. In section \[sec:results\] we present our results for $J_S$ for several dwarf spheroidal galaxies. We conclude with a discussion of our results in section \[sec:conclusion\].
Dark matter distribution function and density profiles {#sec:distribution}
======================================================
In a dSph, the positions in the plane of the sky and the line-of-sight velocities of stars are resolved, leading to a measurement of the projected stellar velocity distribution. For the analysis in this paper, we are interested in the 3D dark matter velocity distribution, which is not necessarily the same as the stellar velocity distribution. To determine this distribution, we use constraints on the gravitational potentials of dSphs, in combination with well-motivated theoretical assumptions.
To calculate the dark matter velocity distribution, we assume the orbits of the dark matter particles are isotropic and the potential is spherically-symmetric. This approximation is justified when examining satellite galaxies in cosmological simulations in which the star particles have ratios of tangential-to-radial velocity anisotropy in the range $\sim 0.8-1.3$ [@Campbell:2016vkb]. Additional studies of the velocity anisotropy profiles of subhalos in dark matter-only simulations are consistent with this range out to the subhalo virial radius [@Vera-Ciro:2014ita]. Under these assumptions, we can use the Eddington formula for the isotropic distribution function $$f_{\textrm{DM}}(\epsilon) = \frac{1}{\sqrt{8}\pi^2} \int_\epsilon^0
\frac{d^2 \rho_{\textrm{DM}}}{d\Psi^2}\frac{d\Psi}{\sqrt{\epsilon - \Psi }} \ ,
\label{eq:eddington}$$ which is a function of energy alone. Here, $\rho_{\textrm{DM}}(r)$ is the dark matter density profile, and the function $\Psi(r) < 0$ is the spherically-symmetric gravitational potential, which depends on the parameters of the dark matter density profile. The gravitational binding energy per mass of a dark matter particle is $\epsilon = v^2/2 + \Psi(r) < 0$, and $v$ is the modulus of the velocity of a dark matter particle. Thus, the quantity $f_{\textrm{DM}}(\epsilon)$ is implicitly a function of $v$ and $r$ and is equivalent to the velocity distribution function $f(r,v) \equiv f_{\textrm{DM}}(\epsilon(r,v))$. The velocity distribution obeys the normalization $$\rho_{\textrm{DM}}(r) = 4\pi \int_0^{v_{\textrm{esc}}} dv \, v^2 f(r,v) \ ,$$ where $v_{\textrm{esc}}(r) = \sqrt{-2\Psi(r)}$ is the maximum velocity obtainable for a gravitationally-bound particle at radius $r$.
We assume a Navarro-Frenk-White (NFW) form of the dark matter density profile: $\rho_{\textrm{NFW}}(r) = \rho_s/[(r/r_s) (1+r/r_s)^2]$. The NFW profile can be a cast as a function of the scale density and scale radius $(\rho_s,r_s)$, or the maximum circular velocity and the radius of maximum circular velocity (${V_\textrm{max}}, {r_\textrm{max}}$). These quantities are related via $${r_\textrm{max}}= 2.16~r_s \ , \qquad {V_\textrm{max}}= 0.465 ~\sqrt{4 \pi G \rho_s r_s^2} \ .
\label{eq:rmaxvmax}$$ Hence, the dark matter velocity distribution $f(r,v)$ depends only on the parameters ($\rho_s,r_s$) of the NFW profile, or (${V_\textrm{max}}, {r_\textrm{max}}$). The properties of dark matter distributions with cuspy NFW profiles have been studied in e.g. Refs. [@Widrow:2000; @Evans:2005tn].
The parameters ($\rho_s,r_s$) or (${V_\textrm{max}}, {r_\textrm{max}}$) can be bound by observations of the average stellar line-of-sight velocity distribution for each dSph. In order to do so, we define the stellar distribution function as $f_\star$. Again under the assumption of spherical symmetry and isotropy for the stellar distribution, $f_\star$ can be calculated from Eq. given a stellar density profile, $\rho_\star$, which we take to be a Plummer profile with the best-fit half-light radius $r_h$ for each dSph [@McConnachie:2012vd]. With our definition of $f_\star$, the average stellar velocity dispersion at a radius $r$ is $$\langle \sigma_\star^2 (r)\rangle =
\frac{\int v_\star^4 f_\star(v_\star,r)\, dv_\star}
{\int v_\star^2 f_\star(v_\star,r)\, dv_\star} \ ,
\label{eq:avgsigma}$$ where $v_\star$ refers to the velocity of the stars. To obtain the quantity that most closely approximates the observed projected velocity dispersion averaged over the entire galaxy, we then calculate $$3\times\langle \sigma_{\star}^2 \rangle =
\frac{\int \langle \sigma_{\star}^2(r) \rangle \rho_{\star} dV}
{\int \rho_{\star} dV} \ .
\label{eq:avgsigmaprojected}$$
---------------- -------------------------------- ------------------------------ --------- --------- --------------- ---------------
dSph Ref. $\langle\sigma_\star\rangle$ $r_h$ $D$ $V_{\rm max}$ $r_{\rm max}$
\[km/s\] \[kpc\] \[kpc\] \[km/s\] \[kpc\]
Coma Berenices [@Simon:2007dq] $4.6$ $0.077$ $44.0$ 9.8 0.38
Ursa Minor [@McConnachie:2012vd] $9.5$ $0.181$ $76.0$ 24.1 1.32
Draco [@McConnachie:2012vd] $9.1$ $0.221$ $76.0$ 17.7 0.86
Segue 1 [@Simon:2010ek] $3.9$ $0.029$ $23.0$ 16.2 0.76
Reticulum II [@Simon:2015; @Walker:2015mla] $3.3$ $0.055$ $32.0$ 7.6 0.28
---------------- -------------------------------- ------------------------------ --------- --------- --------------- ---------------
We use the observed values of $\sigma_{\star}^2$ to constrain the parameters (${V_\textrm{max}}, {r_\textrm{max}}$) of several dSphs. We focus on five of the most promising dSphs for indirect detection: Segue 1, Reticulum II, Coma Berenices, Draco, and Ursa Minor. The stellar velocity dispersions, half-light radii, and the distance to the dSph are summarized in Table \[tb:orbit\]. For these five dSphs, which have the largest ordinary $J$-factors, Fig. \[fig:rmaxvmaxgrid\] shows the (${V_\textrm{max}}, {r_\textrm{max}}$) parameter space that is consistent with the observed average velocity dispersion and its measured uncertainty.
![The observable and theoretically-preferred regions in the ${V_\textrm{max}}- {r_\textrm{max}}$ parameter space for the five dSphs under consideration. Along the solid colored curves are points that match the average stellar velocity dispersion, and the corresponding black dashed curves match the 1$\sigma$ uncertainties. The solid gray curves indicate the median (${V_\textrm{max}},{r_\textrm{max}}$) relation for subhalos [@Martinez:2009jh] in the dark matter-only Aquarius simulation [@Springel:2008cc], and the dashed gray curves represent the scatter in this relation. Dotted gray lines indicate the ordinary $J$-factors, for which the annihilation cross section is assumed to be independent of velocity. The filled circles define the points in parameter space that we use in calculating the Sommerfeld-enhanced $J$-factors.[]{data-label="fig:rmaxvmaxgrid"}](dSph_fv_grid_NFW.pdf)
The range of (${V_\textrm{max}},{r_\textrm{max}}$) parameter space can be further bound by appealing to the results from cosmological simulations. Figure \[fig:rmaxvmaxgrid\] also shows the regions consistent with the (${V_\textrm{max}},{r_\textrm{max}}$) relation for subhalos in the dark matter-only Aquarius simulation (see Fig. 26 of Ref. [@Springel:2008cc]). Specifically, we adopt the relation $\log({r_\textrm{max}}/\mathrm{kpc}) = 1.35\log[{V_\textrm{max}}/(\mathrm{km/s})]-1.75$ from Eq. (16) in Ref. [@Martinez:2009jh], which provides a good description of the Aquarius results, with a uniform scatter of $\sigma_{\log({r_\textrm{max}}/\mathrm{kpc})}=0.22$ for the entire range of ${V_\textrm{max}}$. To approximate the observational uncertainty in the $J$-factor within the context of our isotropic and spherically-symmetric NFW model, we combine the results from both the measured velocity dispersion and theoretical (${V_\textrm{max}}, {r_\textrm{max}}$) relation. Specifically, we consider the area in Fig. \[fig:rmaxvmaxgrid\] that is defined by the intersection of the observed velocity dispersion band and the subhalo (${V_\textrm{max}},{r_\textrm{max}}$) band. From this area, we define five points: the four points that represent the intersection of the outer boundaries of each band, and the central point where the central values of the velocity dispersion and the (${V_\textrm{max}},{r_\textrm{max}}$) lines cross. For most of the dSphs, this area has ${V_\textrm{max}}\lesssim 35$ km/s, corresponding to plausible subhalo hosts of dSphs. The only exception is Ursa Minor, for which we place an upper bound on ${V_\textrm{max}}$ such that ${r_\textrm{max}}< 3$ kpc, which corresponds to an estimate for the dark matter tidal radius. For all dSphs, the central (${V_\textrm{max}}, {r_\textrm{max}}$) values and stellar velocity dispersions are also listed in Table \[tb:orbit\]. In Fig. \[fig:fv\_dSph\] we show the velocity distribution at the half-light radius for the five dSphs, using the central point in Fig. \[fig:rmaxvmaxgrid\].
Note that the values of the $J$-factor indicated in Fig. \[fig:rmaxvmaxgrid\] assume an NFW profile and isotropic orbits. These $J$-factors may differ from previous calculations in the literature [@Martinez:2013els; @Geringer-Sameth:2014yza; @Bonnivard:2015xpq; @Evans:2016xwx; @Sanders:2016eie], which allow for non-NFW profiles, anisotropic stellar velocity dispersions, and assume a Gaussian likelihood for the stellar velocities. In order to consistently determine the impact of the Sommerfeld-enhanced $J$-factors, we must compare them against the $J$-factors represented by the gray dotted curves in Fig. \[fig:rmaxvmaxgrid\]. Although it is possible to consider the impact of both non-NFW and anisotropic models, the case of anisotropic models would require extending beyond the approximation of Eq. .
![Plots of $v^2 f(v)$, evaluated at the half-light radius, for the five dSphs using the central point of Fig. \[fig:rmaxvmaxgrid\]. All curves are normalized as $\int v^2 f(v) dv = 1$.[]{data-label="fig:fv_dSph"}](fv_dSph.pdf)
Sommerfeld-Enhanced Dark Matter Annihilation {#sec:sommerfeld}
============================================
Sommerfeld Enhancement for a Yukawa Potential
---------------------------------------------
We consider two dark matter particles $X$ that interact via the exchange of a light mediator $\phi$ of mass $m_\phi$ with a coupling $g_X=\sqrt{4\pi\alpha_X}$. For a scalar or vector mediator, the attractive force between nonrelativistic dark matter particles is described by a Yukawa potential $$V(r) = -\frac{\alpha_X}{r} e^{-m_\phi r} \ .$$ Although dark matter annihilation can essentially be thought of as a contact interaction, the long range of the potential causes distortion of the incoming dark matter particles’ wave function $\psi(\vec{r})$ (which is asymptotically a plane wave) at nonzero separation $\vec{r}$. As a result, the annihilation cross section is enhanced by a factor $S \equiv |\psi(0)|^2$. We write the annihilation cross section in the absence of the long-range Yukawa interaction as $(\sigma_A v_{\textrm{rel}})_0$, which we assume is non-vanishing in the limit $v_{\textrm{rel}}\rightarrow 0$, where $v_{\textrm{rel}}$ is the relative velocity of the dark matter particles. Thus, the Sommerfeld-enhanced cross section may be written as $(\sigma_A v_{\textrm{rel}}) = (\sigma_A v_{\textrm{rel}})_0 \times S$.
We briefly describe the quantum mechanics behind the Sommerfeld enhancement and refer the reader to Ref. [@ArkaniHamed:2008qn] for a more detailed review. With the central Yukawa potential, the annihilation process is determined by solving a 1D radial Schrödinger equation for the relative motion of the dark matter particles. We recast the physical parameters of the theory into the dimensionless quantities $$\epsilon_v \equiv \frac{v}{\alpha_X} \quad\textrm{and}\quad
\epsilon_\phi \equiv \frac{m_\phi}{\alpha_X m_X} \ ,$$ where $v = v_{\textrm{rel}}/2$ is the velocity of the dark matter particles in the center-of-mass frame. Using the dimensionless variable $x=\alpha_X m_X r$, the radial Schrödinger equation becomes $$\chi^{\prime\prime}(x) + \left[\epsilon_v^2 + V(x) \right] \chi(x) = 0 \ ,
\label{eq:SE}$$ with the potential $V(x) = \exp(-\epsilon_\phi x)/x$. We assume the annihilation is $s$-wave and neglect higher partial-wave contributions. For the boundary conditions $\chi(x) = \exp(i\epsilon_v x)$ and $\chi'(x) = i\epsilon_v \chi(x)$ as $x\to\infty$, the Sommerfeld enhancement is $S = \left|\chi(\infty)/\chi(0)\right|^2$.
![Contour plots of the Sommerfeld enhancement $S$ and the ratio of $S^\textrm{Hul}/S$, where $S^\textrm{Hul}$ is the analytic approximation from the Hulthén potential. The analytic approximation is within 10% of the numerical solution for $\epsilon_v > \epsilon_\phi$. For $\epsilon_v < \epsilon_\phi$, there are discrepancies up to a factor of 2 and beyond (up to many orders of magnitude, represented by the dark red and dark blue arrows in the color bar) that correspond to the misalignment of the resonances in the Hulthén and Yukawa solutions.[]{data-label="fig:sommerfeld"}](sommerfeld.pdf)
The scattering solution for the case of a Yukawa potential is not analytically solvable, although there are a variety of techniques for solving Schrödinger equation numerically in different regimes of parameter space. By approximating the Yukawa potential as the Hulthén potential, the resulting Schrödinger equation can be solved analytically, yielding the Sommerfeld enhancement [@Feng:2010zp] $$S \simeq \frac{\pi}{\epsilon_v}
\frac{\sinh \left(\frac{2 \pi \epsilon_v}{\pi^2 \epsilon_\phi / 6} \right)}
{\cosh \left(\frac{2 \pi \epsilon_v}{\pi^2 \epsilon_\phi / 6} \right)
- \cos \left( 2\pi \sqrt{\frac{1}{\pi^2 \epsilon_\phi / 6}
- \frac{\epsilon_v^2}{(\pi^2 \epsilon_\phi / 6)^2} } \right)} \ .
\label{eq:S}$$ The analytic approximation is typically within $\sim 10\%$ of the result found from the numerical calculation, as seen in Fig. \[fig:sommerfeld\]. Substantial differences for $\epsilon_v < \epsilon_\phi$ arise in narrow regions of parameter space around specific values of $\epsilon_\phi$, due to the location of the resonances of the Hulthén potential (described below) not quite lining up to those of the Yukawa potential. Nonetheless, the analytic approximation exhibits the same generic features as that from the full numerical solution. As expected, in the limit where $\phi$ is heavy ($\epsilon_\phi \gg 1$), we find $S \rightarrow 1$, and there is no Sommerfeld enhancement. In the limit $\epsilon_\phi \ll \epsilon_v$, we find $S \rightarrow \pi/ \epsilon_v = \pi \alpha_X /v$, which is the standard result for Sommerfeld enhancement in the presence of a Coulomb force. In the limit $\epsilon_v \ll \epsilon_\phi$, we have $$S \simeq \frac{12\, \alpha_X m_X}{m_\phi} = \frac{12}{\epsilon_\phi} \ .$$ However, in this regime, there are certain values of $\epsilon_\phi$ for which resonances occur: $$\epsilon_\phi \simeq \frac{6}{\pi^2 n^2}
\quad \textrm{for}\ n \in \mathbb{Z}^+ \ ,$$ where the argument of the cosine in the denominator of Eq. vanishes. Equivalently, the resonances occur for $m_\phi \simeq 6 \alpha_X m_X / (\pi^2 n^2)$, at which $$S \simeq \frac{\alpha_X^2}{v^2 n^2} = \frac{1}{\epsilon_v^2 n^2} \ .$$
In the limit of $v\to 0$, the resonant enhancements become large and unphysical, because we have thus far neglected the effects of zero-energy bound-state formation and decay [@Hisano:2004ds; @Feng:2010zp]. By inserting a $\delta$-function, which is sufficient for $s$-wave processes, into the Schrödinger equation to represent the short-range interaction, these resonances are regularized and yield cross sections that obey partial-wave unitarity bounds [@Blum:2016nrz]. For a perturbative, short-range annihilation cross section $(\sigma_A v_{\textrm{rel}})_0 \ll 4\pi/m_X^2 v_0$, the standard Sommerfeld enhancement $S$ is modified as follows: $$\tilde{S}(v) = \frac{S(v)}{\left|1-i\epsilon_v\alpha_X
\frac{m_X^2}{8\pi}(\sigma_A v_{\textrm{rel}})_0 \left[T(v)+iS(v)\right]\right|^2} \ ,
\label{eq:Smod}$$ where $T$ is another quantity that encodes the effect of the long-range Yukawa force on the wave function and depends on the renormalization of the $\delta$-function. We have neglected the real part of the inverse-scattering length, which corresponds to setting the short-range scattering cross section to be $\sigma_{\textrm{sc},0} = {(\sigma_A v_{\textrm{rel}})_0}^2 (m_X/2)^2 / (4\pi)$. Such an identification may arise in a nonrelativistic theory in which the optical theorem relates the annihilation cross section to the imaginary part of the forward-scattering amplitude [@Hisano:2002fk]. For $\alpha_X \ll 1$, the denominator in Eq. approaches 1, except for the region very close to the resonance. We have verified that for small $\alpha_X$, the correction due to bound states is essentially limited to the peak of the resonance but has little effect otherwise. For our main analysis, we set $\alpha_X=10^{-2}$ and simply use $S$, allowing us to avoid the issue of model-dependence in choosing the form of $(\sigma_A v_{\textrm{rel}})_0$.
It is worth noting that, around the epoch of recombination, it is expected that the typical dark matter particle velocity would be much smaller than it is in the current epoch. One should thus worry that models with Sommerfeld-enhanced annihilation that could potentially be observed with future experiments would already be ruled out by constraints from the Plank experiment on dark matter annihilation in the early Universe [@Ade:2015xua], at which time the Sommerfeld-enhancement could be much larger (see, for example, [@Finkbeiner:2010sm]). Constraints on dark matter annihilation in the early Universe will not be enhanced, relative to constraints arising from observations of dSphs, provided the Sommerfeld enhancement saturates for velocities not far below the typical dark matter velocity in a dSph, which is ${\cal O}(1)~{\rm km} /{\rm s}$. This is equivalent to the constraint $\epsilon_\phi \gtrsim 10^{-6} \alpha_X^{-1}$.
Relating the Sommerfeld Enhancement to the Photon Flux
------------------------------------------------------
If the dark matter particle is its own anti-particle, the differential photon flux produced by dark matter annihilation is $$\frac{d\Phi}{dE_\gamma} =
\frac{1}{4\pi} \frac{dN}{dE_\gamma} \int_{\Delta\Omega} d\Omega \int d\ell
\int d^3 v_1 \frac{f(r(\ell, \Omega), \vec{v}_1)}{m_X}
\int d^3 v_2 \frac{f(r(\ell, \Omega), \vec{v}_2)}{m_X}
\, \frac{(\sigma_A |\vec{v}_1 - \vec{v}_2|) }{2} \ ,$$ where $\ell$ is the distance along the line of sight and $dN/dE_\gamma$ is the photon spectrum produced by a single annihilation process. The angular integration over $\Delta\Omega$ covers a region in the sky encompassing a particular dSph. If the dark matter particle and anti-particle are distinct and equally abundant, this flux would be suppressed by an additional factor of $1/2$.
Expressing the Sommerfeld-enhanced annihilation cross section as $(\sigma_A v_{\textrm{rel}}) = (\sigma_A v_{\textrm{rel}})_0 S(v_{\textrm{rel}}/2)$, where $v_{\textrm{rel}}= |\vec{v}_1 - \vec{v}_2|$, we have $$\frac {d\Phi}{dE_\gamma} = J_S(\Delta\Omega)
\frac{(\sigma_A v_{\textrm{rel}})_0}{8\pi m_X^2} \frac{dN}{dE_\gamma} \ ,$$ where $$J_S (\Delta\Omega) \equiv \int_{\Delta\Omega} d\Omega \int d\ell
\int d^3v_1 f(r(\ell, \Omega), \vec{v}_1)
\int d^3v_2 f(r(\ell, \Omega), \vec{v}_2) \,
S(|\vec{v}_1-\vec{v}_2|/2)$$ is the Sommerfeld-enhanced $J$-factor, which encapsulates all of the dependence of the photon flux on the dark matter distribution of the target. Note that since $S$ is a function of the velocity $|\vec{v}_1-\vec{v}_2|/2$, it depends on the angle between $\vec{v}_1$ and $\vec{v}_2$, as well as the magnitudes $v_1$ and $v_2$. In the limit $S \rightarrow 1$ (*i.e.*, no Sommerfeld enhancement) we recover the ordinary result $$J_S (\Delta\Omega) \rightarrow J(\Delta\Omega) =
\int_{\Delta\Omega} d\Omega \int d\ell \left[\rho(r(\ell, \Omega))\right]^2 \ .$$ Our main goal is to determine $J_S (\Delta \Omega)$ for a variety of dwarf spheroidal galaxies. In general, $J_S$ depends on two parameters of the particle physics model: $\epsilon_\phi$ and $\alpha_X$.
Results {#sec:results}
=======
Sommerfeld-enhanced $J$-factors
-------------------------------
We calculate the Sommerfeld-enhanced $J$-factor for five of the most promising dSphs for indirect detection: Segue 1, Reticulum II, Coma Berenices, Draco, and Ursa Minor. In Fig. \[fig:jfactor\_alpha\], we plot $J_S$ as a function of $\epsilon_\phi$ for Reticulum II for different values of $\alpha_X$, assuming $\Delta \Omega = 2.4 \times 10^{-4}$ (*i.e.*, a cone half-angle of $0.5^\circ$) and nominal values of the NFW parameters given in Table \[tb:orbit\]. As expected, away from the resonances, we find that $J_S$ scales as $\alpha_X$. But near resonances, $S$ scales as $\alpha_X^2 \epsilon_\phi / v^2$; the magnitude of the resonances are thus suppressed for small $\epsilon_\phi$ and essentially disappear for small $\epsilon_\phi$ and $\alpha_X$. In accordance with Sec. \[sec:sommerfeld\], we henceforth focus on the benchmark case $\alpha_X = 10^{-2}$.
![Sommerfeld-enhanced $J$-factors for Reticulum II, using the central point in Fig. \[fig:rmaxvmaxgrid\]. We vary $\alpha_X$ over an order of magnitude and choose $\Delta\Omega \approx 2.4 \times 10^{-4}$, corresponding to a cone with a half-angle of $0.5^\circ$. The left panel shows the values of $J_S$, while the right panel shows the ratio of $J_S$ to the ordinary $J$-factor calculation with no enhancement.[]{data-label="fig:jfactor_alpha"}]({jfactor_Ret2_psi+0.50deg_alpha10all}.pdf)
In Fig. \[fig:jfactor\_all\], we plot $J_S$ as a function of $\epsilon_\phi$ for all five dSphs, assuming $\alpha_X = 10^{-2}$ and $\Delta \Omega = 2.4 \times 10^{-4}$. The width of the bands show the uncertainty in choosing the NFW parameters, as defined by the five points in Fig. \[fig:rmaxvmaxgrid\]. The gray line shows the central point for each dSph, which represents the NFW parameters that match both the average stellar velocity dispersion and the median (${V_\textrm{max}},{r_\textrm{max}}$) relation from simulations. As expected, $J_S \rightarrow J$ in the limit $\epsilon_\phi \gg 1$. However, $J_S / J \sim 10^3$ for small $\epsilon_\phi$, while near resonances $J$ and $J_S$ differ by many orders of magnitude; this corresponds to the factor by which sensitivity to $(\sigma_A v_{\textrm{rel}})_0$ is improved by Sommerfeld enhancement.
![Sommerfeld-enhanced $J$-factors with $\alpha=10^{-2}$ and $\Delta\Omega \approx 2.4 \times 10^{-4}$, corresponding to a cone half-angle of $0.5^\circ$. The width of the bands represents the systematic uncertainties we estimate by using the points in Fig. \[fig:rmaxvmaxgrid\]. The NFW parameters for the gray lines represent the central point.[]{data-label="fig:jfactor_all"}]({jfactor_all_psi+0.50deg_alpha10-2.00}.pdf)
It is interesting to note that the relative order of $J_S$ among the dSphs may change as a function of $\epsilon_\phi$. In particular, in the limit of no Sommerfeld enhancement ($\epsilon_\phi \gg 1$), Reticulum II tends to have a smaller $J$-factor than the other dSphs; however, in the limit of $\epsilon_\phi \ll 1$, the $J_S$-factor for Reticulum II seems relatively higher in comparison. In the left panel of Fig. \[fig:jfactor\_cmp\], we show the $J_S$-factors from the gray lines in Fig. \[fig:jfactor\_all\], plotted together for ease of comparison. To emphasize how the astrophysical uncertainty affects the relative ordering, we choose particular points from those listed in Fig. \[fig:rmaxvmaxgrid\] to show in the right panel of Fig. \[fig:jfactor\_cmp\]. These points represent a scenario in which Reticulum II has the smallest ordinary $J$-factor at large $\epsilon_\phi$, but has the largest $J_S$-factor at small $\epsilon_\phi$. Moreover, Reticulum II maintains its status of having the lowest $J_S$ factor in the valleys between resonance peaks, but settles into a higher $J_S$ factor once $\epsilon_\phi$ is small enough and away from the resonant regime.
Although we have chosen a particular value of $\alpha_X$, the relative order of $J_S$ among the dSphs (for a given set of NFW parameters) is unaffected for a different value of $\alpha_X$ at small and large $\epsilon_\phi$. The $J_S$ dependence on $\alpha_X$ is fairly straightforward, as described at the beginning of this section: for $\epsilon_{\phi} \gtrsim 1$, $J_S$ is independent of $\alpha_X$, while for $\epsilon_{\phi} \ll 1$ (but away from resonances), one instead finds $J_S \propto \alpha_X$. Thus, changing the value of $\alpha_X$ scales $J_S$ for all dSphs by the same amount, resulting in no change in relative ordering among the dSphs outside of the resonant regime.
![Comparison of Sommerfeld-enhanced $J$-factors for the five dSphs. In the left panel, we use the central points in Fig. \[fig:rmaxvmaxgrid\], which are also represented by the gray lines in Fig. \[fig:jfactor\_all\]. In the right panel, we use the central points in Fig. \[fig:rmaxvmaxgrid\] for Reticulum II and Draco and the upper-right points in Fig. \[fig:rmaxvmaxgrid\] \[i.e., points with the largest (${V_\textrm{max}}$, ${r_\textrm{max}}$)\] for the remaining dSphs. This specific combination of NFW parameters for the right panel is a concrete example of Reticulum II having the smallest relative $J_S$-factor at large $\epsilon_\phi$, but the largest relative $J_S$-factor at small $\epsilon_\phi$.[]{data-label="fig:jfactor_cmp"}]({jfactor_cmp_psi+0.50deg_alpha10-2.00}.pdf)
An Analytic Approximation to the Determination of $J$ and $J_S$
---------------------------------------------------------------
Interestingly, it is possible to use analytic results to generate simple expressions which determine if the relative order of $J_S$ at small $\epsilon_\phi$ for two dSphs is different from the relative order of $J$. For this purpose, we focus on comparing two limits: the non-enhanced limit ($\epsilon_\phi \gg 1$) and the limit of a Coulomb-like potential ($\epsilon_\phi \ll 1$).
We assume that the density profile may be expressed in the form $$\rho (r) = \rho_s \times \tilde \rho ( r / r_s) \ ,
\label{eq:rho_form}$$ where $\rho_s $ is an overall density scale, and $\tilde \rho$ is a dimensionless quantity which may be expressed as a function of $\tilde r \equiv r / r_s$ only. Thus, $\tilde \rho (\tilde r)$ is a dimensionless function which is independent of the parameters that characterize the dark matter distribution. For the particular case of an NFW profile, we have $\tilde \rho (\tilde r) = \tilde r^{-1} (1+\tilde r)^{-2}$.
If the integration over a solid angle $\Delta \Omega$ is large enough to essentially encompass the entire region of the dSph in which there is significant dark matter annihilation, the $J$-factor can be expressed in terms of an integral over the radial distance from the center of the dwarf, instead of an integral over the line of sight: $$J^\textrm{total} = \frac{1}{D^2} \int dV \, [\rho (r)]^2
= \frac{4\pi}{D^2} \int dr \, r^2 [\rho (r)]^2 \ ,$$ where we have assumed that $\rho (r)$ is negligible unless $r \ll D$. We then find
$$\begin{aligned}
J^\textrm{total} &= \frac {4\pi \rho_s^2 r_s^3}{D^2} C_J \\
C_J &\equiv \int d\tilde r \, \tilde r^2 [\tilde \rho (\tilde r)]^2 \ ,
\end{aligned}$$
where $C_J$ is a dimensionless quantity that depends on the functional form of the dark matter distribution, but is independent of the parameters ($\rho_s$, $r_s$). For an NFW profile, $C_J = 1/3$.
To determine the Sommerfeld-enhanced $J$-factor, we express the dark matter velocity distribution in a scale-invariant form by utilizing the fact that the gravitational potential can be written as
$$\begin{aligned}
\Psi (r) &= G \rho_s r_s^2 \times \tilde \Psi (\tilde r) \\
\tilde \Psi (\tilde r) &\equiv \int_\infty^{\tilde r} dx \, \frac{1}{x^2}
\int_0^x dy \, (4\pi y^2) \tilde \rho (y) \ ,
\end{aligned}$$
where $\tilde{\Psi}$ is a dimensionless function that is independent of the parameters of the dark matter distribution. We define a scale-invariant velocity $\tilde v \equiv (G \rho_s r_s^2)^{-1/2} v$ and a scale-invariant energy per unit mass $\tilde \epsilon = \tilde v^2 /2 + \tilde \Psi$. In terms of these quantities, the dark matter distribution function may be rewritten as
$$\begin{aligned}
f_{\textrm{DM}}(\epsilon)
&= \rho_s (G \rho_s r_s^2)^{-3/2} \tilde f_{\textrm{DM}}(\tilde \epsilon) \\
\tilde f_{\textrm{DM}}(\epsilon)
&\equiv \frac{1}{\sqrt{8}\pi^2} \int_{\tilde \epsilon}^0
\frac{d^2 \tilde \rho}{d \tilde \Psi^2}
\frac{d \tilde \Psi}{\sqrt{\tilde \epsilon - \tilde \Psi}} \ ,
\end{aligned}$$
where $\tilde{f}_{\textrm{DM}}$ is also a dimensionless quantity that is independent of ($\rho_s$, $r_s$), but is implicitly a function of $\tilde r$ and $\tilde v$.
In the limit $\epsilon_\phi \ll 1$, we approximate the Sommerfeld enhancement factor by $S \sim \pi \alpha_X / v$; that is, we assume the contribution to dark matter annihilation arising from the region of phase space with $\epsilon_v \lesssim \epsilon_\phi$ is negligible. By integrating over an angle which encompasses the entire dwarf, we can write $J_S$ as
$$\begin{aligned}
{J_S}^\textrm{total} &\sim
\left(\frac{4\pi \rho_s^2 r_s^3}{D^2}\right) (G\rho_s r_s^2)^{-1/2} C_{J_S} \\
C_{J_S} &\equiv \int d\tilde r \, \tilde r^2
\int d^3 \tilde v_1 \tilde f (\tilde r, \tilde v_1)
\int d^3 \tilde v_2 \tilde f (\tilde r, \tilde v_2)
\left(\frac{2\pi \alpha_X}{|\overrightarrow{\tilde v}_1
-\overrightarrow{\tilde v}_2|} \right) \ ,
\end{aligned}$$
where $C_{J_S}$ is a dimensionless quantity that depends on the functional form of the dark matter distribution, but not on the parameters ($\rho_s$, $r_s$).
In summary, we find $$\begin{aligned}
J^\textrm{total} &\propto \rho_s^2 r_s^3 / D^2 \ , \nonumber\\
{J_S}^\textrm{total} &\propto \rho_s^{3/2} r_s^2 / D^2 \ .
\label{eq:j_js_join}\end{aligned}$$ Thus, for fixed $J^\textrm{total}$, one increases $J_S^\textrm{total}$ by increasing $\rho_s$ and correspondingly decreasing $r_s$. In other words, if two dSphs have distributions with the same functional form and have the same total $J$-factor, then the dwarf with the smaller scale size will have the larger Sommerfeld-enhanced $J$-factor (in the limit $\epsilon_\phi \ll 1$).
It is important to note that these results do not depend on the assumption of an NFW profile; they apply for any choice of density profile, provided the radial dependence can be expressed entirely in terms of the dimensionless variable $\tilde r = r/r_s$. This result does depend on the assumption that the solid angle encompasses the entire dSph. There are several NFW parameter choices we have studied for which this assumption is not true, implying that the specific results we have found for $\Delta \Omega \approx 2.4 \times 10^{-4}$ need not obey these scaling relations precisely. Nevertheless, they provide useful guidance for the general criteria governing the situations in which the ordering of Sommerfeld-enhanced $J$-factors differs from that of non-enhanced $J$-factors.
With this in mind, in Fig. \[fig:j\_js\_join\] we plot the NFW profile parameter space that we have considered for each of the five dSphs. The shaded region for each dSph encompasses the five benchmark NFW parameters shown in Fig. \[fig:rmaxvmaxgrid\]. The parameter space shown is $({V_\textrm{max}}^4/{r_\textrm{max}}/D^2, {V_\textrm{max}}^3/{r_\textrm{max}}/D^2)$, which is equivalent to the parameter space of interest, $(\rho_s^2 r_s^3 / D^2 , \rho_s^{3/2} r_s^2 / D^2)$ from Eq. . This relation between parameters is clear from dimensional analysis: we have ${r_\textrm{max}}\propto r_s$ from the assumed form of $\rho(r)$ in Eq. ; and since ${V_\textrm{max}}^2 = GM(r={r_\textrm{max}})/{r_\textrm{max}}$, where the mass function $M(r)$ merely involves integrating over $\rho$, we find that ${V_\textrm{max}}^2 \propto G \rho_s r_s^2$. Parameter space points yield larger values of $J^\textrm{total}$ as one moves to the right in Fig. \[fig:j\_js\_join\], and larger values of $J_S^\textrm{total}$ (assuming $\epsilon_\phi \ll 1$) as one moves up. Thus, if there exists a parameter point for one dSph which lies above and to the left of a parameter point of another dSph, then for those choices of parameters, the ordering of $J^\textrm{total}$ will differ from that of $J_S^\textrm{total}$; the point to the upper left will have a smaller $J^\textrm{total}$, but a larger $J_S^\textrm{total}$.
![Regions in the $J^\textrm{total}$–${J_S}^\textrm{total}$ parameter space, as defined in Eq. \[eq:j\_js\_join\], for each of the dSphs. The boundaries of each region, and the respective central points, are defined in Fig. \[fig:rmaxvmaxgrid\].[]{data-label="fig:j_js_join"}]({dSph_J_Js_join}.pdf)
As illustrated by Figure \[fig:j\_js\_join\], there are several pairs of dSphs for which the relative ordering of $J$ can be different from the relative ordering of $J_S$ in the limit $\epsilon_\phi \ll 1$. In particular, there are choices of NFW parameters for which Reticulum II has a smaller $J^\textrm{total}$ than either Coma Berenices, Draco, or Ursa Minor; but has a larger $J_S^\textrm{total}$ than all of the others. But there is no choice of parameters for which Reticulum II has a larger $J_S^\textrm{total}$ than Segue 1. However, for Segue 1, the consistent region of parameter space includes points with a relatively large (${V_\textrm{max}}$, ${r_\textrm{max}}$); such points not only yield a relatively small $J_S^\textrm{total} / J^\textrm{total}$, but also a large angular size (especially since Segue 1 is the closest of these five dSphs). Thus, although Segue 1 may always have a larger $J_S^\textrm{total}$ than Reticulum II, there are points in parameter space for which Reticulum II will have the larger $J_S$ (in the $\epsilon_\phi \ll 1$ limit) when integrated over a cone of half-angle $0.5^\circ$, as shown in Figure \[fig:jfactor\_cmp\].
Conclusion {#sec:conclusion}
==========
In this paper we have self-consistently calculated dSph $J$-factors, using a model for the dark matter phase space distribution and gravitational potentials that are constrained from stellar kinematics. Within the context of our spherically-symmetric and isotropic model for the dark matter phase space distribution, we quantify the astrophysical uncertainty in the $J_S$-factors and show that the relative ordering of the most promising $J_S$-factors can be interchanged relative to the standard velocity-independent $J$-factors. This result may have important implications for the interpretation of possible gamma-ray excesses from a dSph [@Geringer-Sameth:2015lua; @Li:2015kag].
The model that we discuss can be seen as a first step in the self-consistent calculation of astrophysical $J$-factors for velocity-dependent annihilation cross sections. A new step in this analysis could involve the determination of $J$-factors for other types of models with a velocity-dependent annihilation cross section [@Zhao:2016xie]. Although certain forms of the velocity dependence, such as $\sigma_A v \propto v^2$, reduce the cross section below the sensitivity of current experiments, these models may become accessible as the sensitivity improves. This will be particularly true as new dSphs continue to be discovered by the Dark Energy Survey [@Drlica-Wagner:2015ufc], and even further into the future by the Large Synoptic Survey Telescope. Accounting for velocity-dependent cross sections will also be a future requirement of numerical methods that scan the theoretical parameter space given large spectroscopic data sets [@Bonnivard:2015pia; @Chiappo:2016xfs].
From a theoretical perspective, our analysis may be improved by extending beyond NFW profiles and isotropic stellar velocity dispersions. The dark matter potentials of dSphs are at present consistent with both cores [@Walker:2011zu] and cusped profiles [@Breddels:2013qqh; @Richardson:2013lja; @Strigari:2014yea], and the shape of the dark matter phase space distribution is different in both cases, even for isotropic models. For anisotropic models, the dark matter distribution function depends on additional integrals of motion beyond just the energy. Guidance may come from cosmological simulations, which are able to determine the phase space distribution of the dark matter and stars separately [@Campbell:2016vkb].
KB and JK are supported in part by NSF CAREER grant PHY-1250573. LES acknowledges support from NSF grant PHY-1522717. We would also like to acknowledge the Center for Theoretical Underground Physics and Related Areas (CETUP$^\ast$) for hospitality and partial support during the 2016 Summer Program.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: [email protected]
[^5]: We do not consider the enhancement effects of sub-substructure [@Pieri:2009zi], since the results are subject to theoretical extrapolations of the concentration-mass relation.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the implementation of binary projective measurements with linear optics. This problem can be viewed as a single-shot discrimination of two orthogonal pure quantum states. We show that any two orthogonal states can be perfectly discriminated using only linear optics, photon counting, coherent ancillary states, and feedforward. The statement holds in the asymptotic limit of large number of these physical resources.'
address:
- |
[ Quantum Information Technology Group, National Institute of Information and Communications Technology (NICT),\
4-2-1 Nukui-kitamachi, Koganei, Tokyo 184-8795, Japan]{}
- '[ CREST, Japan Science and Technology Agency, 1-9-9 Yaesu, Chuoh-ku, Tokyo 103-0028, Japan]{}'
- |
[ Quantum Information Theory Group, Institute of Theoretical Physics,\
Universität Erlangen-Nürnberg, 91058 Erlangen, Germany]{}
- |
[ Institute for Quantum Computing, University of Waterloo,\
200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada]{}
author:
- Masahiro Takeoka
- Masahide Sasaki
- Norbert Lütkenhaus
title: Binary projective measurement via linear optics and photon counting
---
Projection measurements play an essential role in photonic quantum-information protocols. In these applications, generally, a projection onto superposition states or entangled states of optical fields is required. Physically, it is a highly nontrivial problem how to implement such a measurement.
One plausible approach is to use linear optics and classical feedforward associated with a partial measurement. For example, a universal quantum computation scheme for photonic-qubit states has been proposed, which utilizes only linear optics, photon counting, and highly entangled auxiliary states of $n$ photons generated by probabilistic gate operations [@KLM01]. In principle, it works with unit success probability in the asymptotic limit of large $n$. It is, however, still a nontrivial question how to prepare entangled ancillae even for modest $n$.
In this paper, we discuss the linear optics implementation of a measurement which effects a projection onto two orthogonal states $\{|\Psi\rangle, |\Phi\rangle\}$. This is equivalent to the problem of discriminating two orthogonal quantum signals $\{|\Psi\rangle,|\Phi\rangle\}$ unambiguously [@vanLoock03; @comment1]. We show that, in the asymptotic limit of a large number of partial measurements, one can perfectly discriminate the two states with linear optics, photon counting, and feedforward, but [*without*]{} any non-classical auxiliary states. Even in the worst case, the average error probability of discrimination approaches zero with the scaling factor of $N^{-1/3}$ where $N$ is the number of the partial measurements. Note that the signal space is two-dimensional but $|\Psi\rangle$ and $|\Phi\rangle$ can be any physical states defined in a larger space, e.g. qubit states, continuous variable states, etc.
Before discussing a linear optics implementation, it is worth mentioning a result concerning the distinguishability of two orthogonal multi-partite states via local operations and classical communication (LOCC). The necessary condition for exact local distinguishability is that, after doing a measurement at some local site, every possible remaining states must be orthogonal to each other. Walgate [*et al.*]{} [@Walgate00] showed that there always exists a local projective measurement satisfying this orthogonality condition for any set of two orthogonal states. Thus one can perfectly discriminate them via a series of local projective measurements where the choice of the measurement basis at each local site is conditioned on the previous measurement outcomes. This result means that if one can show a physical scheme that can exactly discriminate any two orthogonal [*single-mode*]{} states, its sequential application can achieve an exact discrimination of any two orthogonal [*multi-mode*]{} states. In the following, therefore, we concentrate on a discrimination of two single-mode states.
An arbitrary set of two orthogonal single-mode states are described by $$\begin{aligned}
\label{eq:two_states}
|\Psi\rangle = \sum^{\infty}_{m=0} c_m |m\rangle_0 , \qquad
|\Phi\rangle = \sum^{\infty}_{m=0} d_m |m\rangle_0 , \end{aligned}$$ where $|m\rangle$ is an $m$-photon number state and $\langle\Psi|\Phi\rangle =
\sum^{\infty}_{m=0} c_m^* d_m = 0$. Figure \[fig:schematic\] is the schematic of the measurement apparatus. The states are equally split into $N$ modes by $N-1$ asymmetric beamsplitters [@vanLoock00], $$\begin{aligned}
\label{eq:N-splitter}
& & \hat{B}_{N-1,0} (\theta_{N-1}) \hat{B}_{N-2,0} (\theta_{N-2})
\cdots \hat{B}_{1,0} (\theta_{1}) |0\rangle^{\otimes N-1} |\Psi\rangle_0
\nonumber\\ & &
= e^{ - \hat{a}_{N-1}^{\dagger} \hat{a}_0 }
\cdots
e^{ - \hat{a}_{1}^{\dagger} \hat{a}_0 }
e^{ \hat{a}_0^{\dagger} \hat{a}_0 \ln \left( 1/\sqrt{N} \right) }
|0\rangle^{\otimes N-1} |\Psi\rangle_0
\nonumber\\ & &
\equiv \hat{N}_{BS} |\Psi\rangle_0 ,\end{aligned}$$ where $\hat{B}_{i,0} (\theta_i) = \exp [ \theta_i(\hat{a}^{\dagger}_i \hat{a}_0
- \hat{a}_i \hat{a}^{\dagger}_0 ) ]$ [@Barnett97] and $\tan\theta_i = 1/\sqrt{N-i}$. The input is symmetrically split to $N$ modes with the effective power reflectance of $1/N$. Then, at each output port, one makes some measurement by using linear optics and photon counters, where the information about the measurement outcome is fed forward to design the next measurement. It should be noted that this is a generalized version of the scheme so-called “Dolinar receiver” [@Dolinar73; @Geremia04; @Takeoka05] which was originally proposed as a physical model attaining the minimum error discrimination of the binary coherent signals $\{|\alpha\rangle, |-\alpha\rangle\}$.
We briefly sketch how two states are discriminated by such a scheme in the limit of $N\to\infty$ and then provide a rigorous proof. Suppose one inserts $|\Psi\rangle$ or $|\Phi\rangle$ into the first beamsplitter. For sufficiently small $1/N$, the reflectance of multi-photons can be neglected. The states after beamsplitting are approximated to be $\hat{B}_{1,0}(\theta_1) |0\rangle_1 |\Psi\rangle_0
\approx |0\rangle_1 |\eta_0\rangle_0 +
N^{-1/2} |1\rangle_1 |\eta_1\rangle_0$, and $\hat{B}_{1,0}(\theta_1) |0\rangle_1 |\Phi\rangle_0
\approx |0\rangle_1 |\nu_0\rangle_0 +
N^{-1/2} |1\rangle_1 |\nu_1\rangle_0$, where, $\langle \eta_0 | \nu_0 \rangle + \langle \eta_1 | \nu_1 \rangle
/N \approx 0 , $ since a beamsplitting operation is unitary. Then mode 1 is measured. The measurement here is required to maintain the orthogonality of any conditional outputs of $|\Psi\rangle$ and $|\Phi\rangle$. The local measurement satisfying this condition is described by a two-dimensional projective measurement, $$\begin{aligned}
\label{eq:approx_projection_vectors}
|\pi_0\rangle & = &
\mathcal{N}_{p0} \left\{ |0\rangle
+ \frac{1}{X^*}\left(1-\sqrt{1+|X|^2}\right) |1\rangle \right\}
\nonumber\\ & = &
\mathcal{N}_{p0} \left\{ |0\rangle
- (X + O(X^2)) |1\rangle \right\}, \\
\label{eq:approx_projection_vectors'}
|\pi_1\rangle & = &
\mathcal{N}_{p1} \left\{ (X^* + O(X^2)) |0\rangle
+ |1\rangle \right\}. \end{aligned}$$ where, $\mathcal{N}_{p0}$ and $\mathcal{N}_{p1}$ are the normalization factors and $$\label{eq:X}
X = \frac{ 2( \langle\nu_0|\eta_1\rangle \langle\eta_1|\nu_1\rangle
- \langle\eta_0|\nu_1\rangle \langle\nu_1|\eta_1\rangle )
}{ \sqrt{N}
(|\langle\eta_0|\nu_1\rangle|^2 - |\langle\eta_1|\nu_0\rangle|^2) }.$$ Here, we have assumed $|\langle\eta_0|\nu_1\rangle|^2
- |\langle\eta_1|\nu_0\rangle|^2 \neq 0$ which implies $X \propto 1/\sqrt{N}$ and thus we can take $|X| \ll 1$ in the limit of large $N$. The other case, i.e. $|\langle\eta_0|\nu_1\rangle|^2
- |\langle\eta_1|\nu_0\rangle|^2 = 0$, will be discussed later. Under this assumption, the projective measurement of Eqs. (\[eq:approx\_projection\_vectors\]) and (\[eq:approx\_projection\_vectors’\]) can be implemented by the displacement operation $\hat{D}(\beta_1/\sqrt{N})$ and photon counting as shown in Fig. \[fig:schematic\](b). Since both the signal and displacement are sufficiently weak, the corresponding measurement vectors are described by $$\begin{aligned}
\label{eq:approx_displaced_pc_0}
\hat{D}^{\dagger}
\left(\frac{\beta_1}{\sqrt{N}}\right) |0\rangle & \approx &
e^{-|\beta_1|^2 /2N}
\left( |0\rangle - \frac{\beta_1}{\sqrt{N}} |1\rangle \right), \\
\label{eq:approx_displaced_pc_1}
\hat{D}^{\dagger}
\left(\frac{\beta_1}{\sqrt{N}}\right) |1\rangle
& \approx &
e^{-|\beta_1|^2 /2N}
\left( \frac{\beta_1^*}{\sqrt{N}} |0\rangle + |1\rangle \right), \end{aligned}$$ which can be same as Eqs. (\[eq:approx\_projection\_vectors\]) and (\[eq:approx\_projection\_vectors’\]) by choosing appropriate $\beta_1$.
The conditional states after the first measurement can be rewritten again as $|\Psi'\rangle = \sum^{\infty}_{m=0} c'_m |m\rangle$ and $|\Phi'\rangle = \sum^{\infty}_{m=0} d'_m |m\rangle$. Since $\hat{N}_{BS}$ splits a state symmetrically, one can repeat the same procedure for the remaining state with the second beamsplitter, the displacement operation $\hat{D}(\beta_2/\sqrt{N})$, where $\beta_2$ is conditioned on the previous measurement outcome, and a photon counter. After repeating the same procedure to modes 1 to $N-1$ with appropriate $\beta_i$’s, the final states at mode 0 contain with dominating weight at most one photon and are still orthogonal to each other. As a consequence, applying the final ($N$-th) displacement and photon counting, one can exactly discriminate $|\Psi\rangle$ and $|\Phi\rangle$ with unit success probability.
![\[fig:schematic\] (a) $N$-splitter, and (b) a measurement apparatus at each step. A displacement operation $\hat{D}(\beta_i/\sqrt{N})$ is realized by combining the signal with a coherent state local oscillator $|\beta_i/\sqrt{N\sin\theta}\rangle$ via a beamsplitter with sufficiently small power reflectance of $\sin^2\theta$. ](fig1.eps){width="0.9\linewidth"}
Now, we discuss the scheme rigorously, i.e. include the effects due to the multi-photon reflections at each beamsplitter, which contribute to the failure of the measurement or giving the incorrect decisions. Here, the input states $|\Psi\rangle$ and $|\Phi\rangle$ are always physical, that is, the average power of them are finite. Moreover, we assume that the probability distribution in photon number of those states decreases exponentially as $c_m \equiv \tilde{c}_m e^{-mx/2}$ where $x$ is a real positive number. The prior probabilities can be set to be equal without loss of generality. Finally we assume that the average powers of local oscillators always satisfy $|\beta_i|^2 \le |C_{\beta_i}|^2 + O(1/N)$ where $C_{\beta_i}$ is a complex constant independent of $N$.
After finishing a whole process of $N$ measurement steps, one can classify the results according to the sequential patterns of detected photon numbers. Let us denote the events in which all the photon counters detect zero or one photon by ‘success’ events and the others by ‘failure’ events. Because of the symmetry of the $N$-beamsplitting, the probability of detecting $k$ photons at the $i$-th measurement [*on average over all possible measurement patterns*]{} is given by [@Barnett97] $$\begin{aligned}
\label{eq:P_k_average}
P_k^{(i)} & = & \left|
{}_i \langle k| \hat{D}_i (\beta_i/\sqrt{N}) \hat{N}_{BS} |\Psi\rangle_0
\right|^2
\nonumber\\
& \le & \frac{
\langle \Psi_{\beta_i} | \hat{a}_0^{\dagger k} \hat{a}_0^k
|\Psi_{\beta_i} \rangle
}{N^k k!}
+ O \left( \frac{1}{N^{k+1}} \right)
\nonumber\\ & \le &
C_k^{\rm max} / N^k + O ( 1/N^{k+1} ), \end{aligned}$$ where $|\Psi_{\beta_i} \rangle \equiv \hat{D}(C_{\beta_i}) |\Psi\rangle$, whose probability distribution still decreases exponentially in number basis (see Appendix A), and $C_k^{\rm max}$ is the maximum value of $\langle \Psi_{\beta_i} | \hat{a}_0^{\dagger k}
\hat{a}_0^k |\Psi_{\beta_i} \rangle / k!$ for all $i$ and possible inputs . The probability of resulting the failure event $P_{fail}$ is then bounded as $$\begin{aligned}
\label{eq:P_failure}
P_{fail}
& \le &
( C_2^{\rm max} / N^2 + O (1/N^3) ) \times N
\nonumber\\ & = &
C_2^{\rm max} / N + O (1/N^2), \end{aligned}$$ which implies that $P_{fail}$ approaches to zero in the limit of large $N$, at least with the order of $1/N$.
Even if the detection is successful, the conditional states get slightly non-orthogonal after each measurement step. To see this, we revisit the first beamsplitter $\hat{B}_{1,0}(\theta_1)$. Let us describe the states after beamsplitting such that the orthogonal and non-orthogonal parts are separated as $$\begin{aligned}
\label{eq:B|0>|psi>}
\hat{B}_{1,0} (\theta_1) |0\rangle |\Psi\rangle & = &
|0\rangle |\eta_0\rangle + N^{-1/2} |1\rangle |\eta'_1\rangle
+ N^{-1} |2\rangle |\eta_2\rangle + \cdots
\nonumber\\ & = &
|0\rangle |\eta_0\rangle
+ N^{-1/2} |1\rangle |\eta_1\rangle
+ N^{-3/2} |1\rangle |\eta_r\rangle
\nonumber\\ & &
+ \sum_{k=2}^\infty N^{-k/2}
|k\rangle|\eta_k\rangle, \\
\label{eq:B|0>|phi>}
\hat{B}_{1,0} (\theta_1) |0\rangle |\Phi\rangle
& = &
|0\rangle |\nu_0\rangle
+ N^{-1/2} |1\rangle |\nu_1\rangle
+ N^{-3/2} |1\rangle |\nu_r\rangle
\nonumber\\ & &
+ \sum_{k=2}^\infty N^{-k/2}
|k\rangle|\nu_k\rangle ,\end{aligned}$$ where the first two terms exactly satisfy the orthogonality $\langle\eta_0|\nu_0\rangle + \langle\eta_1|\nu_1\rangle /N = 0$ and the last terms represent the multi-photon reflection terms. Here, $|\eta_0\rangle = \sum^{\infty}_{m=0} c_m (1-1/N)^{m/2} |m\rangle$, $N^{-1/2} |\eta'_1\rangle
= \sum^{\infty}_{m=1} c_m (m/N)^{1/2} (1-1/N)^{(m-1)/2}
|m-1\rangle$, $N^{-1/2} |\eta_1\rangle
= \sum^{\infty}_{m=1} c_m (1-(1-1/N)^m)^{1/2} |m-1\rangle$, and $N^{-3/2} |\eta_r\rangle
= N^{-1/2}(|\eta'_1\rangle - |\eta_1\rangle)$ ($|\nu_n\rangle$’s are also obtained by replacing $c_m$ with $d_m$). The terms $|\eta_r\rangle$, $|\nu_r\rangle$ and that for multi-photon reflections, which have been neglected in the previous discussion, cause the residual non-orthogonality. Note that the leading terms of all vectors $|\eta_k\rangle$’s and $|\nu_k\rangle$’s are independent of $N$. Denote the $i$-th measurement operation as $$\label{eq:Kraus_op}
\frac{
{}_i \langle k|
\hat{D}_i(\beta_i/\sqrt{N}) \hat{B}_{i,0}(\theta_i) |0\rangle_i
|\Psi\rangle }{|
{}_i \langle k| \hat{D}_i(\beta_i/\sqrt{N}) \hat{B}_{i,0}(\theta_i) |0\rangle_i
|\Psi\rangle |
}
\equiv \hat{E}^{(i)}_k |\Psi\rangle.$$ Then the conditional outputs after detecting zero and one photons at the first measurement are given by $$\begin{aligned}
\label{eq:output_0}
\hat{E}_0^{(1)} |\Psi\rangle
& = &
\mathcal{N}_0
\left\{ |\eta_0\rangle - \frac{\beta_1^*}{N} |\eta_1\rangle
+ \frac{1}{N^2} |\eta_{R_0}^{(1)} \rangle
\right\} ,
\\
\label{eq:output_1}
\hat{E}_1^{(1)} |\Psi\rangle
& = &
\mathcal{N}_1
\left\{ \beta_1 |\eta_0\rangle + |\eta_1\rangle
+ \frac{1}{N} |\eta_{R_1}^{(1)} \rangle
\right\} , \end{aligned}$$ respectively, where $\mathcal{N}_0$ and $\mathcal{N}_1$ are the normalization factors and the third terms $|\eta_{R_i}^{(1)}\rangle$’s ($i=0,1$) come from $|\eta_r\rangle$ and $|\eta_k\rangle$’s for $k\ge2$, and the terms in Eqs. (\[eq:approx\_displaced\_pc\_0\]) and (\[eq:approx\_displaced\_pc\_1\]) whose order is higher than $1/N^{1/2}$. The same outputs are obtained for $|\Phi\rangle$ by replacing $|\eta_n\rangle$ with $|\nu_n\rangle$. The first two terms in Eqs. (\[eq:output\_0\]) and (\[eq:output\_1\]) can be exactly orthogonal to those of $|\Phi\rangle$ by choosing $\beta_1/\sqrt{N} = (1-\sqrt{1+X^2})e^{i\omega}/X$, where X is obtained by substituting $|\eta_0\rangle$, $|\eta_1\rangle$, $|\nu_0\rangle$ and $|\nu_1\rangle$, appearing in Eqs. (\[eq:B|0>|psi>\]) and (\[eq:B|0>|phi>\]), into Eq. (\[eq:X\]). Since $X \propto 1/\sqrt{N}$ as mentioned above, this choice of $\beta_1$ always satisfy the constraint on the average power of the local oscillator, $|\beta_1|^2 \le |C_{\beta_1}|^2 + O(1/N)$. However, we have to care of the fact that, in both events, the [*total*]{} conditional states in Eqs. (\[eq:output\_0\]) and (\[eq:output\_1\]) are no longer orthogonal due to their third terms.
Now, suppose that the same strategy is applied to the choice of $\beta_2$ for the second measurement step. After the second measurement, the states are mapped into the new one with orthogonal and non-orthogonal terms, where the latter has two parts, i.e. contributions from the first and second measurements. Note that the leading order of prefactors of $|\eta_{R_k}^{(1)}\rangle$ with respect to $1/N$ does not change during the measurement process, as also the leading factors of $|\Psi\rangle$ does not change in the mapping in Eqs. (\[eq:output\_0\]) and (\[eq:output\_1\]). Eventually, after repeating $N-1$ measurement steps in a similar way, if all the photon counters detected zero or one photons, one obtains the conditional output consists of the orthogonal term and $N-1$ non-orthogonal terms stemmed from each measurement as $$\begin{aligned}
\label{eq:N-1th_meas}
|\Psi^{(N-1)}\rangle & = &
\hat{E}^{(N-1)} \cdots \hat{E}^{(1)} |\Psi\rangle
\nonumber\\
& = &
|\eta^{(N-1)}\rangle
+ \frac{1}{N^2} \sum_{x=1}^{I^{(N-1)}} |H_0^{(i_x)}\rangle
+ \frac{1}{N} \sum_{y=1}^{J^{(N-1)}} |H_1^{(j_y)}\rangle,
\nonumber\\\end{aligned}$$ where the first term is exactly orthogonal to that of $|\Phi^{(N-1)}\rangle$, while $|H_k^{(l)}\rangle$ is the residual non-orthogonal term coming from $|\eta_{R_k}^{(l)}\rangle$. $I^{(N-1)}$ and $J^{(N-1)}$ are the numbers of the events of detecting zero and one photon, respectively, and thus $I^{(N-1)} + J^{(N-1)} = N-1$.
Let us denote the final $N$-th measurement by $|D_k\rangle \equiv
\hat{D}^{\dagger} (\beta_N/\sqrt{N}) |k\rangle$ ($k=0,1$). Suppose that $\beta_N$ is designed such that $|D_0\rangle$ and $|D_1\rangle$ are the same as the orthogonal terms in $|\Psi^{(N-1)}\rangle$ and $|\Phi^{(N-1)}\rangle$, respectively, up to the order of $1/N^{1/2}$ (the higher order terms contribute to the detection error). Then the error probability $P_{err}^{D_1} = |\langle D_1|\Psi^{(N-1)}\rangle|^2$ is given by $$\begin{aligned}
\label{eq:error_prob1}
P_{err}^{D_1} & = &
\left|
\sum_{x=1}^{I^{(N)}} \frac{\langle D_1|H_0^{(i_x)}\rangle}{N^2}
+ \sum_{y=1}^{J^{(N)}} \frac{\langle D_1|H_1^{(j_y)}\rangle}{N}
\right|^2 . \end{aligned}$$ where $I^{(N)} + J^{(N)} = N$. The leading order of $\langle D_1|H_k^{(j)}\rangle$ is independent of $N$ for every $j$ and $k$.
![\[fig:total\_power\] The original scheme (a) can be transformed into (b) where the total input photon number is the sum of those of two input states. ](fig2.eps){width="1\linewidth"}
One can estimate the order of $J^{(N)}$ by counting the total amount of photons put into the system since the number of the total photon is equal to that of detectors. Although photons are supplied by the input state and $N$ displacements in the original configuration, one can simplify it into the one with only two inputs, $\hat{D}(\beta_0) |\Psi\rangle$ and the coherent state $|\beta_{\rm aux}\rangle$, by adding some linear optics as illustrated in Fig. \[fig:total\_power\]. Here, with the relation $\hat{D}_A (\alpha) \hat{D}_B (\beta) \hat{B}_{AB}(\theta)
= \hat{B}_{AB}(\theta) \hat{D}_A (\alpha\cos\theta - \beta\sin\theta)
\hat{D}_B (\alpha\sin\theta + \beta\cos\theta)$, one finds $|\beta_0|^2 = |\sum_{i=1}^N \beta_i /N|^2$ and $|\beta_{\rm aux}|^2 = \sum_{i=1}^N |\beta_i|^2/N - |\beta_0|^2$, where these are bounded as $|\beta_0|^2 = C_0 + O(1/N)$ and $|\beta_{\rm aux}|^2 = C_{\rm aux} + O(1/N)$ due to the constraint on $|\beta_i|^2$’s. $C_0$ and $C_{\rm aux}$ are constants independent of $N$.
The probability of having $n$ photons in total is given by $P(n) = \sum_{m=0}^n P_{\rm sig} (n-m) P_{\rm aux} (m)
= C_P e^{-n x} + O(1/N)$. Here the photon number statistics of two inputs, $P_{\rm sig}(m)$ and $P_{\rm aux}(m)$ are exponential and Poissonian, which easily implies that $P(n)$ decreases exponentially with resepect to $n$ (see Appendix C). Therefore, one can bound $J^{(N)}$ by some constant $C_J$ with exponentially small exception as $$\begin{aligned}
\label{eq:J}
&&{\rm Prob} \left[ J^{(N)} \le C_J + O(1/N) + N\epsilon \right]
\nonumber\\&&
\ge 1 - C_P \exp[ -(C_J+O(1/N)+N\epsilon)] + O(1/N)
\nonumber\\&&
= 1-\tilde{C}_P e^{-N\epsilon} + O(1/N)\end{aligned}$$ where $\epsilon$ can be arbitrarily small for large $N$. Eventually, substituting it and $I^{(N)} \le N$ into Eq. (\[eq:error\_prob1\]), one obtains $$\begin{aligned}
\label{eq:error_prob1-0}
P_{err}^{D_1} & = &
\left|
\frac{I^{(N)}}{N^2} \langle D_1|H_0\rangle_{\rm av}
+ \frac{J^{(N)}}{N} \langle D_1|H_1\rangle_{\rm av} \right|^2
\nonumber\\ & \le &
C_E/N^2 + O(1/N^3) + \epsilon O(1/N) + \epsilon^2,\end{aligned}$$ where $\langle D_1|H_k\rangle_{\rm av}
= \sum_i \langle D_1|H_k^{(i)}\rangle / L$ ($L=I^{(N)}$ and $J^{(N)}$ for $k=0,\,1$, respectively), and $C_E$ is some constant independent of $N$. In a similar manner, the same bound is derived for $P_{err}^{D_0} = |\langle D_0|\Phi^{(N-1)}\rangle|^2$. Then, summing over all detection patterns, the average error probability is bounded as $$\begin{aligned}
\label{eq:error_prob_av}
P_{err}^{\rm tot} & = &
\sum^{\rm success} P(\sharp) P^{succ}_{err}(\sharp)
+ \sum^{\rm failure} P(\sharp) P_{fail}(\sharp)
\nonumber\\
& \le &
\left( 1 - \frac{C_2^{\rm max}}{N} \right)
\frac{P_{err}^{D_0} + P_{err}^{D_1}}{2}
+ \frac{C_2^{\rm max}}{N} + O \left( \frac{1}{N^2} \right)
\nonumber\\ & \le &
C/N + O ( 1/N^2 )
+ O(1/N) \epsilon + \epsilon^2,\end{aligned}$$ where $C$ is some constant and $P(\sharp)$ is the probability to observe the measurement sequence pattern $\sharp$. As a consequence, in the limit of $N\to\infty$, one can discriminate $|\Psi\rangle$ and $|\Phi\rangle$ with unit probability.
Finally, we discuss the case $|\langle\eta_0|\nu_1\rangle|^2
- |\langle\eta_1|\nu_0\rangle|^2 = 0$ in Eq. (\[eq:X\]), in which the desirable local measurement can not be implemented by a displacement and photon counting. Here, let us consider the projection measurement consisting of slightly perturbed vectors $|\Pi_0\rangle = \sqrt{1-\delta}|\Psi\rangle - \sqrt{\delta}|\Phi\rangle$ and $|\Pi_1\rangle = \sqrt{1-\delta}|\Phi\rangle + \sqrt{\delta}|\Psi\rangle$ with a perturbation parameter $\delta$. One can design such a measurement by the previous strategy with the total error probability of $P_{err}^{\rm tot} =
C/N^{1-2\Delta} + O (1/N^{2-3\Delta})
+ O (1/N^{1-3\Delta/2}) \epsilon + O (N^\Delta) \epsilon^2$, where $\Delta = - \log_N \delta$. This device can discriminate the original states $|\Psi\rangle$ and $|\Phi\rangle$ with the average error probability of $$\begin{aligned}
\label{eq:error_prob_av_special2}
P_{err}^{\rm av} & = & 1 -
(1-P_{err}^{\rm tot})
(|\langle\Pi_0|\Psi\rangle|^2 + |\langle\Pi_1|\Phi\rangle|^2)/2
\nonumber\\ & = &
C_1/N^{\Delta} + O (1/N^{2\Delta})
+ C_2/N^{1-2\Delta} + O (1/N^{2-3\Delta})
\nonumber\\ & &
+ O (1/N^{1-3\Delta/2}) \epsilon + O (N^\Delta) \epsilon^2\end{aligned}$$ In the asymptotic limit of large $N$, this is minimized with $\Delta = 1/3$ and then we obtain $P_{err}^{\rm av} = C/N^{1/3} + O(1/N^{2/3})
+ O(1/N^{1/2}) \epsilon + O(N^{1/3}) \epsilon^2$ which still converges to zero.
In summary, we have proved that arbitrary two orthogonal pure states can be perfectly discriminated by linear optics tools without using any non-classical ancillary states in the asymptotic limit of $N\to\infty$ where $N$ is the number of the detections and feedforwards. It implies that, in principle, one can implement arbitrary projection measurement in any two-dimensional signal space by these tools. The resources discussed here are mostly available with current technology. We also showed a concrete designing strategy of a linear optics circuit to attain this bound for a given $N$ and thus it can be directly applied for various quantum information protocols that require binary projection measurements. The remaining question is whether one can apply a same approach to the problem of more than three states discrimination.
We thank M. Ban, D. Berry, K. Tamaki, and P. van Loock for valuable discussions and comments. M.T. also acknowledges a kind hospitality at the QIT group in Universität Erlangen-Nürnberg. This work was supported by the DFG under the Emmy-Noether program, the EU FET network RAMBOQ and the network of competence QIP of the state of Bavaria.
Photon number statistics of the displaced state
===============================================
In this appendix, we show that if the photon number distribution of the initial state is exponential, then that of its displaced state is also bounded by exponentially decreasing function. For this purpose we use the following three formulae;
[**(1) The number basis components of the displacement operator [@Cahill69];**]{} $$\begin{aligned}
\label{eq:A_1}
\langle n| \hat{D}(\xi) |m\rangle & = &
\sqrt{\frac{m!}{n!}} \xi^{n-m} e^{-|\xi|^2/2} L_m^{(n-m)} (|\xi|^2), \end{aligned}$$ for $(n \ge m)$ and $$\begin{aligned}
\langle n| \hat{D}(\xi) |m\rangle & = &
\sqrt{\frac{n!}{m!}} (-\xi^*)^{m-n} e^{-|\xi|^2/2} L_n^{(m-n)} (|\xi|^2),
\nonumber\\\end{aligned}$$ for $(n \le m)$, where $L_n^{(l)} (x)$ is the associated Laguerre polynomial defined by $$\label{eq:A_1_1}
L_n^{(l)} (x) = \sum_{k=0}^n {n+l \choose n-k}
\frac{(-x)^k}{k!} ,$$ where $L_n^{(0)}(x) = L_n(x)$ is the Laguerre polynomial and $$\label{eq:A_1_2}
\frac{{\rm d}^l}{{\rm d} x^l} L_n (x) = (-1)^l L_{n-l}^{(l)} (x).$$
[*Proof*]{}. We basically follow the proof given in [@Barnett97]. To calculate $\langle n| \hat{D}(\xi) |m\rangle$, it is helpful to see $\langle n| \hat{D}(\xi) |n\rangle$, which is given by $$\begin{aligned}
\label{eq:A_1_3}
\langle n| \hat{D}(\xi) |n\rangle & = &
\langle n| \exp \left( \xi\hat{a}^\dagger - \xi^*\hat{a} \right)
|n\rangle
\nonumber\\ & = &
e^{-|\xi|^2/2} \langle n| e^{\xi\hat{a}^\dagger}
e^{- \xi^*\hat{a}} |n\rangle
\nonumber\\ & = &
\sum_{l=0}^\infty \sum_{m=0}^\infty e^{-|\xi|^2/2}
\frac{\xi^l (-\xi^*)^m}{l!m!}
\langle n| \hat{a}^{\dagger \, l} \hat{a}^m |n\rangle
\nonumber\\ & = &
\sum_{l=0}^\infty \sum_{m=0}^\infty e^{-|\xi|^2/2}
\frac{\xi^l (-\xi^*)^m}{l!m!}
\nonumber\\ &&
\langle n-l| \sqrt{\frac{n!}{(n-l)!}}
\sqrt{\frac{n!}{(n-m)!}} |n-m\rangle
\nonumber\\ & = &
e^{-|\xi|^2/2} \sum_{m=0}^n {n \choose m}
\frac{(-|\xi|^2)^m}{m!}
\nonumber\\ & = &
e^{-|\xi|^2/2} L_n (|\xi|^2). \end{aligned}$$ Then we obtain $$\begin{aligned}
\label{eq:A_1_4}
& & \langle n| \hat{D}(\xi) |n-l\rangle
\nonumber\\ && =
e^{-|\xi|^2/2} \langle n| e^{\xi\hat{a}^\dagger}
e^{- \xi^*\hat{a}} \hat{a}^l |n\rangle
\sqrt{\frac{(n-l)!}{n!}}
\nonumber\\ && =
e^{-|\xi|^2/2} \sqrt{\frac{(n-l)!}{n!}}
\left( -\frac{\partial}{\partial \xi^*} \right)^l
\langle n| e^{\xi\hat{a}^\dagger} e^{- \xi^*\hat{a}} |n\rangle
\nonumber\\ && =
e^{-|\xi|^2/2} \sqrt{\frac{(n-l)!}{n!}}
(-\xi)^l \left( \frac{\partial}{\partial |\xi|^2} \right)^l
L_n (|\xi|^2) .\end{aligned}$$ Therefore, replacing $n-l$ with $m$ in Eq. (\[eq:A\_1\_4\]) with Eq. (\[eq:A\_1\_2\]), we can derive Eq. (\[eq:A\_1\]).
[**(2) Bound on the associated Laguerre polynomials [@HandbookMath];**]{} $$\begin{aligned}
\label{eq:A_2}
\left| L_n^{(a)} (x) \right| \le
{a+n \choose n} e^{x/2}, \end{aligned}$$ where $x \ge 0$ and $a$ is an integer.
[*Proof*]{}. From Eq. (\[eq:A\_1\_3\]), the absolute value of the Laguerre polynomial $L_n (x)$ with $x\ge0$ is bounded by $$\begin{aligned}
\label{eq:A_2_1}
|L_n (x)| & = & e^{x/2}
\left| \langle n|\hat{D}(x^{1/2}) |n\rangle \right|
\nonumber\\ & \le &
e^{x/2} .\end{aligned}$$
To extend it to the associated Laguerre polynomial, we use the relation $$\label{eq:A_2_2}
L_n^{(a)}(x) = \sum_{k=0}^n {a+k-1 \choose a-1} L_{n-k}(x) ,$$ which can be derived as $$\begin{aligned}
\label{eq:A_2_3}
& = &
\sum_{k=0}^n {a+k-1 \choose a-1}
\sum_{l=0}^{n-k} {n-k \choose l} \frac{(-x)^l}{l!}
\nonumber\\ & = &
\sum_{l=0}^n \sum_{k=0}^{n-l}
{a+k-1 \choose a-1} {n-k \choose l} \frac{(-x)^l}{l!}
\nonumber\\ & = &
\sum_{l=0}^n {n+a \choose n-l} \frac{(-x)^l}{l!}
\nonumber\\ & = &
L_n^{(a)} (x) ,\end{aligned}$$ where the formula $$\label{eq:A_2_4}
\sum_{k=0}^{n-l} {a+k \choose a} {n-k \choose l}
= {n+a+1 \choose n-l}$$ has been utilized. Eventually, Eqs. (\[eq:A\_2\_1\]), (\[eq:A\_2\_2\]) and (\[eq:A\_2\_4\]) imply $$\begin{aligned}
\label{eq:A_2_5}
\left| L_n^{(a)}(x) \right| & \le &
\sum_{k=0}^n {a+k-1 \choose a-1} \left| L_{n-k}(x) \right|
\nonumber\\ & \le &
\sum_{k=0}^n {a+k-1 \choose a-1} e^{x/2}
= {a+n \choose a} e^{x/2} . \nonumber\\ \end{aligned}$$
[**(3) Inequality for the binomial distribution [@GenIneq];**]{} $$\begin{aligned}
\label{eq:A_5}
{n \choose \nu} y^\nu (1-y)^{n-\nu} \le
\exp \left[ -2n (y-\nu/n)^2 \right], \end{aligned}$$ where $n > \nu$ and $0<y<1$.
[*Proof*]{}. Define $q=\nu/n$ and $$\begin{aligned}
\label{eq:A_5_1}
f(y) & = & y^\nu (1-y)^{n-\nu} e^{-2n(x-q)^2},
\nonumber\\
F(y) & = & n^{-1} \ln f(x). \end{aligned}$$ Then $$\begin{aligned}
\label{eq:A_5_2}
F(y) & = & q \ln y + (1-q) \ln (1-x) + 2(x-q)^2,
\nonumber\\
F'(y) & = & \frac{(q-y)(1-2y)^2}{y(1-y)}, \end{aligned}$$ and thus $F(y)$ takes its maximum at $y=q$. Also, the same for $f(y)$. Therefore, $f(y) \le f(q)$, i.e. $$\label{eq:A_5_3}
y^\nu (1-y)^{n-\nu} e^{2n(y-q)^2} \le q^\nu (1-q)^{n-\nu},$$ and thus $$\label{eq:A_5_4}
{n \choose \nu} y^\nu (1-y)^{n-\nu} e^{2n(y-q)^2}
\le {n \choose \nu} q^\nu (1-q)^{n-\nu} \le 1,$$ which completes the proof.
[**Derivation of the displaced state.**]{} Now we derive the main statement of this appendix. We assume that $|\Psi\rangle$ can be written as $$|\Psi\rangle = \sum_{m=0}^\infty \tilde{c}_m e^{-mx/2} |m\rangle.$$ Now, let us calculate $\langle n| \hat{D}(\beta) |\Psi\rangle$. $$\begin{aligned}
\label{eq:A_6}
&& \left| \langle n| \hat{D}(\beta) |\Psi\rangle \right| =
\left| \sum_{m=0}^\infty \tilde{c}_m e^{-mx/2}
\langle n| \hat{D}(\beta) |m \rangle \right|
\nonumber\\ && \le
\sum_{m=0}^\infty \left| \tilde{c}_m e^{-mx/2}
\langle n| \hat{D}(\beta) |m \rangle \right|
\nonumber\\ && \le
\sum_{m=0}^n |\tilde{c}_m| e^{-mx/2} \left( \frac{m!}{n!} \right)^{1/2}
|\beta|^{n-m} e^{-|\beta|^2/2}
\left| L_m^{(n-m)} (|\beta|^2) \right|
+ \sum_{m=n+1}^\infty |\tilde{c}_m| e^{-mx/2}
\left( \frac{n!}{m!} \right)^{1/2}
|\beta|^{m-n} e^{-|\beta|^2/2}
\left| L_n^{(m-n)} (|\beta|^2) \right|
\nonumber\\ && \le
|\tilde{c}_{\rm max}| \sum_{m=0}^n e^{-mx/2}
\left( \frac{m!}{n!} \right)^{1/2} |\beta|^{n-m} {n \choose m}
+ |\tilde{c}_{\rm max}| \sum_{m=n+1}^\infty e^{-mx/2}
\left( \frac{n!}{m!} \right)^{1/2} |\beta|^{m-n} {m \choose n}
\nonumber\\ && =
|\tilde{c}_{\rm max}| \sum_{m=0}^n e^{-nx/2}
\left\{ {n \choose m} \frac{(|\beta|^2 e^x)^{n-m}}{(n-m)!} \right\}^{1/2}
+ |\tilde{c}_{\rm max}| \sum_{m=n+1}^\infty e^{-nx/2}
\left\{ {m \choose n} \frac{(|\beta|^2 e^{-x})^{m-n}}{(m-n)!} \right\}^{1/2}
\nonumber\\ && \le
\left\{ n e^{-nx} |\tilde{c}_{\rm max}|^2 \sum_{m=0}^n {n \choose m}
\frac{(|\beta|^2 e^x)^{n-m}}{(n-m)!} \right\}^{1/2}
+ \left\{ n e^{-nx} |\tilde{c}_{\rm max}|^2 \sum_{m=n+1}^\infty {m \choose n}
\frac{(|\beta|^2 e^{-x})^{m-n}}{(m-n)!} \right\}^{1/2} .\end{aligned}$$ The last line follows from the Cauchy-Schwarz inequality.
Introducing a real parameter $q$ which satisfies $e^{-x}<q<1$, the first term of Eq. (\[eq:A\_6\]) is then bounded as $$\begin{aligned}
\label{eq:A_7}
&& \sqrt{n} e^{-nx/2} |\tilde{c}_{\rm max}| \left\{ \sum_{m=0}^n {n \choose m}
\frac{(|\beta|^2 e^x)^{n-m}}{(n-m)!} \right\}^{1/2}
\nonumber\\ && =
\sqrt{n} e^{-nx/2} |\tilde{c}_{\rm max}| \left\{ \sum_{m=0}^n {n \choose m}
q^m (1-q)^{n-m}
\frac{1}{(n-m)!} \left(\frac{q|\beta|^2 e^x}{1-q}\right)^{n-m}
q^{-n} \right\}^{1/2}
\nonumber\\ && \le
\sqrt{n} \left( \frac{e^{-x}}{q} \right)^{n/2} |\tilde{c}_{\rm max}|
\left\{ \sum_{m=0}^n {n \choose m} q^m (1-q)^{n-m}
\exp\left[ \frac{q|\beta|^2 e^x}{1-q} \right] \right\}^{1/2}
\nonumber\\ &&
= \sqrt{n} \left( \frac{e^{-x}}{q} \right)^{n/2} |\tilde{c}_{\rm max}|
\exp\left[ \frac{q|\beta|^2 e^x}{2(1-q)} \right],
\nonumber\\ \end{aligned}$$ and thus it decreases exponentially as $n$ increases. Also, for the second term, one obtains $$\begin{aligned}
\label{eq:A_8}
&& \sqrt{n} e^{-nx/2} |\tilde{c}_{\rm max}| \left\{ \sum_{m=n+1}^\infty
{m \choose n} \frac{(|\beta|^2 e^{-x})^{m-n}}{(m-n)!} \right\}^{1/2}
\nonumber\\ && =
\sqrt{n} e^{-nx/2} |\tilde{c}_{\rm max}| \left\{ \sum_{m=n+1}^\infty
{m \choose n} q^n (1-q)^{m-n}
\frac{1}{(m-n)!} \left(\frac{q|\beta|^2 e^{-x}}{1-q}\right)^{m-n}
q^{-n} \right\}^{1/2}
\nonumber\\ && \le
\sqrt{n} \left( \frac{e^{-x}}{q} \right)^{n/2} |\tilde{c}_{\rm max}|
\left\{ \sum_{m=n+1}^\infty {m \choose n} q^n (1-q)^{m-n}
\exp\left[ \frac{q|\beta|^2 e^{-x}}{1-q} \right] \right\}^{1/2}
\nonumber\\ && \le
\sqrt{n} \left( \frac{e^{-x}}{q} \right)^{n/2} |\tilde{c}_{\rm max}|
\exp\left[ \frac{q|\beta|^2 e^{-x}}{2(1-q)} \right]
\left\{
\sum_{m=n+1}^\infty \exp\left[ -2m \left(q-\frac{n}{m}\right)^2 \right]
\right\}^{1/2} .
\nonumber\\ \end{aligned}$$ Since the last exponential term decreases exponentially as $m$ increase at least in the limit of $m \gg n$, the sum always converges within a finite value, which means that Eq. (\[eq:A\_8\]) itself also decreases exponentially as $n$ increases. As a consequence, these results imply that Eq. (\[eq:A\_6\]) decreases exponentially as $n$ increases.
Derivation of the inequality (\[eq:P\_k\_average\])
===================================================
In this appendix, we derive the inequality (\[eq:P\_k\_average\]). $$\begin{aligned}
\label{eq:P_k_average_appendix}
P_k^{(i)} & = & \left|
{}_i \langle k| \hat{D}_i (\beta_i/\sqrt{N}) \hat{N}_{BS} |\Psi\rangle_0
\right|^2
\nonumber\\ & = &
\left|
{}_i \langle k| \hat{D}_i (\beta_i/\sqrt{N}) \hat{B}_{N-1,0} (\theta_{N-1})
\cdots \hat{B}_{i,0} (\theta_i) \cdots \hat{B}_{1,0} (\theta_1)
|0\rangle^{\otimes N-1} |\Psi\rangle_0
\right|^2
\nonumber\\ & = & \left|
{}_i \langle k| \hat{D}_i (\beta_i/\sqrt{N}) \hat{B}_{N-1,0} (\theta_{N-1})
\cdots \hat{B}_{i+1,0} (\theta_{i+1}) \hat{B}_{i-1,0} (\theta_i)
\cdots \hat{B}_{1,0} (\theta_2) \hat{B}_{i,0} (\theta_1)
|0\rangle^{\otimes N-1} |\Psi\rangle_0
\right|^2
\nonumber\\ & = & \left|
{}_i \langle k| \hat{D}_i (\beta_i/\sqrt{N}) \hat{B}_{i,0} (\theta_1)
|0\rangle_i |\Psi\rangle_0
\right|^2
\nonumber\\ & = & \left|
{}_i \langle k| e^{-|\beta_i|^2/2N}
e^{\beta_i \hat{a}_i^\dagger/\sqrt{N}} e^{-\beta_i^* \hat{a}_i/\sqrt{N}}
e^{- \hat{a}_i^\dagger \hat{a}_0 \tan\theta_1}
e^{- \ln \cos\theta_1
(\hat{a}_i^\dagger \hat{a}_i-\hat{a}_0^\dagger \hat{a}_0) }
e^{\hat{a}_i \hat{a}_0^\dagger \tan\theta_1}
|0\rangle_i |\Psi\rangle_0 \right|^2
\nonumber\\ & = & \left|
{}_i \langle k| e^{-|\beta_i|^2/2N}
e^{\beta_i \hat{a}_i^\dagger/\sqrt{N}}
e^{-\hat{a}_i^\dagger \hat{a}_0 / \sqrt{N-1}}
e^{\beta_i^* \hat{a}_0 / \sqrt{N(N-1)}}
e^{-\beta_i^* \hat{a}_i / \sqrt{N}}
e^{\ln \sqrt{1-1/N} \hat{a}_0^\dagger \hat{a}_0}
|0\rangle_i |\Psi\rangle_0 \right|^2
\nonumber\\ & = & \left|
e^{-|\beta_i|^2/2N} \left\{
\sum_{j=0}^k {}_i \langle k-j| \frac{1}{j!} \sqrt{\frac{k!}{(k-j)!}}
\left( \frac{\beta_i}{\sqrt{N}} - \frac{\hat{a}_0}{\sqrt{N-1}} \right)^j
\right\}
e^{\beta_i^* \hat{a}_0 / \sqrt{N(N-1)}}
e^{\ln \sqrt{1-1/N} \hat{a}_0^\dagger \hat{a}_0}
|0\rangle_i |\Psi\rangle_0 \right|^2
\nonumber\\ & = & \left|
\frac{e^{-|\beta_i|^2/2N}}{\sqrt{k!}}
e^{\hat{a}_0^{\dagger}\hat{a}_0 \ln \sqrt{1-1/N}}
\left( \frac{\beta_i - \hat{a}_0}{\sqrt{N}} \right)^k
e^{\beta_i^* \hat{a}_0 /N}
|\Psi\rangle
\right|^2
\nonumber\\
& \le & \frac{
\langle \Psi_{\beta} | \hat{a}_0^{\dagger k} \hat{a}_0^k
|\Psi_{\beta} \rangle
}{N^k k!}
+ O \left( \frac{1}{N^{k+1}} \right) \end{aligned}$$
where $\cos\theta_1 = \sqrt{1-1/N}$, $\sin\theta_1 = 1/\sqrt{N}$, $|\beta_i|^2 \le |C_{\beta_i}|^2 + O(1/N)$ and $|\Psi_{\beta_i} \rangle = \hat{D}(C_{\beta_i}) |\Psi\rangle$. We have used the relation $e^{\alpha \hat{a}_j} e^{\beta \hat{a}_j^\dagger}
= e^{\beta \hat{a}_j^\dagger} e^{\alpha \hat{a}_j} e^{\alpha\beta}$ from line 5 to 6, and $e^{\phi \hat{a}_j^\dagger \hat{a}_j} \hat{a}_j
e^{-\phi \hat{a}_j^\dagger \hat{a}_j} = \hat{a}_j e^{-\phi}$ from line 7 to 8, where $\alpha$, $\beta$, and $\phi$ are complex numbers. These relations are directly obtained from the commutation relation $[\hat{a}_j, \hat{a}_j^\dagger] = 1$.
The remaining task is to show that $\langle\Psi_{\beta_i}| \hat{a}^{\dagger \, k} \hat{a}^k
|\Psi_{\beta_i}\rangle/k!$ is always finite, i.e. $$\begin{aligned}
\label{eq:B_1}
\frac{\langle\Psi_{\beta_i}| \hat{a}^{\dagger \, k} \hat{a}^k
|\Psi_{\beta_i}\rangle
}{N^k k!}
\le \frac{C_k^{\rm max}}{N^k} ,\end{aligned}$$ with a constant $C_k^{\rm max}$. Here, we replace $\beta_i$ by $\beta$ for simplicity. As shown in Appendix A, the photon number distribution of $|\Psi_\beta\rangle$ decreases exponentially. Denote $$\begin{aligned}
\label{eq:B_2}
|\Psi_\beta\rangle
\equiv \sum_{m=0}^\infty \tilde{b}_m e^{-mx/2} |m\rangle .\end{aligned}$$ The absolute of complex coefficients $\tilde{b}_m$’s are always in between 0 and some constant due to the normalization constraint and let us denote the constant as $|\tilde{b_{\rm max}}|$. Then one has $$\begin{aligned}
\label{eq:B_3}
\hat{a}^k |\Psi_\beta\rangle & = &
\sum_{m=k}^\infty \tilde{b}_m e^{-mx/2} \sqrt{\frac{m!}{(m-k)!}} |m-k\rangle
\nonumber\\ & = &
\sum_{m=0}^\infty \tilde{b}_{m+k}
e^{-(m+k)x/2} \sqrt{\frac{(m+k)!}{m!}} |m\rangle, \end{aligned}$$ and thus $$\begin{aligned}
\label{eq:B_4}
\frac{\langle\Psi_\beta| \hat{a}^{\dagger \, k} \hat{a}^k
|\Psi_\beta\rangle
}{N^k k!} & = &
\frac{e^{-kx}}{N^k k!} \sum_{m=0}^\infty |\tilde{b}_{m+k}|^2
\frac{(m+k)!}{m!} e^{-mx}
\nonumber\\ & \le &
\frac{|\tilde{b}_{\rm max}|^2 e^{-kx}}{N^k k!} \frac{k!}{(1-e^{-x})^{k+1}}
\nonumber\\ & = & \frac{1}{N^k}
\frac{|\tilde{b}_{\rm max}|^2}{1-e^{-x}}
\left( \frac{e^{-x}}{1-e^{-x}} \right)^k .
$$ This bound depends on $\tilde{b}_{\rm max}$ and $x$ i.e. the state $|\Psi_{\beta_i}\rangle$. Therefore, maximizing the rhs of this inequality for all $|\Psi_{\beta_i}\rangle$ and denoting the maximum value as $C_k^{\rm max}/N^k$, one obtains Eq. (\[eq:B\_1\]).
Total photon number statistics
==============================
The exponential and Poissonian distributions are described as $$\label{eq:C_1}
P_E (m) = C_E e^{-mx},$$ and $$\label{eq:C_2}
P_P (m) = \frac{C_P^m}{m!}e^{-C_P},$$ respectively. Then the distribution of the total photon number is given by $$\begin{aligned}
\label{eq:C_3}
P_{\rm tot} (n) & = &
\sum_{m=0}^n P_P (m) P_E (n-m)
\nonumber\\ & = &
e^{-C_P} \sum_{m=0}^n \frac{C_P^m}{m!} C_E e^{-(n-m)x}
\nonumber\\ & = &
C_E e^{-(C_P + nx)} \sum_{m=0}^n \frac{(C_P e^x)^m}{m!}
\nonumber\\ & \le &
C_E e^{C_P (e^x-1)} e^{-nx}, \end{aligned}$$ which decreases exponentially as $n$ increases.
E. Knill, R. Laflamme, and G. J. Milburn, Nature**409**, 46 (2001).
P. van Loock and N. Lütkenhaus, Phys. Rev. A**69**, 012302 (2004).
This is true only for the case when all physical operations during a whole measurement can be described by rank 1 operators. As will be shown in the text, our scheme corresponds to this case.
J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Phys. Rev. Lett.**85**, 4972 (2000).
P. van Loock and S. L. Braunstein, Phys. Rev. Lett.**84**, 3482 (2000).
$e^{\theta_i
(\hat{a}^{\dagger}_i \hat{a}_0
- \hat{a}_i \hat{a}^{\dagger}_0 )} =
e^{-\hat{a}^{\dagger}_i \hat{a}_0 \tan\theta_i}
e^{-\ln \cos\theta_i
(\hat{a}^{\dagger}_i \hat{a}_i - \hat{a}^{\dagger}_0 \hat{a}_0 )}
e^{\hat{a}_i \hat{a}^{\dagger}_0 \tan\theta_i} $ has been used. See S. M. Barnett and P. M. Radmore, [*Methods in Theoretical Quantum Optics*]{} (Oxford University Press, New York, 1997), for example.
S. J. Dolinar, RLE, MIT, QPR No. 111, 1973 (unpublished) p. 115.
J. M. Geremia, Phys. Rev. A**70**, 062303 (2004).
M. Takeoka, M. Sasaki, P. van Loock, and N. Lütkenhaus, Phys. Rev. A**71**, 022318 (2005).
One can show that $C_k^{\rm max}$ takes a finite value, by using the exponential decay of the number distribution of $|\Psi_\beta\rangle$ and the relation $\sum_{m=0}^{\infty} z^m (m+k)!/m! = k!/(1-z)^{k+1}$. (see Appendix B for details).
K. E. Cahill and R. J. Glauber, Phys. Rev. **177**, 1857 (1969).
, ed. M. Abramowitz and I. A. Stegun (Dover Publications Inc., New York, 1965).
, ed. E. F. Beckenbach (Birkhäuser Verlag, Basel, 1978).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Structural chirality can induce counter-intuitive optical forces due to inherent symmetry properties. While optical forces on a single chiral particle in the Rayleigh regime have been well studied, optical forces in coupled chiral particles remain less explored. By using full-wave numerical simulations and analytical methods of source representation and coupled mode theory, we investigated the optical forces induced by a plane wave on two chiral particles coupling with each other via the evanescent near fields. We found that the induced electric and magnetic dipoles of the chiral particles have complicated couplings that give rise to dark and bright modes. The interaction force between the particles can be either attractive or repulsive, and its magnitude can be significantly enhanced by the resonance modes. The attractive force is much stronger if two particles are of opposite handedness compared with the case of same handedness. The electric dipole force and the magnetic dipole force have the same sign for two particles with the same handedness, while they are of different signs for two particles with opposite handedness. The results can lead to a better understanding of chirality-induced optical forces with potential applications in optical manipulations and chiral light-matter interactions.'
author:
- Kah Jen Wo
- Jie Peng
- Madhava Krishna Prasad
- Yuzhi Shi
- Jensen Li
- Shubo Wang
bibliography:
- 'references\_mendeley.bib'
title: Optical Forces in Coupled Chiral Particles
---
\[sec: I. Introduction\]Introduction
====================================
Light carries momentum and can apply optical forces to matter through momentum transfer. Optical forces have been widely used to manipulate small particles [@Ashkin1970; @Grier2003], molecules [@Yang2009], and biological cells [@Ashkin1987]. Recently, there is a growing interest in unusual optical forces such as pulling forces that drag particles towards the light sources [@Chen2011; @Novitsky2011; @Wang2016] and lateral forces that can separate chiral particles [@Wang2014; @Hayat2015; @Rodriguez-Fortuno2015; @Shi2020]. These counterintuitive optical forces can provide novel degrees of freedom to manipulate matter. In contrast to conventional optical forces such as trapping forces which are less sensitive to symmetry properties, unusual optical forces normally arise from special symmetries of light fields or structures. For examples, optical pulling forces can be induced by Bessel beams with conical wavevectors [@Chen2011], and lateral forces can be induced by linearly polarized plane waves on helical particles [@Wang2014]. Surprisingly, microscopic lattice symmetries can change photon pressure at an interface to a tension force [@Wang2016]. Manipulating these symmetry properties, therefore, can generate rich physics of optical forces which we can explore.
Chirality breaks mirror symmetry and can give rise to intriguing optical forces through photonic spin-orbit interactions [@Wang2014; @Bliokh2014; @Sukhov2015; @Kalhor2016; @Alizadeh2016; @Chen2018], where the conversion of light’s spin angular momentum to linear momentum induces asymmetric coupling. Optical forces on chiral particles and chiral structures have been attracting considerable attention not only for the rich physics but also for the potential applications in enantiomer selections [@Cameron2014; @Tkachenko2014; @Bradshaw2014; @Chen2014; @Bradshaw2015; @Zhang2015; @Fernandes2015; @Canaguier-Durand2015; @Bradshaw2015a; @Chen2016; @Zhao2016; @Rahimzadegan2016; @Fernandes2016; @Zhang2017; @Zhao2017; @Cameron2017; @Kamandi2017; @Schnoering2018; @Li2019; @Kazemi2020]. For a single chiral particle in the Rayleigh regime where dipole approximation can be invoked, the particle can be treated as a combination of electric and magnetic dipoles, and the optical force can be well explained by the interaction between the external fields and the dipoles [@Wang2014]. However, it can be complicated if multiple chiral particles couple with each other, in which case the interactions between induced dipoles can dramatically change the scattered fields and hence the optical forces. This scenario is actually more practical as there are usually many chiral particles in enantiomer selections. Understanding the coupling of chiral particles, therefore, is crucial for applications of optical manipulations. In addition, the coupling of chiral particles can generate rich non-Hermitian physics and provide novel degrees of freedom to manipulate light [@Wang2019]. We note that optical forces in various coupled non-chiral structures have been investigated, such as coupled ring cavities [@Wiederhecker2009], coupled waveguides [@Povinelli2005; @Halterman2005; @Li2009; @Roels2009], coupled plates [@Wang2011; @Liu2011a; @Zhang2012], coupled spherical particles [@Burns1989; @Burns1990; @Depasse1994; @Chaumet2001; @Tatarkova2002; @Mohanty2004; @Grzegorczyk2006; @Guillon2006; @Zelenina2007; @Rodriguez2008; @Dholakia2010; @Miljkovic2010; @Liu2011; @Demergis2012; @Frawley2014], and particle-on-slab [@Chaumet2000; @Wang2016a]. However, investigation of optical forces in coupled chiral structures remains in its infancy [@Chen2015; @Shi2020b; @Ahsan2020].
In this paper, we investigated the optical forces induced in coupled chiral particles. The chiral particles have a helical shape and are made of gold whose permittivity is described by a Drude model. We apply full-wave simulations and Maxwell stress tensor to calculate the optical forces acting on each particle and investigate the dependence of the forces on various system parameters including the separation distance, the pitch number, and the handedness. To understand the underlying physics, we employ the source representation and coupled mode theory (CMT) to analytically study the contribution of the induced electric and magnetic dipoles and their mutual interactions. We found that two resonances arise from the coupling of the chiral particles which significantly enhance the interaction force of the two particles. The interaction force can be either attractive or repulsive, and it can be one order of magnitude stronger than the total force of the model system. The electric dipole force always dominates in the interaction force, while the magnetic dipole force can dominate in the total force. In addition, the electric dipole force and the magnetic dipole force have the same sign if the two particles have the same handedness, while they are of different signs if the two particles have opposite handedness.
The paper is organized as follows. In Sec. \[sec: II. MST,SourceRep,CMT\], we introduce the model system and the methodology. In Sec. \[sec: III. Results and Discussion\], we present the numerical results of the scattering cross section and absorption cross section of the model system as well as the numerical results of optical forces. Analytical results based on source representation and CMT will also be presented to explain the phenomena in this section. We then draw the conclusion in Sec. \[sec: IV. Conclusion\].
\[sec: II. MST,SourceRep,CMT\]Maxwell Stress Tensor Method, Source Representation of Force, and Coupled Mode Theory
===================================================================================================================
Our model system consists of two chiral particles with a helical shape, as shown in Fig. \[fig:1\]. The helices have inner radii $r=50$ nm, outer radii $R=150$ nm and pitch $p=220$ nm. The axes of the helices are aligned in z direction. The distance between the centers of the two particles is denoted as $D$. We assume the material of the particles is gold with the relative permittivity described by the Drude model $\varepsilon_r=1-\omega_p^2/(\omega^2+i\omega\gamma)$, where the plasmonic frequency is $\omega_p=1.36\times10^{16}$ rad/s and the damping frequency is $\gamma=7.10\times10^{13}$ rad/s [@Olmon2012]. The two particles are under the illumination of a plane wave propagating in $y$ direction and linearly polarized in $x$ direction: $\mathbf{E}_\mathrm{inc}={\boldsymbol{\hat{\mathbf{x}}}}E_0e^{i\left(ky-\omega t\right)}$. As labelled in Fig. \[fig:1\], we will refer to the two particles as bottom $\left( \mathcal{B}\right)$ and top $\left(\mathcal{T}\right)$ particles.
To numerically calculate the optical forces acting on the chiral particles, we first conduct full-wave simulations of the model system to obtain the total electromagnetic fields by using COMSOL Multiphysics [@COMSOL]. We then evaluate the time-averaged optical force on each particle as where is the time-averaged Maxwell stress tensor [@JohnDavid1998]. Here “\*” denotes the complex conjugate, $\delta_{ij}$ is the Kronecker delta, and $S$ is a closed surface on which we carry out the integral. The surface $S$ completely encloses the particle. The total force acting on the whole system can be obtained by adding up the forces of both particles.
In the long wavelength limit, the fields in Eq. (\[eqn:2\]) can be expressed in terms of multipole sources. The lowest orders of the sources are electric dipole $\mathbf{p}$ and magnetic dipole $\mathbf{m}$ whose Cartesian components are given by the following integrals where $\mathbf{J}$ is the induced current density on the particles which can be obtained from COMSOL simulations, $\mathbf{r}'$ is the position of the current relative to the center of the particles, and $V$ is the volume of the particles. The time-averaged optical force in Eq. (\[eqn:1\]) for one particle can be expressed as a sum of $\mathbf{F^p}$ , $\mathbf{F^m}$ and $\mathbf{F}^\mathrm{int}$ , corresponding to the forces attributed to the electric dipole and magnetic dipole and their interference, respectively. They can be expressed as [@Zhang2015] The force due to high order multipoles including toroidal dipole are negligible in our model system. In Eq. (\[eqn:4\]), $\mathbf{E}$ and $\mathbf{B}$ are the total electric and magnetic fields acting on the particle, which can be expressed as $\mathbf{E}=\mathbf{E}_\mathrm{inc}+\mathbf{E}_\mathrm{sca}$ and $\mathbf{B}=\mathbf{B}_\mathrm{inc}+\mathbf{B}_\mathrm{sca}$. Here $\mathbf{E}_\mathrm{sca}$ and $\mathbf{B}_\mathrm{sca}$ are the fields scattered by the other particle which account for the coupling of the two particles. Both electric dipole and magnetic dipole contribute to the total scattered field such that $\mathbf{E}_\mathrm{sca}=\mathbf{E}_\mathrm{sca}^\mathbf{p}+\mathbf{E}_\mathrm{sca}^\mathbf{m}$ and $\mathbf{B}_\mathrm{sca}=\mathbf{B}_\mathrm{sca}^\mathbf{p}+\mathbf{B}_\mathrm{sca}^\mathbf{m}$. These fields can be directly calculated using the following equations once the dipole moments in Eq. (\[eqn:3\]) are obtained [@JohnDavid1998] where $r=|\mathbf{r}|$ is the radial distance and $\mathbf{n}$ is a unit vector in the direction of $\mathbf{r}$. The Cartesian components of the forces in Eq. (\[eqn:4\]) can be expressed as Here Einstein summation convention has been assumed and $\epsilon_{ijk}$ is the Levi-Civita symbol. In this paper, we will focus on the y component of the forces which is the dominating component in the considered configuration of Fig. \[fig:1\]. As the optical forces are intimately related to the scattering properties of the particles, it is helpful to obtain the normalized scattering cross section using the following expressions [@JohnDavid1998] where $S_\mathrm{inc}=|E_0|^2/(2Z_0)$ denotes the incident intensity. Equations (\[eqn:4\])-(\[eqn:7\]) can be applied to understand the contribution of the electric and magnetic dipoles to the optical forces and scattering properties of the system. However, they cannot provide a intuitive picture to understand the coupling of the dipoles. To this purpose, We apply the temporary coupled mode theory [@Fan2003; @Suh2004] to our model system. Under the source representation, each particle can be approximated as a combination of an electric dipole and a magnetic dipole. The rate equations for the coupled dipoles of the system can be expressed as where $A_\mathbf{p}^\mathcal{B}$, $A_\mathbf{p}^\mathcal{T}$ ($A_\mathbf{m}^\mathcal{B}$, $A_\mathbf{m}^\mathcal{T}$) are the field amplitudes of the electric (magnetic) dipoles of the bottom and top particles, respectively. $S_\mathrm{TE}^\mathcal{B}$ ($S_\mathrm{TM}^\mathcal{B}$) denotes the amplitude of incident wave with TE (TM) polarization. $\omega_0$ is the eigenfrequency of the electric and magnetic dipoles for a standalone particle. $\Gamma_\mathrm{e}=\left( \gamma+\gamma_\mathrm{e}+\gamma_\mathrm{em}\right)/2$ and $\Gamma_\mathrm{m}=\left( \gamma+\gamma_\mathrm{m}+\gamma_\mathrm{em}\right)/2$ denote the total loss of the electric dipole and the magnetic dipole, respectively. $\gamma$ denotes the material loss. $\gamma_\mathrm{e}$, $\gamma_\mathrm{m}$ are the loss of the electric and magnetic dipoles, respectively, due to radiation in the same channel (i.e. polarized in the same direction as the dipoles). $\gamma_\mathrm{em}$ is the loss of the dipoles due to radiation in the cross-polarized channel. $\kappa_\mathrm{ee}$ and $\kappa_\mathrm{mm}$ denote the couplings of the dipoles. The above equation can be re-written as where
In the above formulations, we have assumed that the electric dipoles and the magnetic dipoles have no direct coupling due to the orthogonality of their fields and the subwavelength separation of the particles. This is consistent with the fact that electric dipole and magnetic dipole are “two sides of the same coin”, i.e. they belong to a single mode of the helix which can be considered as a damped LC resonance, since the helix can be modelled as a lumped element consisting of a capacitor, an inductor, and a resistor. We have $\kappa_\mathrm{mm}=\kappa_\mathrm{ee}$ for chiral particles of the same handedness and $\kappa_\mathrm{mm}=-\kappa_\mathrm{ee}$ for chial particles of opposite handedness. Define $\kappa_\mathrm{ee}=\kappa$ for simplicity. The eigenfrequencies of the effective Hamiltonian $H$ can be obtained as $\omega_\mathrm{e}=\omega_0\pm\kappa-i\Gamma_\mathrm{e}$ and $\omega_\mathrm{m}=\omega_0\pm\kappa-i\Gamma_\mathrm{m}$. Assume time convention of $e^{-i\omega t}$, Equation (\[eqn:8\]) or (\[eqn:9\]) can be solved to obtain the field amplitudes $A_\mathbf{p}^\mathcal{B}$, $A_\mathbf{p}^\mathcal{T}$, $A_\mathbf{m}^\mathcal{B}$, $A_\mathbf{m}^\mathcal{T}$: The total scattering cross section of the whole system can be expressed as $\left|\left(\sqrt{\gamma_\mathrm{e}}+\sqrt{\gamma_\mathrm{em}}\right)\left( A_\mathbf{p}^\mathcal{B}+A_\mathbf{p}^\mathcal{T}\right)\right|^2+\left|\left(\sqrt{\gamma_\mathrm{m}}+\sqrt{\gamma_\mathrm{em}}\right)\left( A_\mathbf{m}^\mathcal{B}+A_\mathbf{m}^\mathcal{T}\right)\right|^2$. We note that CMT usually can well reproduce the numerical results if the loss of the system is small [@Fan2003; @Suh2004]. This condition, however, is not fulfilled in our model system due to the material loss of gold and the open boundary conduction, which lead to low quality factors of the modes. Despite this, we will see that the above CMT can qualitatively reproduce the the numerical results and the analytical results given by Eq. (\[eqn:7\]).
\[sec: III. Results and Discussion\]Results and Discussion
==========================================================
\[subsec: III. A. Scattering and absorption cross sections\]Scattering and absorption cross sections
----------------------------------------------------------------------------------------------------
The optical forces in the model system are determined by both the scattered near fields and far fields. To understand the properties of the scattered fields, we first instigated the scattering cross section and absorption cross section of the model system. The full-wave numerical results of the scattering and absorption cross sections are shown in Fig. \[fig:2abc\]. The distance between the particles is fixed at $D=340$ nm. In Fig. \[fig:2abc\](a), we notice that two resonances appear in the considered frequency range due to the coupling of the two particles. The first resonance leads to strong absorption but weak scattering. The second resonance leads to strong scattering but relatively weak absorption. Overall, the scattering of the system is stronger than the absorption in the considered frequency range. We will see that these phenomena are attributed to the dark and bright modes of the model system in the presence of the coupling between the induced dipoles (Sec. ). To understand the effect of pitch number, we then calculated the cross sections from $N=2$ to $N=4$. As shown in Fig. \[fig:2abc\](a) to (c), the resonance frequencies undergo red-shift, therefore, the particles become more subwavelength and have a smaller scattering cross section. The absorption cross section also reduces as the fields’ penetration (i.e. skin depth) relative to the wavelength becomes smaller at lower frequencies. We will discuss the physics of these modes in detail in Sec. .
The scattering and absorption of the model system are also affected by the coupling of the two chiral particles. To understand this effect, we calculated the cross sections for different coupling distance $D=340 \text{ nm},380 \text{ nm}, 460 \text{ nm}$. As shown in Fig. \[fig:3ab\], variation of $D$ leads to a more dramatic change of the absorption cross section compared to the change of the scattering cross section. This is expected as the separation of the two chiral particles are subwavelength and their coupling is mainly attributed to the evanescent near fields which determines the absorption of the system. Increasing the separation distance will reduce the coupling strength and leads to a smaller frequency splitting of the two modes, as shown in Fig. \[fig:3ab\](b). This is consistent with the properties of the total scattering force to be discussed in Sec. . We will provide an intuitive explanation of these phenomena using the analytical model in Sec. .
\[subsec: III. B. Numerical results of optical forces\]Numerical results of optical forces
------------------------------------------------------------------------------------------
The optical force acting on each particle and the total optical force of the model system are calculated using the Maxwell stress tensor approach introduced in Sec. \[sec: II. MST,SourceRep,CMT\]. The results are shown in Figs. \[fig:4abcdefghi\]-\[fig:6abc\]. We focus on the optical forces along $y$ direction. Figure \[fig:4abcdefghi\] (a) shows that the interaction force is attractive at the first resonance and repulsive at the second resonance. In addition, the magnitude of the attractive force is much larger than that of the repulsive force. The total force of the system has two peaks and is always positive (i.e. pushing) since it is a scattering force, as shown in Fig. \[fig:4abcdefghi\](d). Figure 4(a),(b) and (c) show that, when the pitch number increases from $N=2$ to $N=4$, the magnitude of the interaction forces decrease. This is expected as the resonance frequencies undergoes red-shift as $N$ increases, leading to a smaller dipole moments and thus smaller forces. This is consistent with the properties of the scattering cross section in Fig. \[fig:2abc\]. Similar phenomena also exist for the total forces shown in Fig. \[fig:4abcdefghi\](d), (e) and (f). In addition, the frequency splitting of the two peaks increases as the pitch number is increased. The reason is that the coupling due to the evanescent near fields is stronger at lower frequencies.
The dependence of the optical forces on distance $D$ is also studied, and the results are shown in Figure \[fig:5abc\]. The magnitudes of the attractive and repulsive forces at the two resonances are enhanced when $D$ decreases from $D=460$ nm to $D=340$ nm, as shown in Fig. \[fig:5abc\](a) and (b), and meanwhile the frequency splitting of the two resonances are increased. This is expected as the coupling is stronger at smaller distance of $D$. Figure \[fig:5abc\](c) shows that the variation of the total force is small except for an enlarged splitting of the two resonance frequencies, which agrees with the results of scattering cross section in Fig. \[fig:3ab\](a) and confirms the near-field nature of the interaction force and the far-field nature of the total force.
The sign of chirality (i.e. the handedness of the helices) is of critical importance to the optical forces in chiral systems [@Wang2014]. To understand the effect of the handedness of the chiral particles, we set $N=3$, $D=340$ nm and calculated the optical forces for two helices with opposite handedness. The model system is shown by the inset in the corner of Fig. \[fig:6abc\](a). The incident wave remains the same as in Fig. \[fig:1\]. The interaction force acting on individual chiral particle and the total force are shown in Fig. \[fig:6abc\](a) and (b). We notice that the interaction force at the first resonance is attractive as in the case with the same handedness, but it is one order of magnitude stronger. The interaction force at the second resonance is suppressed. The total force at the first resonance is much stronger than the force at the second resonance. These are attributed to the different orientations of the induced dipole moments compared to that in the case with the same handedness, which is directly determined by the handedness of the helices. We will discuss the physics of these phenomena using a coupled dipole picture in the next section.
\[subsec: III. C. Analytical results based on source representation and CMT\]Analytical results based on source representation and CMT
--------------------------------------------------------------------------------------------------------------------------------------
To understand the above numerical results of cross sections and optical forces, we employ a source representation and treat the chiral particles as electric and magnetic dipoles under long wavelength condition. We use a CMT to understand the coupling of the dipoles which helps to explain the different mode properties at the two resonance frequencies. The interaction forces can be understood as a net result of the Coulomb force and the magnetic force induced by the electric and magnetic dipoles.
A standalone chiral particle in Fig. \[fig:1\] has one resonance at $\sim 75$ THz in the considered frequency range with enhanced electric and magnetic dipole moments. When two same chiral particles couple with each other, the degeneracy is broken and two resonances emerge, one at a higher frequency and the other one at a lower frequency. This is confirmed by the dipole moments in Fig. \[fig:7abc\], which are evaluated using Eq. (\[eqn:3\]). Figure \[fig:7abc\](a) and (b) show the amplitudes of the electric dipole moment $p_z$ and magnetic dipole moment $m_z$, respectively, where the blue (red) solid lines denote the dipole moments of the bottom (top) chiral particle. Only the dominating component along $z$ direction (i.e. direction of the helix’s axis) are considered. The orientations of the dipole moments are schematically shown in Fig. \[fig:7abc\](c). At the first resonance, the two electric dipoles align in antiparallel directions and interference destructively, lowering the mode’s energy. The two magnetic dipole moments also align anti-parallelly and lower the mode’s energy. The combined dipoles, therefore, form an electromagnetic dark mode and suppress the scattering of the system. At the second resonance, the two electric dipole moments align in parallel direction and interference constructively, raising the mode’s energy. The magnetic dipole moments also align parallelly and raise the mode’s energy. The combined dipoles, therefore, form an electromagnetic bright mode and enhance the scattering of the system.
The scattering cross sections of the dark and bright modes of the coupled chiral particles can be directly determined using Eq. (\[eqn:7\]) after obtaining the dipole moments. The results are shown in Fig. \[fig:8\] for the contributions of electric dipole $\sigma_\mathrm{sca}^\mathbf{p}$ (dashed purple line) and magnetic dipole $\sigma_\mathrm{sca}^\mathbf{m}$ (dashed green line) and $\sigma_\mathrm{sca}^\mathbf{p}+\sigma_\mathrm{sca}^\mathbf{m}$ (solid blue line). As expected, the electric dipole dominates since the particle is of subwavelength, and the scattering cross section is significantly enhanced at the bright mode at $\sim 82$ THz, while it is much smaller at the dark mode at $\sim 68$ THz. The total contribution of $\sigma_\mathrm{sca}^\mathbf{p}+\sigma_\mathrm{sca}^\mathbf{m}$ quantitatively agrees with the full-wave numerical result (symbol line) obtained using COMSOL. To verify the physical picture of coupled dipoles, we fit $\sigma_\mathrm{sca}^\mathbf{p}+\sigma_\mathrm{sca}^\mathbf{m}$ using the expression of scattering cross section predicted by the CMT in Sec. \[sec: II. MST,SourceRep,CMT\]. The fitting result is denoted by the solid black line in Fig. \[fig:8\], which reasonably well agrees with the analytical result. We notice that the CMT cannot accurately reproduce the analytical result in presence of strong dispersion of gold and the low quality factor of the resonance [@Fan2003].
Using the physical picture of coupled dipoles in Fig. \[fig:7abc\](c), one can immediately predict the sign of the interaction force between the chiral particles. The force is a sum of the Coulomb force due to the electric dipoles and the magnetic force due to the magnetic dipoles. The Coulomb force dominates as a result of the dominance of electric dipoles. For the electromagnetic dark mode, both the electric dipoles and magnetic dipoles induce an attractive force, i.e. a binding force. For the electromagnetic bright mode, both the electric dipoles and magnetic dipoles induce a repulsive force. This agrees with the numerical results in Fig. \[fig:4abcdefghi\]. In addition, we calculated the optical forces using the source representation in Eq. (\[eqn:6\]), and the results are shown in Fig. \[fig:9abc\]. Figure 9(a) and (b) shows the optical forces acting on the top and bottom particles, respectively. We notice that the electric force dominates, as expected, and it is attractive at the first resonance and repulsive at the second resonance. The magnitude of the analytical results (solid blue line) is larger than the numerical result for two reasons: the dipole approximation is not accurate when applied to near-field interactions where evanescent waves with large wavevectors play a dominating role; the size of the chiral particles is comparable to their separation, therefore, neglecting the retardation effect as assumed in the source representation is not appropriate. Despite this, the analytical results can already reproduce the key features of the interaction force, such as the attractive (repulsive) force at the first (second) resonance. The analytical results will approach the full-wave numerical results if the separation of the two chiral particles are larger, in which case the effect of evanescent waves is weak. This is confirmed by the results shown in Fig. 10 for the case of $D=1100$ nm, where the analytical results quantitatively agree with the numerical results for both the individual forces and the total force. Figure \[fig:9abc\](c) shows the comparison between the analytical result and the numerical result for the total optical force on the whole system. The analytical result (solid blue line) agrees very well with the numerical result (symbol red line), since the source representation can well describe the far-field properties of the system under long wavelength condition. The electric dipole force and the magnetic dipole force correspond to the addition of the individual dipole forces in Fig. \[fig:9abc\](a) and (b). Interestingly, the total electric dipole force is negative at the first resonance.
To understand the effect of chirality on the optical forces, we also apply the source representation and CMT to the case where two chiral particles have opposite handedness. Figure \[fig:11abc\](a) and (b) shows the induced electric dipole moment and magnetic dipole moment, respectively. In contrast to the case with the same handedness, the dipole moments are significantly enhanced at the first resonance while are much weaker at the second resonance. The orientations of the dipoles are shown in Fig. \[fig:11abc\](c). We notice that, at the first resonance, the electric dipoles are anti-parallel and give rise to destructive interference, lowering the energy, while the magnetic dipoles are parallel and give rise to constructive interference, raising the energy. The mode is therefore electrically dark but magnetically bright. This mode can be more easily excited as opposed to the electromagnetic dark mode in the case of Fig. \[fig:7abc\], which explains the larger dipole moments. At the second resonance, the electric dipoles are parallel and give rise to constructive interference, raising the energy, while the magnetic dipoles are anti-parallel and give rise to destructive interference, lowering the energy. This mode is therefore electrically bright but magnetically dark, and it is more difficult to excite than the electromagnetic bright mode in the case of Fig. \[fig:7abc\]. The above mechanisms derive from the opposite handedness of the chiral particles and gives rise to the enhanced dipole moments at the first resonance and suppressed dipole moments at the second resonance.
The dipole moments in Fig. \[fig:11abc\] are used to evaluate the scattering cross sections, and the results are shown in Fig. \[fig:12\]. In contrast to the Fig. 8 for the case of same handedness, at the frist resonance, the contribution of the magnetic dipoles (denoted by the dashed green line) is comparable to that of electric dipoles (denoted by the dashed purple line) due to the magnetic bright mode. The electric dipoles dominate in the second resonance since the corresponding mode is electrically bright but magnetically dark. The total scattering cross section of $\sigma_\mathrm{sca}^\mathbf{p}+\sigma_\mathrm{sca}^\mathbf{m}$ (solid blue line) agrees well with the full-wave numerical result (symbol red line). We fit $\sigma_\mathrm{sca}^\mathbf{p}+\sigma_\mathrm{sca}^\mathbf{m}$ using the expression of scattering cross section predicted by the CMT in Sec. \[sec: II. MST,SourceRep,CMT\]. The fitting result is denoted by the solid black line, which qualitatively agrees with the analytical and numerical results. In addition, we found that the orientations of the dipoles predicted by the CMT agree with the physical picture of source representation in Fig. \[fig:11abc\](c).
The optical forces for the coupled chiral particles with opposite handedness can also be understood based on the source representation. As shown in Fig. \[fig:13abc\](a) and (b), at the first resonance, the attractive force due to the electric dipoles (solid blue line) dominates, the resulting interaction force is one order of magnitude larger than the force in the case with the same handedness, as seen by comparing Fig. \[fig:9abc\](a),(b) and Fig. \[fig:13abc\](a),(b). This is due to the larger dipole moments induced at the first resonance. The interaction force at the second resonance is too weak to be observable, in contrast to the case with the same handedness in Fig. \[fig:9abc\](a) and (b). The reason is twofold: the mode is more difficult to excite than the electromagnetic bright mode in the case with the same handedness; the forces due to electric and magnetic dipoles are in opposite directions. The analytical results of the total interaction force (solid blue lines) qualitatively agree with the numerical results (symbol red lines). The agreement is better than the case of Fig. 9(a) and (b) due to the effective larger separation of the electric dipoles (the ends of the helices, where the charges accumulate, point outwards). The net optical force acting on the two particles is shown in Fig. \[fig:13abc\](c). The force is entirely attributed to the scattered far fields which can be well described by the source representation. The analytical result (solid blue line), therefore, quantitatively agrees with the numerical result (symbol red line). The electric dipole force and magnetic dipole force correspond to the addition of the individual forces in Fig. \[fig:13abc\](a) and (b). We notice that the electric dipole force dominates at the first resonance, while at the second resonance it is comparable to the magnetic dipole force.
\[sec: IV. Conclusion\]Conclusion
=================================
In summary, we investigated the optical forces induced by an incident plane wave on two coupled chiral particles. The eigenmodes of the particles in the considered frequency range lead to resonance enhancement of induced electric and magnetic dipoles. The couplings of the dipoles give rise to two new modes, one at a lower frequency and the other at a higher frequency. Using CMT and a source representation, we found that electric dipoles dominate in the near fields and the two modes can be dark or bright depends on the relative orientation of the dipoles. In the case of two particles with the same handedness, the lower frequency mode is an electromagnetic dark mode where both electric and magnetic dipoles align in anti-parallel directions, resulting in suppressed scattering and an attractive force between the particles. The higher frequency mode is an electromagnetic bright mode with both electric and magnetic dipole aligned in parallel direction, resulting in enhanced scattering and a repulsive force. In the case of two particles with opposite handedness, the lower frequency mode is electrically dark but magnetically bright, and the attractive force is significantly enhanced by the resonances of dipoles. In contrast, the higher frequency mode is electrically bright but magnetically dark, where the scattering cross section is dominated by the electric dipoles and the repulsive force is almost negligible.
Light-matter interactions associated with chiral structures have been playing critical roles in nanophotonics and metamaterials. The considered model system is probably the simplest structure with near-field couplings fully taken into account, therefore, it can serve as a basic unit to understand the light-matter interactions in periodic chiral structures such as chiral metamaterials and chiral photonic crystals. The results in this paper can also be applied to understand the optical forces in general chiral structures, which could have applications in optical micromanipulations and reconfigurable optical devices. In addition, engineering the couplings in chiral structures can introduce interesting non-Hermitian physics into the systems, where the integration of structural chirality and the chirality of exceptional points (i.e. a non-Hermitian degeneracy) can generate novel optical forces.
\[sec: V. Acknowledgements\]Acknowledgements
============================================
This work was supported by the National Nature Science Foundation of China (Project No. 11904306) and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project Nos. HKUST C6013-18G and CityU 11306019).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The optical orientation of the exciton spin in an ensemble of self-organized cubic GaN/AlN quantum dots is studied by time-resolved photoluminescence. Under a polarized quasi-resonant excitation, the luminescence linear polarization exhibits no temporal decay, even at room temperature. This demonstrates the robustness of the exciton spin polarization in these cubic nitride nanostructures, with characteristic decay times longer than 10 ns.'
author:
- 'D. Lagarde, A. Balocchi, H. Carrère, P. Renucci, T. Amand, X. Marie'
- 'S. Founta, H. Mariette'
title: |
Room temperature Optical Orientation of Exciton Spin\
in cubic GaN/AlN quantum dots
---
Wide bandgap GaN-based semiconductors and related heterostructures represent promising candidates for spintronic applications[@Beschoten; @Ohno; @Buyanova]. The weak spin-orbit coupling in these materials should yield long electron spin relaxation times [@Krishnamurthy_2003] and their large exciton binding energy ($\sim$26 meV in bulk GaN) should allow the control and manipulation of the exciton spin, even at high temperature.\
Compared to GaAs-based structures, very few measurements of the carrier spin properties have been performed on these wide bandgap nitrides which can crystallize either in the wurtzite (Wz) or zinc-blende cubic (ZB) structure [@Paisley]. Most of these studies deal with the measurement of the electron or hole spin dynamics in Wz GaN or InGaN structures. Beschoten *et al.* measured an *electron* spin lifetime of $\sim$35 ps at room temperature in *n*-doped Wz epilayers [@Beschoten]. A *hole* spin coherence time of $\sim$120 ps was found in *p*-type bulk Wz GaN at low temperature [@Ohno]. However, a very fast *exciton* spin relaxation time, of the order of 1 ps, was deduced from transient reflectivity or spin grating experiments in non-intentionally doped Wz GaN epilayers [@Tackeuchi_2004; @Ishiguro]. In Wz nitride nanostructures (quantum wells or quantum dots), *exciton* spin relaxation times of the order of 200 ps were also reported at $T=300$ K with a rather weak temperature dependence [@Nagahara_2005; @Nagahara_2006]. In all these wurtzite nitride structures, the electronic and spin properties are highly affected by the strong built-in electric field due to the spontaneous and piezoelectric polarizations [@Fonoberov; @Julier].\
In contrast, these polarizations are negligible in zinc-blende GaN structures, due to the high crystal symmetry and much longer carrier spin relaxation times can be expected. Krishnamurthy *et al.* calculated that the electron spin lifetime in pure ZB GaN is about three orders of magnitude larger than in GaAs at all temperatures as a result of the lower spin-orbit interaction and higher conduction band density of states [@Krishnamurthy_2003]. An electron spin lifetime as long as 100 ns is predicted at $T=300$ K in high quality bulk GaN, dropping significantly if the defect density increases. In a ZB GaN epilayer, Tackeuchi *et al.* measured indeed a much longer bound exciton spin relaxation time ($\sim$5 ns at 15 K) compared to Wz structures but surprisingly no spin memory was found above 75 K [@Tackeuchi_2006]. No report has been published to date on the measurement of the spin dynamics in ZB nitride nanostructures, in which the influence of defects on the electronic properties is usually weaker thanks to the carrier confinement.
We present in this letter a detailed time-resolved investigation of the exciton spin dynamics in self-organized ZB GaN/AlN quantum dots (QDs). These optical orientation experiments clearly evidence an exciton linear polarization which persists at room temperature, in contrast to the ususal behaviour in other III-V or II-VI nanostructures [@Paillard; @Kalt; @Henneberger].
The QD sample investigated here consists in 18 self-assembled ZB GaN quantum dot planes in AlN barriers, grown on 3C-SiC(001)/Si pseudosubstrate by molecular-beam epitaxy [@Mariette]. The dots have a typical diameter of 12 nm and a height around 1.5 nm. The sample is nominally undoped and the areal dot density is of the order of $1\cdot10^{11}$ cm$^{-2}$. In time-resolved photoluminescence (PL) experiments, the excitation source is provided by a mode-locked frequency tripled Ti:Sa laser, with a 1.5 ps pulse width and a tunable wavelength in the range 260-310 nm. The laser beam is focused onto the sample to a 100 $\mu$m diameter spot with an average power $P_{exc}=1$ mW [@Puissance]. The PL signal is dispersed by an imaging spectrometer and then temporally resolved by a S20 photocathode streak camera with an overall time resolution of 8 ps. The linear polarization degree of the luminescence is defined as $P_{Lin} = (I^X - I^Y) / (I^X + I^Y)$. Here $I^X (I^Y)$ denotes the $X (Y)$ linearly polarized PL components ($X$ and $Y$ are chosen parallel to the \[110\] and \[1$\overline{1}$0\] crystallographic directions).
![Time-integrated PL spectrum under a quasi-resonant excitation at $T=20$ K. The linear polarization degree following a linearly-polarized ($\blacklozenge$) or circularly-polarized ($\triangle$) excitation is also plotted. Inset: Time-integrated PL spectra of the QD for quasi-resonant excitation ($E_{exc}=4.11$ eV) and non-resonant excitation ($E_{exc}=4.77$ eV).[]{data-label="fig1"}](fig_1){width="50.00000%"}
The inset of Fig. \[fig1\] presents the time-integrated PL spectrum measured at $T=20$ K for a non-resonant excitation energy $E_{exc} = 4.77$ eV. The ground state emission is centered around 4.15 eV with a full width at half maximum of 250 meV which results from the inhomogeneous broadening of the ground state exciton energies due to the size and strain distributions among the QD ensemble. The PL spectrum obtained under Quasi-Resonant (QR) excitation ($E_{exc}=4.11$ eV) is also plotted. QR excitation means here that the laser excitation energy lies within the QD ground state energy range. A much narrower PL emission spectrum is measured in this case due to the spectral selectivity of the excitation [@Heitz]. The linear polarization of the time-integrated[@Integration] exciton PL emission following a linearly-polarized ($\sigma^X$) QR excitation is presented in Fig. \[fig1\]. A clear linear polarization is observed for any detection energy in the spectrum, with a larger value ($P_{Lin}\sim 30$ $\%$) on the high energy part. This linear polarization arises from the optical orientation of the exciton spin (also called exciton optical alignment), as already observed in many III-V or II-VI bulk materials or heterostructures [@Planel; @Orientation; @Paillard; @Kalt]. The circular or linear exciton polarization dynamics can be described in the framework of an effective pseudospin with $S=1/2$ [@Ivchenko]. In this formalism, the exciton states with angular momentum $\left | +1 \right \rangle$ and $\left | -1 \right \rangle$ are equivalent to a pseudospin polarized parallel or anti-parallel to the *z*-axis ($S_z=+\frac{1}{2}$ or $-\frac{1}{2} $). The quantization axis ($Oz$) is chosen along the light propagation direction which is also the sample growth direction. The linear exciton states $\left | X \right \rangle$ and $\left | Y \right \rangle$ are described by a pseudo-spin $S_x=+\frac{1}{2}$ or $-\frac{1}{2} $, respectively. Under a linearly polarized excitation ($\sigma^X$), the initial exciton pseudospin $\overrightarrow{S}(0)$ is parallel to the ($Ox$) axis. The linear exciton PL polarization writes simply $P_{Lin}=2S_x$. The absence of linear polarization following a circularly-polarized excitation ($\sigma^+$), as shown in Fig. \[fig1\], (open triangles) proves that the measured polarization (solid diamonds) does not arise from the valence state mixing induced by the strain or by the QD shape anisotropy, as it was observed in Wz InGaN QDs [@Bimberg; @Angle]. Indeed, such effect would yield a linearly-polarized exciton emission whatever the polarization of the excitation light is. The absence of linear polarization when the laser excitation energy increases (as detailed below) is another confirmation that the observed linear polarization in Fig. \[fig1\] is indeed due to the light-induced optical alignment of excitons in the QDs. Let us furthermore recall that the observation of optical pumping in linear configuration is a clear signature of intrinsic exciton emission [@Planel] : the loss of coherence between electron and hole spins, inherent in the impurity bound excitons or charged excitons, inhibits the optical alignment of these pseudo-particles.
![Temporal dependence of the PL components co-polarized $I^X$ ($\circ$) and counter-polarized $I^Y$ (+) to the $\sigma^X$ polarized excitation laser at $T=20$ K and the corresponding linear polarization degree $P_{Lin}$ ($\blacklozenge$). $E_{exc}=4.11$ eV, $E_{det}=4.02$ eV.[]{data-label="fig2"}](fig_2){width="50.00000%"}
Fig. \[fig2\] displays the time evolution of the co-polarized $I^X$ and counter-polarized $I^Y$ PL intensity components obtained after a linearly-polarized $\sigma^X$ excitation (the detection energy is $E_{det}=4.02eV$). The decay time of the PL intensity is about 350 ps, much shorter than in Wz GaN structures because of the absence of the internal fields [@Simon; @Grandjean]. It is clear from Fig. \[fig2\] that the QD emission exhibits a linear polarization $P_{Lin}\sim 20$ $\%$ which remains strictly constant in time within our experimental accuracy during the exciton emission. This behaviour is the same whatever the detection energy is within the PL spectrum (not shown). Using an exponential fitting procedure, we can infer that the linear polarization decay time is longer than 10 ns. This result differs strongly from the exciton polarization dynamics in Wz-type structures, characterized by a polarization decay time 2 or 3 orders of magnitude shorter [@Tackeuchi_2004; @Nagahara_2006].
The excitonic properties of nitride-based ZB QDs are still poorly understood. However, from similar results on InAs, CdTe or CdSe self-organized QDs [@Paillard; @Kalt; @Henneberger], the experimental data presented above strongly suggest that the eigenstates of the exciton in the GaN QDs are linearly-polarized states, aligned along the \[110\] and \[1$\overline{1}$0\] directions. This exciton fine structure may originate from QD elongation and/or interface anisotropy which yield a significant anisotropic exchange interaction between the electron and the hole forming the exciton. No circular polarization of the PL emission was observed following a circularly-polarized excitation. Similarly, no PL linear polarization was detected after a linearly-polarized excitation along \[100\] or \[010\] (not shown). This is consistent with $\left | X \right \rangle$ and $\left | Y \right \rangle$ linearly-polarized exciton eigenstates in an inhomogeneous QD ensemble. When the polarized excitation creates a coherent superposition of these exciton eigenstates, the anisotropic exchange energy statistical fluctuations among the detected QD ensemble prevent the observation of spin quantum beats [@Paillard].
![PL linear polarization detected at a fixed energy $E_{det}=4.02$ eV as a function of excitation energy at $T=20$ K. Inset: Spectral dependence of the co- ($I^X$) and counter-polarized ($I^Y$) PL components and the associated linear polarization degree for $E_{exc}=4.11$ eV.[]{data-label="fig3"}](fig_3){width="50.00000%"}
![(a) Linear polarization dynamics for $T=20$, 150 and 300 K. (b) Temperature dependence of the linear polarization degree (the full line is a guide to the eyes). For $T=20$ K, the excitation and detection energies are $E_{exc}=4.11$ eV and $E_{det}=4.02$ eV, respectively. For higher temperature, the excitation and detection conditions are detailed in the text.[]{data-label="fig4"}](fig_4){width="50.00000%"}
The relative low value of the linear polarization ($P_{Lin}\sim$20 $\%$) observed in Fig. \[fig2\] is mainly due to the fact that the experiments are not performed under strictly resonant conditions (which could not be performed with our present experimental set-up because of the large laser scattering on the sample surface). In these QR excitation conditions, both the energy relaxation processes and the excitation of exciton states with different symmetries contribute to decrease the measured $P_{Lin}$. Nevertheless, the exciton spin relaxation quenching, evidenced by the absence of decay of the linear polarization, is a striking feature here. Already observed in other self-organized QD structures at low temperature, it proves that neither the electron nor the hole spin relax on the exciton lifetime scale and that once the exciton occupies the ground state of the QD, no transient change of the polarization occurs within the time window under investigation [@Paillard; @Steel; @Henneberger].
Fig. \[fig3\] displays the degree of the PL linear polarization detected at a fixed energy ($E_{det}=4.01$ eV) as a function of the excitation energy. The average $P_{Lin}$ decreases from 20 to 0 $\%$ when the excess energy $E_{exc}-E_{det}$ increases from $\sim$90 to 400 meV. This behavior is consistent with the variation of the linear polarization in the PL spectrum measured in Fig. \[fig1\]. The decrease of $P_{Lin}$ appears as a monotonic function of the excess energy, in agreement with the pioneering work in bulk CdS or recent results in CdSe/ZnSe self-organized QDs [@Planel; @Kusrayev]. Nevertheless, no distinct peaks due to LO-phonon cascade processes are evidenced. The absence of such peaks in Fig. \[fig3\] is probably due to the large inhomogeneous broadening in ZB GaN QDs. We emphasize that for $E_{exc}-E_{det}>$ 400 meV, no linear polarization is measured for any detection energy in the PL spectrum.
Fig \[fig4\](a) presents the dependence of the exciton PL linear polarization dynamics upon the lattice temperature. As the amplitude of the linear polarization depends on the detection energy in the spectrum (Fig. \[fig1\]), we varied the excitation and detection energies for the different temperatures following a simple Varshni law [@Varshni] keeping constant $E_{exc}-E_{det}=90$ meV. This makes possible the measurement of the temperature dependence of the exciton spin properties for the same QDs. It is worth mentioning that we obtained qualitatively the same results for fixed excitation and detection energies (not shown). Contrary to the reported results in ZB bulk GaN [@Tackeuchi_2006], optical orientation of exciton spin is clearly observed in Fig. \[fig4\] above 75 K. Moreover, no temporal decay of $P_{Lin}$ is observed, even for high temperatures. This result is to be compared with those obtained on QDs made of other semiconductor materials (InAs, CdSe, CdTe)[@Paillard; @Kalt; @Henneberger]. In these QDs, the linear polarization decay time, usually longer than the exciton lifetime at T=10 K, drastically drops when the temperature increases a few tens of K [@Paillard; @Kalt]. Thus, the robustness of the exciton linear polarization up to room temperature appears here as a unique property of the ZB GaN QDs. Though we have never observed any temporal decay of $P_{Lin}$ on the exciton lifetime scale, a clear decrease of the amplitude of the linear polarization is evidenced. The uncertainty on the measurements in Fig. \[fig4\](b) does not allow us to extract an accurate thermal activation energy (it lies within the 50-100 meV range which is a plausible value for the exciton binding energy in these structures [@Ramvall; @Xia]).
Finally, we have investigated the possible conversion from optical alignment to optical orientation of excitons when a magnetic field is applied along the growth direction (Faraday geometry) [@Dzhioev; @Paillard]. If the exciton Zeeman splitting $\hbar\Omega_z=g\mu_BB$ is much larger than the anisotropic exchange energy, the QD exciton eigenstates are no longer the exciton $\left | X \right \rangle$ and $\left | Y \right \rangle$ linearly-polarized states but the $\left | +1 \right \rangle$ and $\left | -1 \right \rangle$ circular ones. However, this conversion has not been observed for any magnetic field value up to 4 T (the maximum field available on our experimental set-up), probably because of the very large value of the exchange energy compared to the exciton Zeeman term. Much stronger magnetic fields would be required to evidence these effects.
In conclusion, we have studied the optical orientation of exciton spin in self-organized cubic GaN/AlN quantum dots. We observe no measurable temporal decay of the exciton linear luminescence polarization at any investigated temperature. Even at room temperature, the exciton spin decay time is larger than 10 ns, i.e. 2 or 3 orders of magnitude longer than in wurtzite-type structures. Moreover, these results contrast with the fast exciton spin relaxation previously observed in other III-V or II-VI QD systems at high temperature.
The authors are grateful to EADS Research Foundation and to Institut Universitaire de France for financial support. They are also grateful to the NOVASiC compagny which provides them with the 3C-SiC(001)/Si pseudosubstrates and with the financial support of S.F. PhD fellowship. They thank B. Daudin and B. Gayral (Grenoble) for fruitful discussions.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
by J. Klusoň\
Department of Theoretical Physics and Astrophysics\
Faculty of Science, Masaryk University\
Kotlářská 2, 611 37, Brno\
Czech Republic\
E-mail:
title: 'Proposal for Background Independent Berkovits’Superstring Field Theory'
---
Introduction {#first}
============
It was shown in [@Horowitz] that the fundamental formulation of bosonic open string field theory [@WittenSFT] can be formulated as a pure cubic string field theory action. This action does not contain any BRST operator and it is formally background independent. When we then expand string field around any solution of the equation of motion arising from this pure cubic string field action we obtain exactly Witten’s bosonic field theory with the BRST operator constructed from the field obeying the equation of motion of the pure cubic string field theory. This approach has been further developed in [@Horowitz1], where another solutions of the pure cubic string field theory that do not correspond to any usual BRST operator were found.
Recently there was a great interest in the formulation of the open bosonic string field theory around the closed string vacuum (Vacuum string field theory-VSFT) [@SenV1; @Hata; @Taylor; @Feng; @Feng1; @SenV2; @SenV3; @Gross1; @SenV4; @Muki; @David; @SenV5; @Gross2] that arises as a final point of the tachyon condensation on D25-brane (For a very nice review of the string field theory and its relation to the problem of the tachyon condensation, see [@Ohmori].) VSFT is characterised by BRST operator with trivial cohomology so that there are not any physical open string excitations. This BRST operator is constructed from ghost fields only so that vacuum string field theory is formally background independent. VSFT seems to be a very promising area of research which could lead to better understanding of the string theory.
The problem of the tachyon condensation was studied in superstring theory as well. It seems that the most promising approach to the study of the tachyon condensation in this context is based on Berkovits’ superstring field theory [^1] [@Berkovits2; @Berkovits3]. Two major advantages of Berkovits superstring field theory are that it is manifestly $SO(3,1)$ super-Poincare invariant and that it does not require contact terms to remove tree-level divergences [@BerkovitsD] (For recent review of Berkovits superstring field theory (NSFT), see [@BerkovitsR].) In particular, it was shown that the calculation of the tachyon potential in NSFT is in a very good agreement with Sen’s conjecture [@SenP]. The tachyon kink and lump solution was also analysed (For recent work, see [@Ohmori1], for list of references, see [@Ohmori].) However this theory is formulated for Neveu-Schwarz open string sector only because it is not known how to include Ramond sector in manifestly Lorentz invariant manner.
In this paper we would like to ask the question whether there could be such a formulation of string field theory from which the NSFT arises in a natural way as the Witten’s open string field theory emerges from the pure cubic string field action. Since pure cubic string field theory does not contain any BRST operator it is formally background independent, we can ask the question whether there is such a formulation of the NSFT theory that is background independent as well. In other words, we are looking for a string field theory action that does not depend on any particular matter conformal field theory background. It seems to us that the second formulation of the problem is the appropriate one for Berkovits’ superstring field theory. This can be seen as follows. In Witten’s string field theory the fundamental object is a ghost number one string field. Since the BRST operator carries the same ghost number there is no problem to define BRST operator using string field as has been done in [@Horowitz]. On the other hand, fundamental string field in NSFT is a string field of the ghost number and the picture number zero. At first sight it seems to be impossible to define string field theory action in background independent way. However recent results [@SenV1; @SenV2; @SenV3; @SenV4] suggest that there can exist BRST operator that is constructed purely from the ghost field which is universal for any conformal field theory background. In fact, in a remarkable paper [@BerkovitsBRST] the BRST operator for $N=1$ NSR string theory was given that resembles striking similarity with the terms presented in NSFT action [^2] . It is then natural to presume that the background independent formulation of the NSFT theory could be based on the BRST operator $Q_0$ defined in [@BerkovitsBRST] and explicit form of which will be given below [^3]. Then using recent result [@KlusonNSFT] it is possible to find such a string field that is solution of the equation of motion of the background independent NSFT and that leads to the emergence of the correct BRST operator for any CFT background.
The plan of the paper is follows. In the next section (\[second\]) we briefly review NSFT theory defined on the BPS D-brane.
In section (\[third\]) we review the formulation of the open bosonic string field theory based on the pure cubic action [@Horowitz; @Horowitz1]. We will show how Witten’s open bosonic string field theory action naturally emerges from the pure cubic string field action. We use these elegant calculations in section (\[fourth\]) where we propose a background independent formulation of the Berkovits string field theory. Then we will show that from this action we can obtain any NSFT action for any CFT background.
In conclusion (\[fifth\]) we will outline our results and suggest further extension of this work.
Review of superstring field theory {#second}
==================================
In this section we would like to review basic facts about Berkovits superstring field theory, for more details, see [@Ohmori; @Berkovits1; @Berkovits2; @Sen1; @BerkovitsR]. The general off-shell string field configuration in the [^4] GSO(+) NS sector corresponding Grassmann even open string vertex operator $\Phi$ of ghost number $0$ and picture number $0$ in the combined conformal field theory of a $c=15$ superconformal matter system, and $b,c,\beta,\gamma$ ghost system with $c=-15$. We can also express $\beta,
\gamma$ in terms of ghost fields $\xi,\eta,\phi$ $$\beta=e^{-\phi}\partial \xi , \
\gamma=\eta e^{\phi} \ ,$$ the ghost number $n_g$ and the picture number $n_p$ assignments are as follows $$\begin{aligned}
b: \ n_g=-1,\ n_p=0 \ \ \ c: \ n_g=1, \ n_p=0 \ ; \nonumber \\
e^{q\phi}: \ n_g=0, \ n_p=q \ ; \nonumber \\
\xi: \ n_q=-1 , \ n_p=1 \ \ \ \eta: \ n_q=1, \ n_p=-1 \ . \nonumber \\\end{aligned}$$ The BRST operator $Q_B$ is given $$Q_B=\oint dz j(z)=\int dz
\left\{c(T_m+T_{\xi\eta}+T_{\phi})+c\partial cb+\eta e^{\phi} G_m
-\eta \partial \eta e^{2\phi}b\right\} \ ,$$ where $$T_{\xi\eta}=\partial \xi \eta, \
T_{\phi}=-\frac{1}{2}\partial \phi\partial\phi-\partial^2\phi \ ,$$ $T_{m}$ is a matter stress tensor and $G_m$ is a matter superconformal generator. Throughout this paper we will be working in units $\alpha'=1$.
The string field action is given [@Berkovits1; @Berkovits2] $$\label{NSFTaction}
S=\frac{1}{2}\int\left(
(e^{-\Phi}\star Q_Be^{\Phi})(e^{-\Phi}\star\eta_0
e^{\Phi})-\int_0^1 dt
(e^{-t\Phi}\star\partial_te^{t\Phi})\star\left\{
(e^{-t\Phi}\star Q_Be^{t\Phi}),
(e^{-t\Phi}\star\eta_0e^{t\Phi})\right\}\right) \ ,$$ where $\{A,B\}=A\star B+B\star A$ and $e^{-t\Phi}\star\partial_t
e^{t\Phi}=\Phi$. Here the products and integral are defined by Witten’s gluing prescription of the string. The exponential of string field is defined in the same manner $e^{\Phi}=
1+\Phi+\frac{1}{2}\Phi\star\Phi+\dots$. The basis properties of $Q_B,\eta_0$ which we will need in our analysis (for more details, see [@Ohmori] and reference therein) are $$\begin{aligned}
\label{ax}
Q_B^2=0, \ \eta_0^2=0, \ \{Q_B,\eta_0\}=0 \ ,
\nonumber \\
Q_B(\Phi_1\star\Phi_2)=Q_B(\Phi_1)\star\Phi_2+
\Phi_1\star Q_B(\Phi_2) , \ \nonumber \\
\eta_0(\Phi_1\star\Phi_2)=\eta_0(\Phi_1)\star\Phi_2+
\Phi_1\star\eta_0(\Phi_2) , \ \nonumber \\
\int Q_B(\dots)=0 \ , \int \eta_0(\dots)=0 \ , \nonumber \\\end{aligned}$$ where $\Phi_1,\Phi_2$ are Grassmann even fields.
For our purposes it will be useful to write the BRST operator in the form [@BerkovitsBRST] $$\label{BRSTR}
Q_B=e^{-R}\left(
\frac{1}{2\pi i}\oint dz \gamma^2 b
\right)e^R \ ,$$ where $$R=\frac{1}{2\pi i}\oint dz [c G_m e^{-\phi}e^{\chi}-
\frac{1}{4}\partial (e^{-2\phi})e^{2\chi}c\partial c ]=
\frac{1}{2\pi i}\oint dz r(z) \$$ and $$\begin{aligned}
\beta=e^{-\phi}\partial \xi \ , \gamma=\eta e^{\phi}, \
\xi(z)\eta(w)=\frac{1}{z-w}, \ \phi(z)\phi(w)=-\log (z-w) \ ,
\nonumber \\
\xi=e^{\chi}, \eta=e^{-\chi}, \
\chi(z)\chi(w)=-\log(z-w) \ . \nonumber \\ \end{aligned}$$ It is easy to see that $$\label{BRST0}
Q_0^2=\frac{1}{2}\{Q_0,Q_0\}=
0 \ , Q_0=\frac{1}{2\pi i}\oint \gamma^2 b \$$ and also $$\{\eta_0,Q_0\}=0 \ .$$ Then we immediately obtain $$Q_B^2=e^{-R}Q_0^2e^{R}=0 \ .$$ In other words, $Q_B$ is a nilpotent operator. It was also shown in [@BerkovitsBRST] that $Q_B$ given in (\[BRSTR\]) anticommutes with $\eta_0$ in critical dimensions $D=10$. From the fact that $Q_0$ is constructed from the ghost fields only it is the same for any CFT background so that it seems to be a natural BRST operator for a background independent formulation of NSFT in the similar way as in [@SenV1; @SenV2; @SenV3; @SenV4]. On the other hand, it must be stressed that this operator has not trivial cohomology [^5] so that the background independent formulation of NSFT should not be confused with the vacuum string field theory that describes the bosonic open string field theory after the tachyon condensation. We mean that this seems to be natural fact since we are looking for a background independent formulation of the BPS D-brane so that there is not any tachyon field present and hence the process of the tachyon condensation cannot occur.
In the next section we briefly review the approach given in [@Horowitz]. In particular, we will see how Witten’s open bosonic string field theory naturally emerges from the pure cubic string field theory action.
Pure Cubic Action for String Field Theory {#third}
=========================================
It was proposed in [@Horowitz] that pregeometrical, background independent (at least formally) string field action has a form $$\label{Cubicaction}
S=\frac{1}{3}\int \Phi\star\Phi\star \Phi \ ,$$ where $\Phi$ is Grassmann odd string field of ghost number one. The classical equation of motion is $$\Phi\star\Phi=0 \ .$$ As was shown in [@Horowitz] when we expand around the classical solution $$\Phi=\Phi_0+\phi$$ we get an action for fluctuation $$S=\int \frac{1}{2}\phi\star D_{\Phi_0}\phi+
\frac{1}{3}\phi\star\phi\star\phi \ ,$$ where $$D_{\Phi_0}X=\Phi_0\star X-(-1)^XX\star \Phi_0 \ .$$ It was shown in [@Horowitz] that $D_{\Phi_0}$ is a derivation. In order to recovery Witten’s form of the string field action [@WittenSFT] we must find field $\Phi_0$ solving equation of motion and $$D_{\Phi_0}=Q_B \ ,$$ where $Q_B$ is the BRST operator associated with some background. It was shown in [@Horowitz] that such a field $\Phi_0$ is uniquely given by the relation $$\Phi_0=Q^L\mathcal{I} \ ,$$ where $Q^L \ (Q^R)$ is the BRST charge density integrated over the left (right) half of the string ($Q=Q^R+Q^L$) and $\mathcal{I}$ is the identity operator of the algebra obeying $$\mathcal{I}\star B=B\star \mathcal{I}=B, \ \forall B \ .$$
In Witten’s open string field theory, $Q_B$ represents a reference background and $\phi$ represents the second quantized fluctuation field around that background. As was shown in [@Horowitz], shifting $\phi$ it is possible to eliminate this specific reference to a background. In their second-quantized formulation the backgrounds arise as solutions to the equations of motion.
It is natural to ask the question whether similar formalism works in case of NSFT as well. Although we will not be able to find an analogue of the pure cubic string field action for NSFT, we will see that it is possible (at least formally) to formulate the background independent version of NSFT based on the BRST operator constructed from the ghost field only.
Proposal for background independent NSFT {#fourth}
========================================
Since we will not perform any explicit calculation we will again use abstract Witten’s formalism in string field theory [@WittenSFT]. We would like to find a formulation of the NSFT theory from which a general action (\[NSFTaction\]) emerges in natural way as in [@Horowitz; @Horowitz1], or equivalently, we would like to find a formulation of the NSFT theory that does not depend on any particular CFT background as in case of VSFT. In fact, the second requirement is the more appropriate one for NSFT which can be seen from the following argument. In bosonic string field theory the fundamental object is a string field of the ghost number one so that it is natural that the BRST operator of the same ghost number can emerge from the pure cubic action. On the other hand in NSFT theory the fundamental field is a Grassmann ghost zero field so that it does not seem to be possible construct BRST operator from the action containing fundamental fields only without presence of any ghost number one operator. However, there is certainly fundamental BRST operator that does not depend on any particular background, which is the operator given in [@BerkovitsBRST] $$Q_0=\frac{1}{2\pi i}\oint dz \gamma^2(z)b(z) \ .$$ It is clear that this operator does not depend on any background CFT so it is natural to propose background independent formulation of NSFT theory using this nilpotent operator $$\label{VNSFT}
S=\frac{1}{2}\int\left(
(e^{-\Phi}\star Q_0e^{\Phi})(e^{-\Phi}\star\eta_0
e^{\Phi})-\int_0^1 dt
(e^{-t\Phi}\star\partial_te^{t\Phi})\star\left\{
(e^{-t\Phi}\star Q_0e^{t\Phi}),
(e^{-t\Phi}\star \eta_0e^{t\Phi})\right\}\right) \ .$$ We would like to show, following recent analysis [@KlusonNSFT] that it is possible to find such a solution of the equation of motion arising from the previous action $$\label{eqm1}
\eta_0(e^{-\Phi_0}\star Q_0(e^{\Phi_0}))=0 \$$ that leads to the NSFT action with appropriate BRST operator $Q_B$.
Let us consider any string field $\Phi_0$, corresponding to $G_0=e^{\Phi_0}$, which is a solution of the equation of motion (\[eqm1\]). In order to find a new form of BRST operator, we must study the behaviour of the fluctuation field around this solution [@KlusonNSFT]. For that reason we write general string field containing fluctuation around this solution as $$G=G_0\star h, \ h=e^{\phi}, \ G^{-1}=h^{-1}\star G^{-1}_0 \ .$$ To see that this field really describes fluctuations around solution $G_0$ note that for $\phi=0, G=G_0$. It is also clear that any string field in the form $e^{\Phi_0+\phi'}$ can be always rewritten in the form given above.
Inserting this upper expression in (\[VNSFT\]) we obtain an action for $\phi$. As was argued in [@KlusonNSFT] in order to find a new BRST operator we must ask the question what form of the equation of motion obeys shifted field $h=e^{\phi}$. Then it was shown that the new BRST operator has a form $$\label{newBRST}
Q_B(X)=Q_0(X)+A\star X-(-1)^XX\star A \ ,
A=G_0^{-1}\star Q_0(G_0) \$$ and the string field action for the fluctuation field has exactly the same form as (\[NSFTaction\]) with the BRST operator (\[newBRST\]).
From (\[newBRST\]) we obtain $$(Q_B-Q_0)X=A\star X-(-1)^XX\star A \ .$$ Using results given in [@Horowitz] $$\begin{aligned}
\label{Qhor}
Q^R\mathcal{I}=-Q^L\mathcal{I} \ , Q=Q^R+Q^L, \
\mathcal{I}\star X=X\star\mathcal{I}=X, \ \forall X \ ,
\nonumber \\
(Q^R X)\star Y=-(-1)^XX\star Q^L(Y) , \forall X,Y \ , \nonumber \\
\{Q,Q^L\}=0 \ , \nonumber \\\end{aligned}$$ we see that we can write $A$ as $$\label{A}
A=(Q_B-Q_0)^L\mathcal{I}$$ since then $$\begin{aligned}
(Q_B-Q_0)^L(\mathcal{I})\star X=
-(Q_B-Q_0)^R(\mathcal{I})\star X=
\mathcal{I}\star (Q_B-Q_0)^L(X) \ , \nonumber \\
-(-1)^XX\star (Q_B-Q_0)^L\mathcal{I}=
(Q_B-Q_0)^R(X)\star\mathcal{I} \ , \nonumber \\\end{aligned}$$ after application of (\[Qhor\]). Then we have $$A\star X-(-1)^XX\star A=
(Q_B-Q_0)^L(X) +
(Q_B-Q_0)^R(X)=(Q_B-Q_0)(X) \ .$$ Since we know that $Q_B,\ Q_0$ are correct BRST operators that anticommute with $\eta_0$ [@BerkovitsBRST], we can easily prove that $A$ given in (\[A\]) solves the equation of motion (\[eqm1\]). This can be seen as follows. The fact that $\{\eta_0,G_0\}=
\{\eta_0,Q_B\}$ means that when we express these operators using appropriate currents then OPE between $\eta(z)$ and $j_0(z),j_B(z)$ is non-singular $$\eta (z)j_0(w)\sim O(0) \ ,
\eta(z)j_B(w) \sim O(0) \ ,$$ If we do a contour integral over $z$ in upper expressions we obtain the operator $\eta_0$ and next integration of $j_{0,B}(w)$ over left half of the string we find that the anticommutator is equal to zero $$\{\eta_0,Q_0^L\}=0 \ ,
\{\eta_0,Q_B^L\}=0 \ .$$ Consequently we get from (\[A\]) $$\label{A1}
\eta_0 (A)=\eta_0(Q_B-Q_0)^L\mathcal{I}=
-(Q_B-Q_0)^L(\eta_0(\mathcal{I}))=0 \ ,$$ where we have used $$\eta_0(X)=\eta_0(X\star \mathcal{I})=
\eta_0(X)\star \mathcal{I}+(-1)^XX\star\eta_0(\mathcal{I})
\Rightarrow \eta_0(\mathcal{I})=0 \ .$$ From (\[A1\]) we see that $A$ given in (\[A\]) solves the equation of motion (\[eqm1\]). Now we explicitly determine $\Phi_0$ in $A=e^{-\Phi_0}Q_0(e^{\Phi_0})$.
Let us define the function $$F(t)=e^{-t\Phi_0}Q_0(e^{t\Phi_0})$$ We can make an expansion around the point $t=0$ where $F(0)=Q_0(1)=0$ $$F(t)=\sum_{n=0}^{\infty}\frac{1}{n!}\left.
\frac{d^nF(t)}{dt^n}\right|_{t=0}
t^n \ .$$ The first derivative is equal to $$\frac{dF}{dt}=-e^{-t\Phi_0}\Phi_0 Q_0(e^{t\Phi_0})+
e^{-t\Phi_0}Q_0(\Phi_0 e^{t\Phi_0})=
e^{-t\Phi_0}Q_0(\Phi_0)e^{t\Phi_0} \$$ and the second one $$\frac{d^2 F(t)}{dt^2}=-e^{-t\Phi_0}
\Phi_0 Q_0(\Phi_0)e^{t\Phi_0}+
e^{-t\Phi_0}Q_0(\Phi_0)\Phi_0 e^{t\Phi_0}
=e^{-t\Phi_0}
[Q_0(\Phi_0),\Phi_0]e^{t\Phi_0} \ .$$ Generally we have $$\frac{d^n F(t)}{dt^n}=
e^{-t\Phi_0}
\overbrace{[[Q_0(\Phi_0),\Phi_0],\dots],\Phi_0]}^{n-1}
e^{t\Phi_0}, \ n>1$$ and consequently $$F(t=1)=A=e^{-\Phi_0}Q_0(e^{\Phi_0})=
Q_0(\Phi_0)+\sum_{n=2}^{\infty}
\frac{1}{n!}\overbrace{[[Q_0(\Phi_0),\Phi_0],\dots,]\Phi_0]}^{n-1} \ .$$ We would like to compare this expression with the expression for BRST operator [@BerkovitsBRST] given in (\[BRSTR\]) and with $Q_0$ given in (\[BRST0\]).
As in calculation performed above we define $$F(t)=e^{-Rt}Q_0e^{Rt}=F(0)+\sum_{n=1}^{\infty}
\frac{1}{n!}\frac{d^n F(t)}{dt^n}t^n$$ then $$\begin{aligned}
F(0)=Q_0 , \
\frac{dF}{dt}=-e^{-tR}RQ_0e^{tR}+e^{-tR}Q_0Re^{tR}=
e^{-tR}[Q_0,R]e^{tR}, \ \nonumber \\
\frac{d^2F}{dt^2}=-e^{-tR}R[Q_0,R]e^{tR}+e^{-tR}
[Q_0,R]Re^{Rt}=e^{-tR}[[Q_0,R],R]]e^{tR} \ ,
\nonumber \\
\frac{d^nF}{dt^n}=e^{-tR}\overbrace{[[Q_0,R],\dots,],R]}^ne^{tR}\end{aligned}$$ and consequently $$Q_B=F(1)=Q_0+\sum_{n=1}^{\infty}
\frac{1}{n!}\overbrace{[[Q_0,R],R],\dots ]R]}^{n} \ .$$ Then we obtain from (\[A\]) $$\begin{aligned}
\label{A2}
Q_0(\Phi_0)+\sum_{n=2}^{\infty}
\frac{1}{n!}\overbrace{[[Q_0(\Phi_0),\Phi_0
],\dots,]\Phi_0]}^{n-1}=
\left(\sum_{n=1}^{\infty}
\frac{1}{n!}\overbrace{[[Q_B,R],\dots,],R]}^{n}\right)^L
\mathcal{I} \ . \nonumber \\\end{aligned}$$ Now we must explain more carefully what we mean by the symbols $\mathcal{X}^{L,R}$ for any operator $\mathcal{X}$. Firstly, let us presume that the world-sheet operator $\mathcal{X}$ is defined by [^6] $$\mathcal{X}=\frac{1}{2\pi i} \oint_C dz x(z) \ ,$$ where $C$ is any closed curve encircling the origin of the conformal plane. It is clear that the operator given above can be written as a sum of two operators defined as integrals of world-sheet densities over left half or right half of the string, in particular for $Q,R$ we get $$Q_0=Q_0^R+Q_0^L, \ R=R^L+R^R \ ,$$ where indices $R,L$ correspond to the left, right half of the open string respectively. For example, $Q_L$ can be defined as a contour integral of the holomorphic density over curve $C_L$ that lies in the right half of the complex plane and $Q_R$ as a contour integral over curve $C_R$ that lies in the left half of the complex plane $$Q_0^{R,L}=\frac{1}{2\pi i}\int_{C^R,C^L}dz j(z)_0 \ ,
R^{R,L}=\frac{1}{2\pi i} \int_{C^R,C^L}dz r(z) \ .$$ From comments given above we immediately obtain $$\label{QRRL}
[Q^R_0,R^L]=0$$ since curves $C_R,C_L$ have not common points and hence OPE between holomorphic densities is non-singular. Then for any operator $\mathcal{O}$ that is commutator of two operators $\mathcal{X},\mathcal{Y}$ we get $$\mathcal{O}=\mathcal{O}^R+
\mathcal{O}^L=[\mathcal{X},\mathcal{Y}]=
[\mathcal{X}^L,\mathcal{Y}^L]+
[\mathcal{X}^R,\mathcal{Y}^R]$$ and consequently $$\mathcal{O}^L=[\mathcal{X}^L,\mathcal{Y}^L] \ ,
\mathcal{O}^R=[\mathcal{X}^R,\mathcal{Y}^R] \ .$$ Then we can write $$\label{Q0R}
[Q_0,R]^L\mathcal{I}=[Q_0^L,R^L]\mathcal{I} \ .$$ Using previous results we claim that the solution of the equation (\[A2\]) has a form $$\label{phi0R}
\Phi_0=R^L\mathcal{I} \ .$$ We will show that this expression really leads to the BRST operator $Q_B$. Firstly it is easy to see that (\[phi0R\]) solves the following equation $$\label{Q0R1}
Q_0(\Phi_0)=[Q_0^L,R^L]\mathcal{I} \$$ since the left hand side of (\[Q0R1\]) is equal to $$\begin{aligned}
Q_0(\Phi_0)=(Q_0^R+Q_0^L)(R^L\mathcal{I})=
Q_0^RR^L\mathcal{I}+Q_0^LR^L\mathcal{I}=
R^LQ_0^R\mathcal{I}+Q_0^LR^L\mathcal{I}=
\nonumber \\
=-R^LQ_0^L\mathcal{I}+Q_0^LR^L\mathcal{I}=
[Q_0^L,R^L]\mathcal{I}
\nonumber \\\end{aligned}$$ using (\[QRRL\]) and (\[Qhor\]). The second term in (\[A2\]) is $$\label{A22}
[Q_0(\Phi_0),\Phi_0]=[[Q_0,R],R]^L\mathcal{I}=
[[Q_0,R]^L,R^L]\mathcal{I}=
[[Q_0^L,R^L],R^L]\mathcal{I} \ .$$ In order to see that (\[phi0R\]) really solves upper expression and generally the whole equation (\[A2\]) we must perform lot of calculations in the similar way as in [@Horowitz]. Firstly, as in case of $Q_0$ we can prove $$R(X)=R(X\star \mathcal{I})=R(X)\star \mathcal{I}+
X\star R(\mathcal{I}) \Rightarrow
R(\mathcal{I})=0 \ .$$ This result follows from the contour integration argument. Previous result implies $$\label{RL1}
R^L\mathcal{I}=-R^R\mathcal{I} \ .$$ In the same way as in (\[Qhor\]) we can write $$\label{RL2}
R^R(X)\star Y=-X\star R^L (Y) \ .$$ Note that there is not a factor $(-1)^X$ since $R$ is Grassmann even operator. Then we have $$R^L\mathcal{I}\star R^L\mathcal{I}=
-R^R\mathcal{I}\star R^L\mathcal{I}=
\mathcal{I}\star R^LR^L(\mathcal{I})=
(R^L)^2(\mathcal{I}) \ ,$$ where we have used in the first step (\[RL1\]) and in the second step (\[RL2\]) in the form $$R^R(\mathcal{I})\star R^L(\mathcal{I})=
-\mathcal{I}\star R^L(R^L(\mathcal{I})) \ .$$ We also have $$\begin{aligned}
Q^R_0(R^L\mathcal{I})\star \mathcal{I}=
-R^L\mathcal{I}\star Q^L_0\mathcal{I}\Rightarrow
R^LQ^R_0(\mathcal{I})\star\mathcal{I}=
-R^L\mathcal{I}\star Q^L_0(\mathcal{I})\Rightarrow\nonumber \\
\Rightarrow R^LQ^L_0(\mathcal{I})=R^L(\mathcal{I})
\star Q^L_0(\mathcal{I}) \ ,\end{aligned}$$ where we have firstly used (\[QRRL\]) and then (\[Qhor\]). In the same way we can show that $$\begin{aligned}
R^R(Q_0^L \mathcal{I})\star\mathcal{I}=
-Q_0^L(\mathcal{I})\star R^L\mathcal{I}\Rightarrow
Q_0^LR^R\mathcal{I}=-Q_0^L\mathcal{I}\star R^L\mathcal{I}
\Rightarrow
Q^L_0(R^L\mathcal{I})=Q^L_0\mathcal{I}\star R^L\mathcal{I}
\ ,\nonumber \\
-Q_0^R((R^L)^2\mathcal{I})\star\mathcal{I}=
(R^L)^2\mathcal{I}\star Q_0^L\mathcal{I}
\Rightarrow (R^L)^2 Q_0^L\mathcal{I}=
(R^L)^2\mathcal{I}\star Q^L\mathcal{I} \ , \nonumber \\
R^R(Q_0^L(R^L\mathcal{I}))\star\mathcal{I}=
-Q_0^L(R^L\mathcal{I})\star R^L\mathcal{I}
\Rightarrow \nonumber \\
\Rightarrow
Q_0^L((R^L)^2\mathcal{I})=Q_0^L(R^L\mathcal{I})\star
R^L\mathcal{I} \Rightarrow
Q_0^L(R^L)^2\mathcal{I}=Q_0^L\mathcal{I}\star (R^L)^2
\mathcal{I} \ . \nonumber \\\end{aligned}$$ Generally we obtain $$\begin{aligned}
(R^L)^nQ_0^L\mathcal{I}=(R^L)^n\mathcal{I}
\star Q_0^L\mathcal{I} \ , \nonumber \\
Q_0^L(R^L)^n\mathcal{I}=Q_0^L\mathcal{I}
\star (R^L)^n\mathcal{I} \ . \nonumber \\\end{aligned}$$ We can also write $$\begin{aligned}
(R^L)^nQ_0^L \mathcal{I}\star R^L\mathcal{I}=
-R^R((R^L)^nQ_0^L\mathcal{I})\Rightarrow \nonumber \\
\Rightarrow
(R^L)^nQ_0^L\mathcal{I}\star R^L\mathcal{I}=
-(R^L)^nQ_0^LR^R\mathcal{I} \Rightarrow
(R^L)^nQ_0^L\mathcal{I}\star R^L\mathcal{I}=
(R^L)^nQ_0^LR^L\mathcal{I} ; \ \nonumber \\
R^L\mathcal{I}\star Q_0^L(R^L)^n\mathcal{I}=
-R^R\mathcal{I}\star Q_0^n(R^L)^n\mathcal{I}
=\mathcal{I}\star R^L(Q_0^L (R^L)^n \mathcal{I})=
R^LQ_0^L(R^L)^n\mathcal{I} \ . \nonumber \\\end{aligned}$$ Generally we have $$\begin{aligned}
R^L\mathcal{I}\star (R^L)^nQ_0^L(R^L)^m\mathcal{I}=
(R^L)^{n+1}Q_0^L(R^L)^m\mathcal{I} ; \ \nonumber \\
(R^L)^nQ_0^L(R^L)^m\mathcal{I}\star R^L\mathcal{I}=
(R^L)^nQ_0^L(R^L)^{m+1}\mathcal{I} \ . \nonumber \\\end{aligned}$$ Using these results we can easily calculate $$\begin{aligned}
[Q_0(\Phi_0),\Phi_0]
=[[Q_0^L,R^L]\mathcal{I},R^L\mathcal{I}]=
Q_0^L\mathcal{I}\star R^L\mathcal{I}
\star R^L\mathcal{I}-R^L\mathcal{I}\star Q_0^LR^L
\mathcal{I}-\nonumber \\
-R^L\mathcal{I}\star
Q^L_0R^L\mathcal{I}
+R^L\mathcal{I}\star R^L\mathcal{I}\star Q_0^L\mathcal{I}=
\nonumber \\=
(Q_0^L(R^L)^2-R^LQ_0^LR^L-R^LQ^L_0R^L+
(R^L)^2Q^L_0)\mathcal{I}=
[[Q_0^L,R^L],R^L]\mathcal{I} \ \nonumber \\\end{aligned}$$ which proves (\[A22\]). Generally we have a result $$\overbrace{[[Q_0(\Phi_0),\Phi_0],\dots],\Phi_0]}^{n-1}=
\overbrace{[[Q_0,R],R\dots,],R]^L}^{n}\mathcal{I} \ .$$ We can prove upper relation by mathematical induction. It was shown that this relation holds for $n=1,2$. Let us presume its validity for $n=N-1$. Then we have for $n=N$ $$\begin{aligned}
\overbrace{[[[Q_0(\Phi_0),\Phi_0],\dots],\Phi_0]}^{N-1}=
[\overbrace{[[Q_0,R],R,\dots,]R]^L}^{N-1}\mathcal{I},\Phi_0]=
\nonumber \\
=\overbrace{[[Q_0,R],R],\dots],R]^L}^{N-1}\mathcal{I}\star R^L\mathcal{I}-
R^L\mathcal{I}\star \overbrace{[[Q_0,R],R],\dots],R]^L}^{N-1}\mathcal{I}
=\nonumber \\
=\overbrace{[[Q_0,R],R],\dots],R]^L}^{N-1}R\mathcal{I}-
R\overbrace{[[Q_0,R],R],\dots],R]^L}^{N-1}\mathcal{I}=
\overbrace{[[Q_0,R],\dots],R]^L}^{N}\mathcal{I} \
\nonumber \\\end{aligned}$$ using results given above. We can then claim that $\Phi_0=R^L\mathcal{I}$ solves (\[A2\]). We have also shown above that $A=e^{\Phi_0}Q_B(e^{-\Phi_0})$ solves the equation of motion (\[eqm1\]). In other words, we have found classical field in the background independent NSFT that leads to the NSFT with correct BRST operator corresponding to some particular CFT background. It can be shown in the same way as in [@Horowitz] that for some particular background CFT characterised by $Q_B$ there is unique field $\Phi_0=
R^L\mathcal{I}$.
In this section we have proposed background independent formulation (at least formally) of the NSFT. We have shown that any CFT background arises as a particular solution of the background independent NSFT theory. Of course, our calculation was pure formal as in case of [@Horowitz] so that more detailed analysis should be done. We return to some open questions and suggestions for further work in conclusion.
Conclusion {#fifth}
==========
In this short note we have proposed a background independent formulation of Berkovits’ string field theory [@Berkovits1; @Berkovits2; @Berkovits3]. Our proposal is based on the form of the BRST operator for RNS string theory presented in [@BerkovitsBRST]. Since the BRST operator we started in the background independent NSFT with contains ghost fields only, it is background independent as well. This is the situation similar to the case of vacuum string field theory [@SenV1; @SenV2; @SenV3; @SenV4], where the BRST operator is constructed from the ghost field only and consequently VSFT is background independent. On the other hand, the BRST operator in VSFT has a vanishing cohomology, while in NSFT it has not. We mean that this is not in contradiction since our background independent NSFT theory corresponds to the BPS object so that there is no place for tachyon condensation. This remark also suggests possible limitation of our proposal. We mean that the true background independent formulation of supersymmetric string field theory will have such a form that will allow any solutions corresponding to BPS or non-BPS CFT background. In order to find such a formulation, we think that complete supersymmetric invariant action, including Ramond sector, should be found. We believe that such a formulation will be found in the near future and it will allow us to get new insight in the nature of supersymmetric theory.
We must also stress one important thing. Our calculation was pure formal in the sense that we have worked with Witten’s star product as with an abstract object that does not depend on any background. It would be nice to perform more detailed analysis based on the CFT technique, in the similar way as in the beautiful paper [@SenV4]. We hope to return to this problem in the future.\
\
[**Acknowledgements**]{} We would like to thank N. Berkovits for useful comments and for pointing out the work [@BerkovitsBRST]. This work was supported by the Czech Ministry of Education under Contract No. 1443100006.
[20]{} K. Ohmori, *“A Review on Tachyon Condensation in Open String Field Theories,”* . E. Witten, *“Noncommutative Geometry and String Field Theory,”* . E. Witten, *“Interacting Field Theory of Open Superstrings,”* . N. Berkovits, *“Super-Poincare Invariant Superstring Field Theory,”* , . N. Berkovits, *“A New Approach to Superstring Field Theory,”* . N. Berkovits, *“The Tachyon Potential In Open Neveu-Schwarz String Field Theory,”* , . I. Ya. Aref’eva, P. B. Medvedev and A. P. Zubarev, *“Background Formalism for Superstring Field Theory,”* . I. Ya. Aref’eva, P. B. Medvedev and A. P. Zubarev, *“New Representation for String Fields Solves The Consistency Problem for Open Superstring Field Theory,”* . I. Ya Aref’eva, A. S. Koshelev, D. M. Belov ad P. B. Medvedev, *“Tachyon Condensation in Cubic Superstring Field Theory,”* . N. Berkovits, A. Sen and B. Zwiebach, *“Tachyon Condensation in Superstring Field Theory,”* , . L. Rastelli, A. Sen and B. Zwiebach, *“String Field Theory Around the Tachyon Vacuum,”* . H. Hata and S. Teragushi, *“Test of the Absence of Kinetic Therms around the Tachyon Vacuum in Cubic String Field Theory,”* . I. E. Ellwood and W. Taylor, *“Open string field theory without open strings,”* . B. Feng, Y. He and N. Moeller, *“Testing the Uniqueness of the Open Bosonic String Theory Vacuum,”* . K. Ohmori, *“Tachyon Kink and Lump-like Solutions in Superstring Field Theory,”*, , . I. Ellwood, B. Feng, Y. He and N. Moeller, *“The Identity String Field and the Tachyon Vacuum,”* . L. Rastelli, A. Sen and B. Zwiebach, *“Classical Solutions in String Field Theory Around the Tachyon Vacuum,”* . L. Rastelli, A. Sen and B. Zwiebach, *“Half-strings, Projectors, and Multiple D-branes in Vacuum String Field Theory,”* . D. J. Gross and W. Taylor, *“Split string field theory I,”* . T. Kawano and K. Okuyama, *“Open String Fields As Matrices,”* . L. Rastelli, A. Sen and B. Zwiebach, *“Boundary CFT Construction of D-branes in Vacuum String Field Theory,”* . Y. Matsuo, *“BCFT and Sliver state,”* . J. R. David, *“Excitations on wedge states and on the sliver,”* . T. Nakatsu, *“CLASSICAL OPEN-STRING FIELD THEORY, $A_{\infty}$-Algebra, Renormalisation Group and Boundary States,”* . N. Berkovits and C. T. Echevarria, *“Four-Point Amplitude from Open Superstring Field Theory,”* , . N. Berkovits, *“Review of open superstring field theory,”* . A. Sen,*“Stable non-BPS bound states of BPS D-branes,”* , ;
*“SO(32) spinors of type I and other solitons on brane-antibrane pair,”* ,;
*“Type I D-particle and its interactions,”* ;
*“Non-BPS states and branes in string theory,”* , and reference therein. G. T. Horowitz, J. Lykken, R. Rohm and A. Strominger, *“Purely Cubic Action for String Field Theory,”* . G. T. Horowitz, J. Morrow-Jones and S. P. Martin, *“New Exact Solutions for the Purely Cubic Bosonic String Field Theory,”* . J. N. Acosta, N. Berkovits and O. Chandia, *“A Note on the Superstring BRST Operator,”* . N. Berkovits, private correspondence, unpublished. J. Klusoň, *“Some Remarks About Berkovits’ Superstring Field Theory,”* . L. Rastelli, A. Sen and B. Zwiebach, *“Vacuum String Field Theory,”* . D. J. Gross and W. Taylor, *“Split string field theory II,”* .
[^1]: For another interesting formulation of superstring field theory, see [@Zubarev1; @Zubarev2; @Zubarev3].
[^2]: We are very thankful to N. Berkovits for emphasising this form of the BRST operator and pointing out the reference [@BerkovitsBRST].
[^3]: Similar suggestion has been proposed in [@Berkovitsunpub].
[^4]: In this paper we will consider NSFT for BPS D-branes only.
[^5]: We would like to thank N. Berkovits for stressing this point to us.
[^6]: It is convenient to use “double trick” for open strings. We trade the holomorphic and antiholomorphic components of any field, defined in upper half plane, for a single holomorphic field defined in the whole complex plane.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present freshly evaluated B$(E2\uparrow;0^+\rightarrow2^+)$ values across the even-even Sn-isotopes which confirm the presence of an asymmetric behavior as well as a dip in the middle of the full valence space. We explain these features by using the concept of generalized seniority. The dip in the B$(E2)$ values near $^{116}$Sn is understood in terms of a change in the dominant orbits before and after the mid shell, which also explains the presence of asymmetric peaks in the B$(E2)$ values. This approach helps in deciding the most active valence spaces for a given set of isotopes, and single out the most useful truncation scheme for Large Scale Shell Model (LSSM) calculations. The LSSM calculations so guided by generalized seniority are also able to reproduce the experimental data on B$(E2)\uparrow$ values quite well.'
address:
- 'Department of Physics, Indian Institute of Technology, Roorkee-247667, India.'
- 'Department of Physics and Astronomy, McMaster University, Hamilton, Ontario-L8S 4M1, Canada.'
author:
- Bhoomika Maheshwari
- Ashok Kumar Jain
- and Balraj Singh
title: 'Asymmetric behavior of the B$(E2 \uparrow; 0^+ \rightarrow 2^+)$ values in $^{104-130}$Sn and Generalized Seniority'
---
`2_1^+ states, Sn-isotopes, Generalized seniority, Even-tensor E2 transitions`
Introduction
============
The Sn-isotopes present the longest available isotopic chain between two doubly magic nuclei from $^{100}$Sn to $^{132}$Sn, and beyond. They provide a very useful data set to explore nuclear structural properties across the $N=50$ to $82$ shell, and also to test the realistic effective interactions [@simpson; @maheshwari]. The concept of seniority [@racah] has been successfully used in the past as an important tool to understand the behavior of the semi-magic nuclei, from the particle number independent energies to the parabolic B$(E2)$ variation [@casten; @maheshwari1]. We have recently presented a simple microscopic scheme for the generalized seniority in multi-j degenerate orbits, where we find that the electric transitions, for both the even and the odd tensors, behave similar to each other. This enabled us to find for the first time a new kind of seniority isomers due to the odd-tensor $E1$ transitions in the Sn-isotopes [@maheshwari1], and gave us chance to explain the behavior of $E3$ transitions in Sn-isotopes as well [@maheshwari2]. We use the same approach in this paper to understand the behavior of the B$(E2 \uparrow; 0^+ \rightarrow 2^+)$ values in the Sn-isotopes.
The nearly constant energy of the first $2^+$ states from $^{100}$Sn to $^{130}$Sn suggests that seniority may be a good quantum number in these isotopes. In a theoretical study, Sandulescu $et$ $al.$ [@sandulescu] compared the energies of the yrast generalized seniority states with shell model states for $^{104-112}$Sn, and concluded that a model space with seniority greater than 2 is probably necessary. Several groups, both theoretical and experimental, have been studying the B$(E2 \uparrow; 0^+ \rightarrow 2^+)$ variation in the Sn-isotopes. It was expected that the B$(E2)$ values would show a parabolic variation with a peak in the middle as predicted by the seniority scheme applied to the full valence space. Earlier measurements also supported the same. However, Jungclaus $et$ $al.$ [@jungclaus] have recently reported new measurements which deviate from the expected parabolic behavior in the middle at $^{116}$Sn, where they find a dip rather than the expected peak [@ekstrom], and attributed it to a reduced collectivity in the middle. Morales $et$ $al.$ [@morales], however, explained this minimum in terms of the different rates of filling of the orbits by using a generalized seniority approach [@arima; @talmi; @shlomo].
More recently, Doornenbal $et$ $al.$ [@doornenbal] have reported a new measurement of B$(E2)$ in $^{104}$Sn; they have also compared the results from the large scale shell model (LSSM) and quasiparticle random phase approximation (QRPA), and other theoretical calculations [@morales; @ansari; @ansari1; @back; @bader] with the available experimental data in the Sn-isotopes [@jungclaus; @ekstrom; @radford; @allmond; @banu; @cederkall; @vaman; @doornenbal1; @kumar; @guastalla; @bader].
Iudice $et$ $al.$ [@iudice] have tried to reproduce the asymmetry of B$(E2)$ parabolas by using the quasi-particle phonon model (QPM), and explore its origin in the evolution of single particle energies and polarization of the $N=Z=50$ core. On the other hand, Jiang $et$ $al.$ [@jiang] also presented a theoretical framework of nucleon pair approximation (NPA), where they used two sets of single particle energies, two-body interaction parameters and effective charges, one each for A $\le 116$ and A $\ge 116$, in order to reproduce the B$(E2)$ data with two asymmetric parabolas and a minimum at $^{116}$Sn. In fact, some doubts have also been raised about the existence of the minimum in the experimental B$(E2)$s as several measurements with significant variations exist.
The purpose of this paper is to perform a systematic study of the B$(E2)$ values in even-even Sn-isotopes for the first excited $2^+$ state in the framework of generalized seniority in multi-j environment. We found it necessary to re-evaluate the measured B$(E2)$ data for Sn-isotopes, in view of the large number of measurements differing from each other, even though a 2015 update of the B$(E2)$ values has just been published [@pritychenko]. We then use a simple microscopic approach based on quasi-spin scheme [@kerman; @helmers] applied to the generalized seniority in many-j degenerate orbits, and calculate the reduced transition probabilities in the Sn-isotopes by a simple formula, which has been presented in our recent paper [@maheshwari1]. These generalized seniority calculations, which are very handy in nature, reproduce the overall experimental trend with a minimum in the middle, and provide direct information about the orbits involved before and after the mid-shell. The seniority-guided shell model calculations are then used to support these results, which also reproduce the minimum at the middle and explain the asymmetric peaks in the B$(E2)$ values quite well. Our generalized seniority and seniority guided LSSM calculations, hence, support the understanding of two asymmetric parabolas in terms of filling of different orbits, before and after the middle, and support the results of Morales $et$ $al.$ [@morales].
The paper is organized in four sections. We present the evaluated B$(E2)$ values for Sn-isotopes in section II. Section III presents a brief theoretical framework about the generalized seniority scheme, including the formulas used in the calculations. Section IV gives all the details of the calculations and results for both generalized seniority and seniority guided LSSM calculations. Section V summarizes the present work.
Evaluated B(E2) data
====================
Different experimental techniques often lead to the different B$(E2)$ values in $^{104-130}$Sn isotopes. A recently published table of B$(E2)$ values presents the evaluated data [@pritychenko], available from all the sources, and lists the recommended values. Allmond $et$ $al.$ [@allmond1] recently published their B$(E2)$ measurements in Sn-isotopes, which were not available at the time of this update. Besides this, the table [@pritychenko] also missed a paper [@orce] having measurements for $^{116,118}$Sn isotopes. Other experimental works are already included and listed in [@pritychenko]. We have, therefore, reevaluated the new/missing data to incorporate these changes and presented the evaluated B$(E2 \uparrow; 0^+ \rightarrow 2^+)$ values in Table I, based on the data available until now. Most of the values are found to be almost consistent with the recent update of B$(E2)$ values [@pritychenko] with slight modifications, except for $^{128,130}$Sn which remain unchanged.
[c c]{} Isotope & B$(E2)\uparrow$ $e^2b^2$\
\
$^{104}$Sn &0.176(26)\
$^{106}$Sn &0.209(39)\
$^{108}$Sn &0.224(19)\
$^{110}$Sn &0.226(22)\
$^{112}$Sn &0.2362(70)\
$^{114}$Sn &0.2212(73)\
$^{116}$Sn &0.2030(45)\
$^{118}$Sn &0.2055(34)\
$^{120}$Sn &0.2002(37)\
$^{122}$Sn &0.1894(55)\
$^{124}$Sn &0.1631(33)\
$^{126}$Sn &0.1273(72)\
$^{128}$Sn &0.0771(38)$\star$\
$^{130}$Sn &0.023(5)$\star$\
In this evaluation, we have not used the experimental values from the $(e,e')$ method and any other model-dependent method. We have also used a minimum uncertainty of 4$\%$ in a certain experiment. All the values are weighted averaged values. Note that the values from Allmond $et$ $al.$ [@allmond1] and Jungclaus $et$ $al.$ [@jungclaus], are systematically different from each other, particularly in the middle, though both the groups show a dip in the middle and support the presence of two asymmetric parabolas. So far, no B$(E2)$ measurement exists for $^{102}$Sn.
Theoretical Framework
=====================
We use the generalized seniority scheme for multi-j degenerate orbits to calculate the B$(E2)$ values in the Sn-isotopes, already presented in our recent paper [@maheshwari1]. The B$(E2)$ transition probabilities between the initial $J_i$ and final $J_f$ states in multi-j case can be obtained, by defining $\tilde{j}=j \otimes j' ....$ with the total pair degeneracy $\Omega= \frac{1}{2}(2 \tilde{j} +1)= \frac{1}{2} \sum \limits_j (2j+1)$, by using the formula, $$\begin{aligned}
B(E2) \uparrow=\frac{5}{2J_i+1}|\langle \tilde{j}^n v l J_f || \sum_i r_i^2 Y^{2}(\theta_i,\phi_i) || \tilde{j}^n v' l' J_i \rangle |^2\end{aligned}$$ where the reduced matrix elements in the $\tilde{j}^n$ configuration can be reduced to the configuration for seniority changing $\Delta v = 2$ transitions from the equation [@maheshwari1], $$\begin{aligned}
\langle \tilde{j}^n v l J_f ||\sum_i r_i^2 Y^{2}|| \tilde{j}^n v\pm 2 l' J_i \rangle = \Bigg[ \sqrt{\frac{(n-v+2)(2\Omega+2-n-v)}{4(\Omega+1-v)}} \Bigg] \nonumber\\ \langle \tilde{j}^v v l J_f ||\sum_i r_i^2 Y^{2}|| \tilde{j}^v v\pm 2 l' J_i \rangle \end{aligned}$$ Note that the coefficients in the square brackets in the multi-j case, are similar to the well known single-j case, due to the simple incorporation of the multi-j by defining $\tilde{j}=j \otimes j' ....$. The coefficient decides the behavior of the B$(E2)$ values, which depends only on the particle number $n$, the generalized seniority $v$ and the corresponding total pair degeneracy $\Omega$ in the multi-j environment.
This means that the B$(E2)$ transition probabilities will show a parabolic behavior with a maximum at the mid-shell for $\Delta v = 2$, seniority changing transitions as is the situation in $(0^+ \rightarrow 2^+)$ transitions. We use the generalized seniority formula in the following to calculate the B$(E2 \uparrow)$s. We also obtain the B$(E2 \uparrow)$s by using the LSSM calculations, using the valence spaces guided by the generalized seniority scheme. These LSSM calculations support the generalized seniority results very well, with a minimum in the middle and two asymmetric parabolas. We present details of the calculations in the next section.
Calculations and discussion
===========================
We plot our calculated results in Fig. \[fig:be2\], using dashed and dash-dotted lines for generalized seniority and seniority guided LSSM calculations, respectively. For the generalized seniority calculations, we divide available valence space of g$_{7/2}$, d$_{5/2}$, h$_{11/2}$, d$_{3/2}$ and s$_{1/2}$ orbits in the Sn-isotopes into two parts: (a) $\tilde{j}= g_{7/2} \otimes d_{5/2} \otimes d_{3/2} \otimes s_{1/2}$, $\Omega=10$ and (b) $\tilde{j}= d_{5/2} \otimes d_{3/2} \otimes s_{1/2} \otimes h_{11/2}$, $\Omega=12$. We take $^{100}$Sn as a core for $\Omega=10$, a natural choice, while $^{108}$Sn as a core for $\Omega=12$ due to the g$_{7/2}$ orbit which is now full. For dividing the valence space, we use the fact that the h$_{11/2}$ orbit mainly dominates after the mid-shell $(^{116}Sn)$, while the g$_{7/2}$ orbit completely freezes on reaching $^{116}$Sn. We then calculate the B$(E2)$ values by using the generalized seniority scheme, and obtain two asymmetric parabolas as shown in Fig. \[fig:be2\].
It is interesting to note that the two parabolas would cross over each other at $^{116}$Sn (if extrapolated in Fig. \[fig:be2\]), and naturally support a minimum at the middle. Therefore, the minimum of B$(E2)$ values at $^{116}$Sn surely does not correspond to any reduced collectivity as suggested earlier by Jungclaus $et$ $al.$ [@jungclaus]. It only describes the change in the filling of the orbits before and after the middle, and also corresponds to the location, where g$_{7/2}$ freezes out, and h$_{11/2}$ takes over. These calculations, hence, explain the overall experimental trend quite well, and also provide a direct cue for the configuration involved and their influence on the B$(E2)$ values. It may also be concluded that proper configuration mixing is required to understand the origin of the excited states in the semi-magic nuclei; the first excited $2^+$ states of the Sn-isotopes in the present case.
We further use this generalized seniority scheme to single out the active valence spaces and resultant truncations for LSSM calculations. We divide the LSSM calculations also in two parts: first we take $^{100}$Sn as a core with the multi-j configuration $\tilde{j}= g_{7/2} \otimes d_{5/2} \otimes d_{3/2} \otimes s_{1/2}$ (LSSM1) and second we choose $^{108}$Sn as a core with the multi-j configuration $\tilde{j}= d_{5/2} \otimes d_{3/2} \otimes s_{1/2} \otimes h_{11/2}$(LSSM2). The neutron single particle energies have been taken as $-10.6089$, $-10.2893$, $-8.7167$, $-8.6944$, $-8.8152$ MeV for the available 0g$_{7/2}$, 1d$_{5/2}$, 1d$_{3/2}$, 2s$_{1/2}$ and 0h$_{11/2}$ valence orbits. The harmonic oscillator potential was chosen with an oscillator parameter of $\hbar \omega =45A^{-1/3}-25A^{-2/3}$. Only two sets of the neutron effective charge have been used, $e_\nu=1.2$ in first valence space and $e_\nu=1$ in the second valence space. We have used the Nushell code [@brown] for the large scale shell model calculations along with the SN100PN [@brown1] effective interaction. The seniority guided shell model calculations reproduce the experimental data quite well, as shown in Fig. \[fig:be2\]. This also suggests that the breaking of $Z=N=50$ core is not so crucial to understand the origin of first excited $2^+$ states in Sn-isotopes. These calculations strongly validate the interpretation provided by generalized seniority about the truncations and configuration mixings needed in the generation of these $2_1^+$ states. The generalized seniority scheme applied to B$(E2 \uparrow; 0^+ \rightarrow 2^+)$ values thus provides a simple way to handle the LSSM calculations, particularly where the dimensions become very large.
For comparison, we have also calculated the B$(E2)$ values by using $\Omega=7$ with $^{100}$Sn core and $\Omega=9$ with $^{114}$Sn core in the generalized seniority and plotted these results in Fig. \[fig:be21\], along with the evaluated experimental trend. Here $\Omega=7$ and $9$ correspond to $\tilde{j}= g_{7/2} \otimes d_{5/2}$ and $\tilde{j}= d_{3/2} \otimes s_{1/2} \otimes h_{11/2}$ mixed configurations, respectively. The calculated results are quite far from the experimental data, and exhibit an obvious gap at the $^{114}$Sn isotope. One can, therefore, infer that the mixing of other orbits (as included in $\Omega=10$ and $12$) is essential for a complete understanding of these B$(E2)$ values and the asymmetric parabolas. Though the contribution of the d$_{3/2}$ and s$_{1/2}$ remains less than that for the $g_{7/2}$ and $d_{5/2}$ orbits before $^{116}$Sn, it does affect the values of the reduced transition probabilities and also the corresponding wave functions of the first excited $2^+$ states, while $h_{11/2}$ can be ignored. On the other hand, the major contribution of the $h_{11/2}$ starts after the mid-shell along with the $d_{5/2}$, $d_{3/2}$ and $s_{1/2}$ orbits, while $g_{7/2}$ can be frozen out. Though the isotopes around $^{116}$Sn may need a mixing of both the $g_{7/2}$ and $h_{11/2}$ orbits in the resultant wave functions, this results in a very large dimension in the LSSM calculations and could not be carried out by us.
Hence, the seniority generalized for multi-j orbits, provides a direct understanding of the configuration mixing, and the truncations in the Sn-isotopes. The seniority guided shell model calculations confirm the same. We, therefore, conclude that the dip in the middle, neither corresponds to the reduced collectivity, nor supports any core-excited structure in any significant way. The dip is only due to the different sets of active orbits and their different rate of mixings, which finally leads to the asymmetric B$(E2)$ parabolas. These results also agree with the conclusions of Morales $et$ $al.$ [@morales].
Conclusion
==========
In conclusion, we have re-evaluated the B$(E2 \uparrow; 0^+ \rightarrow 2^+)$ transition probabilities in the Sn-isotopes and provided a set of most acceptable values. The evaluated data confirm the existence of a dip in the middle of full valence space. We, then, use them in the generalized seniority scheme for multi-j degenerate orbits. These simple calculations reproduce the experimental data quite well along with the dip in the middle. This also works as a guide to fix the orbits involved in configuration mixing. This information turns out to be very useful in deciding the truncations in the LSSM calculations, which again reproduce the experimental trend very well. The dip in the middle, therefore, corresponds to a competition between the orbits in $N=50-82$ valence space, and pinpoints the location where the role of the $g_{7/2}$ orbit is taken over by the $h_{11/2}$ orbit. It is remarkable that the calculated values from our formulation match with the LSSM results very well. This confirms the effectiveness of our formulation based on the generalized seniority. We also conclude that the minimum in the middle does not correspond to a reduced collectivity as also pointed out by Morales $et$ $al.$ [@morales].
 Comparison of the experimental B$(E2 \uparrow; 0^+ \rightarrow 2^+)$ values and the generalized seniority and seniority guided large scale shell model (LSSM) calculations in even-even Sn-isotopes. Evaluated data are based on the experimental measurements [@pritychenko; @allmond1; @orce].](BE2systematics.eps){width="13cm" height="11cm"}
 Same as Fig. \[fig:be2\], but the generalized seniority calculations using $\Omega = 7$, and $9$ values, corresponding to $g_{7/2} \otimes d_{5/2}$ and $h_{11/2} \otimes d_{3/2} \otimes s_{1/2}$ valence spaces, respectively, in Sn-isotopes. ](BE2systematics1.eps){width="13cm" height="11cm"}
Acknowledgments {#acknowledgments .unnumbered}
===============
Financial support from the Ministry of Human Resource Development (Government of India) is gratefully acknowledged.
[50]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Recently much attention is paid to the role of the orbital degrees of freedom in transition metal oxides as it remains unclear whether they can remain in a quantum disordered state at zero temperature. Discrete symmetry of the orbital sector counteracts the quantum melting, but especially in doped systems there are signs of dynamical frustration involving the spin-, charge-, and orbital sector simultaneously. We discovered that even the simple Kugel-Khomskii model, describing $e_g$ degenerate Mott-insulators, is characterized by a point of perfect dynamical frustration on the classical level, reached in the absence of Hund’s rule and electron-phonon couplings. This frustration is lifted on the quantum level, and the true nature of the ground state is still unknown. At present there are two proposals: the KCuF$_3$ phase, stabilized by an order-out-of-disorder mechanism, or spin-orbital valence bond phases. It will be argued that at least in the Cu based systems of this kind, the electron-phonon coupling is primarely responsible for driving the systems away from the special point in the phase diagram.'
address:
- ' Lorentz Institute for Theoretical Physics, Leiden University, P.O.B. 9506, NL-2300 RA Leiden, The Netherlands '
- |
Institute for Theoretical Physics, Utrecht University, Princetonplein 5, NL-3584 CC Utrecht, The Netherlands, and\
Philips Research Laboratories, Prof. Holstlaan 4, NL-5656 AA Eindhoven, The Netherlands
- |
Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Kraków, Poland, and\
Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Federal Republic of Germany
author:
- Jan Zaanen
- Louis Felix Feiner
- 'Andrzej M. Oleś'
title: ' Classical Frustration and Quantum Disorder in Spin-Orbital Models '
---
[2]{}
Introduction
============
The paradigm of correlated electron physics is based on the idea that for a certain category of systems one better starts out with the electronic structure of the atoms, treating the delocalization of the electrons in the solid as a perturbation. Any student of physics has to struggle through the theory of atomic multiplets, which is rather complicated because of the intricacies associated with orbital angular momentum. At first sight it is therefore remarkable that these orbital degrees of freedom are completely neglected in the main stream of correlated electron physics. Recently the interest in ‘orbitals’ has been reviving, especially since they appear to be relevant in one way or another in the colossal magnetoresistance (CMR) manganites. In the wake of this development, questions are asked on the relevancy of these orbitals in the context of seemingly settled problems like the metal-insulator transition in V$_2$O$_3$ [@Bao98]. In this contribution we will review yet another recent development. Even in the Mott-insulating limit, where the physics simplifies considerably, the interplay of orbital and spin degrees of freedom poses a problem of principle.
There are two limits where the role of orbital degeneracy is well understood: (i) The ‘band structure limit’, which is based on the assertion that electron correlations can be neglected. In any modern local density approximation (LDA) band structure calculation, orbitals are fully taken into account on the one particle level, in so far as the atomic limit is of any relevance. These translate into various bands, giving rise to multi-sheeted fermi surfaces, etcetera. (ii) The localized, orbital and spin ordered case which we will refer to as the ‘classical limit’. In Mott-insulators, orbital degrees of freedom acquire a separate existence in much the same way as the spins of the electrons do. The orbitals can be parametrized by pseudospins and these form together with the physical spins a low energy sector which is described by generalizations of the Heisenberg spin-Hamiltonian [@Jan93]. These are the spin-orbital models, like the Kugel-Khomskii (KK) model for $e_g$ degenerate cubic cuprates [@Kug82]. The ‘classically’ ordered states, becoming exact in the limit of infinite dimensions ($d\rightarrow\infty$) and/or large (pseudo) spin ($S\rightarrow
\infty$), define what is usually meant with orbital and spin order.
The question arises if there are yet other possibilities. We started to study this problem quite some time ago [@Crete], well before the subject revived due to the manganites. Our motivation was actually related to a theoretical development flourishing in the 1980’s: large $N$ theories [@Aue94]. By enlarging the symmetry, say from $SU(2)$ to $SU(N)$ with $N$ large, new saddle points (ordered states) appear which correspond to the fluctuation dominated (non-perturbative) limit of the large $S$/large $d$ theories. For a single correlated impurity, orbital degeneracy leads in a natural way to these large $N$ notions. We asked the question if these large $N$ notions could become of relevance in lattice problems. We focussed on the simple problem of the $e_g$ Jahn-Teller degenerate Mott-insulator, rediscovering the KK Hamiltonian [@Kug82]. We tried to tackle this problem using the techniques invented by Arovas and Auerbach for the $SU(N)$ symmetric Heisenberg model [@Aro88]. We found that the $SU(4)$ symmetry is so badly broken that the large $N$ techniques were of little help, which is another way of saying that the physics of the KK model is not controlled by large global symmetry. However, we did find a special approximate solution which revealed that the quantum fluctuations are actually enhanced, and this motivated us to study these fluctuations in more detail starting from the large $S$ limit. In this process we discovered that the enhancement of the fluctuations is due to the control exerted by a point in parameter space which can be either called an infinite order quantum-critical point, or a point of perfect [*dynamical frustration*]{} in the classical limit [@Fei97].
This phenomenon will be discussed in the next section. It poses a rather interesting theoretical problem. So much is clear that the ground state degeneracy of the classical limit is lifted by quantum fluctuations and the question is on the character of the true ground state. As will be discussed, either the classical spin-orbital order might survive, stabilized by an order-out-of-disorder mechanism, or quantum-incompressible valence-bond like states might emerge. In Section III the role of electron-phonon coupling will be addressed, emphasizing the rather counter-intuitive result of LDA+U electronic structure calculations that phonons play a rather secondary role despite the fact that the lattice deformations are large. Finally, the situation in the manganites will be shortly discussed in Section IV.
The Kugel-Khomskii model and dynamical frustration
====================================================
Consider a Mott-insulator which is characterized by orbital degeneracy, besides the usual spin degeneracy. Different from pure spin problems, these spin-orbital problems are rather ungeneric and depend on the precise system under consideration. A simple problem is a cubic lattice of $3d$-ions in a $d^9$ configuration: the Kugel-Khomskii problem, which directly applies to Cu perovskites like KCuF$_3$ or K$_2$CuF$_4$ [@Kug82]. The large Mott gap in the charge excitation spectrum simplifies matters considerably and one derives an effective Hamiltonian by insisting on one hole per unit cell, deriving superexchange-like couplings between the spin and orbital degrees of freedom by integrating out virtual charge fluctuations.
The spins are described as usually in terms of an $su(2)$ algebra ($\vec{S}_i$). The orbital degrees of freedom are the $e_g$ cubic harmonics $x^2-y^2 \sim |x\rangle$ and $3z^2-r^2 \sim |z\rangle$, which can be parametrized in terms of pseudospins as $|x\rangle ={\scriptsize\left( \begin{array}{c} 1\\ 0\end{array}\right)},\;
|z\rangle ={\scriptsize\left( \begin{array}{c} 0\\ 1\end{array}\right)}$. Pauli matrices $\sigma^u$ ($u = x, y, z$) are introduced acting on these states. Different from the spins, the $SU(2)$ symmetry associated with the pseudospins is badly broken because the orbitals communicate with the underlying lattice. Although the $e_g$ states are degenerate on a single site, this degeneracy is broken by the virtual charge fluctuations, which take place along the interatomic bonds, i.e., in a definite direction with respect to the orientation of the orbitals. It is therefore convenient to introduce operators which correspond to orbitals directed either along or perpendicular to the three cubic axes $\alpha=a,b,c$, given by $(\tau^{\alpha}_j-\frac{1}{2})$ and $(\tau^{\alpha}_j+\frac{1}{2})$, where $$\tau^{a(b)}_i =\frac{1}{4}( -\sigma^z_i\pm\sqrt{3}\sigma^x_i ),
\hskip 1cm
\tau^c_i = \frac{1}{2} \sigma^z_i \;.
\label{orbop}$$ In terms of these operators, the Kugel-Khomskii Hamiltonian can be written as ($J=t^2/U$ and $t$ is the hopping along the $c$-axis) [@Fei97], $$\begin{aligned}
\label{kk1}
H_1 = &J& \sum_{\langle ij\rangle,\alpha} \left[ 4(\vec{S}_i\cdot\vec{S}_j )
(\tau^{\alpha}_i - \frac{1}{2}) (\tau^{\alpha}_j - \frac{1}{2})\right.
\nonumber \\
& & \hskip 1.0cm + \left. (\tau^{\alpha}_i+\frac{1}{2})(\tau^{\alpha}_j
+ \frac{1}{2}) - 1 \right] ,\end{aligned}$$ neglecting the Hund’s rule splittings $\propto J_H$ of the intermediate $d^8$ states ($J_H$ is the singlet-triplet splitting). Including those up to order $\eta=J_H/U$ yields in addition, $$\begin{aligned}
\label{kk2}
H_2 = & J\eta & \sum_{\langle ij\rangle,\alpha}
\left[ (\vec{S}_i\cdot\vec{S}_j)
(\tau^{\alpha}_i + \tau^{\alpha}_j - 1 ) \right. \nonumber \\
&+& \left. \frac{1}{2}(\tau^{\alpha}_i-\frac{1}{2})
(\tau^{\alpha}_j-\frac{1}{2})
+ \frac{3}{2} (\tau^{\alpha}_i \tau^{\alpha}_j - \frac{1}{4})\right] . \end{aligned}$$ Eq.’s (\[kk1\],\[kk2\]) are rather unfamiliar: they describe a regular Heisenberg spin problem coupled into a Pott’s like orbital problem (choose two out of three possibilities $\sim x^2-y^2, \sim y^2-z^2, \sim z^2-x^2$).
The oddity of Eq.’s (\[kk1\],\[kk2\]) becomes clear when one studies the classical limit. As usually, the $\vec{S}$’s and the $\vec{\tau}$’s are treated as classical vectors. In order to draw a phase diagram we introduced another control parameter, $$\label{kk3}
H_3 = - E_z \sum_i \tau^z_i,$$ a “magnetic field” for the orbital pseudo-spins, loosely associated with a uniaxial pressure along the $c$-axis. The classical limit phase diagram as function of $\eta$ and $E_z$ is shown in Fig. 1.
(8,8) (0.,0.)
For a detailed discussion of the various phases we refer to Ref. [@Fei97]. To give some feeling, for large positive $E_z$ the $x^2-y^2$ orbitals are occupied, forming $(a,b)$ planes of antiferromagnetically coupled spins (AFxx). This is nothing else than the situation realized in, e.g. La$_2$CuO$_4$. For large negative $E_z$ the $3z^2-r^2$ orbitals condense, forming a 3D spatially anisotropic Heisenberg antiferromagnet \[AFzz with stronger exchange coupling along the $c$-axis than in the $(a,b)$ planes\]. Finally, the MOFFA, MOAFF and MOAAF phases are variations of the basic Kugel-Khomskii spin-orbital order [@Kug82] obtained by rotating the magnetic and orbital structure by $\pi/2$. For the MOFFA phase at $E_z =0$, the orbitals have a two-sublattice structure in the $(a,b)$-planes ($x^2-z^2$ and $y^2-z^2$ on the A- and B-sublattice, respectively). Along the $c$-axis strong antiferromagnetic spin-spin couplings are found, while the spin couplings in the $(a,b)$ planes are ferromagnetic with a strength $\sim\eta$.
The anomaly occurs at the origin $(E_z,\eta)=(0,0)$ of the phase diagram: a 3D antiferromagnet (AFzz), a 2D antiferromagnet (AFxx) and a quasi-1D A-type antiferromagnet (MOFFA/MOAFF/MOAAF) become degenerate! The emphasis on the ‘uniaxial pressure’ $E_z$ is misleading in the sense that the full scope of the problem is not visible directly from this phase diagram: at the origin of Fig. 1 an [*infinity*]{} of classical phases become degenerate. This is trivial to understand. In the absence of Hund’s rule exchange, the Hamiltonian Eq. (\[kk1\]) becomes the full story. Assuming a 3D classical antiferromagnet, $\vec{S}_i \cdot \vec{S}_j = -1/4$, and inserting this in Eq. (\[kk1\]) yields, $$H_{eff} = J\sum_{\langle ij\rangle,\alpha}
\left( \tau_i^{\alpha} + \tau_j^{\alpha} - 1 \right) .
\label{3ddeg}$$ The orbital degrees of freedom are completely decoupled and all $2^N$ orbital configurations have the same energy ($\sum_{\alpha}\tau_i^{\alpha}=0$)! In addition, this infinity of different 3D spin systems has the same energy as the MOFFA/MOAFF/MOAAF phases. It is actually so that at any finite temperature the 3D antiferromagnet becomes stable because of the entropy associated with the decoupled orbital sector [@janup].
This ‘gauge’ degeneracy is clearly a pathology of the classical limit. We continued by studying the stability of the classical phase diagram with respect to Gaussian quantum fluctuations. As discussed in more detail in Ref. [@Foz98] this is a somewhat subtle affair. Intuitively, one could be tempted to think that the orbitals and spins can be excited independently. This is however not the case. The dynamical algebra of relevance to the problem is an $so(4)$ algebra, and this implies that modes will occur which excite at the same time the spins and the orbitals: the spin-and-orbital waves (SOW)’s. Next to a (longitudinal) sector of pure orbital excitations, a ‘transversal’ sector is found corresponding with spin-excitations which are mixed with spin-and-orbital excitations, except for the acoustic modes at long wavelength which become pure spin-waves as imposed by the Goldstone theorem.
We found that upon approaching the infinite critical point, the mass gap associated with the discrete symmetry in the orbital sector collapses. The (mixed) transverse modes give the dominating contribution to the renormalization of energy and magnetic order parameter. In the AFxx (AFzz) phase the lowest transverse mode softens along $\vec{k}=(\pi,0,k_z)$ \[$\vec{k}=(k_x,0,0)$\], and equivalent lines in the Brillouin zone (BZ), regardless how one approaches the critical lines. Thus, these modes become dispersionless along particular (soft-mode) lines in the BZ, where we find [*finite*]{} masses in the perpendicular directions, $$\begin{aligned}
\omega_{\rm AFxx}(\vec{k}) \rightarrow & \Delta_x &
+ B_x \left( k_x^4 + 14k_x^2k_y^2 + k_y^4 \right)^{1/2}, \nonumber \\
\omega_{\rm AFzz}(\vec{k}) \rightarrow & \Delta_z &
+ B_z \left( k_y^2 + 4k_z^2 \right),
\label{mass0}\end{aligned}$$ with $\Delta_i=0$ and $B_i\neq 0$ at the $M$ point, and the quantum fluctuations diverge logarithmically, $\langle\delta S^z\rangle\sim
\int d^3k/\omega(\vec{k})\sim\int d^2k/(\Delta_i+B_ik^2)\sim\ln\Delta_i$, if $\Delta_i\rightarrow 0$ at the transition. We found that the quantum correction to the order parameter $\langle S^z\rangle$ becomes large, well before the critical point is reached. In Fig. 1 the lines are indicated where $|\langle \delta S^z\rangle|=\langle S^z\rangle$: in the area enclosed by the dashed and dotted lines classical order cannot exist, at least not in gaussian order.
If the classical limit is as sick as explained in the previous paragraphs, what is happening instead? [*A priori*]{} it is not easy to give an answer to this question. There are no ‘off the shelf’ methods to treat quantum spin problems characterized by classical frustration, and the situation is similar to what is found in, e.g. $J_1-J_2-J_3$ problems [@Pre88]. A first possibility is quantum order-out-of-disorder [@Chu91]: quantum fluctuations can stabilize a particular classical state over other classically degenerate states, if this particular state is characterized by softer excitations than any of the other candidates. Khaliullin and Oudovenko [@Kha97] have suggested that this mechanism is operative in the present context, where the AFzz 3D anisotropic antiferromagnet is the one becoming stable. Their original argument was flawed because of the decoupling procedure they used, which violates the $so(4)$ dynamical algebra constraints [@Foz98]. However, Khaliullin claims to have found an ‘$so(4)$ preserving’ self-consistent decoupling procedure which does yield order-out-of-disorder [@Kha98]. Nevertheless, there is yet another possibility: valence-bond (VB) singlet (or spin-Peierls) order, which at the least appears in a more natural way in the present context than is the case in higher dimensional spin-only problems, because it is favored by the directional nature of the orbitals.
The essence of a (resonating) valence bond \[(R)VB\] state is that one combines pairs of spins into singlets. In the short-range (R)VB states these singlets involve nearest-neighbor spin pairs. Subsequently, one particular covering of the lattice with these ‘spin-dimers’ might be favored (VB or spin-Peierls state), or the ground state might become a coherent superposition of many of these coverings (RVB state). On a cubic lattice the difficulty is that although much energy is gained in the formation of the singlet pairs, the bonds between the singlets are treated poorly. Nevertheless, both in 1D spin systems (Majumdar-Ghosh [@Maj69], AKLT-systems [@Aff87]) and in the large $N$ limit of $SU(N)$ magnets in 2D, ground states are found characterized by spin-Peierls/VB order [@Read].
It is straightforward to understand that the interplay of orbital- and spin degrees of freedom tends to stabilize VB order. Since the orbital sector is governed by a discrete symmetry, the orbitals tend to condense in some classical orbital order. Different from the fully classical phases, one now looks for orbital configurations optimizing the energy of the spin VB configurations. The spin energy is optimized by having orbitals $3\zeta^2-r^2$ on the nearest-neighbor sites where the VB spin-pair lives, with $\zeta$ directed along the bond. This choice maximizes the overlap between the wave functions, and thereby the binding energy of the singlet. At the same time, this choice of orbitals minimizes the unfavorable overlaps with spin pairs located in directions orthogonal to $\zeta$. The net result is that VB states are much better variational solutions for the KK model, as compared to the standard Heisenberg spin systems.
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Adressing this systematically, we found that two families of VB states are most stable: (i) The ‘staggered’ VB states like the PVBA and PVBIc states of Fig. 2. These states have in common that the overlap between neighboring VB pairs is minimized: the large lobes of the $3\zeta^2-r^2$ wave functions of different pairs are never pointing to each other. (ii) The ‘columnar’ VB states like the VBc (or VBa) state of Fig. 2. In the orbital sector, this is nothing else than the AFzz state of Fig. 1 ($3z^2-r^2$ orbitals on every site). Different from the AFzz state, the spin system living on this orbital backbone is condensed in a 1D spin-Peierls state along the $z$-direction which is characterized by strong exchange couplings. The spins in the $a(b)$-directions stay uncorrelated, due to the weakness of the respective exchange couplings as compared to the VB mass gap.
The energies of these VB states and the classical states dressed up with quantum fluctuations are quite close together. A key issue is if the true ground state is compressible (dressed classical state), or characterized by a dynamical mass-gap (VB states). This will most likely depend on subtleties beyond the reach of the relatively crude variational Ansätze presented here [@notekhalu]. So the nature of the ground state of the Kugel-Khomskii problem for small Hund’s-rule coupling is still an open problem.
Electron-phonon coupling in KCF$_3$
=====================================
In the previous Section we discussed the orbital order as driven by the electron-electron interactions. However, one can think quite differently about the real systems: the deformations found in KCuF$_3$ (or LaMnO$_3$) could in principle be entirely caused by phonon-driven collective Jahn-Teller effects. This subject has been intensely studied in the past and is well understood. It starts out neglecting electron-electron interactions, and the focus is instead on the electron-phonon coupling. In case that the ions are characterized by a Jahn-Teller (orbital) degeneracy, one can integrate out the (optical) phonons, and one finds effective Hamiltonians with phonon mediated interactions between the orbitals. In the specific case of $e_g$ degenerate ions in a cubic crystal, these look quite similar to the KK Hamiltonian, except that the spin dependent term is absent[@KKphon]. Any orbital order resulting from this Hamiltonian is now accompanied by a lattice distortion of the same symmetry.
The size of the quadrupolar deformation in the $(a,b)$ plane of KCuF$_3$ is actually as large as 4 % of the lattice constant ($a$). It is therefore often argued that the orbital order is clearly phonon-driven, and that the physics of the previous section is an irrelevancy. Although appealing at first sight, this argument is flawed: large displacements do not necessarily imply that phonons do all the work.
The deformations of the lattice and the orbital degrees of freedom cannot be disentangled using general principles: they constitute an irreducible subsector of the problem. The issue is therefore a quantitative one, and in the absence of experimental guidance one would therefore like to address the issue with a quantitative electronic structure method. The LDA+U method is the method of choice. It is constructed to handle the physics of electronic orbital ordering, keeping the accurate treatment of the electron-lattice interaction of LDA intact. According to LDA+U calculations the total energy gained by the deformation of the lattice is minute as compared to the energies involved in the electronic orbital ordering [@Lie95]. At the same time, the phonons are important on the macroscopic scale and they contribute to driving KCuF$_3$ away from the infinite-critical point of the phase diagram Fig. 1.
We start out with the observation that according to LDA KCuF$_3$ would be an undistorted, cubic system: the energy increases if the distortion is switched on (see Fig. 3). The reason is that KCuF$_3$ would be a band metal according to LDA (the usual Mott-gap problem) with a Fermi-surface which is not susceptible to a band Jahn-Teller instability. LDA+U yields a drastically different picture [@Lie95]. LDA can be looked at as unpolarized LDA+U, and by letting both the orbitals and the spins polarize an energy is gained of order of the band gap, i.e., of the order of 1 eV. The orbital- and spin polarization is nearly complete and the situation is close to the strong coupling limit underlying the spin-orbital models of Section II. Also when the cubic lattice is kept fixed, the correct orbital and spin ordering (MOFFA of Fig. 1) is found, with spin-exchange constants which compare favorably with experiment [@Lie95]. Because the orbital order has caused the electron density to become highly unsymmetric, the cubic lattice is unstable. Further energy can be gained by letting the lattice relax. The lattice distortion calculated in LDA+U ($\sim$ 3% of $a$) comes close to the actual distortion of KCuF$_3$ ($\sim$ 4 %). However, despite the fact that the distortion is large, the energy gained by the lattice relaxation is rather minute: $\sim 50$ meV (see Fig. 3)! Obviously, in the presence of the electronic orbital order the cubic lattice becomes very soft with regard to the quadrupolar distortions and even a small electron-phonon coupling can cause large distortions.
(8,8) (0.,0.)
Although the energy gained in the deformation of the lattice is rather small, the electron-phonon coupling is quite effective in keeping KCuF$_3$ away from the physics associated with the origin of the phase diagram (Fig. 1). Since the ferromagnetic interactions in the $(a,b)$ plane of KCuF$_3$ are quite small ($J_{ab}=-0.2$ meV, as compared to the ‘1D’ exchange $J_c=17.5$ meV [@Ten95]), one might argue that the effective Hund’s rule coupling $J\eta$ as of relevance to the low energy theory is quite small. Although this still needs further study, it might well be that in the absence of the electron-phonon coupling KCuF$_3$ would be close to the origin of Fig. 1. However, the electron-phonon coupling can be looked at as yet another axis emerging from the origin. In principle, the electron-phonon coupling introduces two scales: (i) a retardation scale, which is governed by the ratio of the phonon frequency and the electronic scale set by $J\sim 20$ meV. Since $J$ is relatively small, KCuF$_3$ is close to the anti-adiabatic limit where the lattice follows the electronic fluctuations, (ii) in the anti-adiabatic limit the phonons are high energy modes which can be integrated out, causing the effective orbital-orbital couplings we earlier referred to. These couplings destroy the cancellations leading to Eq. (\[3ddeg\]), thereby driving the system away from the point of classical degeneracy. The typical scale for the phonon induced effective orbital interactions is at most of the order of the LDA+U lattice relaxation energy. However, as the latter ($\sim 50$ meV) is quite a bit larger than $J$, the effective interaction will likely be able to put KCuF$_3$ well outside the ‘dangerous’ region near the origin of the phase diagram.
In summary, although further work is needed it might be that phonons are to a large extent responsible for the stability of KCuF$_3$’s classical ground state. In any case, one cannot rely on the sheer size of the lattice deformations to resolve this issue!
How about the manganites ?
==========================
Given the discussion so far, the search for interesting quantum effects in orbital degenerate Mott-insulators should not be regarded as hopeless. Unfortunately, the insulating parent compounds of the CMR manganites, such as LaMnO$_3$, are [*not*]{} candidates for this kind of physics. The reason is not necessarily phonons: also in the manganites the ‘Jahn-Teller’ lattice distortions are sizable, but this does not necessarily imply that the phonons are dominating. Two of us derived a Kugel-Khomskii-type model of relevance to this regime, and we did find a dynamical frustration of $e_g$-superexchange at $J_H\simeq 0$ [@Fei98]. However, the system is driven away from this point by two effects: (i) the manganites are in the Hund’s rule dominated regime, with a large splitting between the lowest energy high-spin state at $U-5J_H$ (with $J_H=0.69$ eV [@Miz95]), and the low-spin states at energies $\sim U$; (ii) the additional $t_{2g}$-superexchange between the $S=3/2$ cores favours an antiferromagnetic order in all three spatial directions. The net outcome is that the ferromagnetic interaction between the [*total $S=2$ spins*]{} in the $(a,b)$ planes is of order of the $c$-axis exchange, signalling that the manganites are in the Hund’s rule stabilized regime of the phase diagram.
The mysteries of the manganites relate to what happens when quantum-mechanical holes are added to the orbital/spin ordered insulator. This is undoubtedly a problem with its own characteristics, which cannot be reduced to a variation on the far simpler problems encountered in the insulators. Nevertheless, we do believe that the study of the insulating limit might be of some help in better appreciating what is going on in the doped systems. It is tempting to think about orbital degrees of freedom as being spins in disguise. This is not quite the case. Orbitals are far less quantum-mechanical – they are more like Ising spins than Heisenberg spins. Secondly, orbitals carry this unfamiliar property that depending on their specific orientation in internal space, overlaps increase in particular real space directions, while they diminish in orthogonal directions. Our valence-bond constructions illustrate this peculiar phenomenon in the case of spins, but the same logic is at work when the hole is delocalizing. This intimate connection between internal symmetry and the directionality of delocalization causes the dynamical frustration which has been highlighted in this communication. This motive seems also at work in the doped system, witness the many near degenerate states found both in mean-field calculations [@Miz95; @Nag98] and in experiment [@Tok94]. Further work is needed on this fascinating problem.
[*Acknowledgements*]{}. We thank A. I. Lichtenstein for helpful discussions. AMO acknowledges support by the Committee of Scientific Research (KBN) of Poland, Project No. 2 P03B 175 14.
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For instance, one can argue that our columnar VB states are quite like the ‘order-out-of-disorder’ states of Ref. [@Kha97], with the only difference that we have imposed a spin-Peierls order in the chain direction.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
R. S. Ward[^1]\
Department of Mathematical Sciences,\
University of Durham,\
Durham DH1 3LE
title: Planar Skyrmions at High and Low Density
---
Introduction
============
Topological solitons are of interest both mathematically and in many areas of physics (as models for particles, topological defects in condensed-matter physics, etc). A particularly important question is the nature of, and transition between, the high-density and low-density phases of such solitons. Ultimately, one would like to understand the thermodynamics of these systems; the starting-point for this is to investigate their zero-temperature behaviour. At its most basic, this amounts to considering a fixed number $N$ of solitons confined to a finite volume, and investigating the way in which static classical $N$-soliton solutions depend on the volume (or density). Alternatively, one may use the dimensionless ratio [*(size of soliton)/(size of space)*]{} as a parameter.
The typical situation is as follows. At large density, there is a high degree of symmetry and uniformity, and in particular it may not possible to identify individual solitons; in Skyrme models, for example, one gets a periodic crystal-like structure of half-Skyrmions. At low densities, by contrast, solitons (or multi-solitons) become localized in space, and there is less symmetry. The localized multi-solitons may be ‘chunks’ of the high-density crystal, or may have a quite different (for example shell-like) shape — this depends on the details of the system.
The picture for the three-dimensional Skyrme model may be summarized as follows. Let $N$ denote the Skyrme number, and $E$ the normalized energy-per-Skyrmion (so the Bogomolny-Faddeev bound is $E\geq1$). At high density, the ground state is a triply-periodic lattice of half-Skyrmions [@K85; @GM87; @KS88; @W88; @KS89; @CJJVJ89], with energy $E=1.038$ at its most favourable density. At low density (for example for an isolated multi-Skyrmion in ${{\bf R}}^3$), and for relatively low values of $N$ (in particular for $N\leq22$), the minimal-energy static Skyrmions take the form of polyhedral shells [@BS02]. Their normalized energy $E$ is a decreasing function of $N$, and it appears [@BS98] that these polyhedra have energy $E\approx1.06$ for large $N$. Since this is larger than the energy of the Skyrme crystal, one expects that there is a critical value $N_{{\rm c}}$ such that for $N>N_{{\rm c}}$, the minimal-energy Skyrmion resembles a chunk of the Skyrme crystal. Such a lattice chunk will have energy $E \approx 1.038 + kN^{-1/3}$, with the second term being a surface contribution. There have been attempts [@B96] to estimate the constant $k$, and hence to determine $N_{{\rm c}}$, but these have not been definitive. There are also other possibilities for the shape of large-$N$ Skyrmions, for example a multi-shell structure [@MP01].
The present paper deals with the two-dimensional O(3) Skyrme system, which one may view as an analogue of the three-dimensional case, as well as being of interest in its own right. The 2-D Skyrme system contains, as a limiting case, the two-dimensional O(3) sigma model; the sigma-model crystal, obtained by imposing periodic boundary conditions, has been studied both as a model quantum field theory [@RR83] and in connection with the dynamics of (classical) solitons [@CZ97; @S98]. But sigma-model solitons do not have a fixed size, and in particular tend to shrink and decay; so in that sense they are not true solitons. We shall consider only the case where both the Skyrme term and a potential term $V$ are present, and the soliton size is consequently fixed. The interest here is in the nature of the low-density and the high-density configurations, and the transition between them; as we shall see, the details of these depend crucially on the choice of potential $V$. Two different systems, corresponding to two different choices of $V$, will be investigated in detail.
Two-Dimensional Skyrmions
=========================
The two-dimensional Skyrme system involves a unit 3-vector field ${\vec\phi}=(\phi_1,\phi_2,\phi_3)$ defined on (2+1)-dimensional space-time. We are interested only in static configurations, so ${\vec\phi}$ depends on the spatial variables $x^j=(x^1,x^2)$, thought of as (local) coordinates on a two-dimensional space $S$. If $S$ is a compact surface, then ${\vec\phi}$ has a winding number $N$, which we think of as the number of Skyrmions. One special case is where space is the plane ${{\bf R}}^2$ with boundary condition ${\vec\phi}\to(0,0,1)$ as $r\to\infty$ — this corresponds (by conformal invariance) to the 2-sphere $S=S^2$. Another case is that of periodic boundary conditions ${\vec\phi}(x^1,x^2)={\vec\phi}(x^1+L,x^2)={\vec\phi}(x^1,x^2+L)$, which corresponds to the 2-torus $S=T^2$. In either of these cases, we may regard the spatial metric as being the flat (Euclidean) metric, and we shall do so in what follows.
The integer $N$ can be either positive or negative; without loss of generality, we shall assume $N$ to be positive. The energy density of a configuration ${\vec\phi}$ is $$\label{Enden}
{\cal E} = {{\scriptstyle\frac{1}{2}}}({\partial}_j{\vec\phi})\cdot({\partial}_j{\vec\phi})
+ {{\scriptstyle\frac{1}{2}}}\alpha\Omega^2 + {{\scriptstyle\frac{1}{2}}}\alpha V({\vec\phi}),$$ where $\Omega = {\vec\phi}\cdot{\partial}_1{\vec\phi}\times{\partial}_2{\vec\phi}$, $V$ is some potential function, and $\alpha$ is a dimensionless constant. The length-scale in this system is determined by the ratio of the coefficients of the $\Omega^2$ and $V$ terms, and so is fixed by choosing these coefficients to be equal as in (\[Enden\]). Static multi-Skyrmion solutions are critical points of the (normalized) energy functional $$\label{En}
E = \frac{1}{4\pi N} \int {\cal E} \,d^2x.$$ There is a topological (Bogomolny) lower bound on the energy, namely $$\label{Bogbound}
E \geq 1 + \frac{\alpha}{4\pi} \int_{S^2} \sqrt{V({\vec\phi})} \,d\omega,$$ where $d\omega$ is the usual area element on the space $S^2$ of unit vectors (in other words, $\int d\omega = 4\pi$). Under certain circumstances, this bound can be saturated; the following statement is adapted from [@IRPZ92], see also [@IW01]. Write $z=x^1+{{\rm i}}x^2$, and let $W=(\phi_1+{{\rm i}}\phi_2)/(1-\phi_3)$ denote the stereographic projection of ${\vec\phi}$. Let $W(z)$ be a complex-analytic function satisfying a first-order ordinary differential equation of the form $dW/dz=F(W)$ for some function $F$. Then the configuration $W(z)$ saturates the bound (\[Bogbound\]), and hence is a static Skyrmion solution, provided that the potential $V$ is given by $$\label{Bogpot}
V = \frac{16\,|F(W)|^4}{(1+|W|^2)^4}.$$ The simplest example of this is where $F$ is a constant; the corresponding system (on ${{\bf R}}^2$) has been investigated in some detail, for example in [@LPZ90b; @Sut91]. In this case, there is a repulsive force between Skyrmions, and consequently there are no static multi-Skyrmion solutions.
For any system of the form (\[Enden\]), the nature and shape of multi-Skyrmion solutions depend on the choice of $V$, and many different choices have been considered [@IRPZ92; @ESZ00]. The following two sections will deal with two possible choices. They are both analogous to the three-dimensional Skyrme system, in that they allow both crystal-like and ring-like solutions (the latter being the counterpart of the polyhedral shells mentioned earlier).
To conclude this section, it is worth mentioning yet another case which has been extensively investigated, namely $V({\vec\phi}) = 1-\phi_3$ [@PSZ95a; @PSZ95b]. In this case, the static multi-Skyrmion solutions on ${{\bf R}}^2$ appear to form a lattice-like structure [@PSZ95a; @ESZ00] — for example, for even values of $N$ one gets a lattice of double-Skyrmions as the lowest-energy state. There are also other local minima of the energy, but there do not appear to be any ring-like solutions (except for $N=2$). The thermodynamics of this system has been studied numerically [@SW02].
The System $V = 1-\phi_3^2$
===========================
In this section, the potential function $V$ appearing in (\[Enden\]) is taken to be $V({\vec\phi}) = 1-\phi_3^2$. Let us discuss, first, the localization-delocalization transition for this system. A useful order parameter in this regard is the quantity $$\langle\phi_3\rangle = \frac{1}{A} \int_S \phi_3\,d^2x,$$ where $A$ is the area of the 2-space $S$. At high density, there is a high degree of symmetry, and we expect to have $\langle\phi_3\rangle=0$; while at low density, the field will localize in space and we will have $\langle\phi_3\rangle\neq0$. The transition between these two phases occurs at some critical density $\rho_{{\rm c}}$. This transition was investigated by taking space to be a sphere ($S=S^2$) in [@IW01]; in particular, this paper looked at $\langle\phi_3\rangle$, as a function of $A$, for a single skyrmion on $S^2$. Using approximate analytic expressions for the Skyrmion indicated that the transition occurs at $A=4\pi\sqrt{5}$, [*ie*]{} the critical density is $$\label{NBScrit1}
\rho_{{\rm c}}=1/(4\pi\sqrt{5})=0.036\,.$$ Numerical simulations gave results that supported this estimate.
Let us now compare this to the situation on the torus $T^2$. So we are looking for doubly-periodic solutions — periodic in both $x^1$ and $x^2$ with period $L$. If $\alpha=0$ ([*ie*]{} for the O(3) sigma-model), then solutions correspond to elliptic functions, and these have topological charge $N=2$ in the unit cell. So we expect the minimal-energy crystal in the Skyrme case also to have an $N=2$ cell. Such solutions can be found by numerical minimization of the energy functional. It turns out there are two different crystals, namely one with (approximate) symmetry $A$:
A1.
: $(x,y)\mapsto(-x,y)$, $(\phi_1,\phi_2,\phi_3)\mapsto(\phi_1,-\phi_2,\phi_3)$,
A2.
: $(x,y)\mapsto(y,x)$, $(\phi_1,\phi_2,\phi_3)\mapsto(-\phi_1,\phi_2,\phi_3)$,
A3.
: $(x,y)\mapsto(x+L/2,y)$, $(\phi_1,\phi_2,\phi_3)\mapsto(\phi_3,-\phi_2,\phi_1)$;
and one with symmetry $B$:
B1.
: $(x,y)\mapsto(-x,y)$, $(\phi_1,\phi_2,\phi_3)\mapsto(\phi_1,-\phi_2,\phi_3)$,
B2.
: $(x,y)\mapsto(y,x)$, $(\phi_1,\phi_2,\phi_3)\mapsto(-\phi_3,\phi_2,-\phi_1)$,
B3.
: $(x,y)\mapsto(x+L/2,y)$, $(\phi_1,\phi_2,\phi_3)\mapsto(-\phi_1,-\phi_2,\phi_3)$.
In the $\alpha=0$ system, symmetries $A$ and $B$ are equivalent; but this degeneracy is broken by the potential term $V$. The situation is illustrated in Figure \[fig1\], which plots the normalized energy $E$ for the two solutions, and the quantity $\Psi=\langle\phi_3\rangle_A$ for the type-$A$ solution, as functions of the periodicity $L$.
The value of the parameter $\alpha$ used in obtaining these graphs is $\alpha=0.1$. The Bogomolny bound (\[Bogbound\]) for this system is $E\geq1+\pi\alpha/4=1.0785$. The lowest crystal energy is attained when $L=L_0=6.0$, [*ie*]{} at density $\rho=0.056$; this solution has symmetry of type $B$, and normalized energy $E_{{\rm crys}}=1.088$. The type-$B$ solution exists for lower density, and is always delocalized, [*ie*]{} has $\langle\phi_3\rangle=0$.
At relatively high density, in fact for $L<L_1=7.0$, the solution with type-$A$ symmetry also has $\langle\phi_3\rangle=0$; but it is unstable, and its energy (which is not plotted in Figure \[fig1\]) is higher than that of the type-$B$ solution. When perturbed, it decays to the type-$B$ solution. For $L>L_1$, the type-$A$ solution delocalizes ($\Psi\neq0$) and becomes stable. (It no longer has the symmetry $A$ — in fact $A3$ is invalid — but we still refer to it as the type-$A$ solution.) This is illustrated in the right-hand diagram of Figure \[fig1\]; the numerical results indicate that $$\lim_{L\to L_1+} \Psi \approx 0.06,$$ but this limit is difficult to study owing to the instability which sets in below $L_1$. In the range $L_1<L<L_2=7.35$, the energy of the type-$A$ solution is greater than that of type $B$ — in other words, there are two local mimina of the energy, one of which (type $B$) is more symmetric than the other. For $L>L_2$, the minimal-energy $N=2$ solution is the localized type-$A$ one.
We see, therefore, that there is a transition between a dense, highly-symmetric homogeneous phase and a low-density, less homogeneous phase. The latter appears at $L_1=7.0$, corresponding to a density $\rho=2/7^2=0.041$, and it becomes the minimal-energy state at $L_2=7.35$, corresponding to the critical density $$\label{NBScrit2}
\rho_{{\rm c}}=2/7.35^2=0.037\,.$$ The close agreement between (\[NBScrit2\]) (for $N=2$ Skyrmions on $T^2$) and (\[NBScrit1\]) (for $N=1$ Skyrmions on $S^2$) is remarkable.
Let us now consider the zero-density limit $L\to\infty$. The only known static solutions on ${{\bf R}}^2$ are rotationally-symmetric rings [@We99]. The energy of these goes like $E\approx\beta+\gamma/N^2$, where $\beta$ and $\gamma$ are constants. The significant fact here is that $\beta$ is lower than $E_{{\rm crys}}$; this has been checked numerically for $\alpha\leq1$. For example, for $\alpha=0.1$, we get $\beta=1.082$, compared with $E_{{\rm crys}} = 1.088$. The implication of this is that, unlike the case for three-dimensional Skyrmions, isolated crystal chunks are not energetically favourable, even for large values of $N$. In fact, it seems likely that the minimum-energy configuration on ${{\bf R}}^2$ is, for all values of $N$, a single O(2)-symmetric ring (other local minima may, however, exist). In this respect, the system is different from the three-dimensional Skyrme model.
The System $V =(1-\phi_3^2)(1-\phi_1^2)$
========================================
The motivation for the potential $V$ studied in this section comes from requiring that the Bogomolny bound (\[Bogbound\]) be saturated by a doubly-periodic solution. As a consequence of this, the energy of the corresponding Skyrme crystal will be as low as it possibly can be; and so an isolated multi-Skyrmion will, for $N$ large enough, take the form of a crystal chunk rather than a ring-like configuration. In this respect, it will be a better analogue of the three-dimensional Skyrme system.
Our assumption, therefore, is that the system should admit a solution $W(z)$ which is an elliptic function. One particularly simple choice is to take $W(z)=2\wp(z)$, where $\wp$ is the Weierstrass p-function with parameters $g_2=1$ and $g_3=0$. So the function $F$ appearing in (\[Bogpot\]) is given by $F(W)^2=2W(W^2-1)$; and the corresponding potential function is easily seen to be $$\label{BBS}
V({\vec\phi}) = 16\,(1-\phi_3^2)(1-\phi_1^2).$$ For the rest of the section we shall use this $V$.
The fundamental cell of the lattice is a square with sides of length $L_0=2\,K(0.5)=3.708$, where $K(m)$ is the complete elliptic integral of the first kind with parameter $m$. This cell contains two units of topological charge (that is, the map from $T^2$ to $S^2$ has winding number $N=2$); the distribution of charge- and energy-density indicates that it should be thought of as a square lattice of half-Skyrmions. It has the symmetry of type $A$ defined in the previous section. If we scale the solution to fit a lattice with edge-length $L$, then we get a solution with normalized energy $$\label{Elat}
E(L) = 1 + 2.3729\times\frac{\alpha}{2}
\left(\frac{L}{L_0}+\frac{L_0}{L}\right).$$ If $L=L_0$, the solution saturates the lower bound (\[Bogbound\]), which in this case is $E \geq 1 + 2.3729\,\alpha$.
For large $L$ (low density) one expects that this solution (a scaled Weierstrass function) will no longer be the minimal-energy doubly-periodic $N=2$ solution. In fact, the minimal solution will correspond to an isolated $N=2$ lump at some point on $T^2$ — and so will have $\langle\phi_3\rangle\neq0$.
![Energy $E$ for the two $N=2$ solutions. \[fig2\]](fig2.eps)
The result of a numerical investigation of this phenomenon is presented in Figure \[fig2\]; it was obtained by finding local minima of the energy with $N=2$ for a range of values of the periodicity $L$. The value of $\alpha$ was taken to be $\alpha=0.1$. For $L<5.15$, there is only one solution, namely the Weierstrass-type one with $\langle\phi_3\rangle=0$. Its energy grows fairly rapidly with $L$, but it remains a stable solution ([*ie*]{} a local minimum), at least up to $L=20$. For $L\geq5.15$, another solution (local minimum) appears — a spatially-localized one with $\langle\phi_3\rangle\neq0$. Its energy grows less rapidly with $L$, and for $L>5.73$ it is the lowest-energy solution. In this sense, there is a phase transition at $L_1=5.73$, corresponding to the critical density $\rho_{{\rm c}}=2/5.73^2=0.061$. It would be interesting to study this transition in other ways, for example by examining the system on a sphere (as in [@M87; @IW01]) or by Monte-Carlo simulations at finite temperature (as in [@SW02]).
Let us now look at the $L\to\infty$ limit, and consider finite-$N$ solutions on the infinite plane ${{\bf R}}^2$, with the boundary condition $\phi_3=1$ at spatial infinity. First, the system admits ring-like solutions. For these, we have $\phi_3=-1$ at a single point $O$, and $\phi_3=0$ on a deformed ring centred at $O$. Because of the factor $(1-\phi_1^2)$ in (\[BBS\]), this ring is not quite rotationally-symmetric about $O$; its symmetry group is a subgroup of the rotation group O(2), namely the dihedral group $D_{2N}$. In other words, these solutions resemble polygonal rings, analogous to the polygonal shells of the three-dimensional Skyrme system. A numerical investigation (for the case $\alpha=0.1$) reveals that their normalized energy $E$ is very well fitted by the function $$\label{Ering}
E=1.262+0.227/N^2$$ for $N\geq2$. For example, the energy of the $N=8$ ring is $E=1.266$; for comparison, the topological lower bound is $E=1.2373$. This $N=8$ ring is depicted in the left-hand plot of Figure \[fig3\]; it is a stable static solution with $D_{16}$ symmetry, and is best thought of as a ring of sixteen half-Skyrmions.
The other localized solution illustrated in Figure \[fig3\] corresponds to a clump of the Skyrme crystal — in the right-hand plot, we see a $2\times2$ clump of crystal, again containing sixteen half-Skyrmions. The energy of such a clump is approximately equal to the (bulk) energy of the crystal plus a surface contribution. If the circumference of the clump has length $ML_0$, where $M$ is a positive integer, then one expects that the normalized energy will have the form $$\label{Eclump}
E \approx E_0 + M\,\delta E\,/N$$ where $E_0 = 1 + 2.373\,\alpha$ and where $\delta E$ is the contribution from a single edge of lattice-cell. One may estimate $\delta E$ as follows (this is analogous to the discussion in [@B96]). Start with a single cell, with coordinate range $0\leq x\leq L_0$, $0\leq y\leq L_0$. Now stretch the cell in the positive $x$-direction, so that the $x$-range becomes $0\leq x<\infty$. The boundary conditions are that the field ${\vec\phi}$ remains periodic in the $y$-direction, maintains its original lattice-configuration on the edge $x=0$, and tends to its asymptotic value $(0,0,1)$ as $x\to\infty$. We can then relax this configuration, minimizing its energy numerically to obtain a mimimum energy $E_1$. The deformed cell still contains two units of Skyrme charge, so the resulting estimate for $\delta E$ is $$\label{deltaE}
\delta E \approx E_1 - 2E_0.$$ For example, if $\alpha=0.1$, one gets $\delta E=0.026$. Consequently, our estimate for the energy of a localized square lattice clump of charge $N$, with $\alpha=0.1$, is $$\label{Eclumpp}
E \approx E_0 + 2\sqrt{2}\,\delta E/\sqrt{N} = 1.237+0.074/\sqrt{N};$$ but for low $N$ this will be an underestimate, since there are also corner effects to be considered. These corner effects presumably give an additional contribution to $E$ of the form $C/N$, where $C$ is a constant.
If we take a single lattice cell ($N=2$), then it resembles a square array of four half-Skyrmions, but this amounts to the same thing as a ring of four half-Skyrmions — indeed, the $N=2$ solution on ${{\bf R}}^2$ has a unique shape. For $N=8$, however, there is the possibility of the two solutions depicted in Figure \[fig3\], and each of these is in fact realized, as a local minimum of the energy. The formulas (\[Eclumpp\]) and (\[Ering\]) suggest that the energy of the $N=8$ ring should be higher than that of the square clump, by an amount $\Delta E = 0.0027$. The numerical result is that $\Delta E = -0.0035$, which gives an estimate for the corner contribution which has to be added to (\[Eclumpp\]), namely $0.05/N$.
For larger crystal chunks, the ring will be less favourable. For example, we expect, from (\[Ering\]) and the modified version of (\[Eclumpp\]), that the minimal-energy $N=18$ solution will correspond to a $6\times6$ square array of half-Skyrmions; and in particular, that the energy of the $N=18$ ring will be higher than that of the $6\times6$ clump by an amount $\Delta E = 0.0057$. It should be emphasized, however, that there are likely to be many local minima, of which the square chunk and the ring are just two examples. All this is very similar to the situation for the three-dimensional Skyrme system.
In conclusion, this particular planar system, in addition to admitting an explicit crystal solution which saturates the topological lower bound on the energy, is a good model for three-dimensional Skyrmions.
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[^1]: email: [email protected]
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present the first high-resolution near-infrared images of the edge-on silhouette circumstellar disk, Orion 114-426, made using NICMOS on the [*Hubble Space Telescope*]{}. Images taken against the bright nebular background of the ionized hydrogen Pa$\alpha$ line at 1.87 show the major axis of the disk to be approximately 20% smaller than at 0.6, from which we deduce the structure of the edge of the disk. Continuum images of diffuse polar lobes above and below the plane of the disk show a morphology and evolution with wavelength consistent with predictions for reflection nebulae in a diffuse envelope with large polar cavities, surrounding a thin, massless, Keplerian disk, centered on an otherwise hidden central star. We make use of our observations and reasonable assumptions about the underlying disk structure to show that the disk mass is at least 10 and plausibly $\geq 5\times 10^{-4}$.'
author:
- 'Mark J. McCaughrean[^1], Hua Chen[^2] John Bally[^3] Ed Erickson[^4] Rodger Thompson, Marcia Rieke, Glenn Schneider, Susan Stolovy, and Erick Young'
title: 'High-resolution near-infrared imaging of the Orion 114-426 silhouette disk[^5] '
---
6.25in 8.9in
Subject headings: accretion, accretion disks — circumstellar matter — ISM: individual (Orion Nebula) — stars: formation, pre-main sequence — infrared: ISM: continuum, lines and bands
Introduction
============
The discovery of a family of circumstellar disks seen as dark silhouettes in projection against the Orion Nebula using the [*Hubble Space Telescope*]{} provided strong confirmation of the disk paradigm of star formation (O’Dell 1993; O’Dell & Wen 1994; McCaughrean & O’Dell 1996 \[MO96\]). The disks range in diameter from 50–1000AU, and thus the $\sim$50AU (0.1 arcsec at 450pc) resolution of the HST observations was sufficient to examine their structure directly at optical wavelengths. An important finding was that the radial surface density profiles appear to be abruptly truncated at some outer radius, perhaps due to external effects from the surrounding region and dense cluster (MO96), and more detailed examination of this transition zone should lead to a greater understanding of the evolution of disks in harsh environments.
The discovery images were obtained over a relatively narrow wavelength range (5007–6585Å), and further insight should be possible through HST observations at shorter and longer wavelengths. In the blue/near-UV ($\sim$2000–4000Å), the spatial resolution approaches $\sim$15AU, while increased dust opacity at these wavelengths should also allow more tenuous structures to be traced to larger radii. Conversely, the considerable [*reduction*]{} in dust opacity at near-IR wavelengths should allow us to trace structures to smaller radii, albeit with commensurately poorer spatial resolution. Consequently, we are conducting follow-up HST studies from the near-UV to near-IR (0.3–2.5), and in the present paper, we report preliminary near-IR observations using NICMOS of one silhouette disk, Orion 114-426. The largest of the sample at $\sim$1000AU diameter, this disk is seen near edge-on, and while the central star is not directly visible at optical wavelengths, its presence is betrayed by two polar nebulosities believed to be illuminated by it.
Observations
============
A comprehensive General Observer program (McCaughrean : GO7367) studying the Orion silhouette disks with NICMOS, STIS, and WFPC2 is being carried out during HST Cycle 7. Early Release Observations using NICMOS were subsequently proposed by the Instrument Development Team (Erickson : SM2/ERO7114) for scientific verification and media use. Due to this overlap, the ERO data were reduced and analysed collaboratively, resulting in studies of 114-426 (presented here) and of the 182-413/183-419 field (Chen 1998).
NICMOS observations of the 114-426 field were obtained on 19 April 1997 during the Servicing Mission Orbital Verification following installation in the HST. Images were taken through broad-band, narrow-band, and polarimetric filters between 1 and 2.1 as summarized in Table 1. Data reduction combined standard ground-based near-IR imaging techniques with parts of the NICMOS calibration pipeline. Multiple read-outs combined with multiple positions on the sky were used to reject cosmic-ray events; electronic offsets were removed with on-orbit dark images; quantum efficiency variations were removed with flat fields taken on-orbit where possible, otherwise from ground tests. Finally, mosaics were made registering the multiple images using stars or HST pointing information. Detailed photometric calibration was not attempted, but ground-based near-IR magnitudes for stars in the field were used to calibrate within $\pm$.
Despite integration times significantly shorter than those planned for the GO program, important preliminary results were nevertheless obtained from the narrow-band imaging against the bright Pa$\alpha$ background at 1.87, broad-band imaging at 1.1 and 1.6, and the polarization imaging at 2.0. The three polarizer position images were combined to form a 2 continuum image, but due to remaining uncertainties in the correct analysis techniques for NICMOS polarimetry and incomplete on-orbit polarization calibration, the polarization results themselves are deferred to a future paper. The remaining narrow-band images did not provide useful additional information and are not further discussed.
Results
=======
Narrow-band imaging in Pa$\alpha$
---------------------------------
The highest S/N images of the silhouettes obtained by MO96 were through a narrow-band H$\alpha$ ($\lambda$6565Å) filter, admitting the full emission line flux from the bright Orion Nebula region, while minimizing continuum emission from the central stars, or in the case of 114-426, its polar lobes. The brightest near-IR counterpart is the Pa$\alpha$ line at 1.87, which cannot be detected from the ground due to atmospheric absorption. For typical region ionization parameters (10$^4$K, 10$^4$cm$^{-3}$, Case B) and $A_V$$\sim$ foreground to the nebula, the detected photon flux at Pa$\alpha$ should be $\sim$60% of that at H$\alpha$: the brightest equivalent line available to ground-based observers (Br$\gamma$ at 2.16) would be a further factor of ten fainter (Osterbrock 1989).
The Pa$\alpha$ 1.87 image of 114-426 is shown in Figure 1 with the H$\alpha$ ($\lambda$6565Å) image from MO96. The S/N in the P$\alpha$ image is poor (${\mbox{$\mathrel{\mathpalette{\lower0.5ex\vbox{\baselineskip=0pt\lineskip=0.2ex
\ialign{$\mathsurround=0pt <\hfil##\hfil$\crcr}\crcr\sim\crcr}}}$}}$5:1) since the integration time was short (288 sec), and the NIC1 image scale of 0.0432 arcsec/pixel over-resolved the 0.19 arcsec FWHM diffraction-limited resolution of the telescope at 1.87. Nevertheless, the silhouette is clearly detected, allowing a preliminary measurement of its size. The data were binned by a factor of two to better match the appropriate pixel size ( 2 pixels per FWHM) and then averaged across the minor axis. The resulting 1D major axis profile had high enough S/N to show the two ends of the disk as sharp dips separated by 1.8 arcsec. As discussed in detail by MO96, the apparent size and structure of a silhouette disk is a convolution of its real form with the instrumental point spread function, and following MO96, we adjusted the parameters of a model edge-on disk convolved with a model HST+NICMOS PSF calculated using the TinyTim software (Krist & Hook 1997) until the major axis length was reproduced. The resulting best-fit model disk has a major axis size of $\simeq$800AU at 1.87, $\sim$20% less than the 1012AU measured at 0.6 (MO96). The same procedure used on the 2 continuum image (Section \[sec:continuum\]) yielded the same result to within 5%. Finally, we verified the overall procedure by degrading the high S/N \[OIII\] image from MO96 to the same spatial resolution and S/N as the Pa$\alpha$ image, then performing the same fitting process, before retrieving the correct size for the disk at optical wavelengths.
Continuum imaging of the polar lobes {#sec:continuum}
------------------------------------
The optical continuum image of 114-426 showed faint polar lobes, interpreted as reflection nebulae of tenuous dust above and below the plane of the disk, illuminated by the otherwise unseen central star (MO96). Similar reflection nebulae are seen above and below the plane of an edge-on disk in the HH30 system (Burrows 1996). The wavelength dependent morphology and polarization structure of the lobes in 114-426 should allow us to probe the underlying form of the disk, the geometry of polar cavities, the grain size, and scattering function.
The near-IR broad-band continuum images, along with the F547M image from MO96, are shown in Figure 2 in grayscale and contour forms. As the wavelength increases, three effects are seen. First, the initially fainter SE polar lobe increases in brightness until it equals then outshines the intensity of the NW lobe. The peak intensities in the lobes (after background subtraction) are in the ratios 7.3:1, 2.2:1, 1.2:1, and 0.85:1 for the NW:SE lobes, at 0.57, 1.1, 1.6, and 2.0 respectively. Second, the nebulae move closer together, as the reduced extinction allows us to probe closer to the disk midplane: the separations of the peak pixels in the two lobes are at 0.64, 0.43, 0.40, and 0.32 arcsec at 0.57, 1.1, 1.6, and 2.0 respectively. Third, the nebulae appear to flatten from conical to slab-like.
These features can be compared to model disks and envelopes ( Lazareff, Pudritz, & Monin 1990; Whitney & Hartmann 1992 \[WH92\], 1993 \[WH93\]; Fischer, Henning, & Yorke 1996 \[FHY96\]). The general broad fan shape and increasing flatness of the nebulae are best reproduced by model SH of FHY96, a thin, massless disk (in comparison to the central star) in Keplerian rotation, with an envelope and broad polar cavities. Models with thick, massive disks have more polar material and result in images with too much elongation perpendicular to the disk. The same is true of models with just a narrow, cylindrical polar hole rather than a broader, so-called “streamline” cavity (WH93).
Since the central star in 114-426 is not seen, the disk must lie within a few degrees of edge-on, as the thin disk of the SH model does not occult the central star unless this condition is met (FHY96). This degree of alignment is also argued for on the grounds that the two lobes have nearly equal brightness in the near-IR. The asymmetry between the two lobes in the optical continuum (Figure 2a) can probably explained by asymmetries in the outer, more diffuse parts of the envelope, perhaps due to external effects in the region.
Discussion
==========
Disk structure {#sec:structure}
--------------
The disk appears $\sim$20% smaller at 1.87 than at 0.6, and thus we are clearly resolving structure in its outer parts. In order to understand the implications of these observations, we need to examine the theoretical expectations. For a thin, massless, Keplerian disk that is hydrostatically supported and vertically isothermal (Shakura & Sunyaev 1973; Pringle 1981; Lazareff 1990), the density $\rho$ as a function of radius, $r$, and height above the midplane, $z$, is: $$\rho(r,z) = \rho_d \left( \frac{r}{r_d} \right)^{-15/8}
\exp \left[ -\frac{\pi}{4}
\left( \frac{z}{h(r)} \right)^2 \right]
\label{eq:one}
$$ where $\rho_d$ is the midplane density at the outer disk radius, $r_d$, and $h(r)$ is the disk scale-height: $$h(r) = z_d \left( \frac{r}{r_d} \right)^{9/8}
\label{eq:two}
$$ where $z_d$ is the scale-height at $r_d$. The disk surface density is then: $$\Sigma (r) = 2 \rho_d z_d \left( \frac{r}{r_d} \right)^{-3/4}
\label{eq:three}
$$
For typical Orion Nebula dust grains (R=5; Cardelli, Clayton, & Mathis 1989), the extinction at 1.87 is one sixth of that at 0.6, and thus achieving the same effective optical depth requires six times higher column density at the longer wavelength. For a face-on disk with unity optical depth at 0.6 at 506AU radius, Eq.\[eq:three\] shows that the equivalent optical depth at 1.87 would occur at 46AU, the disk would appear much smaller in the near-IR. For an edge-on disk, the calculation is harder: we are probing the midplane density of the disk, not the surface density, and the line-of-sight through the disk at a given “impact parameter” ( the distance off-center) integrates over different densities at different radii. Assuming the disk is truncated at some outer radius, we can integrate the total column density through the midplane as a function of the impact parameter, $a$. We have calculated profiles for disks in which the midplane density scales as $r^\alpha$, with $\alpha = -1, -2,$ and $-3$ (Figure 3). These values were chosen since simple analytical integrals exist in each case, but they also closely correspond to plausible disk models: $\alpha = -9/8 \simeq -1$ would yield a surface density independent of radius; $\alpha = -15/8 \simeq -2$ yields the canonical $\Sigma(r) \propto r^{-3/4}$; $\alpha = -21/8 \sim -3$ yields $\Sigma(r) \propto r^{-3/2}$, a commonly assumed density law for more massive circumstellar disks ( Adams, Shu, & Lada 1988).
Rough power laws can be fit to the inner section of each curve in Fig.3, with the column density increasing as $a^\gamma$, and $\gamma \simeq -1/4,
-1$, and $-2$ corresponding to $\alpha = -1, -2,$ and $-3$ respectively. In the canonical case of $\alpha = -2 \simeq -15/8$, a disk with unity optical depth at 506AU at 0.6 would have the same optical depth at 84AU at 1.87. Thus, even though an edge-on disk will shrink less with wavelength than a face-on one, it is clear that for 114-426, the observed decrease in size of only 20% ( $-0.1$dex) for an increase in density of six ( 0.78dex) would require $\gamma\sim -8$, and thus $\alpha\sim -9$ and $\Sigma(r)\propto r^{-8}$, quite inconsistent with conventional understanding of disk structure.
The exception comes near the edge of a disk, where the integrated column density rises sharply as the path length increases most rapidly. Indeed, Fig.3 shows that in the $\alpha = -2$ case, the density [*does*]{} increase by the required factor of six within 80% of the outer radius, as observed. Thus, the observed shift between the optical edge at 506AU and the infrared edge at 400AU could still be consistent with the midplane density following the canonical power law for a thin, massless disk, [ *as long as*]{} the disk is physically truncated somewhere near the optical edge. As discussed by MO96, it seems plausible that the disk may be truncated due to physical processes present in the Orion Nebula region and/or Trapezium Cluster, and it may be possible to ascertain which particular process is responsible from our future HST observations, which will trace the radial structure of 114-426 over a full order of magnitude in limiting column density.
The disk mass revisited
-----------------------
The mass of the 114-426 disk remains unknown: in particular, does it exceed the $\sim$0.01 of the minimum mass solar nebula (MMSN), is it massive enough to form a planetary system similar to our own? O’Dell & Wen (1994) and MO96 discuss how the mass of a silhouette disk can be roughly estimated, noting that the minimum intensity seen towards it is typically $\sim$10% of that of the background region. Assuming this to be background emission attenuated by the disk, a line-of-sight column density is calculated for each point in the silhouette, and summing over the whole disk area, a total mass is estimated. However, virtually all of this “disk light” is an artifact, as the the background emission is blended into the disk by the instrumental PSF (MO96). Thus the usefulness of the “transmission technique” is diminished, but it at least provides an absolute [*lower-limit*]{} mass, an important counterpoint to the results from millimeter interferometry which are, as yet, generally provide only [*upper-limit*]{} masses for disks in the Trapezium Cluster (Mundy, Looney, & Lada 1995; Lada 1996).
MO96 used the transmission technique to estimate lower-limit masses for the six silhouette disks based on their optical images. The masses were small, typically several orders of magnitude less than the MMSN. The most massive was 114-426 at 0.002, but we must report here that an error was made in calculating the area and thus mass of 114-426 in MO96: its minimum mass should be revised downwards to $2\times 10^{-4}$. (The masses for the other disks given in Table 2 of MO96 remain unaffected by this error).
In principle, the transmission technique could be applied to our new near-IR images of 114-426, resulting in a significantly [*higher*]{} estimate of the minimum mass. Since the disk is only marginally smaller at 1.87, the reduced near-IR extinction should lead to a higher mass, on the order of the factor of six in $A_V/A_{1.87\mu{\rm m}}$. However, the present images are not good enough to make such estimates: the Pa$\alpha$ image has very poor S/N, making a reliable measure of the disk/background flux ratio impossible, and while the continuum images have much higher S/N, the silhouette is contaminated by the reflection nebulae, making it difficult to measure the underlying disk/background flux ratio and the true silhouette area. Such calculations must wait until higher-quality Pa$\alpha$ data are available.
In the interim, alternative approaches to estimating the disk mass can be taken. The non-detection of the central star in the POL 2 continuum image can be used to set a lower limit on the total column density through the disk midplane, depending on the intrinsic brightness of the star. The integrated flux from the two reflection nebulae is =, which clearly only represents a fraction of the flux we would see from the central star if it had no disk. Some estimate of this fraction can be made from WH92 and WH93, who tabulate $F/F_{\star}$ (the total flux observed as a fraction of the flux that would be seen if the star had no disk or envelope) for a variety of models. While $F/F_{\star}$ ranges from 0.01–10% depending on the disk and envelope structure, inclination angle, grain properties, and total opacity, $\sim$1% is typical for models inferred for 114-426 ( near edge-on flared disk, WH92 models 3,4 with $\mu < 0.3$, perhaps with a tenuous envelope with broad polar cavities, WH93 models 5–8). The inferred intrinsic brightness of the central star is then $\sim$, corresponding to a 1Myr old, $\sim$1.5 star at 450pc (D’Antona & Mazzitelli 1994). Since the star is undetected in the 2 image to a 5$\sigma$ point source detection limit of =, the implied line-of-sight extinction towards it is $\geq$ at 2, or $A_V \geq {\mbox{$ 60^m$}}$. Assuming the disk is exactly edge-on, the corresponding column density through the disk midplane to the central star is $\geq 1.1\times 10^{23}$cm$^{-2}$ (assuming 1$A_V \equiv 1.9\times 10^{21}$cm$^{-2}$).
The disk outer radius, $r_d = 506$AU, is known from the data of MO96. The same data can be used to determine $z_d$, the scale-height at $r_d$: we fit a 1D average minor axis profile across the \[OIII\] image of 114-426 (Fig.4 of MO96) with a Gaussian characterized by Eq.\[eq:two\], yielding $z_d = 72$AU. Finally, we assume the disk inner radius, $r_i = 1$AU: since the surface density increases towards the center with a power $< 1$, our results are not too sensitive to this assumption. Integrating Eq.\[eq:one\] (with $\alpha = -15/8$) through the midplane, we calculate the midplane density at the outer edge, $\rho_d \geq 5.8 \times 10^4$cm$^{-3}$. Then integrating Eq.\[eq:three\] between $r_d$ and $r_i$, the total number of particles (HI+2H$_2$) is $\geq 3.4 \times 10^{52}$, yielding a mass estimate of $\geq 5.7 \times 10^{28}$g or $\geq 10$.
Although it is clear that very little mass is required to render the central star invisible, the disk mass is likely to be significantly greater. Using the analytical solution for the total column density seen through the midplane of an edge-on disk in the $\alpha = -2 \simeq -15/8$ case (Section \[sec:structure\]), and assuming $\rho_d = 5.8 \times 10^4$cm$^{-3}$ at an outer radius of 506AU, we obtain values $\sim 2\times 10^{20}$cm$^{-2}$ or $A_V\sim{\mbox{$ 0^m \!\!\!.\!\,\, 1$}}$ just inside the edge. Yet the disk must have a significantly higher column density there to be [*seen*]{} as an edge: the results of MO96 show that the attenuation at the edge of the disk is at least 85%, equivalent to $A_V\geq{\mbox{$ 2^m$}}$, in turn implying $\rho_d \geq 1.2\times 10^{6}$cm$^{-3}$. This is a plausible value if (for example) the location of the disk edge is set by pressure balance with the surrounding region: assuming a temperature and density for the latter of 10$^4$K and 10$^4$cm$^{-3}$ and an outer disk temperature of 10–100K, densities at the disk edge of $10^6$–$10^7$cm$^{-3}$ would be predicted.
Inserting $\rho_d \geq 1.2\times 10^{6}$cm$^{-3}$ into Eq.\[eq:three\] along with our other derived disk parameters yields a disk mass of $\geq 1.1\times 10^{30}$g or $\geq 5\times 10^{-4}$. While still considerably less than the MMSN, this is a lower limit, and the disk might be more massive. However, our present simple calculations based on lower-limit extinction measurements alone cannot be pushed much further. Burrows (1996) have shown for HH30 that more realistic mass estimates can be made by fitting the structure of the polar reflection nebulae, in particular the width of the dark lane of obscuration between them, and such an approach will be taken with 114-426 when our higher S/N and resolution Cycle 7 HST data are available. Ultimately, the most meaningful estimate of disk mass must come from measurements of optically thin tracers at millimeter wavelengths: such observations have been obtained for 114-426 and will be discussed in a future paper.
Conclusions
===========
We have presented preliminary NICMOS images of the Orion 114-426 circumstellar disk. The disk appears to be roughly 20% smaller at 1.87 than at at 0.6, which we attribute to a standard radial power-law density distribution inside a heavily truncated edge to the disk at $\sim$500AU radius. The general wavelength-dependent morphology of the polar lobes is consistent with models of a thin, massless disk surrounded by a tenuous envelope with broad polar cavities, and their flux was used as an indirect way of estimating the mass of the central star to be 1.5. The non-detection of the star at 2 was then used to estimate the minimum total column density through the disk midplane, and thus a minimum disk mass of $\geq 10$, assuming that it follows the standard form for a massless disk in Keplerian rotation. The hard edge of the disk was used to improve the lower-limit estimate under similar assumptions, resulting in a mass $\geq 5\times 10^{-4}$.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the University of Arizona, Ball Aerospace, Rockwell, NASA GSFC, and the STScI for developing, managing, and commissioning NICMOS. MJM thanks Abi Saha for brokering the agreement between the NICMOS IDT ERO and Cycle 6/7 GO teams, Chris Skinner and Eddie Bergeron for data reduction help, and Matthew Bate, Olaf Fischer, Bob O’Dell, Peter Schilke, and Hans Zinnecker for useful discussions. Finally, we would like to thank the referee, Karl Stapelfeldt, for his insightful comments.
Adams, F. C., Shu, F. H., & Lada, C. J. 1988, , [**326**]{}, 865 Burrows, C. J. 1996, , [**473**]{}, 437 Cardelli, J. A., Clayton, G. C. & Mathis, J. S. 1989, , [**345**]{}, 245 Chen, H., Bally, J., O’Dell, C. R., McCaughrean, M. J., Thompson, R. I., Rieke, M. J., Schneider, G., & Young, E. T. 1998, , in press D’Antona, F., & Mazzitelli, I. 1994, , [**90**]{}, 467 Fischer, O., Henning, T., & Yorke, H. 1996, , [**308**]{}, 863 (FHY96) Krist, J., & Hook, R. 1997, TinyTim User Manual Version 4.3, [http://scivax.stsci.edu/$\sim$krist/tinytim.html]{} Lada, E. A., Dutrey, A., Guilloteau, S., & Mundy, L. G. 1996, [*BAAS*]{}, [**189**]{}, \#53.01 Lazareff, B., Pudritz, R. E., & Monin, J.-L. 1990, , [**358**]{}, 170 McCaughrean, M. J., & O’Dell, C. R. 1996, , [**111**]{}, 1977 Mundy, L. G., Looney, L. W., & Lada, E. A. 1995, , [**452**]{}, L137 O’Dell, C. R., Wen, Z., & Hu, X. 1993, , [**410**]{}, 696 O’Dell, C. R., & Wen, Z. 1994, , [**436**]{}, 194 Osterbrock, D. E. 1989, [*Astrophysics of gaseous nebulae and active galactic nuclei,*]{} (Mill Valley: University Science Books), p84 Pringle, J. E. 1981, , [**19**]{}, 137 Shakura, N. I., & Sunyaev, R. A. 1973, , [**24**]{}, 337 Whitney, B. A., & Hartmann, L. 1992, , [**395**]{}, 529 (WH92) Whitney, B. A., & Hartmann, L. 1993, , [**402**]{}, 605 (WH93)
[^1]: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany; [email protected]
[^2]: Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721
[^3]: Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309–089
[^4]: NASA Ames Research Center, MS 245–6, Moffett Field, CA 94035–1000
[^5]: To appear in the special NICMOS/STIS ERO edition of the [*Astrophysical Journal (Letters),*]{} January 1998
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Fast development of sharing services has become a crucial part of the cyber-enabled world construction process, as sharing services reinvent how people exchange and obtain goods or services. However, privacy leakage or disclosure remains as a key concern during the sharing service development process. While significant efforts have been undertaken to address various privacy issues in recent years, there is a surprising lack of review for privacy concerns in the cyber-enabled sharing world. To bridge the gap, in this study, we survey and evaluate existing and emerging privacy issues relating to sharing services from various perspectives. Differing from existing similar works on surveying sharing practices in various fields, our work comprehensively covers six branches of sharing services in the cyber-enabled world, and selects solutions mostly from the recent five to six years. We conclude the issues and solutions from three perspectives, namely, from users’, platforms’ and service providers’ perspectives. Hot topics and less discussed (cold) topics are identified, which provides hints to researchers for their future studies.'
author:
- 'Ke Yan, [*Member*]{}, [*IEEE*]{}, Wen Shen, Qun Jin, [*Senior Member*]{}, [*IEEE*]{}, Huijuan Lu [^1]'
bibliography:
- 'root2.bib'
title: 'Emerging Privacy Issues and Solutions in Cyber-Enabled Sharing Services: From Multiple Perspectives'
---
Cyber Technology; Sharing Service; Privacy; Crowdsourcing; Collaborative Consumption.
Introduction
============
Cyberization is transforming our physical living world into a virtual computerized world by leveraging the Internet and computational methodologies [@ma2016cybermatics; @ma2016perspectives]. In the virtual computerized world, or more specifically, the cyber-enabled world, people are connected via Internet regardless of physical distances. Cyber-enabled sharing services, or in short, sharing services, which provide information, goods, and services in a shared form to multiple individuals, who know or do not know each other, are essential and necessary components of cyber-world development and probably the most exciting cyber-related concept in the current stage of cyberization. Sharing services encourage people to share both virtual and physical assets through the Internet using cyber-enabled clients, including mobile phones, all kinds of computers and similar digital devices. Sharing services contribute to the fast development of cyber technology, where the control, responsibility for the common good, earnings, capitalization, information, and efforts are all shared among the participants or distributed to peer members [@schor2014debating]. In recent years, cyberized sharing service companies, such as Uber, Airbnb, Etsy and Amazon Family Library, have been overwhelmingly popular and enjoyed incredible growth [@zervas2014rise; @belk2014you].
There are various reasons for people to participate in sharing practices. For instance, no single entity or person can control the whole market or economy, although some participants have more regulatory power than others. All participants share the responsibility of making the market to operate healthily. This form of collaborative economy or peer-to-peer (P2P) sharing leads to more efficient resource allocations and more sustainable lifestyles. However, any participant in the sharing practice, regardless whether it is a user or service provider, can be a potential attacker who compromises legitimate users’ privacy. Therefore, to attract more people to share, it is necessary to build trust, establish reputation, protect privacy and guarantee security for both the user and service provider [@bella2011enforcing]. Personal privacy concern is the main factor that hinders the development of sharing services in the cyber-enabled world [@du1985data; @daswani2003open]. On one hand, people are reluctant to adopt sharing practices due to privacy disclosure concerns [@feeney2015ridesharing; @gobble2015regulating; @li2015privacy]; on the other hand, sharing service providers insist that personal data is part of the necessary information in user experience analysis for improving service quality. While only privacy protection is explored in this paper, the authors would like to note that privacy is relevant and closely related to trust, reputation and security. Users need to trust the service provider, which implies that the service provider must have a good reputation that the users can trust. Reputations are established through the interactions between the users and service providers. However, during the interaction process, privacy issues arise, since private information pieces from both parties are inevitably revealed to each other [@Dimitriou2012Multi; @Dimitriou2014Multi].
Unfortunately, due to the fast development pace of sharing service technology, privacy issues were not well addressed before sharing services were widely spread over the physical world [@katz2015regulating]. For example, in the ridesharing service practice, although the business model exists for quite a while, there are still many privacy leakage concerns, including location privacy concerns, driver/customer’s personal information leakage concerns, physical privacy concerns and etc. [@christin2016privacy]. Cyber-technologies that can be used to protect various aspects of privacy are urgently desired to prohibit both the user and service provider from revealing each other’s sensitive information. In the starting stage of the sharing economy, some service providers intentionally neglect the privacy issues to survive in the highly competitive business environment. In other words, profit is usually the highest priority for most starter-level sharing service companies. In this study, we surveyed over one hundred research works from recent years that are closely related to the privacy issues with the newly developed sharing service technologies and observed that the privacy protection level is highly related to the number of users who participate in the sharing service, which affects the final profit of the service providers. In addition, from the user’s perspective, increasing the self-awareness of privacy disclosure is an important task for the users to protect themselves in the current stage of cyberization.
In summary, the emerging privacy issues of sharing services in the cyber-enabled world and the available solutions are reviewed comprehensively. From the literature, we summarize the sharing services in the current stage of the cyber-enabled world into two categories [@schor2014debating; @botsman2011s]:
- [**Crowdsourcing**]{} employs collective intelligence or power to fulfil tasks or achieve goals. Concrete examples of crowdsourcing are Internet crowdsourcing marketplaces, crowdfunding, and crowdtesting [@doan2011crowdsourcing]. For a typical crowdsourcing practice, there are, in general, three roles involved: the task requester, the platform and the worker. The task requester posts tasks on the platform and attracts workers to finish the job in a crowdsourcing way.
- [**Collaborative consumption**]{} allows consumers to use products or services without full ownership. Concrete examples of collaborative consumption include collaborative online shopping, ridesharing, and homesharing practices [@belk2014you]. For a typical collaborative consumption model, there are again three roles involved: the host, the platform and the customer. Differing from the crowdsourcing practice, the host provides P2P sharing of goods or services to customers through an online platform.
In this study, we refer to the combination of task requesters and hosts as service providers, and the combination of workers and customers as users. The review of privacy issues and solutions follows the above two outlines and reveals the main concerns in the literature, which include the requester’s data protection, the balance between privacy protection and sacrifice, data encryption, unreliable data analysis, location privacy and physical privacy. Figure \[fig:overall2\] lists a taxonomy of important works that are surveyed for privacy issues and solutions in crowdsourcing and collaborative consumption practices.
{width="7.3in"}
Although there are similar works concerning privacy in sharing practices from the literature, e.g., [@androutsellis2002survey; @aggarwal2008general; @fung2010privacy; @rahman2015survey; @heurix2015taxonomy], they focused on traditional privacy protection methods. Traditional privacy protection techniques, including k-anonymity [@samarati2001protecting; @sweeney2002k], l-diversity [@machanavajjhala2007diversity] and t-closeness [@li2007t], have been heavily reviewed in the past few decades. In contrast, our work focuses on privacy protection technique development in recent years, skips the traditional approaches and covers technologies comprehensively in the area of cyber-enabled sharing services. Most surveyed works in this study were published in past five to six years. The sources of the reviewed papers include the most popular databases, such as ACM Digital Library, IEEE Xplore Digital Library, Springer Link and ScienceDirect. The searched keywords include ‘sharing service’, ‘privacy issue’, ‘privacy protection‘, ‘crowdsourcing privacy’, ‘collaborative consumption privacy’, ‘crowdfunding privacy’ and etc.
The main contributions of this work include 1) categorizing recent research studies working on privacy issues of sharing services into trends, 2) identifying the hot/cold research topics, and 3) finding the research gaps for real-world sharing services to better protect people’s privacy. For example, from Fig. \[fig:overall2\], we found that data reliability analysis and location privacy are two hot topics for collaborative consumption, whereas the physical privacy issue in homesharing practice is less discussed. Moreover, there are research works indicating that physical privacy is also largely concerned by users in the sharing service practices. Those less discussed topics require more attention in future studies.
The remaining parts of this work are organized as follows: The emerging privacy issues and solutions of crowdsourcing are analyzed in Section \[sec:Crowdsourcing\]. The emerging privacy issues and solutions of collaborative consumption are reviewed in Section \[sec:collaborative\]. In Section \[sec:emerging\], all six branches discussed in Sections \[sec:Crowdsourcing\] and \[sec:collaborative\] are summarized from three perspectives, namely, user, platform and service provider perspectives. Section \[sec:openIssues\] raises open research issues for each branch of the sharing services and from the three perspectives mentioned in Section \[sec:emerging\]. In Section \[sec:conclusion\], several conclusions are drawn regarding cyber technology development to predict the future trends in the development of cyber-enabled sharing technologies.
Privacy Issues and Solutions in Crowdsourcing Practices {#sec:Crowdsourcing}
=======================================================
Crowdsourcing refers to the distribution of tasks that cannot be easily accomplished in a traditional way to a large group of online workers [@howe2006rise] (Figure \[fig:crowdsouring\]). The tasks are usually difficult problems or issues that cannot easily be resolved by small groups of users or individuals. Despite its many advantages, crowdsourcing brings increasing risks of information leakage and privacy violation, which limits its development and application potential.
{width="5.5in"}
There are two types of users in a crowdsourcing platform: the worker (or the employee) and the task requester (or the employer). The task requester provides incentives and tasks, while the worker performs the tasks to receive the incentives. The interaction between them gives rise to the risks of information leakage and privacy violation, which is either unidirectional or bidirectional. In other words, either the worker or the requester, or both, have the possibility to leak sensitive information or violate the privacy agreement.
We next identify potential privacy leaking risks in three key applications of crowdsourcing: Internet crowdsourcing marketplaces, crowdfunding, and crowdtesting. For each application, we consider the privacy protection issues in the process of sharing practice and survey the existing solutions in the literature.
Crowdsourcing Marketplace {#sec:internetmarketplace}
-------------------------
An online crowdsourcing marketplace provides a platform for matching the task requesters and the task performers for mutual benefits. Numerous crowdsourcing marketplaces have been developed during the past few years, e.g., the Amazon Mechanical Turk (MTurk) [@kittur2008crowdsourcing], which enable individuals and business entities to use their own intelligence to perform tasks that are ‘difficult’ for automated computerized programs. Requesters post jobs or work in the form of human intelligence tasks on the MTurk platform, while workers browse the tasks and complete them to earn monetary incentives from the requesters.
Data privacy concerns limit the spreading speed of crowdsourcing because many users refuse to participate in crowdsourcing if personal data cannot be not securely protected. For example, when a requester evaluates the design of a particular artefact, it is likely that the requester desires to prevent exposure of the artefact. Similarly, a testing organization usually requires test takers not to disclose the content of the test. However, unlike a testing organization, which has the power to penalize test takers who violate the confidentiality agreement, the requester does not always have the power or effective methods to penalize workers who leak sensitive data or extract information for other purposes. What makes it worse is that the workers are sometimes unreliable and usually not identifiable. Therefore, it is challenging to protect the privacy of the requesters.
Generally, there are two approaches tackling the privacy protection problem for the requesters. The first solution, which is introduced by Varshney, distorts sensitive data directly using random perturbations to conceal private information [@varshney2012privacy]. A series of extensions were introduced by the same group of researchers for completing the framework based on coding theories [@vempaty2014reliable; @vempaty2014coding; @wang2005distributed]. The coding theory successfully hides the sensitive information from the workers. However, it loses the task performance quality when random perturbations are added to the original data. A mathematical model was used to analyze the tradeoffs between privacy, reliability, and cost, by considering five insight elements: error-correcting codes, reliability, perturbation, decoding and collusion attacks [@varshney2014assuring].
The second approach is the instance clipping protocol (ICP), which was introduced by Little and Sun [@little2011human] and Chen [*et al.*]{} [@chen2012shreddr]. Kajino [*et al.*]{} [@kajino2014instance] proposed a quantitative analysis framework (QAF) based on the instance clipping protocol. The QAF evaluates the instance privacy-preserving protocols and protects the target privacy, which is defined as contextual information. The instance-privacy preserving protocols preserve instance privacy at the cost of task performance. For instance, in Figure \[fig:clipping\], a task (represented by a 2D shape) is clipped by clipping windows which are marked by red boxes. Each worker is only allowed to access one clipping window for his/her task result. The ICP preserves privacy but decreases the quality of the task results. Similar to Varshney’s work, there is a tradeoff between privacy preservation and task quality. The instance clipping protocol clips an instance by a moving window, which preserves the data privacy by limiting the data that each worker acquires.
{width="1.5in"}
Celis [*et al.*]{} [@celis2016assignment] improved the clipping protocol by introducing a collusion network. The requested task is partitioned into pieces; and each piece of task is assigned to different individuals with minimal privacy leakage. Moreover, a framework is proposed with three operations: PULL, PUSH and Tug Of War (TOW). PULL and PUSH are two usual operations that represent a worker choosing tasks and a requester choosing workers, respectively. The TOW operation is used as an intermediate layer for information leakage minimization, which captures workers’ personal information, such as social networks, financial information, task history and etc. However, information leakage is still possible from the workers’ side.
Amor [*et al.*]{} [@amor2016discovering] developed a social relationship management system based on clustering algorithms, named ‘SocialCrowd’, to manage competition and collaboration in crowdsourcing practices. Experimental results showed that the data leakage was effectively prevented using SocialCrowd. Since the first version of SocialCrowd uses global search algorithms, the main concern in Amor [*et al.*]{}’s work is the computational complexity problem. While a heuristic random search method is used in later versions, it can still be trapped into local extremes in worst case scenarios.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
& 2012, 2014 & [Studying the tradeoffs between privacy, reliability, and cost]{} & An improved coding scheme by considering five insight elements & Unable to solve the collision between privacy and task performance quality\
[Kajino [*et al.*]{} [@kajino2014instance]]{} & 2014 & [Protecting the requesters’ privacy defined as contextual information]{} & Quantitative analysis framework based on instance clipping protocol & Making tradeoff between task performance and privacy\
[Celis [*et al.*]{} [@celis2016assignment]]{} & 2016 & [Partitioning the task with minimal privacy leaks]{} & The collusion network & Information leakage from the worker side\
[Amor [*et al.*]{} [@amor2016discovering]]{} & 2016 & [Increasing the privacy awareness]{} & SocialCrowd & Using heuristic function for optimal solution search, which can be trapped in worst case scenarios\
**********
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:branch1\]
Table \[table:branch1\] lists all the references that we have discussed in this section, including their main objectives, proposed solutions, and weaknesses. In summary, while most recent studies of privacy protection in crowdsourcing marketplaces consider coding schemes or clipping protocols, new technologies, such as SocialCrowd, are proposed to help improve the data security. The common problem for the coding schemes and clipping protocols is that the manipulation of the original data decreases the task performance quality. Moreover, the extra time complexity that is added to the original data transmission and storage process is a notable issue for those efforts on privacy protection. In addition, while traditional works focus on protecting the requesters’ data in a fundamental way, other issues are raised for improving users’ awareness of privacy leakage during crowdsourcing practices. This will be further discussed in Section \[sec:emerging1\].
Crowdfunding
------------
Crowdfunding has undergone fast development recently [@belleflamme2014crowdfunding; @mollick2014dynamics]. It enables founders of various ventures to fund their projects by collecting funds or other resources from a large group of individuals through an online platform, such as Kickstarter [@kuppuswamy2014crowdfunding] or Indiegogo [@blogindiegogo]. While most works focus on economic aspects of crowdfunding, few address privacy issues [@bradford2012new]. To bridge the gap between privacy concerns and practical use of crowdfunding, in this subsection, we review several existing works on privacy concerns in crowdfunding practices.
In the practice of crowdfunding, a fundraiser (the requester) proposes a project with a plan on an online platform and convinces users or supporters to invest small amounts of money in the project. The modern crowdfunding platforms, such as Indiegogo, allow users to customize their security level and conceal their personal information, such as their name and the amount of their contribution. However, our surveyed works suggest that revealing a certain amount of private information can be helpful in crowdfunding practice. For example, concealing the contribution amount of the prior contributor discourages followers from contributing more to the crowdfunding project [@burtch2013empirical]. Moreover, a fundraiser may choose to reveal more of his/her personal information to attract crowdfunders [@snyder2016crowdfunding].
Burtch [*et al.*]{} [@burtch2013empirical; @burtch2014experiment] conducted a series of experiments on a large-scaled customized crowdfunding platform to test the relationship between the privacy protection level and the results of users’ contribution histories. An econometric model was constructed where the dependent variables included the likelihood of information hiding and contribution amount from crowdfunders. The independent variables included the privacy control of the fundraiser’s platform, elapsed time of fundraising, and fundraiser’s reputation. Six hypotheses were formulated: the privacy concern effect (H1), exposure effect (H2), extremity effect (H3), self-contribution effect (H4), anchor effect (H5) and censorship effect (H6). The econometric model is depicted in Figure \[fig:econometric\], where the likelihood of information hiding and the amount of contribution from crowdfunders are affected by the six hypotheses, as shown with arrows. Although the econometric model provided valuable suggestions on privacy protection, it did not consider other factors that influence the crowdfunders’ decisions, such as wording, information regulation, transaction mechanism design and etc.
{width="5.5in"}
In 2015, Burtch [*et al.*]{} [@burtch2015hidden] conducted another online experiment to study the hidden cost of protecting crowdfunders’ privacy by utilizing modern techniques, such as invisible transaction information. Their result indicated that privacy protection increased the net funding in overall, but decreased the contribution amount from each individual. The main insufficiency of [@burtch2015hidden] is that all experiments and simulations were conducted in a randomized manner. Moreover, the users were given complete freedom for their fund contributions, which made the experimental result relatively unreliable.
Zheng [*et al.*]{} [@Zheng2016] analyzed the importance of trust management for crowdfunding practices. A research model was constructed for verification of five hypotheses. Experimental results showed that effective trust management techniques significantly improve the fundraising performance. Nevertheless, some important factors, such as funding information and presentation format of funding description, were not considered in the research model, which weakened the reliability of their conclusions.
Kang [*et al.*]{} [@kang2016understanding] introduced a structural equation modeling technique to analyze the true motivations of fundraisers for crowdfundings. Three factors are considered to examine the trustworthiness of a crowdfunding project. The fundraisers’ credentials were deeply analyzed by a bootstrapping method that is formed based on historical investment experiences. The main insufficiency of Kang [*et al.*]{}’s work is that the proposed method was not validated via any cross-sectional surveys.
All reviewed works for privacy issues in crowdfunding practices are listed in Table \[table:branch2\] Each reviewed work is accompanied by its reference, year, main objective, proposed solution and major insufficiencies. Certain levels of privacy protection, as well as sacrifices, are hidden key factors for successful crowdfunding practices. With a well-established privacy protection protocol, crowdfunders are more willing to contribute because of a safer environment. However, in some situations, a certain degree of acceptable and controllable privacy sacrifice can be helpful for a successful crowdfunding practice. The fundraisers and platforms have to realize that the net funding is directly proportional to their reputations. One open problem is to develop a more sophisticated platform for protecting the funder information. For example, a hierarchical encryption system can be built to serve the basic crowdfunding purposes and allow the fundraisers to select different levels of information sharing with the public for various purposes. Another future research direction is to explore an appropriate degree of fundraisers’ privacy disclosure that maximizes the probability of reaching a fundraising goal. Existing works showed that a certain degree of fundraiser’s privacy disclosure encourages the funding contributions from users [@wen2018information]. However, the most appropriate degree of fundraisers’ privacy disclosure remains as an open problem for crowdfunding practices. Existing works showed that a certain degree of fundraiser’s privacy disclosure encourages the funding contributions from users [@shen2018information]. Generally speaking, while crowdfunding is a relatively new concept to people in the cyber-enabled world and is directly related to assets, privacy issues are more emerging and are considered one of the most crucial research topics in the development process of the cyber-enabled world.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
&2013, 2014& [Studying the relationship between security and willingness]{} & An econometric model & Not taking the full consideration for factors that influence the crowdfunders’ decisions\
[Burtch [*et al.*]{} [@burtch2015hidden]]{} &2015& [Showing the hidden cost of protecting crowdfunders’ privacy utilizing modern techniques]{} & Online randomized experiments & Experiment users are given complete freedom for their fund contributions\
[Zheng [*et al.*]{} [@Zheng2016]]{} &2016& [Analyzing the importance of trust management]{} & A research model based on the elaboration likelihood model & Focusing on the trust management and ignoring other highly influential factors\
[Kang [*et al.*]{} [@kang2016understanding]]{} &2016& [Revealing the fundraiser’s true motivation for crowdfunding]{} & A structural equation modeling technique & The survey dataset is small in size and limited to only one country\
**********
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- -- -- --
\[table:branch2\]
Crowdtesting
------------
Crowdtesting employs crowdsourcing technology to employ a large group of testers for software or products testing at low costs [@zhang2017crowdsourced], which is reported to be more reliable, more cost-effective, and faster than traditional user-testing mechanisms [@vukovic2009crowdsourcing; @riungu2010research]. One popular crowdtesting platform is well-known as PyBossa [@pybossa2015], where customized crowdsourcing tasks can be posted, which require human cognition, knowledge or intelligence. The ultimate objectives of a crowdtesting practice include testing usability, acceptability, task performance and the quality of the results.
In a crowdtesting practice, both requesters and workers post crowdsourced data on an online platform, i.e., tasks and results. Part of the crowdsourced data can be privacy related, e.g., the data can include the requester’s confidential data and tester’s private information. The top priority for privacy preservation in crowdtesting is to protect user privacy in the data collection process. Harkous [*et al.*]{} [@harkous2014c3p] found that users usually had difficulties in accessing the privacy levels of their shared data. A context-aware framework was proposed to identify the privacy risk of shared data on a cloud server. Simulations on synthetic data were performed to show the effectiveness of their method, where data privacy levels were automatically assigned without user interaction. The main limitation of their work is that the proposed system only identifies the risky data items without proposing solutions. Moreover, there is no policy or computational technique proposed in [@harkous2014c3p].
Existing data protection schemes focus on encryption algorithms. Kandappu [*et al.*]{} [@kandappu2013exposing] showed how easily privacy leakage can occur with online survey platforms, such as MTurk and Google Consumer Surveys [@mcdonald2012comparing], which are commonly used in crowdtesting practices. A customized survey platform called Loki was developed to let users choose their preferred security level before proceeding with the online survey. The actual survey results were masked by noises before been evaluated. There are two important insufficiencies in [@kandappu2013exposing]. First, the result quality decreased because of the additional noises. Second, there was no guidance for the user to choose the most appropriate security level, which decreases the overall survey quality.
Li [*et al.*]{} [@li2016privacy] explored the privacy issues in crowdsourcing-based site survey systems utilizing WiFi fingerprint-based localization techniques. In a site survey practice, multiple suppliers were required to visit different locations and send back WiFi signals in a crowdsourced manner, which is similar to a crowdtesting practice. The main shortcoming of the work in [@li2016privacy] is that the location privacy protection of the suppliers is achieved by encryption and adding noises. The homomorphic encryption can distort the original measurement signal.
Although the crowdtesting service provides an innovative way for services/products to be tested by a large group of testers at low costs, the privacy issues were never well addressed to protect the sensitive information from both the requesters and the testers. Three specific applications of the crowdtesting practices are surveyed in this subsection: shared data protection on the cloud servers [@harkous2014c3p], online surveys [@kandappu2013exposing] and indoor site survey practice [@li2016privacy]. The objectives, solutions and main insufficiencies are listed in Table \[table:branch3\]. Almost all reviewed works demonstrate that user privacy can be easily breached by the service providers and platforms in crowdtesting practices. Various techniques were proposed to identify risky shared data and protect those sensitive information pieces. However, encryption or masking of the original data affects the usability of the final testing results, which limits the use of these cyber technologies.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
&2014& [Identifying risking data pieces on cloud server]{} & A context-aware framework based on item response theory (IRT) & Not discussing the way to protect privacy\
[Kandappu [*et al.*]{} [@kandappu2013exposing]]{} &2013& [Allowing users to choose security level for online surveys]{} & A customized survey platform called Loki & No guidance for the user to choose appropriate security level\
[Li [*et al.*]{} [@li2016privacy]]{} &2016& [Hiding the location information of the suppliers in indoor site survey practices]{} & A homomorphic encryption scheme & The original measurement signal was distorted\
**********
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:branch3\]
Summary and Discussion
----------------------
In conclusion, in crowdsourcing practices, there are always three roles in the model: user, requester and platform. On one hand, the requester has the responsibility to protect workers’ privacy. On the other hand, the requester designs mechanisms or protocols that discourage workers from leaking sensitive data of the tasks; and the workers are responsible for following the privacy agreements of tasks. The platform serves as a mediator that protects the privacy of both parties. Both the task requester and the users must understand that there are always tradeoffs between privacy and interests (e.g., incentives, task quality, funds). Both entities must sacrifice part of their privacy to enjoy a quality crowdsourcing practice. For example, in the crowdfunding practice, a reliable platform protects the privacy from both the users’ and requesters’ perspective, which increases the trust between both parties and further increases the chance of successfulness of the crowdfunding campaign [@wen2018information; @liang2018why].
While most of the works that are surveyed in this section focus on cyber technology development on the platform for protecting the privacy of both the task requesters and workers, some policy/regulation works are mentioned as supplementary materials. Although the business models of these three crowdsourcing practice branches are different, raising the privacy protection level is always helpful to both the workers and task requesters in achieving their goals.
In general, on a crowdsourcing platform, users should be allowed to retrieve information from the database of a sharing service provider while the queries are maintained privately. In addition, to increase the security level of data protection for users, data de-identification methods are available in most cases [@graham1977dynamic; @uzuner2007evaluating; @gilbert2001identification; @el2009globally]. Traditional methods, such as $k-$anonymity, $l-$diversity models, etc., can also be used to avoid linkage attacks [@samarati2001protecting; @samarati1998generalizing; @bayardo2005data].
Privacy Issues and Solutions in Collaborative Consumption Practices {#sec:collaborative}
===================================================================
Unlike crowdsourcing-based sharing services, which combine the power of a large group of individuals to perform tasks, collaborative consumption allows individuals to access goods or services through P2P sharing that is coordinated by online platforms [@belk2014you; @botsman2011s]. In collaborative consumption practices, hosts provide shared goods or services through a collaborative consumption platform to customers. The sharing methods can be selling, borrowing, trading and sharing. Typical examples of collaborative consumption platforms include eBay (collaborative Online shopping), Uber (ridesharing) and Airbnb (homesharing) (Figure \[fig:collaborate\]). Collaborative consumption has many benefits, such as greenhouse gas emissions reduction, cost saving, unaffordable goods access and decentralization [@hamari2015sharing; @botsman2011s]. Although collaborative consumption has many advantages, it suffers from privacy concerns. In this section, we review problems and solutions related to privacy issues in collaborative consumption.
{width="4.5in"}
Collaborative Online Shopping
-----------------------------
Online shopping is probably the first successful model in which cyber technology has changed our living world. In the first stage of online shopping development, people found that it was more convenient and economical to purchase goods over Internet. In the process of cyber technology development, the concept of collaborative consumption was gradually embedded into the online shopping experience. People started to sell small items, trade services, share cars and borrow items through online shopping websites [@botsman2011s].
On the other hand, online shopping websites have received many criticisms due to their notorious privacy policies despite their popularity [@wright2001controlling; @light2013sure; @bowie2006privacy; @valentine2000privacy]. Although it is illegal to reveal user information to third parties without user consent, online platforms are not subject to a penalty for analyzing user data. These platforms rely on third-party organizations for data analysis, which deteriorate customers’ privacy. The privacy policy terms are supposed to be accepted by customers without negotiations, which are in some sense unfair to the customers. Except for limited government regulations, these marketplaces are self-regulated or autonomous, which makes it difficult to protect consumer’s privacy. Moreover, these platforms suffer from data leakage due to cyber attacks or intrusion. These factors contribute to the vulnerability of consumers’ privacy.
Miyazaki and Fernandez [@miyazaki2001consumer] surveyed about online shopping fears on a set of U.S. Internet users from different age groups, economical classes and educational backgrounds. The survey results indicated that the untrusted security system is the largest fear of the customers. Malhotra [*et al.*]{} [@malhotra2004internet] systematically analyzed Internet users’ information privacy concerns (IUIPCs) through two separate surveys of 742 household respondents. They designed a theoretical framework for studying IUIPCs and proposed a causal model that predicts the reaction of online customers to privacy threats from shopping websites. Tsai [*et al.*]{} [@tsai2011effect] studied how the privacy concerns of customers affected their decisions in the online shopping process. They conducted an experiment to test the shopping decisions that were made by customers after displaying their personal information on the shopping websites. Their results demonstrate the customers’ willingness to pay a premium for extra privacy protection (from a more expensive shopping website). All of the above mentioned works reveal the fact that the privacy concern is the main fear in online shopping experiences. However, these works do not present a deep analysis on how to build privacy protection trust between online shopping websites and customers using regulation policies or cyber technology.
Shiau and Luo [@shiau2012factors] built a research model using partial least squares (PLS) method to analyze the relationship between consumer satisfaction, intention of online group buying and user beliefs (Figure \[fig:shiau\]). The PLS results show that consumer satisfaction highly depends on trust, followed by reciprocity. It is the first work to draw an overall picture of the different factors that affect the online shopping decisions. Moreover, it is also the first work to clearly identify privacy concern as the first priority for online shopping security. Following Shiau and Luo’s work, Bergstr[ö]{}m [@bergstrom2015online] built an analytic system with different groups of people concerning various privacy issues in online shopping experiences. Both the customers and the privacy concerns were partitioned into different dimensions to interpret the links between socialization, Internet experience, trust, politics, and security understanding. Their analysis result clearly indicated that the trust is the major concern of people who worry about the misuse of personal data. Although these research models go one step further than the simple survey results, they still do not provide a clear solution for protecting the customers’ privacy in online collaborative shopping practices.
{width="4.5in"}
Preibusch [*et al.*]{} [@preibusch2016shopping] studied and reported a concrete example of privacy leakage in online shopping practices. They performed online tracking and found that online shopping websites send unnecessary personal information to payment providers, such as Paypal. Therefore, there is an on-going risk for customers who shop online. The most effective method for changing this situation is to facilitate relevant legislation. However, the lack of government regulation of online shopping websites exists globally. Moreover, it remains unclear what rules can be added and how they can be enforced. Although there are existing regulations (Directive 95/46/EC by the European Union [@poullet2006eu] and USA Patriot Act [@kerr2003internet]), existing studies have shown that those regulations are usually ignored due to insufficient government monitoring.
One solution to protect users’ privacy in collaborative online shopping practices is to install third-party privacy protection software in the web browser. Available software on Internet includes the Tor Browser [@macrina2015tor], the Privacy Bird [@vu2016user] and the Platform for Privacy Preferences [@perera2015end]. These third-party software programs or plugins identify untrusted shopping websites and mask personal information for the customers. However, third-party software is usually not formally authorized or registered by the government, which potentially raises other concerns of privacy leakage.
Lee [*et al.*]{} [@lee2013pibox] proposed a $\pi$-box mobile app to control the sensitive data transmission between different users and from users to service providers. The $\pi$-box extends the user apps and was built based on the cloud services that were supplied by large companies, such as Google. Two separate channels were designed: the sharing channel, which controls the data transmission between users and the aggregate channel; and the aggregate channel, which controls the data transmission from users to the service provider. The structure of $\pi$-box is illustrated in Figure \[fig:pibox\]. All channels are internally monitored by a centralized system. The limitation of the proposed $\pi$-box is that it does not universally apply to any app in market. According to a user survey conducted by Lee [*et al.*]{} [@lee2013pibox], only 48% of paid apps support $\pi$-box, which limits its usage on privacy protection.
{width="4.5in"}
Kokolakis [@kokolakis2017privacy] studied the conflict between the customer’s high demand for privacy protection and the customer’s willingness to sacrifice privacy for the exchange of goods or services in the online shopping practice. Kokolakis concluded that this inconsistency represents a collision between a customer’s attitude and behaviour, which is known as the privacy paradox [@Norberg2007The]. A large volume of works was surveyed to justify the existence of the privacy paradox; however, most of them are surveys or experimental works that do not involve theoretical model.
Bilge and Polat [@bilge2013comparison] introduced a method for improving the online shopping experience by collecting customers’ personal information, such as ratings and comments for a particular service, in a privacy-preserving manner. A number of clustering methods were integrated into the collaborative filtering service. The system filtered out customized information by training on encrypted user data using clustering methods. The main insufficiency of the work in [@bilge2013comparison] is that, due to the encryption of the users’ data, the recommendation error rates increased. In addition, the clustering methods introduced extra computational costs to the recommendation system.
The reviewed works, which are listed in Table \[table:branch4\], identified two privacy threats in collaborative online shopping practice. The first threat comes from the service provider, where unreliable platforms may misuse customers’ data for marketing analysis. This threat can be prevented by refining government regulations [@preibusch2016shopping], masking customers’ data before sending them out [@bilge2013comparison] or separating communication channels on the cloud server [@lee2013pibox]. Other possible solutions to prevent such malicious behaviours include utilizing trusted computing [@Santos2009Towards] or building services based on a trusted provider [@Paladi2017Providing; @Paladi2014Domain]. The second threat comes from the customer side, where most customers realize that they must sacrifice a certain degree of privacy to enjoy the collaborative shopping experience [@kokolakis2017privacy]. It is difficult for them to choose a trustworthy service provider, products [@bergstrom2015online], and most importantly, the kinds of permissions to grant [@ismail2015crowdsourced]. The second threat can be alleviated by increasing the overall privacy awareness of the users, which will be extensively discussed in Section \[sec:emerging1\].
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
, [Malhotra [*et al.*]{} [@malhotra2004internet]]{}, [Tsai [*et al.*]{} [@tsai2011effect]]{} &2001, 2004, 2011& [Pointing out the biggest fear for online shopping experiences]{} & Surveys on Internet users & Lacking a deep analysis to build the privacy protection trust between the online shopping websites and the customers\
[Shiau and Luo [@shiau2012factors], Bergstr[ö]{}m [@bergstrom2015online]]{} &2012, 2015& [Learning the largest privacy concern in online shopping practices]{} & Drawing the overall online shopping fears relationships by research models & No clear solution for protecting the online customers’ privacy\
[Preibusch [*et al.*]{} [@preibusch2016shopping]]{} &2016& [Pointing out the need of raising government regularization for online shopping globally]{} & A concrete example of the privacy leakage in online shopping practices & What rules to be added and how to add are two big questions\
[Lee [*et al.*]{} [@lee2013pibox]]{} &2013& [Separating the data transmission from user to user and from user to service provider]{} & A mobile app called $\pi$-box & Not supporting all paid apps\
[Kokolakis [@kokolakis2017privacy]]{} &2017& [Revisiting the conflict known as ‘privacy paradox’]{} & A survey covering related existing works & No theoretical model was discussed\
[Bilge and Polat [@bilge2013comparison]]{} &2013& [Protecting privacy in user information collection process for a recommend system]{} & Masking sensitive data and using clustering methods for data analysis & Losing analysis accuracy\
**********
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:branch4\]
Ridesharing
-----------
Real-time ridesharing or dynamic carpooling is a transportation service that allows commuters to share rides on very short notice through mobile apps [@chan2012ridesharing; @agatz2011dynamic; @shen2015managing; @shen2016online; @shen2017regulating]. Successful ridesharing platforms, such as Uber, are available in most major cities in the world. When a user needs a ride, he/she may simply use a mobile app to request a ride by entering the destination. The app provides the estimated cost and assigns a driver to the passenger. The payment is generally made with the credit card or other digital payment methods that are associated with his/her account. In the end, both the passenger and the driver will rate each other.
It is well known that the mobile apps can track the customers’ location information and travel information for better service quality. The driver has to access to the rider’s travel information, such as riders’ names, trip starting points and destinations to provide services. Under current privacy policies, riders have to share part of their private information to receive ridesharing services. The platforms have limited regulatory power over the drivers because the drivers are contractors rather than employees of the ridesharing companies. Moreover, drivers’ names and license plate information are also subject to disclosure. Concerns have been raised about the internal misuse of user data within the ridesharing companies. For instance, staffs in the ridesharing companies have the access to data for tracking the movements of customers. Taking Uber as an example, in its user agreement terms, it is clearly stated that user information, such as the geo-location, is recorded and internally used by the company for research development purposes. However, the purposes of internal research are not defined explicitly. Customers worry about how their private data is used. Additionally, Uber can access, use, preserve, transfer and disclose user information to prevent, discover or investigate violations of the privacy policy or the user agreements as determined necessary or appropriate. However, customers do not know what information is necessary or appropriate.
Location privacy has been studied extensively in recent decades because of the pervasiveness of geo-location related software and mobile apps [@beresford2003location; @mitrokotsa2014location; @shahandashti2015reconciling; @bettini2005protecting; @barkhuus2003location]. While location-aware applications track customers’ location or other data online, they generate a huge amount of potentially sensitive data. The privacy of location data depends on the regulation of data access. It is neither necessary nor possible to forbid all accesses because the systems must access the data for analysis purposes. Moreover, access permissions should be given to authorized persons and should never be exposed to others. In other words, the data and the access should be tightly controlled and data should be accessed only with legal authorization [@beresford2003location].
Kido [*et al.*]{} [@kido2005protection] proposed one of the first techniques for concealing the actual locations of customers in location-based services, including ridesharing practices. When a user sends an inquiry to the server, he/she sends his/her actual location, together with two false positions called ‘dummies.’ The dummy nodes in the tracking system are carefully generated such that an observer cannot easily identify the actual location of the user; however, the location-based server (LBS) can find the difference through optimized algorithms with external information such as road navigation service (RNS) data. The obvious shortcoming of Kido [*et al.*]{}’s work is that the real location is not completely concealed (by using dummies). There is still a chance that the observer will identify the actual location.
Yao [*et al.*]{} [@yao2010clustering] provided an effective encryption service for ridesharing customers using the clustering $k$-anonymity (CK) scheme [@samarati2001protecting]. The CK scheme encrypts the user location information by utilizing a cloaked spatial-temporal boundary (CSTB) that involves $K$ users. The spatial and temporal constraints, which determine the resolution of the encryption, can be customized by users. However, the use of CSTB decreases location information resolution, and consequently, degrades the service quality of ridesharing.
Pan and Meng [@pan2013preserving] extended Yao [*et al.*]{}’s work using a $p$-anti-conspiration model for location privacy protection. Various techniques were introduced, including methods that provide LBS without knowing the actual locations of the customers. It is a large advancement for the ridesharing companies in protecting the user locations. A follow-up work done by the same group of authors in [@pan2016protecting] showed that the approach proposed in [@pan2013preserving] lacks protection on sensitive information during the data transmission process.
Jagwani and Kaushik [@jagwani2012defending] intended to prevent location information leaks using the concept of Zero knowledge proof (ZKP). The construction process of the authentication scheme based on ZKP was introduced; and the possible applications of ZKP in the location-based service domain were discussed. The main shortcoming of the ZKP approach is that an authentication scheme is always required to coordinate between customers and hosts.
Gao [*et al.*]{} [@gao2013trpf] introduced trajectory privacy in the ridesharing practices. The trajectory privacy contains spatial-temporal information, which is an important addition to the location privacy protection scheme. In their study, they proposed a mixed-zone graph model to protect the trajectory privacy. The actual implementation relies on a third party middleware, where the actual location information leakage exists.
In recent years, online social networks or geosocial information have started to be used in ridesharing services. It is preferable to use a friend’s car rather than stranger’s. Based on this motivation, Elbery [*et al.*]{} [@elbery2013carpooling] proposed a social Vehicular Ad-Hoc Network (S-VANET) carpooling recommendation system. They embedded friendship locality, preference locality, and travel locality information into the ridesharing recommendation system, which requires a large amount of privacy information from both the requester and his/her friends.
Ni [*et al.*]{} [@ni2016ama] suggested that customers’ true identities can be hidden by incorporating an anonymous mutual authentication (AMA) protocol into the carpooling recommendation system. A real-time navigation system is proposed for concealing the drivers’ privacy [@ni2016privacy]. One important feature of their application system is the false information traceability, where the trusted third party authority can trace incorrect information, either from a user or a driver. The main limitation of their work is that a trusted third-party middleware is still required.
A[ï]{}vodji [*et al.*]{} [@aivodji2016meeting] proposed a privacy-preserving local computational method for determining the meeting point of a driver and a rider in a ridesharing system, which does not require third-party middleware. Multimodal routing algorithms are used to compute a mutually interested meeting point for both the driver and rider. However, the current developed system was only designed to accommodate one driver and one rider. A more sophisticated system that can include multiple drivers and riders for ridesharing practices is left as a future work.
Shokri [*et al.*]{} [@shokri2016privacy] concluded that the current location privacy protection approaches can be concluded on three trends, which are perturbing the actual location, tracing the perturbed location, and evaluating the privacy-preserving methods. While most existing works only focus on encrypting the customer’s current location, strategies were employed by attackers to trace down the actual location of the customer. Useful private information pieces, such as recently visited locations, frequently visited places and nearby landmark buildings, become potential clues for the attackers in estimating the current location of the customer. In [@shokri2016privacy], a comprehensive Bayesian security game is designed to simulate various cases in which a strategic attacker traces the actual location of a customer. Four different scenarios were studied. However, it was difficult to predict the intelligence level of the attacker; and the whole simulation system is too complex in most of the real-world scenarios.
Vergara-Laurens [*et al.*]{} [@vergara2017privacy] categorized privacy preserving systems into approaches for two processes: the tasking process, where tasking devices (such as mobile phones) collect data in certain areas; and the reporting process, where distributed devices report sensed data to the platform. Both processes exist in ridesharing practices. Three open problems were raised for crowdsensing (CS) researchers in the field of location privacy preservation, which are 1) privacy-preserving mechanisms for tasking processing, 2) privacy-preserving mechanisms for reporting process and 3) selecting the most appropriate privacy-preserving mechanism.
Wang [*et al.*]{} [@wang2018truthful] proposed a two-stage auction algorithm taking both trust degree and privacy sensibility into consideration for mobile crowdsourcing systems, such as ride-sharing practices. The $k$-anonymity scheme is integrated with $\epsilon$-differential scheme to add Gaussian white noise to the actual locations of users. The proposed scheme was proven to be trustful and can inspire more users to participate in the mobile croudsourcing systems. Insufficiency exists while the added Gaussian white noise increases the computational complexity and consequently weakens the service quality for mobile crowdsourcing systems.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
&2005& [Protecting location privacy using dummies]{} & An anonymous communication technique & The actual location is not completely concealed\
[Yao [*et al.*]{} [@yao2010clustering]]{} &2010& [Encrypting the user location information]{} & Clustering K-anonymity (CK) scheme & Decreasing location information resolution and degrading the QoS\
[Pan and Meng [@pan2013preserving]]{} &2013& [Providing location-based services without knowing the exact location]{} & The $p$-anti-conspiration privacy model & lacking protecting sensitive information\
[Jagwani and Kaushik [@jagwani2012defending]]{} &2012& [Removing the dependency of using third party software]{} & Zero knowledge proof & An authentication scheme is required\
[Gao [*et al.*]{} [@gao2013trpf]]{} &2013& [Protecting the trajectory privacy]{} & A trajectory privacy-preserving framework & The exact location must be revealed to a third party middleware\
[Ni [*et al.*]{} [@ni2016ama; @ni2016privacy]]{} &2016& [Concealing both customers and drivers’ sensitive information]{} & An anonymous mutual authentication (AMA) protocol & A trusted third party middleware is required\
[A[ï]{}vodji [*et al.*]{} [@aivodji2016meeting]]{} &2016& [Computing the mutually interested meeting point]{} & Multimodal routing algorithms & A more complicated system involving multiple drivers and riders are left for future exploration\
[Shokri [*et al.*]{} [@shokri2016privacy]]{} &2016& [Considering strategic attackers for customer’s location privacy]{} & A comprehensive Bayesian security game & The complexity is not necessary for most of the real-world scenarios\
[Vergara-Laurens [*et al.*]{} [@vergara2017privacy]]{} &2017& [Surveying privacy-preserving mechanisms]{} & A survey of all existing works on location privacy preservation & Only location privacy is heavily surveyed\
[Wang [*et al.*]{} [@wang2018truthful]]{} &2018& [Inspiring more users to participate in the mobile croudsourcing systems]{} & Integrating $k$-anonymity scheme with $\epsilon$-differential scheme & The addition of Gaussian white noise weakens the crowdsourcing service quality\
**********
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:branch5\]
All reviewed papers are summarized in Table \[table:branch5\]. Similar to other sharing services, customers realize that a certain degree of their privacy must be sacrificed to enjoy better service quality. Taking the Uber service as an example, the platform (Uber app) usually records the customers’ private information, including current location, destination, phone number, recent trips and so on, to serve them better. However, the customers sacrifice their privacy to enjoy the Uber service. The conflict between the disclosure of private information and the service quality becomes more obvious in the ridesharing practices, which is also mentioned in most of the surveyed works, such as [@gao2013trpf; @ni2016privacy; @aivodji2016meeting; @shokri2016privacy].
Compared to other fields of sharing service, ridesharing is a relatively new technology. Few regulations have been established in this area; and most privacy concern solutions are on technical aspect. Despises the variety of technologies proposed by the existing works, only location privacy is extensively discussed. Ridesharing services include direct interpersonal interactions (IPIs), e.g., the conversation between the rider and the driver when they are travelling [@Vertommen1980The; @Andersen2002The; @cho2011interpersonal]. Computerized technologies, which are designed to be embedded in the online platform, can be helpless in IPI; and physical privacy concerns exist at this stage [@christin2016privacy]. Physical privacy concerns, which were first defined by Belk, occur when the driver or passage’s personal space is invaded, where we refer to the remaining privacy concerns as online privacy concerns [@belk1988possessions; @belk2010sharing]. For future works in this field, we would like to note that physical privacy protections for both the riders and drivers are demanded in the ridesharing practice.
Homesharing
-----------
Homesharing is a business model that connects hosts and travellers through an online marketplace platform and enables transactions without the platform owning any rooms itself. It does not provide the rental services directly. Instead, it matches hosts who have extra rooms for rent and travellers who need a room for stay [@panda2015emergence; @gaikar2013first]. One of the most famous homesharing platforms is Airbnb [@jefferson2014airbnb].
The face-to-face e-commerce model makes the physical privacy issue more serious for homesharing practices compared with online sharing model. The host and traveller usually meet each other before a deal was made and both of them have the possibility to reveal the privacy of each other to the public. For example, a host may install a hidden camera in an Airbnb room to monitor travellers. A traveller may take pictures to reveal the details of the room or other parts of the house to the public. The online platform records sensitive information of both the hosts (e.g., names, travel plans) and the travellers (e.g., names, home locations).
Kamal [@kamal2016trust] realized that the largest inhibitor of homesharing services is the fear of privacy disclosure. They argued that additional background checks are always necessary for participants in homesharing activities, with the possibility of additional security measures, such as certificates and safety insurance. However, we would like to point out that the cost comparison between the additional security checks and the actual accommodation is not discussed in [@kamal2016trust].
Morosan and DeFranco [@morosan2015disclosing] determined the level of willingness of travellers to disclose their personal information to hotel apps. An extended version of the privacy calculus model was adopted. The experimental results indicated that personal information disclosure was indeed helpful for the hotel business, i.e., to choose the best customers. But the willingness of disclosing such information was related to privacy concerns, trust, emotions and etc. The main insufficiency of this work is that the study data was collected from U.S. customers who were involved in a relatively safe environment with reliable network security, regulations and hotels. The experimental results may not be applicable to third-world countries.
Ert [*et al.*]{} [@ert2015trust] designed an experiment that used mixed-logit analysis to determine the relationship between the posting of a host’s photo in the advertisement and the booking likelihood. The results show that both the trustworthiness and attractiveness of the host’s photo increase the likelihood of the house being booked. Similar to crowdfunding and crowdtesting, an appropriate degree of private information disclosure from the hosts’s side increases the probability of success for the entire practice/business. However, on the other hand, the leakage of the hosts’ privacy, including posting of the host’s photo and identity information, is another issue in homesharing practices, which is deeply discussed by Hooshmand [@hooshmand2015risks].
Lutz [*et al.*]{} [@lutz2017role] explicitly divided the privacy concerns into physical privacy concerns (e.g., physical damages of private assets) and online privacy concerns (e.g., personal identity leakage). They conducted a survey on MTurk involving 389 participants; and most of them were hosts on Airbnb. The survey results showed that physical privacy concerns are more crucial than online privacy concerns in the homesharing business. The main shortcoming of their work is that the survey is limited to Airbnb hosts and does not include any customers. Thus, the survey results may be biased towards the hosts’ preferences.
We list all reviewed works for security concerns of homesharing in Table \[table:branch6\]. Similar to crowdsourcing practices, certain degrees of private information disclosure from the hosts side positively influence the trust from the customers side and consequently attract more customers. Moreover, compared to other sharing services, homesharing involves more interpersonal interactions. Concerns about physical privacy are heavily studied in this field. Most of our surveyed works agreed that the hosts are more concerned about their privacy leakage than the travellers. Future studies can focus more on the development of privacy protection schemes for hosts. In the current stage of homesharing, while it is unlikely to solve the privacy issue with a single method, it is quite possible to provide a general privacy-preserving environment for both hosts and travellers through the joint efforts of hosts, travellers, platforms, and governments.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
&2016& [Building trust in homesharing practices]{} & Proposing additional security checks & Not discussing the cost of additional security measurements\
[Morosan and DeFranco [@morosan2015disclosing]]{} &2015& [Checking the willingness of hotel customers to disclose personal information]{} & An extended version of the privacy calculus model & Collecting data only based on U.S. customers\
[Ert [*et al.*]{} [@ert2015trust]]{} &2015& [Showing the relationship of posting a host’s photo and the booking likelihood]{} & Mixed-logit analysis & The relationship between trust and privacy is of interest but not discussed\
[Lutz [*et al.*]{} [@lutz2017role]]{} &2017& [Investigating the impact of physical privacy concerns to homesharing]{} & A survey on MTurk & Only hosts were surveyed\
**********
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:branch6\]
Summary and Discussion
----------------------
Collaborative consumption collects extra information resources and distributes them to people who do not have access to them. Users are subject to privacy leakage due to the exchange of data and improper use of user data by internal staffs or the platforms. Similar to crowdsourcing practice, both hosts and customers must understand that certain degrees of their privacy have to be sacrificed for better service quality. The most appropriate degrees for private information disclosure from both hosts and customers side are left as open problems to maximize the service quality and net profit. While it is difficult to provide an absolute privacy-safe environment without sacrificing service quality, it is possible to increase the protection levels of privacy through a joint effort of all participants, platforms and governments [@bergstrom2015online; @preibusch2016shopping; @lee2013pibox; @kokolakis2017privacy; @bilge2013comparison].
Compared with collaborative online shopping, both ridesharing and homesharing involve more interpersonal interactions (IPIs). For ridesharing, location privacy is separated from the general concept of privacy and is extensively studied and discussed. For homesharing, the general form of privacy is further divided into online privacy (electronic forms of personal information) and physical privacy (human body, house, furniture, etc.) [@christin2016privacy]. There are works showing that the physical privacy concerns are more important than online privacy concerns for homesharing [@belk2010sharing; @lutz2017role]. We believe that the concept of physical privacy will be considered in privacy protection studies in other areas in near future, such as ridesharing.
It is noted that there are other methods available for privacy preservation in collaborative consumption practices in the early years. Milberg [*et al.*]{} [@milberg1995values] studied various aspects that affected the customers’ willingness to participate in collaborative consumption in the early 1990s. The study shows some early efforts and results from governments in designing suitable regulations for protecting the customers’ privacy. Luo [*et al.*]{} [@luo2002trust] examined several mechanisms to demonstrate the close relationship between trust and privacy preservation. Nissenbaum [@nissenbaum2009privacy] discussed privacy from the perspective of contextual integrity in technology, policy, and social life.
Summarizing Emerging Privacy Issues from the User, Platform and Service Provider Perspectives {#sec:emerging}
=============================================================================================
Fast development of cyber technology facilitates the invention of novel sharing practices in the cyber-enabled world. While traditional privacy problems have either been solved or at least realized by the government and society, privacy issues in cyber-enabled sharing services are less understood. In all six branches of the taxonomy in Figure \[fig:overall2\], there are always interactions between users, platforms, and service providers.
In this section, all existing privacy issues are further discussed from three perspectives, namely, users’, platforms’ and service providers’ perspectives. we argue that all privacy issues from different applications are internally related. Users concern with their own privacy and always demand high quality reliable sharing services. Service providers realize that privacy security level is a key element towards a successful achievement. Anybody involved in the sharing service can be a potential attacker to compromise other people’s privacy. The linkages between the privacy concerns from the three perspectives are shown in Figure \[fig:threePers\]. The concluded emerging issues of the cyber-enabled sharing services are: increasing users’ privacy awareness from their perspective, protecting shared information from the platforms’ perspective and making privacy concerns the top priority from the service providers’ perspective. Works that are surveyed in this section are listed in Figure \[fig:overall3\] and summarized from the three perspectives.
{width="5.5in"}
{width="6.5in"}
From Users’ Perspective: Increasing Privacy Awareness {#sec:emerging1}
-----------------------------------------------------
Although most websites, software and mobile apps provide user agreements for user privacy awareness, only a negligible portion of users read through the tedious clauses carefully. The first emerging privacy issue for cyber-enabled sharing services is to maximize users’ awareness of privacy leakage, e.g., to provide an online tool for users to trace down entities that may reveal their personal information. The transparent information tracing system will increase the confidence of users in participating in sharing practices on Internet, as well as facilitating the service providers to improve their reputations.
For example, in the crowdsourcing marketplace, it is not sufficient to protect only requesters’ data privacy because workers also value their privacy equally. Workers are commonly afraid of the leakage of their location data or the identity information (e.g., age, contact, hobbies, activities) [@wang2013respecting; @to2014framework]. According to a survey that was performed by the U.S. Federal Trade Commission [@federalprotecting], more than $85\%$ of users were too impatient to read the user agreements regarding privacy settings carefully. They were surprised that mobile phone apps sent their approximate or precise location, phone’s unique ID to service providers. Some apps even have control of the camera flashlight and audio settings. Although these privileges were authorized by users, they did not know when or where they give the authorizations, because they never read the articles about the privacy settings. Some efforts have been made to solve the above problem.
Malandrino [*et al.*]{} [@malandrino2011supportive; @malandrino2013privacy] proposed a privacy awareness software named ‘NoTrace’ to provide privacy recommendations to the users, such as privacy protection level settings, private information transmission warnings and unnoticeable privacy leaks warnings. The graphical user interface of ‘NoTrace’ clearly displays the private information pieces that are received by the service provider. The main shortcoming of Malandrino [*et al.*]{}’s work is that they did not provide a deep analysis of which private information pieces are necessary for the service quality and therefore, could not provide proper recommendations on selective disclosure of personal information for users.
Omoronyia [*et al.*]{} [@omoronyia2013engineering] proposed an adaptive privacy framework to assist automatic privacy disclosure decision making for various applications. The framework is designed following the famous MAPE (Monitor, Analyse, Plan and Execute) loop, and is focused on three aspects: application attributes, potential privacy threats and derived benefits from privacy disclosure. One important insufficiency of their work is that it does not categorize privacy protection requirements according to different service functions, which makes the automatic privacy disclosure decision making relatively unreliable [@meis2016computer].
Amini [*et al.*]{} [@amini2014analyzing; @amini2013mobile] developed a software called ‘AppScanner’ to help users better understand the functionalities of mobile applications. The software provides an informative description of what mobile apps are actually doing under a crowdsourcing environment. The transparency and detailed analysis of the mobile apps help make users aware of privacy leakage when using mobile apps for crowdsourcing. AppScanner only categorizes the mobile app behaviors as normal or abnormal. A detailed categorization according to the behaviors purposes, e.g., advertising and social networks, can be used to enhance the decision making ability for users [@wang2015using].
Zhu [*et al.*]{} [@zhu2014mobile] implemented a mobile app recommendation system with security and privacy awareness. The proposed system first analyzes the mobile application with detection and diagnosis of the security risks from insecure data access permissions. The recommendation system then provides suggestions to the user on whether to continue using the mobile app according to the app’s popularity and user settings. The recommendation is based on modern portfolio theory. The main insufficiency of Zhu [*et al.*]{}’s work is that the security risks are only evaluated based on the permissions that the mobile apps request.
Hartmann [*et al.*]{} [@hartmann2016privacy] summarized six main threats of mobile apps to make the users aware of potential privacy risks: insufficient control features, excessive data mining, data theft, surveillance, information leakage and social engineering. They also proposed eight recommendations for guarding against these privacy threats: privacy dashboard, privacy policy, data handling guidelines, user permissions, anonymization, IT infrastructure security, encryption, and relationship. All the guidelines are valuable for future privacy-aware mobile application development. However, most importantly, immediate solutions for all conflicts are missing from both regulation and cyber technology perspectives.
Chandramohan [*et al.*]{} [@chandramohan2016new] concluded that over 90% of users accept user agreements unconsciously, without knowing that their personal information can be misused. They described a complete privacy-preserving scheme called Petri-net Privacy-Preserving framework that was installed on a cloud server. However, the practicability and the scalability of their algorithm are still questionable.
Similar to traditional websites that force users to accept user agreements, the mobile apps mitigate the privacy risks to the users by requesting resource access permissions. Quay-de la Vallee [*et al.*]{} [@quay2016per] developed two app systems that help users find privacy-respective apps and manage the apps’ permissions in their mobile phones. The main shortcoming of Quay-de la Vallee [*et al.*]{}’s work is that the two systems only provide privacy management assistance after the apps have been installed, instead of providing the assistance during the installations process.
Ismail [*et al.*]{} [@ismail2015crowdsourced] studied the privacy threats from mobile apps that require access to sensitive resources during the processes of installation or updating. A crowdsourcing strategy that identifies the minimal number of permissions to keep the mobile apps fully functioning for a diverse range of users was proposed. A user study that involved 26 participants and the popular mobile app ‘Instagram’ showed the effectiveness of their approach. However, the survey size was relatively small; and the method was only tested on a single mobile app. The usability of the proposed crowdsourcing strategy requires further justification.
Zhou [*et al.*]{} [@zhou2017control] accessed the gap between users’ desire of privacy control and the actual privacy setting functions provided by mobile app systems. Through a simple lab survey consisting of 26 users, three important facts had been concluded: 1) personal privacy protection is still an important factor that influences the users to choose their smartphones; 2) although smartphone nowadays provides more functions protecting user privacy through complex user interface, people are not well adapted to those new functions; and 3) Sorting methods, as well as recommendation systems are still useful to assist users to protect their private data. The shortcomings of Zhou [*et al.*]{}’s study is that the number of participated user is relatively small. Moreover, there’s no specific solution has been proposed to increase the users’ awareness of privacy protection.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
&2011, 2013& [Measuring revealed data by service provider and privacy leakage to third-party websites]{} & ‘NoTrace’ software & Lacking of analysis on necessary information disclosure for a known service\
[Omoronyia [*et al.*]{} [@omoronyia2013engineering]]{} &2013& [Assisting privacy disclosure decisions made by applications]{} & An adaptive privacy framework & Lacking systematical privacy requirements listing for a given set of service functions\
[Amini [*et al.*]{} [@amini2014analyzing; @amini2013mobile]]{} &2013, 2014& [Helping users better understand the functionality of mobile applications]{} & AppScanner & A detailed categorization according to the behaviors purposes can be more helpful\
[Zhu [*et al.*]{} [@zhu2014mobile]]{} &2014& [Recommending mobile apps to users with security and privacy awareness]{} & A mobile app recommendation system &The security risks are only evaluated based on the permissions that the apps request\
[Hartmann [*et al.*]{} [@hartmann2016privacy]]{} &2016& [Addressing threats of mobile apps and proposing solutions]{} & Eight recommendations for six main threats & No immediate solution is provided\
[Chandramohan [*et al.*]{} [@chandramohan2016new]]{} &2016& Protecting user privacy on cloud & Petri-net Privacy-Preserving Framework & The practicability and real-time applicability of their algorithm need further discussion\
[Quay-de la Vallee [*et al.*]{} [@quay2016per]]{} &2016& Managing apps’s access permissions & Two management apps & The privacy management assistance was only provided after the apps been installed\
[Ismail [*et al.*]{} [@ismail2015crowdsourced]]{} &2015& Identifying the minimal number of permissions to keep the mobile apps fully functioning & A crowdsourcing strategy & The proposed strategy is only tested on one single mobile app\
[Zhou [*et al.*]{} [@zhou2017control]]{} &2017& accessed the gap between users’ desire of privacy control and the actual privacy setting functions provided by mobile app systems & A simple lab survey consisting of 26 users & No specific solution was proposed\
**********
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:em1\]
In summary, all the above mentioned works, which we list in Table \[table:em1\], suggest that privacy leakage on some level is unavoidable for users to enjoy the sharing service. However, users’ awareness of privacy leakage can be improved by listing threats from third-party websites/applications [@malandrino2011supportive; @malandrino2013privacy; @amini2014analyzing; @amini2013mobile; @hartmann2016privacy], recommending safe decisions to users [@omoronyia2013engineering; @zhu2014mobile] and using cyber-technologies [@chandramohan2016new; @quay2016per; @ismail2015crowdsourced]. Although various techniques are proposed to raise the users’ awareness level, most sharing service platforms only provide user agreement terms to warn about possible privacy leakage. There is still a large gap between forcing users to agree to terms, granting access permissions to sensitive data and motivating users to actively protect their own privacy. Platform and service providers should be encouraged to use the existing cyber-technology to maximize users’ awareness of privacy issues. Future works and surveys can be conducted in this direction.
{width="6in"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
&2005, 2011, 2013& [Providing efficient and secure classifier using cloud computing technology with privacy preserved]{} & A random space encryption (RASP) approach & Updating the encrypted database is not an easy task\
[Hong [*et al.*]{} [@hong2013survey]]{} &2013& [Preserving privacy under distributed environment]{} & Surveying existing privacy protection strategies & Mainly focusing on time-series data mining\
[Dong [*et al.*]{} [@dong2014achieving; @dong2015seco]]{} &2014, 2015& [Suggesting a privacy-preserving data security policy]{} & A series of encryption techniques & Resulting in key escrow problems\
Han [*et al.*]{} [@han2016security] & 2016 & privacy-preserved data outsourcing under cloud environment & ABE based privacy protected data access control scheme & Requiring efficiency improvements\
[Le [*et al.*]{} [@le2014consistency]]{} &2014& [Ensuring the enforceability for multi-access to stored data in cloud servers]{} & An inconsistency checking and removing algorithm & Requiring pre-defined rule regulations\
[Wang [*et al.*]{} [@wang2011hierarchical; @wang2013secure; @wang2014security; @liu2014time]]{} &2011, 2013, 2014& [Keeping the shared data confidential against untrusted cloud service providers]{} & The hierarchical attribute-based encryption scheme & lacking user revocation and was restricted by the same domain condition\
[Rahman [*et al.*]{} [@rahman2015survey]]{} &2015& [Protecting shared data on cloud]{} & An information protection model combining incident handling strategy and digital forensics principles & The surveyed works were only up to the year 2014\
**********
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:em2\]
From the Platforms’ Perspective: Protecting Shared Information
--------------------------------------------------------------
Although users can agree to share part of their personal information on the intermediate platform, the shared information/data still faces various potential attacks without proper regulation protocol setups or cyber technology implementations. Data analysis for different purposes exists in almost all third-party platforms [@Reddy2018Comparative]. The main purpose of data analysis is to achieve better service quality. However, privacy concerns make users reluctant to share sensitive information. In this section, several recent existing works for privacy protection from the platforms’ perspective are surveyed.
Chen [*et al.*]{} [@chen2005privacy; @chen2011rasp; @chen2013perturboost] presented a random space encryption (RASP) scheme that produces secure privacy protection on the cloud. RASP provides service to transfer the analyzing data into an encrypted space with a two-stage encoding algorithm. The way of updating the encrypted database is another important challenge for their work.
Hong [*et al.*]{} [@hong2013survey] surveyed several existing privacy protection strategies under the distributed data sharing environment. The proposed privacy protection techniques were simultaneously applied to the database, queries or aggregation. The main insufficiency of their work is that they only focused on privacy-preserving schemes for time series data processing.
Dong [*et al.*]{} [@dong2014achieving; @dong2015seco] suggested a security policy based on existing encryption techniques. The proposed framework allows the users to dynamically access their own personal data freely. Both attribute based encryption (ABE) and identity based encryption(IBE) were used to minimize the key management overhead; however, the proposed method resulted in key escrow problems [@sajid2016data].
Following Dong [*et al.*]{}’s work, Han [*et al.*]{} [@han2016security] provided a promising solution for privacy-preserved data outsourcing under the cloud environment. They proposed an attribute-based encryption (ABE) based control scheme on two major problems for data accessing privacy protection on the cloud. However, the time complexities of both the encryption and decryption processes in the proposed method were not optimized for real-world applications.
Le [*et al.*]{} [@le2014consistency] assumed that there were pre-defined rule regulations in the data processing scenarios. An inconsistency checking and removing algorithm was designed to ensure the enforceability for multi-access to stored data in cloud servers. The main concern of their work is that the pre-defined regulations can be not applicable under extreme conditions or worst case scenarios.
Wang [*et al.*]{} [@wang2011hierarchical; @wang2013secure; @wang2014security; @liu2014time] proposed a hierarchical encryption scheme to maintain access controls for different levels of users (Figure \[fig:habe\]). Each domain master generates keys to a specific group of users in the next sub-level. In addition, they also proposed a scalable revocation scheme for users to access their own personal data. The proposed scheme lacked user revocation and was only applicable to the situation that all attributes were administered by the same domain authority.
Rahman [*et al.*]{} [@rahman2015survey] reviewed 139 works from 2009 to 2014 regarding information security in cloud computing. The cyber technology of incident handling strategy (IHS) is heavily discussed, which is an important tool for protecting data in a shared cloud service system. They pointed out that although IHS setup is straightforward on a personal computer, it becomes complicated when cloud computing allows multiple computers to access the same data on the same hard-disk. The main insufficiency of their work is that the survey was done in 2014 and only covered IHS techniques proposed before that year.
In summary, a list of the surveyed works can be found in Table \[table:em2\]. From the platforms’ point of view, there are mainly two parts of the data sharing practice can be worked on to provide more secure sharing services: the data transmission process and the data storage on the cloud server. To protect sensitive data during the data transmission process, data encryption is usually utilized [@dong2014achieving; @dong2015seco]. For data protection on the cloud server, encryption scheme [@chen2005privacy; @chen2011rasp; @chen2013perturboost], a hierarchical data-accessing scheme [@han2016security; @le2014consistency; @wang2011hierarchical], and other cyber technologies [@rahman2015survey] were used. We believe that establishing an effective protocol in the platform is beneficial for both users and service providers. Although data analysis is necessary for service quality improvement, the part of the user data that must be revealed to the analyzer to obtain the full functionality of the sharing service remains questionable.
{width="5in"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[Reference]{} & **[Year]{} & **[Main objective]{} & **[Proposed solution]{} & **[Important insufficiency]{}\
& 2014 & [determining the relation between different service quality aspects (including privacy protection) and the final profit]{} & Identifying the four most important aspects for service quality enhancement & The survey results are only limited to a single country (i.e. Thailand)\
Hartono [*et al.*]{} [@hartono2014measuring] & 2014 & Identifying the most important dimensions of perceived security for online shopping & A second-order structural model on perceived security & Only responses from Korea are used\
Ingham [*et al.*]{} [@ingham2015shopping] & 2015 & Examining the internal relationship between trust, perceived risks, and customers’ acceptance & The technology acceptance model (TAM) nomological network & Lacking ways to gain the customers’ trusts\
Wang and Lin [@Wang2016Perceived] & 2016 & Studying the internal linking of service quality and intention of continuous usage of location-based services & A research conceptual framework & The survey was only conducted in Taiwan\
**********
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
\[table:em3\]
From the Service Providers’ Perspective: Making Privacy the Top Priority
------------------------------------------------------------------------
As the last but important participant, the service provider has to learn the importance of protecting user privacy. Numerous studies have shown that privacy protection/security level is an important component of the overall service quality, and therefore influences the final profit of the company [@cases2010web; @zhao2013designing; @thaichon2014development]. More specifically, the enhancement of privacy protection quality by the service provider potentially attracts more customers to pay for the service [@carlos2009importance]. Service providers must give the privacy protection issue the highest priority in a successful business model.
Thaichon [*et al.*]{} [@thaichon2014development] surveyed the relationships between various aspects of service quality and the perceived value by customers. They identified the four most important service quality dimensions that influence the final profit of the company, which include privacy concerns. The limitation of their work is that the survey is conducted in the context of a single country (Thailand).
Hartono [*et al.*]{} [@hartono2014measuring] further identified the most important dimensions of perceived security for online purchases as confidentiality, integrity, availability, and non-repudiation. They validated that these four aspects significantly impact the customer’s willingness to participate e-commerce services by using a second-order structural model of perceived security. In their experiment, only responses from Korea were used, which reduces the generalization of the study results.
Ingham [*et al.*]{} [@ingham2015shopping] examined the internal relationships among trust, perceived risks and customers’ acceptance in e-shopping practices. The technology acceptance model (TAM) nomological network is deeply discussed to measure the values in a different dimensions. The testing results are analyzed by the meta-analytical path approach. This was a comprehensive survey paper that searched for potential ways to promote e-commerce to achieve better sales. However, regulation or cyber technology solutions for enhancing the trusts gained from the customers are missing.
Wang and Lin [@Wang2016Perceived] established a conceptual research framework for studying the internal links between service quality and user experience of location-based services (LBS) (Figure \[fig:wangLin\]). Based on a survey with 1399 participants, Wang and Lin identified positive and negative influences between factors, such as service quality and privacy trust in using LBS. Cultural bias exists in their results since the survey was conducted only in Taiwan.
All surveyed works from the service providers’ perspective are listed in Table \[table:em3\]. The internal relationship between the privacy protection and the net profit is heavily studied. The privacy protection level is an essential component in service quality evaluation and significantly impacts the customers’ willingness to participate, customers’ trust and net profit. And certain degrees of privacy disclosure from the service providers’ side can also increase the willingness of the customers to trust the sharing services. In conclusion, it is important for the service providers to consider privacy issues the top priority of their commercial strategies, provide a more secure servicing environment and build more successful business models.
Open Research Issues {#sec:openIssues}
====================
In the first part of this study, the cyber-enable sharing services are divided into six branches, which are crowdsourcing marketplace, crowdfunding, crowdtesting, collaborative online shopping, ridesharing and homesharing. In this section, we summarize the main open research issues from the above six branches and list them as follows:
1. [**Improving task performance quality and efficiency with privacy-preserving protocols (crowdsourcing marketplace).**]{} For Internet crowdsourcing marketplace, existing data manipulation approaches, such as coding theory and clipping protocols, decrease the task performance quality and efficiency. More efficient and effective mechanisms are demanded to better preserve the privacy from task requesters’ perspective.
2. [**Degree of privacy sacrifice for the requesters towards a successful crowdfunding campaign (crowdfunding).**]{} Trust is the key component for a successful crowdfunding campaign [@wen2018information; @liang2018why]. However, the most appropriate degree of privacy sacrifice for the requesters remains as an open problem to attract more funding contributions.
3. [**Tradeoff between data encryption and testing result quality (crowdtesting).**]{} Data encryption is a commonly used technique for protecting user privacy in crowdtesting practices, which unfortunately appear to decrease the testing result quality [@fabio2018understanding]. The way of balancing the tradeoff between data encryption and testing results quality is an important future working direction for crowdtesting practices.
4. [**An integrated approach to prevent misuse of customers’ data (collaborative online shopping).**]{} Data misuse is the main threat for customers who participate in the collaborative online shopping practice. Although there are solutions from both regulation and technical side, an integrated approach is demanded to better protect the users’ privacy.
5. [**Conflict between location privacy and service based on location (ridesharing).**]{} Location privacy is one of the hot topics in the field of location based services, such as ridesharing. However, there is always a conflict between hiding customers’ real locations and utilizing the location information to serve customers better. A better solution to balance the conflict remains as an open problem in the field.
6. [**Physical privacy protection for hosts (homesharing).**]{} For homesharing practices, existing works focus on mechanisms of protecting customers’ privacy. However, from our study, homesharing involves lots of interpersonal interactions, where the physical privacy violation is also a potential threat for the hosts. A well-regulated scheme to better protect the physical privacy for hosts involved in homesharing practices remains open.
In the second part of this work, the emerging privacy issues of the sharing services are further analyzed from three perspectives, namely, the users’, platforms’ and service providers’ perspectives. The open problems from the three individual perspectives are:
- [**From users’ perspective: motivating users to protect their own privacy.**]{} While most of surveyed works use cyber technologies to protect users from potential privacy leakage, we pointed out that those techniques can only be used against unnoticeable threats. Utilizing cyber techniques to motivate the users to actively protect their own privacy is still the main solution and must be further emphasized in future works.
- [**From platforms’ perspective: establishing effective protocol for data analysis.**]{} Encryption is a mature and commonly used cyber technique to protect user information during the data transmission and storage in sharing service platforms. Our study shows that, on top of the data encryption, a more sophisticated protocol is demanded for platform companies to access the necessary data for analysis in order for them to provide better services.
- [**From service providers’ perspective: enhancing the awareness of the importance of privacy protection using cyber technology.**]{} From service providers’ perspective, the surveyed works indicated that the privacy protection level is directly co-related to the net profit. However, the way of enhancing service providers’ awareness for the importance of protecting users’ privacy using cyber technology remains as an open problem for future studies.
Conclusions {#sec:conclusion}
===========
{height="3.2in"}
Privacy issues will sooner or later become the main barriers for both users and service providers who participate in the sharing economy. Over the past few years, great research efforts have been devoted to address various privacy issues existed in sharing service practices. Figure \[fig:statistics\] shows the yearly distribution of the number of all surveyed works from Table \[table:branch1\] to Table \[table:em3\]. Different colors are used to indicate various types of sharing services. It can be clearly seen that a substantial part of the works published in the recent five years, i.e., starting from 2013 to 2017 and later, is surveyed in this study.
{height="3.2in"}
{height="2.8in"}
The cyber-enabled sharing services were divided into two categories: crowdsourcing and collaborative consumption. Crowdsourcing is further divided into three branches: Internet crowdsourcing marketplace, crowdfunding and crowdtesting. In Internet crowdsourcing marketplace practices, we tackled the privacy protection problem for task requesters. Two approaches were surveyed: the coding theory and the instance clipping protocol. In crowdfunding practices, modern crowdfunding platforms, such as Indiegogo, allow users to select their preferred security level and conceal their personal information privately, such as their names and contribution amounts. However, the surveyed works suggest that a certain level of privacy sacrifice can be helpful in crowdfunding practice. For crowdtesting practices, three real-world applications were surveyed, including shared data protection on a the cloud server [@harkous2014c3p], online surveys [@kandappu2013exposing] and indoor site survey practice [@li2016privacy]. The main difficulties in protecting the privacy in crowdtesting practices are identified, which leads to one of the future research directions in the crowdtesting field. In collaborative consumption, the three sub-categories are: collaborative online shopping, ridesharing and homesharing. Collaborative online shopping, as a new generation of online shopping experience, raises two potential privacy concerns. The first privacy concern is the misuse of user data for marketing analysis, which can be prevented by refining government regulation [@preibusch2016shopping], masking customers’ data before sending them out [@bilge2013comparison] or separating communication channels on the cloud server [@lee2013pibox]. The second privacy concern is related to users’ awareness of privacy leakage in online shopping, which was further discussed in later sections. In ridesharing practice, it is important to note that revealing the passenger’s information, such as location, is necessary for the user to utilize the service. For homesharing, the surveyed works reveal that the hosts are actually more concerned about their privacy leakage than the travellers. Most of the privacy concerns are physical privacy issues.
In summary, Figure \[fig:distribution\] shows the distribution of all listed surveyed works from Table \[table:branch1\] to \[table:branch6\], including 37 works in total. In overall, the topics of privacy issues in collaborative online shopping and ridesharing are heavily discussed, whereas the topics of privacy issues in crowdtesting are less noticed. Although the surveyed works in this study do not include all works discussing the privacy issues of sharing services in the literature, the distribution reflects some aspects of the hotness/coldness of each mentioned topic, which provides potential directions to researchers for their future studies.
The above six branches of privacy concerns in the cyber-enabled sharing world are further summarized at the later part of this work from three perspectives. From the user perspective, users have started to realize that they have to sacrifice a certain degree of personal information to enjoy the sharing services. Therefore, the emerging issue is to increase the privacy awareness of the users. From the platform perspective, it is necessary for the third party platform to analyze the user’s shared data to improve the service quality. The emerging issue from the platform perspective is to develop an effective protocol for identifying and protecting sensitive data during the transmission process, as well as the storage on the cloud server. From the service provider perspective, privacy must be recognized as the most important issue in the business model, which potentially impacts the perceived security and trust as well as the final profit.
Figure \[fig:distribution2\] shows the distribution of all surveyed works from Table \[table:em1\] to \[table:em3\], including 28 works in total. In overall, most existing works focus on privacy protection solutions from user and platform perspectives. There are only a few works mentioning that the privacy protection level can be improved by making the service providers realize the importance of protecting user privacy for their businesses. The privacy protection solution from the service providers’ perspective deserves more attentions in future studies.
Table \[table:technicalCompare\] covers the main cyber techniques surveyed in this work to protect privacy in sharing service practices. Each of these works was carefully evaluated to summarize its advantages/disadvantages compared with the remaining methods. All methods listed in Table \[table:technicalCompare\] provide important solutions to protect privacy in different sharing service practices. Some of these methods can be more preferable under particular contexts or scenarios. For example, for privacy protection in crowdsourcing marketplace, SocialCrowd is preferred if the computational speed is not the main concern [@amor2016discovering]. Otherwise, a collusion network, proposed by Celis [*et al.*]{} [@celis2016assignment], can be more preferable to minimize the privacy leakage.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- -- -- --
**[Method]{} & **[Reference]{} & **[Main technique]{} & **[Advantage]{} & **[Disadvantage]{}\
& [[@varshney2012privacy], [@vempaty2014reliable], [@vempaty2014coding], [@wang2005distributed]]{} & [Add random perturbations to sensitive data]{} & [Hide sensitive information from the workers]{} & [Lose the task performance quality]{}\
[Instance clipping protocol (ICP)]{} & [[@little2011human], [@chen2012shreddr], [@kajino2014instance]]{} & [Clip the task into pieces]{} & [Allow each work only to access one clipped piece]{} & [Decrease the quality of the task results]{}\
[Collusion network]{} & [[@celis2016assignment]]{} & [Integrate ICP with three operations]{} & [ICP with minimal privacy leakage (better than ICP)]{} & [Information leakage from the workers’ side]{}\
[SocialCrowd]{} & [[@amor2016discovering]]{} & [Clustering algorithms with heuristic function]{} & [Data leakage was effectively prevented]{} & [High computational complexity]{}\
[Encryption scheme for crowdtesting]{} & [[@kandappu2013exposing], [@li2016privacy], [@bilge2013comparison]]{} & [Add noises to the test results]{} & [Sensitive data cannot be revealed easily]{} & [Original data can be distorted]{}\
[$\pi$-box]{} & [[@lee2013pibox]]{} & [Develop a third-party app to control sensitive data transmission]{} & [Separated channels are designed to control data transmission]{} & [Not all paid apps support $\pi$-box]{}\
[Clustering $k$-anonymity (CK) scheme]{} & [[@yao2010clustering], [@pan2013preserving]]{} & [Encrypt the user location information]{} & [Help ridesharing companies in protecting the user location privacy]{} & [Decrease location information resolution; lack protection on sensitive data transmission]{}\
[Anonymous mutual authentication (AMA) protocol]{} & [[@ni2016ama; @ni2016privacy]]{} & [Develop AMA protocol for real-time navigation system]{} & [Conceal both customers and drivers’ sensitive information (better than CK scheme)]{} & [A trusted third party middleware is required]{}\
[Two-stage auction algorithm]{} & [[@wang2018truthful]]{} & [Integrate $k$-anonymity scheme with $\epsilon$-differential scheme]{} & [Conceal users’ locations in a more sophisticated way]{} & [Add Gaussian white noise to actual locations]{}\
[NoTrace]{} & [[@malandrino2011supportive; @malandrino2013privacy]]{} & [Provide privacy recommendations to the users]{} & [Display the private information pieces received by service provider]{} & [Lack analysis on necessary information disclosure for a known service]{}\
[AppScanner]{} & [[@amini2014analyzing; @amini2013mobile]]{} & [Let users better understand the functions of mobile apps]{} & [Provide informative description of what mobile apps do]{} & [Only categorize the mobile app behaviors as normal or abnormal]{}\
[Mobile app recommendation systems]{} & [[@zhu2014mobile], [@quay2016per]]{} & [Analyze security risks and request resource access permissions from users]{} & [More complete system design with detailed permission levels (better than NoTrace and AppScanner)]{} & [Security risks are hard to be measured during the installation process]{}\
[Random space encryption (RASP) approach]{} & [[@chen2005privacy; @chen2011rasp; @chen2013perturboost]]{} & [Perform privacy protection on the cloud]{} & [Provide an encrypted space with a two-stage encoding algorithm on the cloud]{} & [Difficult to update an encrypted database]{}\
\
[Attribute based encryption (ABE) scheme]{} & [[@dong2014achieving; @dong2015seco], [@han2016security]]{} & [Encrypt data on the cloud]{} & [Allow users to dynamically access personal data freely]{} & [Time complexities are not optimized for encryption and decryption processes]{}\
\
[Hierarchical encryption scheme]{} & [[@wang2011hierarchical; @wang2013secure; @wang2014security; @liu2014time]]{} & [Maintain access controls for different levels of users]{} & [Each domain master generates keys to next sub-level users (better than ABE scheme)]{} & [Not applicable to the situations that attributes were administered by different domain authorities]{}\
**********
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- -- -- --
\[table:technicalCompare\]
In conclusion, we would like to point out that the solutions for emerging privacy issues in the cyber-enabled world include many different aspects, such as developing a more sophisticated encryption scheme for masking the user data, proposing a more reliable recommendation system for user privacy management, implementing a more secure transmission protocol and etc. All these issues/solutions represent the future research directions for privacy protection in sharing service practices.
[**Conflict of Interests**]{}\
All authors declare that there is no conflict of interest regarding the publication of this manuscript.\
Acknowledgment {#acknowledgment .unnumbered}
==============
This study is supported by National Natural Science Foundation of China (No. 61850410531), Zhejiang Provincial Natural Science Foundation of China under Grant No. LY19F020016 and National Natural Science Foundation of China (No. 61602431).
[^1]: $^*$Corresponding author: Qun Jin (Email address: [email protected]). Author’s addresses: K. Yan, H. Lu [and]{} Q. Jin, College of Information Engineering, China Jiliang University, 258 Xueyuan Street, Hangzhou, China, 310018, emails: [email protected], [email protected] (K. Yan) and [email protected] (H. Lu); W. Shen, Department of Informatics, University of California Irvine, Irvine, CA 92697, email: [email protected]; Q. Jin, Department of Human Informatics and Cognitive Sciences, Waseda University, 2-579-15 Mikajima, Tokorozawa, 359-1192, Japan.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in \[D. Chae, P. Constantin, J. Wu, *to appear in* IUMJ (2014)\] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time-decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space $L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.
**AMS MSC:** 35Q35, 76D03, 35A01, 35B06, 35B40, 35R11, 86A10
**Keywords:**[ Active scalar equations, Global well-posedness, Decay of solutions, Symmetry, spaces]{}
author:
- |
**Lucas C. F. Ferreira**\
[Universidade Estadual de Campinas, IMECC- Departamento de Matemática,]{}\
[Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil.]{}\
[`email: [email protected]`]{}\
**Lidiane S. M. Lima**\
[Universidade Estadual de Campinas, IMECC- Departamento de Matemática,]{}\
[Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil.]{}\
[`email: [email protected]`]{}
title: 'Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings'
---
Introduction
============
We are concerned with the initial value problem (IVP) for a family of dissipative active scalar equation, which reads as $$\begin{cases}
\frac{\partial\theta}{\partial t}+\kappa\left( -\Delta\right) ^{\gamma
}\theta+u\cdot\nabla_{x}\theta=0, & \qquad x\in\mathbb{R}^{n}\ ,\ t>0,\ \\[3mm]\theta(x,0)=\theta_{0}(x), & \qquad x\in\mathbb{R}^{n},\
\end{cases}
\label{dase}$$ where $n\geq2$, $\kappa\geq0$ and $\gamma>0$. The fractional laplacian operator $(-\Delta)^{\gamma}$ is defined by $$\widehat{\lbrack(-\Delta)^{\gamma}f]}(\xi)=|\xi|^{2\gamma}\widehat{f}(\xi),$$ where $\widehat{f}=\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}f(\xi)d\xi$ stands for the Fourier transform of $f.$ The velocity field $u$ is determined by the active scalar $\theta$ by means of the multiplier operators $$u=P[\theta]=(\widetilde{P}_{1}[\theta],...,\widetilde{P}_{n}[\theta]),
\label{velocity}$$ such that $\nabla\cdot u=0$, and $$u_{j}=\widetilde{P}_{j}[\theta]=\sum_{i=1}^{n}a_{ij}\mathcal{R}_{i}\Lambda^{-1}P_{i}[\theta],\;\;\text{for}\ 1\leq j\leq n, \label{ujotas}$$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $\mathcal{R}_{i}=-\partial_{i}(-\Delta)^{-\frac{1}{2}}$ is the $i$-th Riesz transform, $a_{ij}$’s are constant and
$$\widehat{P_{i}[\theta]}(\xi)=P_{i}(\xi)\widehat{\theta}(\xi)\text{.}
\label{Pi´s}$$
Denoting $I=\sqrt{-1},$ it follows that $$\text{ }\widehat{\widetilde{P}_{j}[\theta]}(\xi)=\widetilde{P}_{j}(\xi)\widehat{\theta}(\xi)\text{ with }\widetilde{P}_{j}(\xi)=\sum_{i=1}^{n}a_{ij}\frac{\xi_{i}I}{\left\vert \xi\right\vert ^{2}}P_{i}(\xi),$$ and the vector field $u$ can be expressed in Fourier variables in the shorter form $$\widehat{u}=\widehat{P[\theta]}=P(\xi)\widehat{\theta}(\xi)\text{ where }P(\xi)=(\widetilde{P}_{1}(\xi),...,\widetilde{P}_{n}(\xi)). \label{p-Fourier}$$ Throughout this manuscript the symbol $P_{i}(\xi)$ in (\[Pi´s\]) is assumed to belong to $C^{[\frac{n}{2}]+1}(\mathbb{R}^{n}\backslash\{0\})$ with $$\left\vert \frac{\partial^{\alpha}P_{i}}{\partial\xi^{\alpha}}(\xi)\right\vert
\leq C|\xi|^{\beta-|\alpha|}, \label{Pi-cond}$$ for all $\alpha\ \in\ (\mathbb{N}\cup\{0\})^{n}$, $|\alpha|\leq\lbrack\frac
{n}{2}]+1$ and $\xi\neq0$, where $\beta\geq0$. The brackets $[\cdot]$ stands for the greatest integer function. In particular, for $\alpha=0$ it follows from (\[p-Fourier\]) and (\[Pi-cond\]) that $$\left\vert \widehat{u}(\xi)\right\vert \leq C\left\vert \xi\right\vert
^{\beta-1}\left\vert \widehat{\theta}(\xi)\right\vert ,\text{ for all }\xi
\neq0. \label{est-field-1}$$ Concerning the criticality of (\[dase\])-(\[ujotas\]), there is an interplay between the field $u$ and fractional viscosity $\left(
-\Delta\right) ^{\gamma}$ expressed by means of three basic cases: sub-critical $\beta<2\gamma,$ critical $\beta=2\gamma,$ and super-critical $\beta>2\gamma.$
We could consider an arbitrary $\kappa>0,$ nevertheless $\kappa=1$ is assumed for the sake of simplicity. The IVP (\[dase\])-(\[ujotas\]) can be converted into the integral equation $$\theta(t)=G_{\gamma}(t)\theta_{0}+B(\theta,\theta)(t), \label{mild}$$ where $$B(\theta,\varphi)(t)=-\int_{0}^{t}G_{\gamma}(t-s)(\nabla_{x}\cdot
(P[\theta]\varphi))(s)ds \label{termo bilinear}$$ and $G_{\gamma}(t)$ is the convolution operator with kernel given in Fourier variables by $\hat{g}_{\gamma}(\xi,t)=e^{-t|\xi|^{2\gamma}}$. Solutions of (\[mild\]) are called mild ones for (\[dase\])-(\[ujotas\]).
Assuming that $P_{i}$’s are homogeneous functions of degree $\beta$, we have formally that $$\theta_{\lambda}=\lambda^{2\gamma-\beta}\theta(\lambda x,\lambda^{2\gamma}t)$$ verifies (\[dase\])-(\[ujotas\]), for all$\ \lambda>0,$ provided that $\theta$ does so. It follows that $$\theta\rightarrow\theta_{\lambda}=\lambda^{2\gamma-\beta}\theta(\lambda
x,\lambda^{2\gamma}t),\ \text{for}\ \lambda>0, \label{scaling}$$ is the scaling map for (\[dase\])-(\[ujotas\]). Also, making $t\rightarrow0^{+}$ in (\[scaling\]), one obtains the scaling for the initial data $$\theta_{0}\rightarrow\lambda^{2\gamma-\beta}\theta_{0}(\lambda x).
\label{scaling initial}$$ In view of (\[Pi-cond\]), even when $P_{i}$ is not homogeneous, we can consider (\[scaling\]) as an intrinsic scaling for (\[dase\])-(\[ujotas\]) in the sense that it is useful to identify threshold indexes for functional settings and properties of solutions. One of our aims is to provide a global well-posedness result for (\[dase\])-(\[ujotas\]) in a scaling invariant framework outside the $L^{2}$-space.
Active scalar equations like (\[dase\])-(\[ujotas\]) arise in a large number of physical models in fluid mechanics and atmospheric science. Examples of those are 2D surface quasi-geostrophic equation (SQG) $u=\nabla^{\perp
}((-\Delta)^{-1/2}\theta)$ ($\beta=1$), Burgers equation $u=\theta$ ($\beta
=1$)$,$ 2D vorticity equation $u=\nabla^{\perp}(-\Delta)^{-1}\theta$ ($\beta=0$). SQG is a famous model with a lot of papers concerning existence, uniqueness, regularity and asymptotic behavior of solutions in the inviscid case $\kappa=0$ or in the subcritical ($1/2<\gamma<1$), critical ($\gamma
=1/2$) and supercritical ($\gamma\in(0,1/2)$) ranges. Without making a complete list, we would like to mention [@Caff-Vass], [@Car-Fer1],[@Const2],[@Const3],[@Const4],[@Cordoba1],[@Dong-1],[@Gancedo1],[@Ju],[@Kiselev1],[@Kiselev2],[@Kiselev3],[@NS],[@SS], and their references. In the case $u=\theta$, see [@Kiselev3] and [@Dong-3] for results on blow-up, global existence and regularity of solutions. One dimensional active scalar models have also attracted the attention of many authors, see e.g. [@Car-Fer3],[@Cordoba2],[@Dong-2],[@Li-Ro] where the reader can find global existence, finite-time singularity and asymptotic behavior results with velocity coupled via singular integral operators that are zero-order multiplier ones.
In the case of SQG, notice that $u$ can be written by using Riesz transform as $$u=(-\mathcal{R}_{2}\theta,\mathcal{R}_{1}\theta) \label{Riesz-field}$$ and then the velocity is coupled to the active scalar via zero-order multiplier operators. The model (\[dase\])-(\[ujotas\]) was introduced in [@Chae1],[@Chae2] and covers positive-order couplings when $\beta>1$ (see (\[est-field-1\])). In this range, the operator $P[\cdot]$ behaves morally like a positive derivative of $(\beta-1)$-order and produces more difficulties in comparison with SQG ($\beta=1$, zero-order) and $\beta<1$ (negative-order).
The paper [@Chae1] deals mainly with the inviscid case $\kappa=0$, while [@Chae2] with the dissipative one $\kappa>0$. This last work is our main motivation since we also focus in the dissipative model. The authors of [@Chae2] showed existence of global solutions in $L^{\infty}((0,\infty);Y)$ for (\[dase\])-(\[ujotas\]) where $Y=L^{1}\cap L^{\infty
}\cap B_{q,\infty}^{s,M}$ with $s>1$ and $2\leq q\leq\infty.$ The index $M=\{M_{j}\}_{j\geq-1}$ is a sequence and the space $B_{q,\infty}^{s,M}$ is an extension of the classical Besov space $B_{q,\infty}^{s}$ where the $B_{q,\infty}^{s,M}$-norm increases according to the growth of $M$. The results of [@Chae2] consider couplings $P[\cdot]$ in (\[velocity\]) such that $P_{i}\in C^{\infty}(\mathbb{R}^{n}\backslash\{0\}),$ $P_{i}$ is radially symmetric, $P_{i}=P_{i}(\left\vert \xi\right\vert )$ is nondecreasing with $\left\vert \xi\right\vert $, and a technical growth hypothesis involving $P_{i}(\xi)$ and the sequence $M$. Applying their results to the special case $$u=\nabla^{\perp}(\Lambda^{\beta-2}\theta)=\Lambda^{\beta-1}(-\mathcal{R}_{2}\theta,\mathcal{R}_{1}\theta) \label{field1}$$ with $n=2$, $0\leq\beta<2\gamma<1$ (within the sub-critical range), $\kappa>0$ and $M_{j}=j+1$, they obtained well-posedness of solutions with initial data in $L^{1}\cap L^{\infty}\cap B_{q,\infty}^{s,M}.$ Roughly speaking, the technique employed in [@Chae2] for constructing solutions relies on ** a successive approximation scheme together *a priori* estimates involving Besov norms. The field (\[field1\]) corresponds to the modified SQG that interpolates 2D vorticity equation and SQG by varying the parameter $\beta$ from $0$ to $1.$ This model has been studied for instance in [@Chae1],[@Const1],[@Kiselev3],[@May1],[@Miao1],[@Miao-2] where one can find existence and regularity results with data in Sobolev spaces $H^{m}$ with $m\geq0$. The conditions $\kappa>0$, $\beta
\in\lbrack0,1]$ and $\beta=2\gamma$ were assumed in [@Const1],[@Kiselev3],[@May1],[@Miao1]; $\kappa>0$ and $1\leq\beta
<2\gamma<2$ in [@Miao-2]; and $\kappa=0$ and $\beta\in\lbrack1,2]$ in [@Chae1]. In this last work, local well-posedness of $H^{m}(\mathbb{R}^{2})$-solutions was proved for (\[dase\])-(\[field1\]) with $m\geq4$.
In this paper we prove the global-in-time well-posedness of (\[dase\])-(\[ujotas\]) in the Lebesgue space $L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ without smallness conditions (see Theorem \[teoglobal\]). This is the unique $L^{r}$-space whose norm is invariant by the scaling (\[scaling initial\]), that is, $L^{\frac{n}{2\gamma-\beta}}$ is the critical one in the scale of Lebesgue spaces. We can consider initial data outside the $L^{2}$-framework and, due to the inclusion $L^{1}\cap L^{\infty
}\subset L^{\frac{n}{2\gamma-\beta}}$, our initial data class is larger than that of [@Chae2]. In comparison with [@Chae2], some new symbols $P_{i}(\xi)$ are considered here (e.g. non-radially symmetric ones). Even for a singular initial data $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$, the global solution $\theta\in BC([0,\infty);L^{\frac
{n}{2\gamma-\beta}}(\mathbb{R}^{n}))$ is instantaneously $C^{\infty}$-smoothed out and verifies (\[dase\])-(\[ujotas\]) classically, for all $t>0$. Here we focus in the range $\beta\geq1$ and consider the sub-critical case $\beta<2\gamma$. More precisely, we assume $$1\leq2\beta-1<2\gamma<\min\{\frac{2}{3}(n+\beta+1),(n+1)\}. \label{cond-1}$$ The range $0\leq\beta<2\gamma$ with $\beta<1$ also can be treated with an adaptation on the proofs (see Remark \[rem1\]).
Also, we show some decay properties in $L^{q}$-norms (see Theorem \[teoglobal\]). Precisely, for $\frac{n}{2\gamma-\beta}\leq q\leq\infty$ and $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}),$ we obtain the time-polynomial decay $$\left\Vert \theta(\cdot,t)\right\Vert _{L^{q}}\leq Ct^{-(\frac{2\gamma-\beta
}{2\gamma}-\frac{n}{2\gamma q})}\,,\ \text{for all }t>0\text{.} \label{decay1}$$ Assuming further $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})\cap L^{1}(\mathbb{R}^{n}),$ the solution $\theta$ belongs to $BC([0,\infty);L^{1}(\mathbb{R}^{n}))$ and the estimate (\[decay1\]) is improved to$$\left\Vert \theta(\cdot,t)\right\Vert _{L^{q}}\leq Ct^{-(\frac{2\gamma-\beta
}{2\gamma}-\frac{n}{2\gamma q})-(\frac{n+\beta}{2\gamma}-1)}\,,\text{ for all
}t>0, \label{decay2}$$ where $1\leq q\leq\infty.$ Notice that the decay in (\[decay2\]) is faster than those of (\[decay1\]) due to the condition $2\gamma<\frac{2}{3}(n+\beta+1)<n+\beta$.
In view of the $L^{p}$-$L^{q}$ estimate (\[est linear2\]) for the semigroup $G_{\gamma}(t)$, it is not expected that (\[decay1\]) holds true for $q<\frac{n}{2\gamma-\beta}$ and an arbitrary $\theta_{0}\in L^{\frac
{n}{2\gamma-\beta}}(\mathbb{R}^{n}).$ Thus $\theta(\cdot,t)$ may not be a $L^{2}$-solution when $2<\frac{n}{2\gamma-\beta}$ although $\theta(\cdot,t)\in
C^{\infty}(\mathbb{R}^{n})$, for all $t>0.$ Even in the subcritical case, this fact seems to prevent an adaptation from previous techniques based on $L^{2}$-frameworks (see e.g. the famous papers [@Caff-Vass; @Kiselev1]) in order to obtain global well-posedness of $L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$-solutions. Roughly speaking, the approach employed here relies on time-weighted Kato type norms, scaling arguments, and arguments of the type parabolic De Giorgi-Nash-Moser. These ingredients also were used in [@Car-Fer1] in order to analyze SQG ($\beta=1$). However, due to the coupling between $\theta$ and $u$ being via a positive-order operator, the model (\[dase\])-(\[ujotas\]) requires more involved arguments and further care in comparison with SQG. For instance, since $P[\cdot]$ is not continuous from $L^{p_{1}}$ to $L^{p_{2}}$ when $\beta>1$, we need to employ an auxiliary time-weighted Kato-type norm based on homogeneous Sobolev spaces $\dot{H}_{q}^{s}$ with $q>\frac{n}{2\gamma-\beta}$ in order to control the nonlinear term in (\[dase\])-(\[ujotas\]). So, different from SQG, Sobolev norms play here a crucial role for the local existence and extension of solutions with data in Lebesgue spaces (see e.g. (\[aux-prop1\]) and (\[aux-def1\])-(\[aux-ext-10\]), respectively). Let us also mention that there is no maximum principle for $\dot{H}_{q}^{s}$-norms when $s>0$; and consequently there is a lack of global-in-time control on these norms (see (\[Contituity-Hq\])).
In view of the structure of (\[dase\]), it is natural to wonder about symmetry properties of solutions under symmetry conditions for the symbols $P_{i}(\xi)$ and initial data $\theta_{0}.$ In Theorem \[Teo-sym\], we show that the global solution given in Theorem \[teoglobal\] is radially symmetric, for all $t>0$, provided that $\theta_{0}$ and $div_{\xi}(P(\xi))$ present this same property. Moreover, results on odd and even symmetry of solutions are obtained under parity conditions for $\theta_{0}$ and $P_{i}$’s. In Remark \[rem2\], we also comment about conditions for solutions to be non-symmetric.
Let us also comment on *log*-type couplings which are interesting ones covered by (\[dase\])-(\[ujotas\]). Ohkitani [@Okitani] has presented numerical evidences that, even with $\kappa=0,$ (\[dase\]) with $n=2$ and $$u=\nabla^{\perp}(\log(I-\Delta))^{\chi}\theta,\text{ }\chi>0,
\label{log-field-1}$$ may be globally well-posed. The authors of [@Chae1] have proved local well-posedness of $H^{4}(\mathbb{R}^{2})$-solutions for (\[dase\])-(\[log-field-1\]) with $\kappa>0$. As pointed in [@Chae1], the field (\[log-field-1\]) is of order higher (at least logarithmically) than derivatives of order $1$ and in particular than (\[Riesz-field\]). Another examples are $$\begin{aligned}
P_{i}(\xi) & =\left\vert \xi\right\vert ^{\sigma}(\log(1+\left\vert
\xi\right\vert ^{2}))^{\chi},\text{ }\chi\geq0,\text{ }\label{log-field-2}\\
P_{i}(\xi) & =\left\vert \xi\right\vert ^{\sigma}(\log(1+\log(1+\left\vert
\xi\right\vert ^{2})))^{\chi},\chi\geq0, \label{log-field-3}$$ which are indeed of order higher than (\[log-field-1\]) when $\sigma>1$. These couplings are also treated in [@Chae2] with $\sigma=\beta$ and $\chi\geq0.$ When $\sigma=0$ and $n=2,$ (\[log-field-2\]) and (\[log-field-3\]) correspond to *log* and *log-log* Navier-Stokes which are intermediate models between 2D vorticity equation and SQG. See [@Chae3] for further details and global existence results in the case $\kappa=0,$ $0\leq\chi\leq1$ and data $\theta_{0}\in L^{1}\cap L^{\infty
}\cap B_{q,\infty}^{s}$, where $B_{q,\infty}^{s}$ stands for an inhomogeneous Besov space with $s>1$ and $q>2$. An interest in *log*-type couplings has also arisen in connection with other fluid mechanics models (see [@Chae5]).
Finally, we remark that our results cover the couplings (\[field1\]), (\[log-field-1\]), (\[log-field-2\]) and (\[log-field-3\]). The condition (\[Pi-cond\]) is clearly satisfied by (\[field1\]), and if $\beta\in\lbrack1,2]$ and $2\beta-1<2\gamma<\min\{2+\frac{2\beta}{3},3\}$ then (\[cond-1\]) holds true. Also, (\[log-field-1\]) verifies (\[Pi-cond\]) with $\beta=1+\varepsilon,$ for any $\varepsilon>0,$ and we have (\[cond-1\]) when $\frac{1}{2}<\gamma<\frac{4}{3}$ and $0<\varepsilon
<\gamma-\frac{1}{2}$. By considering $\beta=\sigma+\varepsilon,$ conditions analogous to the ones for (\[log-field-1\]) can be obtained for (\[log-field-2\]) and (\[log-field-3\]) with $\chi>0.$ The cases (\[log-field-2\]) and (\[log-field-3\]) with $\chi=0$ are similar to (\[field1\]).
This manuscript is organized as follows. In the next section we recall some estimates in $L^{q}(\mathbb{R}^{n})$ and Sobolev homogeneous spaces for Fourier multiplier operators and the semigroup $\{G_{\gamma}(t)\}_{t\geq0}$. Our results are stated in section 3 in two theorems, namely Theorems \[teoglobal\] and \[Teo-sym\]. Estimates for the bilinear operator (\[termo bilinear\]) are obtained in section 4. Local well-posedness and some properties of solutions are proved in subsection 5.1. The proofs of Theorems \[teoglobal\] and \[Teo-sym\] are performed in subsections 5.2 and 5.3, respectively.
Preliminaries
=============
In this section we recall some estimates for the fundamental solution of the linear part of (\[dase\]) in $L^{p}(\mathbb{R}^{n})$ and $\dot{H}_{p}^{s}(\mathbb{R}^{n})$, whose norms will be denoted by $\Vert
\cdot\Vert_{p}$ and $\Vert\cdot\Vert_{\dot{H}_{p}^{s}}$, respectively.
We remember that given $s\in\mathbb{R}$ and $1<p<\infty$, the homogeneous Sobolev space $\dot{H}_{p}^{s}(\mathbb{R}^{n})$ is the space of all $u\in\mathcal{S}^{\prime}/\mathcal{P}$ such that $(-\Delta)^{\frac{s}{2}}u\in$ $L^{p}(\mathbb{R}^{n})$. In other words, $\dot{H}_{p}^{s}=(-\Delta)^{-\frac
{s}{2}}L^{p}$ and it is a Banach space with norm $$\Vert u\Vert_{\dot{H}_{p}^{s}}=\Vert(-\Delta)^{\frac{s}{2}}u\Vert_{p}.$$ The following Sobolev type embedding holds true $$\dot{H}_{p_{2}}^{s_{2}}(\mathbb{R}^{n})\subset\dot{H}_{p_{1}}^{s_{1}}(\mathbb{R}^{n}), \label{Sobolev}$$ for $1<p_{2}\leq p_{1}<\infty$ and $s_{1}-\frac{n}{p_{1}}=s_{2}-\frac{n}{p_{2}}$. The reader is refereed to [@Grafakos chapter 6] for further details on these spaces.
The next lemma gives estimates for certain multiplier operators acting in $\dot{H}_{p}^{s}(\mathbb{R}^{n})$ (see e.g. [@Kozo1])
\[multiplier\] Let $m$, $s\in\mathbb{R}$, $1<p<\infty$ and $F(\xi)\in
C^{[\frac{n}{2}]+1}(\mathbb{R}^{n}\backslash\{0\})$, where $[\cdot]$ stands for the greatest integer function. Assume that there is $L>0$ such that $$\left\vert \frac{\partial^{\alpha}F}{\partial\xi^{\alpha}}(\xi)\right\vert
\leq L|\xi|^{m-|\alpha|}, \label{est fjotas}$$ for all $\alpha\in(\mathbb{N}\cup\{0\})^{n}$, $|\alpha|\leq\lbrack\frac{n}{2}]+1$, and $\xi\neq0$. Then the multiplier operator $F(D)$ on $\mathcal{S}^{\prime}/\mathcal{P}$ is bounded from $\dot{H}_{p}^{s}$ to $\dot{H}_{p}^{s-m}$. Moreover, the following estimate holds true $$\Vert F(D)f\Vert_{\dot{H}_{p}^{s-m}}\leq C\Vert f\Vert_{\dot{H}_{p}^{s}},
\label{fjotas}$$ where $C>0$ is independent of $f$.
The next lemma gives estimates for $\{G_{\gamma}(t)\}_{t\geq0}$ on spaces $L^{p}(\mathbb{R}^{n})$ and $\dot{H}_{p}^{s}(\mathbb{R}^{n})$.
\[sol fund\] Let $n\geq2,$ $0<\gamma<\infty$, $1\leq p\leq q\leq\infty$ and $k\in(\mathbb{N}\cup\{0\})^{n}.$ Then $$\Vert\nabla_{x}^{k}G_{\gamma}(t)f\Vert_{q}\leq C\ t^{-\frac{\left\vert
k\right\vert }{2\gamma}-\frac{n}{2\gamma}(\frac{1}{p}-\frac{1}{q})}\Vert
f\Vert_{p}\text{,} \label{est linear2}$$ for all $f\in L^{p}(\mathbb{R}^{n})$. Now, let $s_{1}\leq s_{2}$, $s_{i}\in\mathbb{R}$ and $1<p_{1}\leq p_{2}<\infty$. There is a constant $C>0$ such that $$\Vert G_{\gamma}(t)f\Vert_{\dot{H}_{p_{2}}^{s_{2}}}\leq Ct^{-\frac
{(s_{2}-s_{1})}{2\gamma}-\frac{n}{2\gamma}(\frac{1}{p_{1}}-\frac{1}{p_{2}})}\Vert f\Vert_{\dot{H}_{p_{1}}^{s_{1}}}. \label{est linear}$$ for all $f\in\dot{H}_{p_{1}}^{s_{1}}$. Moreover, given $f\in L^{\frac
{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ with $1\leq\beta<2\gamma\leq n+\beta$ and $\frac{n}{2\gamma-\beta}<q<\infty$, then $$\sup_{0<t<T}t^{\eta_{q}}\Vert G_{\gamma}(t)f\Vert_{\dot{H}_{q}^{\beta-1}}\leq
C\Vert f\Vert_{\frac{n}{2\gamma-\beta}}\qquad\mbox{and}\qquad\lim
_{t\rightarrow0^{+}}t^{\eta_{q}}\Vert G_{\gamma}(t)\theta_{0}\Vert_{\dot
{H}_{q}^{\beta-1}}=0 \label{zeroPrincipal}$$ where $\eta_{q}=\frac{2\gamma-1}{2\gamma}-\frac{n}{2\gamma q}$ and $C$ is independent of $f$ and $0<T\leq\infty$.
**Proof.** The estimate (\[est linear2\]) is well-known (see e.g. [@Car-Fer1] for a proof). Also, (\[est linear2\]) still holds true by replacing $\nabla_{x}^{k}$ by $(-\Delta)^{\frac{\left\vert k\right\vert }{2}}.$ In view of the latter comment and $(-\Delta)^{\frac{s_{2}}{2}}=(-\Delta)^{\frac{s_{2}-s_{1}}{2}}(-\Delta)^{\frac{s_{1}}{2}}$, we obtain (\[est linear\]) from (\[est linear2\]) because $G_{\gamma}(t)$ commutates with $(-\Delta)^{\frac{s_{1}}{2}}.$ The estimate in (\[zeroPrincipal\]) comes from (\[est linear\]) with $p_{2}=q,$ $s_{2}=\beta-1,$ $p_{1}=\frac
{n}{2\gamma-\beta}$ and $s_{1}=0.$ Due to (\[est linear\]), it is easy to see that the limit in (\[zeroPrincipal\]) holds true for every $\theta
_{0}\in L^{\frac{n}{2\gamma-\beta}}\cap\dot{H}_{q}^{\beta-1}.$ This fact together with $\overline{L^{\frac{n}{2\gamma-\beta}}\cap\dot{H}_{q}^{\beta-1}}^{\left\Vert \cdot\right\Vert _{\frac{n}{2\gamma-\beta}}}=L^{\frac{n}{2\gamma-\beta}}$ and the estimate in (\[zeroPrincipal\]) yield the limit in (\[zeroPrincipal\]), for every $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}).$
[ ]{}
Results
=======
This section is devoted to the statements of the results whose proofs will be performed in section 5.
\[teoglobal\] (Global solutions) Assume the condition (\[cond-1\]) and let $\eta_{q}=\frac{2\gamma-1}{2\gamma}-\frac{n}{2\gamma q}$ and $\tilde{\eta
}_{q}=\frac{2\gamma-\beta}{2\gamma}-\frac{n}{2\gamma q}.$ If $\theta_{0}\in
L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ then there is a unique global solution $\theta\in BC([0,\infty);L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}))$ for (\[dase\])-(\[ujotas\]) such that $$\begin{aligned}
t^{\tilde{\eta}_{q}}\theta & \in BC\left( (0,\infty),L^{q}(\mathbb{R}^{n})\right) ,\text{ for all }\frac{n}{2\gamma-\beta}<q\leq\infty
,\label{Decay-Lq-1}\\
t^{\eta_{q}}\theta & \in C((0,\infty);\dot{H}_{q}^{\beta-1}(\mathbb{R}^{n})),\text{ \ for all }\frac{n}{2\gamma-\beta}<q<\infty,
\label{Contituity-Hq}$$ where the limits of $t^{\tilde{\eta}_{q}}\theta$ in (\[Decay-Lq-1\]) and $t^{\eta_{q}}\theta$ in (\[Contituity-Hq\]) are zero as $t\rightarrow0^{+}.$ Moreover, if $\theta_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ and $1<q\leq\infty$, then $\theta
\in\ BC([0,\infty);L^{1}(\mathbb{R}^{n})\cap L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}))$ and $$t^{\tilde{\eta}_{q}+\frac{n+\beta}{2\gamma}-1}\theta\in BC((0,\infty
);L^{q}(\mathbb{R}^{n})). \label{Decay-Lq-2}$$
\[ContDepend\](Continuous dependence on initial data) The proof of Theorem \[teoglobal\] also gives that the solution $\theta$ depends continuously on the data $\theta_{0}$ in finite time intervals $[0,T]$. More precisely, if $\theta_{k,0}\rightarrow\theta_{0}$ in $L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ then $\theta_{k}\rightarrow\theta$ in $C([0,T];L^{\frac
{n}{2\gamma-\beta}}(\mathbb{R}^{n}))$, for all $0<T<\infty$, where $\theta
_{k}$ is the solution with initial data $\theta_{k,0}.$
\[rem1\]One can treat the range $0\leq\beta<1$ by modifying the estimates of section 4 (particularly (\[bili-1\]) and (\[bili-4\])). For that matter, one should replace $\sup_{0<t<T}t^{\eta_{l}}\left\Vert \theta
(\cdot,t)\right\Vert _{\dot{H}_{l}^{\beta-1}}$ by $\ \sup_{0<t<T}t^{\tilde{\eta}_{l}}\left\Vert \theta(\cdot,t)\right\Vert _{l}$ into those estimates $(l=q,r)$. In fact, due to Hardy-Littlewood-Sobolev inequality, this case is easier to handling than $\beta>1$ and it is not necessary to use norms based on homogeneous Sobolev spaces in order to prove existence of solutions.
Before proceeding, we recall the concept of even and odd symmetry. A function $h$ is said to be even (resp. odd) when $h(x)=h(-x)$ (resp. $h(x)=-h(-x)$).
\[Teo-sym\](Symmetry) Under the hypotheses of Theorem \[teoglobal\].
(i)
: *The solution* $\theta(x,t)$* is odd (resp. even) for all* $t>0,$ ** provided that $\theta_{0}$ and $P_{i}$’s are odd (resp. even).
(ii)
: Let $P(\xi)$ be as in (\[p-Fourier\]). If $\theta_{0}$ and $div_{\xi}(P(\xi))$ *are radially symmetric then* $\theta
(x,t)$* is radially symmetric for all* $t>0$.
\[rem2\]*(non-symmetry)* Adapting the arguments in the proof of Theorem \[Teo-sym\], one also can prove the following non-symmetry results: if $\theta_{0}$ is odd (resp. even) and $P_{i}$’s are even (resp. odd) then $\theta(x,t)$ is not odd (resp. not even). Also, if $\theta_{0}$ is nonradial and $div_{\xi}(P(\xi))$ is radial, then $\theta(x,t)$ is not radially symmetric. The detailed verification is left to the reader.
Bilinear Estimates
==================
This part of the article is devoted to estimates for the bilinear term (\[termo bilinear\]).
\[LemaBiliq\] Let $0<T\leq\infty,$ $n\geq2,$ $1\leq\beta<2\gamma<\infty,$ and let $1<q<\infty$ be such that $\frac{\beta-1}{n}<\frac{1}{q}<\frac
{2\gamma-1}{n}.$ Denote $\eta_{q}=\frac{2\gamma-1}{2\gamma}-\frac{n}{2\gamma
q}$ and $\tilde{\eta}_{q}=\frac{2\gamma-\beta}{2\gamma}-\frac{n}{2\gamma q}$.
- If $\frac{2\gamma-(\beta+1)}{n}-\frac{1}{q}<\frac{1}{r}\leq\frac
{1}{q^{\prime}}$ and $q^{\prime}\leq p\leq\infty$ then there are positive constants $K_{1},K_{2},K_{3},$ independent of $\theta,\phi$ and $T,$ such that $$\begin{aligned}
\sup_{0<t<T}t^{\tilde{\eta}_{r}}\Vert B(\theta,\phi)\Vert_{r}\text{ } & \leq
K_{1}\sup_{0<t<T}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\,\sup_{0<t<T}t^{\tilde{\eta}_{r}}\Vert\phi\Vert_{r},\label{bili-1}\\
\sup_{0<t<T}\Vert B(\theta,\phi)\Vert_{p}\text{ } & \leq K_{2}\sup
_{0<t<T}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\,\sup
_{0<t<T}\Vert\phi\Vert_{p},\label{bili-2}\\
\sup_{0<t<T}\Vert B(\theta,\phi)\Vert_{1} & \leq K_{3}T^{\frac{2\gamma
-1}{2\gamma}-\eta_{q}}\sup_{0<t<T}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\,\sup_{0<t<T}\Vert\phi\Vert_{q^{\prime}}. \label{bili-3}$$
- If $\frac{2\gamma-2}{n}-\frac{1}{q}<\frac{1}{r}<\frac{n+\beta
-1}{n}-\frac{1}{q}$ then $$\sup_{0<t<T}t^{\eta_{r}}\Vert B(\theta,\phi)\Vert_{\dot{H}_{r}^{\beta-1}}\text{ }\leq K_{4}\sup_{0<t<T}t^{\eta_{r}}\Vert\theta\Vert_{\dot{H}_{r}^{\beta-1}}\,\sup_{0<t<T}t^{\eta_{q}}\Vert\phi\Vert_{\dot{H}_{q}^{\beta
-1}}, \label{bili-4}$$ where $K_{4}>0$ is a constant independent of $\theta,\phi$ and $T.$
**Proof.**
*Proof of part (i):* Let $p_{1}=p$ and $p_{2}=r$. Using Lemma \[sol fund\] and Hölder inequality, we estimate $$\begin{aligned}
\Vert B(\theta,\phi)\Vert_{p_{i}} & \leq\int_{0}^{t}\Vert\nabla_{x}G_{\gamma}(t-s)(P[\theta]\phi)(s)\Vert_{p_{i}}\text{ }ds\nonumber\\
& \leq C\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}-\frac{n}{2\gamma q}}\,\Vert(P[\theta]\phi)(s)\Vert_{\frac{p_{i}q}{p_{i}+q}}\text{ }ds\nonumber\\
& \leq C\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}-\frac{n}{2\gamma q}}\,\Vert
P[\theta(s)]\Vert_{q}\,\Vert\phi(s)\Vert_{p_{i}}\text{ }ds\nonumber\\
& \leq C\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}-\frac{n}{2\gamma q}}\,\Vert\theta(s)\Vert_{\dot{H}_{q}^{\beta-1}}\,\Vert\phi(s)\Vert_{p_{i}}\text{
}ds \label{Bili}$$ where $i=1,2$ and in the third line we have used Lemma \[multiplier\] in order to infer $$\Vert P[\theta]\Vert_{q}\leq C\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}.$$ Therefore $$\begin{aligned}
\Vert B(\theta,\phi)\Vert_{p} & \leq C\,I_{1}(t)\,\sup_{0<t<T}\Vert
\phi(t)\Vert_{p}\,\sup_{0<t<T}t^{\eta_{q}}\Vert\theta(t)\Vert_{\dot{H}_{q}^{\beta-1}},\label{aux-bili-1}\\
\Vert B(\theta,\phi)\Vert_{r} & \leq C\,I_{2}(t)\,\sup_{0<t<T}t^{\tilde
{\eta}_{r}}\Vert\phi(t)\Vert_{r}\sup_{0<t<T}t^{\eta_{q}}\Vert\theta
(t)\Vert_{\dot{H}_{q}^{\beta-1}}, \label{aux-bili-2}$$ where the integrals $I_{1}(t)$ and $I_{2}(t)$ can be computed as $$\begin{aligned}
I_{1}(t) & =\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}-\frac{n}{2\gamma q}}s^{-\eta_{q}}\text{ }ds=\int_{0}^{1}(1-s)^{\eta_{q}-1}s^{-\eta_{q}}\text{
}ds=C<\infty,\label{aux-int-1}\\
I_{2}(t) & =\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}-\frac{n}{2\gamma q}}s^{-\eta_{q}-\tilde{\eta}_{r}}\,ds=t^{\eta_{q}-1-\eta_{q}-\tilde{\eta}_{r}+1}\int_{0}^{1}(1-s)^{\eta_{q}-1}s^{-\eta_{q}-\tilde{\eta}_{r}}\,ds\,=\,C\,t^{-\tilde{\eta}_{r}}. \label{aux-int-2}$$ The estimates (\[bili-1\]) and (\[bili-2\]) follows from (\[aux-bili-2\]) with (\[aux-int-2\]), and (\[aux-bili-1\]) with (\[aux-int-1\]), respectively.
Moreover, we have that $$\begin{aligned}
\Vert B(\theta,\phi)(t)\Vert_{1} & \leq\int_{0}^{t}\Vert\nabla_{x}G_{\gamma
}(t-s)(P[\theta]\phi)(s)\Vert_{1}\text{ }ds\\
& \leq C\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}}\,\Vert P[\theta]\Vert
_{q}\,\Vert\phi\Vert_{q^{\prime}}ds\\
& \leq C\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}}\,\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\,\Vert\phi\Vert_{q^{\prime}}ds\\
& \leq C\int_{0}^{t}(t-s)^{-\frac{1}{2\gamma}}\,s^{-\eta_{q}}ds\sup
_{0<t<T}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\,\sup
_{0<t<T}\Vert\phi\Vert_{q^{\prime}}\\
& \leq CT^{1-\frac{1}{2\gamma}-\eta_{q}}\sup_{0<t<T}t^{\eta_{q}}\Vert
\theta\Vert_{\dot{H}_{q}^{\beta-1}}\,\sup_{0<t<T}\Vert\phi\Vert_{q^{\prime}},\end{aligned}$$ which gives (\[bili-3\]).
*Proof of part (ii):* Consider $$\frac{1}{h}=\frac{1}{r}+\frac{1}{q}-\frac{\beta-1}{n}\text{ and }\delta
=\frac{n}{h}-\frac{n}{r}. \label{aux-param1}$$ Note that $\frac{\beta+\delta}{2\gamma}<1$ because $\frac{1}{q}<\frac
{2\gamma-1}{n}$. We employ the continuous inclusion $\dot{H}_{h}^{\beta-1+\delta}\subset\dot{H}_{r}^{\beta-1}$, (\[est linear\]) and afterwards (\[fjotas\]) to obtain $$\begin{aligned}
\Vert B(\theta,\phi)\Vert_{\dot{H}_{r}^{\beta-1}} & \leq\int_{0}^{t}\Vert
G_{\gamma}(t-s)[\nabla\cdot(P[\theta]\phi)(s)]\Vert_{\dot{H}_{r}^{\beta-1}}\text{ }ds\nonumber\\
& \leq\int_{0}^{t}\Vert G_{\gamma}(t-s)[\nabla\cdot(P[\theta]\phi
)(s)]\Vert_{\dot{H}_{h}^{\beta-1+\delta}}\text{ }ds\nonumber\\
& \leq\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\Vert\nabla
\cdot(P[\theta]\phi)(s)]\Vert_{\dot{H}_{h}^{-1}}\text{ }ds\nonumber\\
& \leq\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\Vert P[\theta
]\phi(s)\Vert_{h}\text{ }ds. \label{bilinormh3}$$ In view of (\[aux-param1\]), we can choose $1<l<\infty$ in such a way that $l>q$, $\frac{1}{h}=\frac{1}{r}+\frac{1}{l}$ and $\frac{1}{l}=\frac{1}{q}-\frac{\beta-1}{n}$. Then, Hölder inequality, (\[fjotas\]), and Sobolev embedding (\[Sobolev\]) imply that $$\begin{aligned}
\Vert P[\theta]\phi\Vert_{h} & \leq\Vert P[\theta]\Vert_{r}\Vert\phi
\Vert_{l}\nonumber\\
& \leq\Vert\theta\Vert_{\dot{H}_{r}^{\beta-1}}\Vert\phi\Vert_{\dot{H}_{q}^{\beta-1}}. \label{holdersobolev}$$ Inserting (\[holdersobolev\]) into (\[bilinormh3\]), we get $$\begin{aligned}
\Vert B(\theta,\phi)\Vert_{\dot{H}_{r}^{\beta-1}} & \leq C\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\Vert\theta\Vert_{\dot{H}_{r}^{\beta-1}}\Vert\phi\Vert_{\dot{H}_{q}^{\beta-1}}\text{ }ds\\
& \leq C\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\text{ }s^{-\eta
_{r}-\eta_{q}}ds\sup_{0<t<T}t^{\eta_{r}}\Vert\theta(t)\Vert_{\dot{H}_{r}^{\beta-1}}\sup_{0<t<T}t^{\eta_{q}}\Vert\phi(t)\Vert_{\dot{H}_{q}^{\beta-1}}\\
& \leq t^{-\frac{\beta+\delta}{2\gamma}-\eta_{r}-\eta_{q}+1}\int_{0}^{1}(1-s)^{-\frac{\beta+\delta}{2\gamma}}s^{-\eta_{r}-\eta_{q}}\text{ }ds\sup_{0<t<T}t^{\eta_{r}}\Vert\theta(t)\Vert_{\dot{H}_{r}^{\beta-1}}\sup_{0<t<T}t^{\eta_{q}}\Vert\phi(t)\Vert_{\dot{H}_{q}^{\beta-1}}\\
& \leq Ct^{-\eta_{r}}\sup_{0<t<T}t^{\eta_{r}}\Vert\theta(t)\Vert_{\dot{H}_{r}^{\beta-1}}\sup_{0<t<T}t^{\eta_{q}}\Vert\phi(t)\Vert_{\dot{H}_{q}^{\beta-1}},\end{aligned}$$ which is equivalent to (\[bili-4\]). [ ]{}
Proofs
======
Local in Time Solutions
-----------------------
We start by recalling an abstract lemma in Banach spaces which is useful in order to avoid extensive fixed point computations (see e.g. [@Lewis Theorem 9]).
\[genlem\] Let $X$ be a Banach space with norm $\Vert\cdot\Vert_{X}$, and $B:X\times X\rightarrow X$ be a continuous bilinear map, i.e., there exists $K>0$ such that $$\Vert B(x_{1},x_{2})\Vert_{X}\leq K\text{ }\Vert x_{1}\Vert_{X}\text{ }\Vert
x_{2}\Vert_{X},$$ for all $x_{1},x_{2}\in X$. Given $0<\varepsilon<\frac{1}{4K}$ and $y\in X$ such that $\Vert y\Vert_{X}\leq\varepsilon$, there exists a solution $x\in X$ for the equation $x=y+B(x,x)$ such that $\Vert x\Vert_{X}\leq2\varepsilon$. The solution $x$ is unique in the closed ball $\left\{ x\in X:\left\Vert
x\right\Vert _{X}\leq2\varepsilon\right\} .$ Moreover, the solution depends continuously on $y$ in the following sense: If $\Vert\tilde{y}\Vert_{X}\leq\varepsilon$, $\tilde{x}=\tilde{y}+B(\tilde{x},\tilde{x})$, and $\Vert\tilde{x}\Vert_{X}\leq2\varepsilon$, then $$\Vert x-\tilde{x}\Vert_{X}\leq\frac{1}{1-4K\varepsilon}\Vert y-\tilde{y}\Vert_{X}.$$
\[rem-seq\](Picard sequence) The solution given by Lemma \[genlem\] can be obtained as the limit in $X$ of the Picard sequence $\{x_{k}\}_{k\in
\mathbb{N}}$ where $x_{1}=y$ and $x_{k+1}=y+B(x_{k},x_{k}),$ for all $k\in\mathbb{N}$. Moreover, $\left\Vert x_{k}\right\Vert _{X}\leq2\varepsilon$ for all $k\in\mathbb{N}$.
The following proposition shows that (\[dase\])-(\[ujotas\]) is locally in time well-posed for $L^{\frac{n}{2\gamma-\beta}}$-data.
\[Proplocal-1\](Local in time solutions) Assume (\[cond-1\]) and let $q$ be such that $$\max\left\{ \frac{\beta-1}{n},\frac{\gamma-1}{n}\right\} <\frac{1}{q}<\min\left\{ \frac{2\gamma-\beta}{n},\frac{n+\beta-2\gamma}{n},\frac
{n+\beta-1}{2n}\right\} .$$ If $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ then there exists $T>0$ such that (\[dase\])-(\[ujotas\]) has a unique mild solution $\theta$ in the class $$t^{\eta_{q}}\theta\in BC((0,T);\dot{H}_{q}^{\beta-1}(\mathbb{R}^{n}))\text{
and }\lim_{t\rightarrow0^{+}}t^{\eta_{q}}\left\Vert \theta\right\Vert
_{\dot{H}_{q}^{\beta-1}}=0. \label{loc-sol-1}$$ Moreover, $\theta\in BC([0,T);L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})).$
**Proof.**
*Step 1:* For $T>0$, let us define the Banach space $$\mathcal{E}_{T}=\left\{ \theta\text{ measurable};\text{ }t^{\eta_{q}}\theta\in BC((0,T);\dot{H}_{q}^{\beta-1}(\mathbb{R}^{n}))\right\}$$ with norm given by $$\left\Vert \theta\right\Vert _{\mathcal{E}_{T}}=\sup_{0<t<T}t^{\eta_{q}}\Vert\theta(\cdot,t)\Vert_{\dot{H}_{q}^{\beta-1}}. \label{norm1}$$ Due to (\[zeroPrincipal\]) in Lemma \[sol fund\] and $\theta_{0}\in
L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}),$ for any $\varepsilon>0$, there exists a $T>0$ such that $$\sup_{0<t<T}t^{\eta_{q}}\Vert G_{\gamma}(t)\theta_{0}\Vert_{\dot{H}_{q}^{\beta-1}}\leq\varepsilon. \label{aux-lin}$$ Take $0<\varepsilon<\frac{1}{4K_{4}}$ where $K_{4}$ is as in (\[bili-4\]) with $r=q$. In view of (\[aux-lin\]) and (\[bili-4\]), we can apply Lemma \[genlem\] in $\mathcal{E}_{T}$ to obtain a unique solution $\theta(x,t)$ for (\[mild\]) such that$$\ \sup_{0<t<T}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\leq2\varepsilon. \label{aux-prop1}$$
Using (\[zeroPrincipal\]) and an induction argument, one can shows that every element $\theta_{k}$ of the Picard sequence $$\begin{aligned}
\theta_{1}(x,t) & =G_{\gamma}(t)\theta_{0}(x),\label{seq1}\\
\theta_{k+1}(x,t) & =\theta_{1}(x,t)+B(\theta_{k},\theta_{k}),\text{ }k\in\mathbb{N}, \label{seq2}$$ satisfies $\lim_{t\rightarrow0^{+}}t^{\eta_{q}}\left\Vert \theta
_{k}\right\Vert _{\dot{H}_{q}^{\beta-1}}=0.$ Then the second property in (\[loc-sol-1\]) follows from the fact that the fixed point $\theta$ is the limit in $\mathcal{E}_{T}$ of $\{\theta_{k}\}_{k\in\mathbb{N}}$ (see Remark \[rem-seq\]). Further details are left to the reader.
*Step 2:* In what follows we show that $\theta\in$ **** $BC([0,T);L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}))$. We have that the recursive sequence (\[seq1\])-(\[seq2\]) satisfies (see Remark \[rem-seq\]) $$\sup_{0<t<T}t^{\eta_{q}}\Vert\theta_{k}\Vert_{\dot{H}_{q}^{\beta-1}}\leq2\varepsilon,\text{ for all }k\in\mathbb{N}. \label{aux-seq-1}$$ Using Lemma \[sol fund\], (\[bili-2\]) with $p=\frac{n}{2\gamma-\beta}$, and (\[aux-seq-1\]), we get $$\sup_{0<t<T}\Vert\theta_{1}(t)\Vert_{\frac{n}{2\gamma-\beta}}\leq C\Vert
\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}\text{,}$$ and $$\begin{aligned}
\sup_{0<t<T}\Vert\theta_{k+1}(t)\Vert_{\frac{n}{2\gamma-\beta}} & \leq
C\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+K_{2}\sup_{0<t<T}t^{\eta_{q}}\Vert\theta_{k}(t)\Vert_{\dot{H}_{q}^{\beta-1}}\sup_{0<t<T}\Vert\theta
_{k}(t)\Vert_{\frac{n}{2\gamma-\beta}}\nonumber\\
& \leq C\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+2\varepsilon K_{2}\sup_{0<t<T}\Vert\theta_{k}(t)\Vert_{\frac{n}{2\gamma-\beta}},\text{ for all
}k\in\mathbb{N}. \label{aux-seq-3}$$ By reducing $T>0$ in (\[aux-lin\]) if necessary, we can consider $0<\varepsilon<\min\{\frac{1}{4K_{4}},\frac{1}{2K_{2}}\}.$ Since $2K_{2}\varepsilon<1,$ an induction argument in (\[aux-seq-3\]) shows that $\{\theta_{k}\}_{k\in\mathbb{N}}$ is uniformly bounded in $L^{\infty
}((0,T);L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}))$ and then there exists a subsequence of $\{\theta_{k}\}_{k\in\mathbb{N}}$ (denoted in the same way) that converges toward $\widetilde{\theta}$ weak$-\ast$ in that space and consequently in $\mathcal{D}^{\prime}(\mathbb{R}^{n}\times\lbrack0,T)).$ Because $\theta_{k}\rightarrow\theta$ in $\mathcal{E}_{T}$, which implies convergence in $\mathcal{D}^{\prime}(\mathbb{R}^{n}\times\lbrack0,T)),$ the uniqueness of the limit in the sense of distributions yields $\theta
=\widetilde{\theta}\in L^{\infty}((0,T);L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}))$. The time-continuity of $\theta$ follows from standard arguments by using that $\theta$ verifies (\[mild\]), $\theta\in L^{\infty
}((0,T);L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}))\cap\mathcal{E}_{T}$, and the second property in (\[loc-sol-1\]) (see e.g. [@Ka1; @Ka2]).
[ ]{}
In the next proposition we investigate the $L^{1}$-persistence of the solutions obtained in Proposition \[Proplocal-1\].
\[Proplocal-2\]Under hypotheses of Proposition \[Proplocal-1\]. There exists $T>0$ such that the solution $\theta$ belongs to $BC\left(
[0,T);L^{1}(\mathbb{R}^{n})\right) $ provided that $\theta_{0}\in
L^{1}(\mathbb{R}^{n})\cap L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$.
**Proof.** Let $q$ be such that $1<q^{\prime}<\frac{n}{2\gamma-\beta}.$ From interpolation, we have that $\theta_{0}\in L^{q^{\prime}}(\mathbb{R}^{n}).$ Employing (\[aux-seq-1\]) and the estimate (\[bili-2\]) with $p=q^{\prime}$ , we get $$\sup_{0<t<T}\Vert\theta_{1}(t)\Vert_{q^{\prime}}\leq C\Vert\theta_{0}\Vert_{q^{\prime}}\text{,}$$ and $$\begin{aligned}
\sup_{0<t<T}\Vert\theta_{k+1}(t)\Vert_{q^{\prime}} & \leq C\Vert\theta
_{0}\Vert_{q^{\prime}}+K_{2}\sup_{0<t<T}t^{\eta_{q}}\Vert\theta_{k}(t)\Vert_{\dot{H}_{q}^{\beta-1}}\sup_{0<t<T}\Vert\theta_{k}(t)\Vert
_{q^{\prime}}\\
& \leq C\Vert\theta_{0}\Vert_{q^{\prime}}+2K_{2}\varepsilon\sup_{0<t<T}\Vert\theta_{k}(t)\Vert_{q^{\prime}},\text{ for all }k\in\mathbb{N}.\end{aligned}$$ Again reducing $T>0$ if necessary, we can consider $2K_{2}\varepsilon<1$ and proceed similarly to the end of the proof of Proposition \[Proplocal-1\] to obtain that $$\theta\in BC\left( [0,T);L^{q^{\prime}}(\mathbb{R}^{n})\right) .
\label{loc-sol-2}$$ Now we use (\[est linear2\]), (\[bili-3\]), (\[loc-sol-1\]), (\[loc-sol-2\]) to estimate $$\begin{aligned}
\sup_{0<t<T}\Vert\theta(t)\Vert_{1} & \leq C\Vert\theta_{0}\Vert_{1}+\sup_{0<t<T}\left\Vert B(\theta,\theta)\right\Vert _{1}\\
& \leq C\Vert\theta_{0}\Vert_{1}+K_{3}T^{1-\frac{1}{2\gamma}-\eta_{q}}\sup_{0<t<T}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\,\sup
_{0<t<T}\Vert\theta\Vert_{q^{\prime}}<\infty,\end{aligned}$$ as required.
[ ]{}
The existence time $T$ in Propositions \[Proplocal-1\] and \[Proplocal-2\] may depend on index $q.$ In the next proposition we show that indeed one can take a same small time $T>0$ for all $q.$
\[extnormas\] Under hypotheses of Proposition \[Proplocal-1\]. Let $\theta$ be the solution given by Proposition \[Proplocal-1\] with data $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$. There is a $T>0$ such that $$\begin{aligned}
t^{\eta_{r}}\theta & \in BC((0,T);\dot{H}_{r}^{\beta-1}(\mathbb{R}^{n})),\text{ for all }\frac{n}{2\gamma-\beta}<r<\infty,\label{solutionL}\\
\text{ }t^{\tilde{\eta}_{r}}\theta & \in BC((0,T);L^{r}(\mathbb{R}^{n})),\text{ for all }\frac{n}{2\gamma-\beta}<r<\infty. \label{solutionL2}$$
**Proof.** Let $q$ be fixed and as in Proposition \[Proplocal-1\]. Given $\frac{n}{2\gamma-\beta}<r<\infty$ verifying $\frac{2\gamma-2}{n}-\frac{1}{q}<\frac{1}{r}<\frac{n+\beta-1}{n}-\frac{1}{q},$ we can use (\[bili-4\]) instead of (\[bili-1\]) and proceed just like as in step 2 of the proof of Proposition \[Proplocal-1\] to obtain (reducing $T>0$ if necessary) $$t^{\eta_{r}}\theta\in BC((0,T);\dot{H}_{r}^{\beta-1}(\mathbb{R}^{n})).
\label{aux-H1}$$ Now let $\frac{2\gamma-2}{n}-\frac{1}{q}<\frac{1}{r}<\frac{2\gamma-1}{n}-\frac{1}{q},$ and consider $r<\tilde{r}<\infty$ . Taking $\frac{1}{h}=\frac{1}{q}+\frac{1}{z}=\frac{1}{q}+\frac{1}{r}-\frac{\beta-1}{n}$ and $\delta=\frac{n}{h}-\frac{n}{\tilde{r}}$, it follows that $\delta>0,$ $\frac{\beta+\delta}{2\gamma}<1,$ and $\eta_{q}+\eta_{r}<1$. Then, we can estimate $$\begin{aligned}
\Vert\theta\Vert_{\dot{H}_{\tilde{r}}^{\beta-1}} & \leq Ct^{-\eta_{\tilde
{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+\int_{0}^{t}\Vert
G_{\gamma}(t-s)\nabla_{x}\cdot(P[\theta]\theta)(s)\Vert_{\dot{H}_{\tilde{r}}^{\beta-1}}\text{ }ds\nonumber\\
& \leq Ct^{-\eta_{\tilde{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+C\int_{0}^{t}\Vert G_{\gamma}(t-s)\nabla_{x}\cdot(P[\theta]\theta
)(s)\Vert_{\dot{H}_{h}^{\beta-1+\delta}}\text{ }ds\nonumber\\
& \leq Ct^{-\eta_{\tilde{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+C\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\Vert\nabla_{x}\cdot(P[\theta]\theta)(s)\Vert_{\dot{H}_{h}^{-1}}\text{ }ds,
\label{aux-proof2}$$ where above we have used Sobolev embedding and afterwards (\[est linear\]).
Now we employ (\[fjotas\]), Hölder inequality and Sobolev embedding in order to estimate $$\begin{aligned}
& \text{R.H.S. of (\ref{aux-proof2})}\nonumber\\
& \leq Ct^{-\eta_{\tilde{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+C\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\Vert P[\theta
]\theta(s)\Vert_{h}\text{ }ds\nonumber\\
& \leq Ct^{-\eta_{\tilde{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+C\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\Vert P[\theta]\Vert
_{q}\Vert\theta(s)\Vert_{z}\text{ }ds\nonumber\\
& \leq Ct^{-\eta_{\tilde{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+C\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}\Vert\theta\Vert_{\dot
{H}_{q}^{\beta-1}}\Vert\theta\Vert_{\dot{H}_{r}^{\beta-1}}\text{
}ds\nonumber\\
& \leq Ct^{-\eta_{\tilde{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+C\int_{0}^{t}(t-s)^{-\frac{\beta+\delta}{2\gamma}}s^{-\eta_{q}-\eta_{r}}\text{ }ds\left( \sup_{0<t<T}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}\right) \left( \sup_{0<t<T}t^{\eta_{r}}\Vert\theta\Vert
_{\dot{H}_{r}^{\beta-1}}\right) \label{aux-T}\\
& \leq Ct^{-\eta_{\tilde{r}}}\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}+Ct^{-\eta_{\tilde{r}}}\int_{0}^{1}(1-s)^{-\frac{\beta+\delta}{2\gamma}}s^{-\eta_{q}-\eta_{r}}\text{ }ds\nonumber\\
& \leq Ct^{-\eta_{\tilde{r}}},\nonumber\end{aligned}$$ and then we arrive at (\[aux-H1\]) with $\tilde{r}$ in place of $r$, and with the same existence time $T>0$. From interpolation, notice that (\[aux-H1\]) also holds true for every $r=l$ such that $\frac{n}{2\gamma-\beta}<l<\tilde{r}.$ Since $\tilde{r}>r$ is arbitrary, we obtain (\[aux-H1\]) with $r=l$ ( and the same $T>0$), for all $\frac{n}{2\gamma-\beta}<l<\infty,$ which gives (\[solutionL\]).
The proof of (\[solutionL2\]) can be performed in a similar way by using (\[bili-1\]) instead of (\[bili-4\]). [ ]{}
Proof of Theorem \[teoglobal\]
------------------------------
### Step 1: Local smoothness and maximum principle
The solutions obtained in Proposition \[Proplocal-1\] are instantaneously $C^{\infty}$-smoothed for any $t>0$ belonging to the existence interval $(0,T).\,$ This smooth effect holds for several parabolic equations in several frameworks, like e.g. $L^{p},$ weak-$L^{p}$, Morrey, Besov-Morrey, when mild solutions are constructed by using time-weighted norms of Kato type (see [@Ka1]). Precisely, adapting arguments from [@Ka1] (see also [@Car-Fer2]), one can obtain that the solution verifies$$\partial_{t}^{m}\nabla_{x}^{k}\theta(x,t)\in C\left( (0,T);L^{\frac
{n}{2\gamma-\beta}}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n})\right) ,
\label{derivadas}$$ for all $\frac{n}{2\gamma-\beta}<q<\infty,$ $m\in\{0\}\cup\mathbb{N}$ and multi-index $k\in(\{0\}\cup\mathbb{N)}^{n},$ where $T>0$ is the existence time given in Proposition \[extnormas\]. In particular, it follows that $\theta(x,t)\in C^{\infty}(\mathbb{R}^{n}\times(0,T))$ and $\theta(t)\in
L^{\infty}(\mathbb{R}^{n})$ with $$\left\Vert \theta(t)\right\Vert _{\infty}\leq C\left\Vert \theta(t)\right\Vert
_{\frac{n^{2}}{2\gamma-\beta}}^{\alpha}\left\Vert \nabla_{x}\theta
(t)\right\Vert _{\frac{n^{2}}{2\gamma-\beta}}^{1-\alpha},\text{ }
\label{Aux-inf}$$ for all $0<t<T,$ where $\alpha=\frac{n+\beta-2\gamma}{n}$. If further $\theta_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ then $q$ in (\[derivadas\]) can be taken in the range $1<q<\infty.$
Due to (\[derivadas\]) we have that $\theta$ verifies (\[dase\])-(\[ujotas\]) in the classical sense and $\partial_{t}^{m}\nabla_{x}^{k}\theta(x,t)\rightarrow0$ when $|x|\rightarrow\infty,$ for all $0<t<T.$ In view of $\nabla\cdot$ $P[\theta]=0$, we can integrate by parts to obtain $$\begin{aligned}
\frac{\partial}{\partial t}\Vert\theta(t)\Vert_{p}^{p} & =p\int
_{\mathbb{R}^{n}}\theta(t)^{p-1}\frac{\partial}{\partial t}\theta
(t)dx\nonumber\\
& =p\int_{\mathbb{R}^{n}}\theta(t)^{p-1}\left( -(-\Delta)^{\gamma}\theta-\nabla_{x}\cdot(P[\theta]\theta)\right) dx\nonumber\\
& =-p\int_{\mathbb{R}^{n}}\theta(t)^{p-1}(-\Delta)^{\gamma}\theta dx\leq
-\int_{\mathbb{R}^{2}}\left\vert (-\Delta)^{\frac{\gamma}{2}}(\theta^{\frac
{p}{2}})\right\vert ^{2}dx, \label{ener}$$ for all $t\in(0,T)$. The last inequality in (\[ener\]) can be found in [@Const3; @Cordoba1] (see also [@Ju]).
In view of the estimate (\[ener\]), we have that $L^{p}$-norms of $\theta(t)$ are non-increasing in $(0,T).$ If $\theta_{0}\in L^{\frac
{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ and $\theta_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n}),$ we obtain respectively $$\Vert\theta(t)\Vert_{\frac{n}{2\gamma-\beta}}\leq\Vert\theta(t_{0})\Vert_{\frac{n}{2\gamma-\beta}}\text{ and }\Vert\theta(t)\Vert_{1}\leq
\Vert\theta(t_{0})\Vert_{1}, \label{aux-max-1}$$ for $0<t_{0}\leq t<T.$
Making $t_{0}\rightarrow0^{+}$ in (\[aux-max-1\]), it follows that the solution $\theta(x,t)$ satisfies $$\Vert\theta(t)\Vert_{\frac{n}{2\gamma-\beta}}\leq\Vert\theta_{0}\Vert
_{\frac{n}{2\gamma-\beta}}\text{ and }\Vert\theta(t)\Vert_{1}\leq\Vert
\theta_{0}\Vert_{1}, \label{max-prin}$$ for all $t\in(0,T),$ when $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ and $\theta_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{\frac
{n}{2\gamma-\beta}}(\mathbb{R}^{n}),$ respectively.
### Step 2: Extension of the local solution
We start by making the following observation: if $\frac
{n}{2\gamma-\beta}<q<\infty$ and $\theta_{0}\in L^{q}(\mathbb{R}^{n})$ then $$t^{\eta_{q}}\Vert G_{\gamma}(t)\theta_{0}\Vert_{\dot{H}_{q}^{\beta-1}}\leq
Ct^{\tilde{\eta}_{q}}\Vert\theta_{0}\Vert_{q}\rightarrow0\text{ when
}t\rightarrow0^{+}. \label{aux-ext-1}$$ Therefore, for $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})\cap
L^{q}(\mathbb{R}^{n})$, the existence time $T>0$ obtained in Proposition \[Proplocal-1\] can be taken depending on the norm $\Vert\theta_{0}\Vert
_{q}.$ Indeed it can be chosen as $$T=\left( \frac{\varepsilon}{C\Vert\theta_{0}\Vert_{q}}\right) ^{\frac
{1}{\tilde{\eta}_{q}}}, \label{aux-T-1}$$ where $0<\varepsilon<\frac{1}{4K_{4}}$ and $C>0$ is as in (\[aux-ext-1\]).
Now let $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$. From Proposition \[Proplocal-1\], there exist $M_{0}>0$, $T_{0}>0$ and a unique mild solution for (\[dase\])-(\[ujotas\]) in $(0,T_{0})$ such that $$\sup_{0<t<T_{0}}t^{\eta_{q}}\Vert\theta(t)\Vert_{\dot{H}_{q}^{\beta-1}}\leq
M_{0}\text{ and }\sup_{0<t<T_{0}}\Vert\theta(t)\Vert_{\frac{n}{2\gamma-\beta}}\leq\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}, \label{aux-proof1}$$ where (\[max-prin\]) has been used in the second inequality in (\[aux-proof1\]).
Let us now denote $$T=\sup\left\{ \tilde{T}>0;\theta\in C((0,\tilde{T});\dot{H}_{q}^{\beta
-1}(\mathbb{R}^{n})\cap L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})),\sup_{0<t<\tilde{T}}t^{\eta_{q}}\Vert\theta\Vert_{\dot{H}_{q}^{\beta-1}}<\infty,\sup_{0<t<\tilde{T}}\Vert\theta\Vert_{\frac{n}{2\gamma-\beta}}\leq\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}}\right\} . \label{aux-def1}$$ We desire to prove that $T=\infty.$ Suppose by contradiction that $T<\infty$, and let $a=\theta(T-\varepsilon)$ where $0<\varepsilon<T$ will be choose later. Proposition \[extnormas\] gives that $a\in L^{\frac{n}{2\gamma-\beta
}}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n}),$ for all $\frac{n}{2\gamma
-\beta}<q<\infty$. Moreover, if $\varepsilon<\frac{T}{2}$ we get $\Vert
\theta(T-\varepsilon)\Vert_{q}\leq\Vert\theta(\frac{T}{2})\Vert_{q}$. Therefore, taking $\varepsilon<T/2$ and $a$ as initial data, we have that given $0<M_{1}<\frac{1}{4K_{4}}$ there exist $T_{1}>0$ and a unique mild solution $\tilde{\theta}$ for (\[dase\])-(\[ujotas\]) such that$$\tilde{\theta}\in C((T-\varepsilon,T_{1}+T-\varepsilon);L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})\cap\dot{H}_{q}^{\beta-1}(\mathbb{R}^{n}))$$ and $$\begin{aligned}
\sup_{T-\varepsilon<t<T_{1}+T-\varepsilon}(t-(T-\varepsilon))^{\eta_{q}}\left\Vert \tilde{\theta}(t)\right\Vert _{\dot{H}_{q}^{\beta-1}} &
\leq2M_{1}\label{aux-ext-10}\\
\sup_{T-\varepsilon<t<T_{1}+T-\varepsilon}\Vert\tilde{\theta}(t)\Vert
_{\frac{n}{2\gamma-\beta}} & \leq\Vert\theta(T-\varepsilon)\Vert_{\frac
{n}{2\gamma-\beta}}.\nonumber\end{aligned}$$ From uniqueness part of Proposition \[Proplocal-1\], it follows that $\theta=\tilde{\theta}$ in $(T-\varepsilon,T).$ In view of (\[aux-T-1\]), we can choose $T_{1}=\min\left\{ \left( \frac{M_{1}}{C\Vert\theta(\frac{T}{2})\Vert_{q}}\right) ^{\frac{1}{\tilde{\eta}_{q}}},T\right\} $. Taking $0<\varepsilon<\min\{\frac{T}{2},T_{1}\}$ and $T_{2}=T_{1}+T-\varepsilon,$ we have that $T<T_{2}$ and get a solution$$\theta\in C((0,T_{2});L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})\cap\dot
{H}_{q}^{\beta-1}(\mathbb{R}^{n}))$$ such that $$\sup_{0<t<\tilde{T}}t^{\eta_{q}}\left\Vert \theta(t)\right\Vert _{\dot{H}_{q}^{\beta-1}}<\infty\text{ and }\sup_{0<t<\tilde{T}}\Vert\theta
(t)\Vert_{\frac{n}{2\gamma-\beta}}\leq\Vert\theta_{0}\Vert_{\frac{n}{2\gamma-\beta}},$$ for all $0<\tilde{T}<T_{2},$ which contradicts the maximality of $T$ in (\[aux-def1\]). Therefore $T=\infty$ and we are done.
### Step 3: Global $L^{q}$-decay of solutions
We will prove only the part of the statement corresponding to the case $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$. The estimate (\[Decay-Lq-2\]) for $\theta_{0}\in
L^{1}(\mathbb{R}^{n})\cap L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ follows similarly to the first one by using $\left\Vert \theta(t)\right\Vert _{1}\leq\left\Vert \theta_{0}\right\Vert _{1}$ and the sequence $q_{k}=2^{k}$ instead of $\left\Vert \theta(t)\right\Vert _{\frac{n}{2\gamma-\beta}}\leq\left\Vert \theta_{0}\right\Vert _{\frac{n}{2\gamma-\beta}}$ and $q_{k}=\frac{n}{2\gamma-\beta}2^{k}$.
Since we have extended the solution $\theta$, it follows that (\[derivadas\]) and (\[Aux-inf\]) hold true for $T=\infty$. Then $$\left\Vert \theta(\cdot,t)\right\Vert _{\infty}<\infty,\text{ for all }t>0.
\label{Aux-inf-2}$$
Now we proceed as in [@Car-Fer1] and [@Ka2]. In view of the Gagliardo-Nirenberg inequality, we have that $$\left\Vert \phi\right\Vert _{2}\leq C\left\Vert \phi\right\Vert _{1}^{\alpha
}\left\Vert (-\Delta)^{\frac{\gamma}{2}}\phi\right\Vert _{2}^{1-\alpha}\text{
with }\alpha=\frac{2\gamma}{n+2\gamma}. \label{Ga-Ni}$$ Taking $\phi=\theta^{\frac{q}{2}}$ in (\[Ga-Ni\]), it follows that $$\left\Vert \theta\right\Vert _{q}^{q(\frac{n+2\gamma}{n})}\leq C\left\Vert
\theta\right\Vert _{\frac{q}{2}}^{\frac{2q\gamma}{n}}\left\Vert (-\Delta
)^{\frac{\gamma}{2}}(\theta^{\frac{q}{2}})\right\Vert _{2}^{2}\text{.}
\label{aux-Global1}$$ Denoting $\psi_{q}(t)=$ $\Vert\theta(t)\Vert_{q}^{q}$ , we obtain from (\[ener\]) and (\[aux-Global1\]) that $$\frac{\partial}{\partial t}\psi_{q}\leq-C(\psi_{\frac{q}{2}})^{-\frac{4\gamma
}{n}}\psi_{q}^{\frac{n+2\gamma}{n}}. \label{ineq}$$ The differential inequality (\[ineq\]) can be solved by an induction procedure. In fact, using the first inequality in (\[max-prin\]) and considering the sequence $q_{k}=\frac{n}{2\gamma-\beta}2^{k}$ for $k\geq0,$ we arrive at $$\psi_{q_{k}}(t)\leq M_{q_{k}}t^{-\frac{n}{2}(\frac{2^{k}-1}{\gamma})}\,,\,
\label{mx}$$ where $$M_{q_{0}}=\left\Vert \theta_{0}\right\Vert _{\frac{n}{2\gamma-\beta}}\text{
and }\,M_{q_{k}}=\left( \frac{n(2^{k}-1)}{2C\gamma}\right) ^{\frac
{n}{2\gamma}}M_{\frac{q_{k}}{2}}^{2}\text{, \ \ for }k\in\mathbb{N}.$$ It follows that $$\begin{aligned}
M_{q_{k}}^{\frac{1}{q_{k}}} & =\left( \frac{n(2^{k}-1)}{2C\gamma}\right)
^{\frac{n}{2\gamma q_{k}}}M_{q_{k-1}}^{\frac{1}{q_{k-1}}}=\left(
\frac{n(2^{k}-1)}{2C\gamma}\right) ^{\frac{n}{2\gamma2^{k}q_{0}}}\left(
\frac{n(2^{k-1}-1)}{2C\gamma}\right) ^{\frac{n}{2\gamma2^{k-1}q_{0}}}M_{q_{k-2}}^{\frac{1}{q_{k-2}}}=...\\
& =\left[ \Pi_{i=1}^{k}\left( \frac{n(2^{i}-1)}{2C\gamma}\right)
^{\frac{n}{2^{i}2\gamma q_{0}}}\right] (M_{q_{0}})^{\frac{1}{q_{0}}},\text{
for all }k\in\mathbb{N},\end{aligned}$$ and then $$\begin{aligned}
\Vert\theta(t)\Vert_{2^{k}q_{0}} & \leq M_{q_{k}}^{\frac{1}{q_{k}}}\,t^{-\frac{n}{2}(\frac{2^{k}-1}{\gamma q_{k}})}\nonumber\\
& =\left( \Pi_{i=1}^{k}\left( \frac{n}{2}\frac{2^{i}-1}{C\gamma}\right)
^{\frac{n}{2^{i}2\gamma q_{0}}}\right) (M_{q_{0}})^{\frac{1}{q_{0}}}\,t^{-\frac{n}{2\gamma q_{0}}(\frac{2^{k}-1}{2^{k}})}, \label{aux-global3}$$ where $q_{0}=\frac{n}{2\gamma-\beta}.$ In view of (\[Aux-inf-2\]), we can make $k\rightarrow\infty$ in (\[aux-global3\]) to obtain $$\Vert\theta(t)\Vert_{\infty}\leq C\Vert\theta_{0}\Vert_{\frac{n}{2\gamma
-\beta}}^{\frac{2\gamma-\beta}{n}}\,t^{-\frac{2\gamma-\beta}{2\gamma}}.
\label{aux-global4}$$ Interpolating the first inequality in (\[max-prin\]) with (\[aux-global4\]), the result is $$\Vert\theta(t)\Vert_{q}\leq C\,t^{-\tilde{\eta}_{q}},\text{ for all }\frac
{n}{2\gamma-\beta}\leq q\leq\infty,$$ as required.
The uniqueness statement follows from the local uniqueness property in Proposition \[Proplocal-1\].
[ ]{}
Proof of Theorem \[Teo-sym\]
----------------------------
**Part (i):** We will prove only the odd part of the statement since the even one follows similarly. Let $\theta$ be the solution of Proposition \[Proplocal-1\] with existence time $T>0$. From step 2 of the proof of Theorem \[teoglobal\], $\theta$ can be extended by using Proposition \[Proplocal-1\] and solving (\[dase\])-(\[ujotas\]) consecutively with initial data $\theta(\frac{T}{2}),$ $\theta(\frac{T}{2}+T_{1}),$ $\theta(\frac{T}{2}+2T_{1})$ and so on, where $T_{1}=\min\{\left( \frac{\varepsilon}{C\Vert\theta(\frac{T}{2})\Vert_{q}}\right)
^{\frac{1}{\tilde{\eta}_{q}}},T\}$, $\varepsilon=\frac{1}{8K_{4}}$, and $C$ as in (\[aux-ext-1\]). Because of that, it is sufficient to show the following claim: if $\theta_{0}\in L^{\frac{n}{2\gamma-\beta}}$ is odd then so is the solution $\theta(x,t)$ given by Proposition \[Proplocal-1\], for all $t\in(0,T)$. In fact, notice that we can use this claim repeatedly to show that the global solution $\theta(x,t)$ is odd, for all $t>0.$
Let $\psi(x,t)=G_{\gamma}(t)\theta_{0}.$ We have that $\theta_{0}(-x)=-\theta_{0}(x)$ is equivalent to $$-\widehat{\theta_{0}}(\xi)=[\theta_{0}(-x)]^{\wedge}(\xi)=\widehat{\theta_{0}}(-\xi)\text{ in }\mathcal{S}^{\prime}(\mathbb{R}^{n})\text{.}
\label{aux-sym1}$$ It follows from (\[aux-sym1\]) that $$\begin{aligned}
\lbrack\psi(-x,t)]^{\wedge}(\xi) & =e^{-t\left\vert \xi\right\vert
^{2\gamma}}\widehat{\theta_{0}}(-\xi)\\
& =-e^{-t\left\vert \xi\right\vert ^{2\gamma}}\widehat{\theta_{0}}(\xi)=-\widehat{\psi(x,t)}(\xi),\end{aligned}$$ which shows that $G_{\gamma}(t)\theta_{0}$ is odd, for each fixed $t>0.$ Also, if $\theta$ is odd then $\nabla\theta$ is even, because $$\nabla(\theta(x,t))=\nabla(-\theta(-x,t))=(\nabla\theta)(-x,t)).$$ Recall that $A=(a_{ij})$ and $$P(\xi)=(\widetilde{P}_{1}(\xi),...,\widetilde{P}_{n}(\xi))$$ where $$\widetilde{P}_{j}(\xi)=\sum_{i=1}^{n}a_{ij}\frac{\xi_{i}I}{\left\vert
\xi\right\vert ^{2}}P_{i}(\xi).$$ It follows that$$\widehat{u}(-\xi)=(\widehat{u_{1}}(-\xi),...,\widehat{u_{n}}(-\xi
))=P(-\xi)\widehat{\theta}(-\xi,t)\text{ \ \ }$$ with $$\begin{aligned}
P(-\xi) & =\frac{I}{\left\vert \xi\right\vert ^{2}}[-\xi_{1}P_{1}(-\xi),...,-\xi_{n}P_{n}(-\xi)]\text{ }A\\
& =\frac{I}{\left\vert \xi\right\vert ^{2}}[\xi_{1}P_{1}(\xi),...,\xi
_{n}P_{n}(\xi)]\text{ }A=P(\xi),\end{aligned}$$ because $P_{i}$’s are odd. Therefore $u=P[\theta]$ is odd when $\theta$ is odd, and then $(u\cdot\nabla\theta)=(P[\theta]\cdot\nabla\theta)$ is odd too. Hence if $\theta$ is odd then so is $B(\theta,\theta)$.
So, employing an induction argument, one can prove that each element $\theta_{k}$ of the Picard sequence (\[seq1\])-(\[seq2\]) is odd. Since $\theta_{k}\rightarrow\theta$ in the norm (\[norm1\]), then the sequence (\[seq1\])-(\[seq2\]) also converges (up to a subsequence) to $\theta$ a.e. $x\in\mathbb{R}^{n}$, for all $t\in(0,T)$. It follows that $\theta(x,t)$ is odd, for each fixed $t\in(0,T),$ because pointwise convergence preserves odd symmetry. This shows the desired claim.
**Part (ii):** From the same reasons given in *part (i)*, we need only to prove that the local solution of Proposition \[Proplocal-1\] is radially symmetric whenever $\theta_{0}$ and $div_{\xi}(P(\xi))$ are too. For that matter, we first observe that $G_{\gamma}(t)\theta_{0}$ is radial because $\theta_{0}$ and the kernel $\hat{g}_{\gamma}(\xi,t)=e^{-|\xi|^{2\gamma}t}$ are radial, for all $t>0.$ Also, for $\theta$ radially symmetric, we have that $$\begin{aligned}
(u\cdot\nabla\theta) & =\sum_{j=1}^{n}u_{j}\partial_{x_{j}}\theta
=\frac{\theta^{\prime}(r)}{r}\sum_{j=1}^{n}u_{j}x_{j}\nonumber\\
& =I\frac{\theta^{\prime}(r)}{r}\sum_{j=1}^{n}(\partial_{\xi_{j}}\widehat{u_{j}})^{\vee}=I\frac{\theta^{\prime}(r)}{r}\sum_{j=1}^{n}\left(
\partial_{\xi_{j}}\widetilde{P}_{j}(\xi)\widehat{\theta}\right) ^{\vee
}\nonumber\\
& =I\frac{\theta^{\prime}(r)}{r}\left( \widehat{\theta}(\xi,t)\left(
div_{\xi}(P(\xi)\right) \right) ^{^{\vee}}. \label{aux-rad1}$$ It follows from (\[aux-rad1\]) that if $\theta$ and $\left( div_{\xi}(P(\xi)\right) $ are radial then so is $(u\cdot\nabla\theta)$. Using that $G_{\gamma}(t)$ preserves radiality, we obtain that $B(\theta,\theta)$ defined in (\[termo bilinear\]) is radially symmetric, for each $t\in(0,T)$, whenever $\theta$ is too. Analogously to *part (i)*, we now can use induction in order to show that each function $\theta_{k}$ defined in (\[seq1\])-(\[seq2\]) is also radially symmetric. Since $\theta_{k}$ converges (up to a subsequence) to $\theta$ a.e. $x\in$ $\mathbb{R}^{n}$, for each $t\in(0,T)$, we obtain the required conclusion.
[ ]{}
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Boaz Barak[^1]'
- 'Siu On Chan[^2]'
- 'Pravesh Kothari[^3]'
bibliography:
- 'refs.bib'
title: Sum of Squares Lower Bounds from Pairwise Independence
---
Acknowledgements {#acknowledgements .unnumbered}
================
Thanks to Ryan O’Donnell, Li-Yang Tan, and David Steurer for fruitful discussions and the anonymous reviewers for their valuable comments and suggestions on a previous version of this paper.
[^1]: Microsoft Research New England
[^2]: Microsoft Research New England
[^3]: University of Texas, Austin. Work done while an intern at Microsoft Research New England.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This article surveys the *System Level Synthesis* framework, which presents a novel perspective on constrained robust and optimal controller synthesis for linear systems. We show how SLS shifts the controller synthesis task from the design of a controller to the design of the entire closed loop system, and highlight the benefits of this approach in terms of scalability and transparency. We emphasize two particular applications of SLS, namely large-scale distributed optimal control and robust control. In the case of distributed control, we show how SLS allows for localized controllers to be computed, extending robust and optimal control methods to large-scale systems under practical and realistic assumptions. In the case of robust control, we show how SLS allows for novel design methodologies that, for the first time, quantify the degradation in performance of a robust controller due to model uncertainty – such transparency is key in allowing robust control methods to interact, in a principled way, with modern techniques from machine learning and statistical inference. Throughout, we emphasize practical and efficient computational solutions, and demonstrate our methods on easy to understand case studies.'
author:
- 'James Anderson, John C. Doyle, Steven Low, Nikolai Matni'
bibliography:
- 'Distributed\_new.bib'
- 'lqr.bib'
title: System Level Synthesis
---
Introduction {#sec:intro}
============
Prior Work {#sec:prior}
----------
Paper Structure
---------------
The paper is organized as follows. We begin with a gentle warmup in Section \[sec:sys\_resp\_intro\], wherein we consider full-state-feedback problems over a finite time horizon – by restricting ourselves to this simplified setting, we are able to build up nearly all of the core machinery of SLS with minimal technical overhead. We show how several known optimal and robust control synthesis problems can be posed in the SLS framework, essentially turning (robust) optimal control problems into (robust) optimization problems over system responses.
In Section \[sec:LTIinfinite\_horizon\] we formally present the general optimal control problem formulation, and briefly review classical solution approaches. With this information in hand, the finite time horizon results of Section \[sec:sys\_resp\_intro\] are extended to the infinite horizon setting in Section \[sec:state\_feedback\_SLS\]. The notion of locality is introduced in Section \[sec:locality\] and then used to show how to decompose the synthesis problem into uncoupled subproblems of reduced dimension in Section \[sec:scalability\]. We end with a robust extension of the previous results, and show how it can be used to guarantee near-optimal performance of computationally tractable solutions to SLS problems. Finally, in Section \[sec:output\], we extend these results to the output-feedback setting. We end with conclusions in Section \[sec:conclusion\].
Working with System Responses {#sec:sys_resp_intro}
=============================
To introduce the notion of a system response, and to illustrate some of its benefits, we begin by considering linear optimal control problems over a finite time horizon with full state-feedback. We show how the corresponding controller synthesis task can be cast as an optimization over the closed loop behavior of the system, i.e., over system responses, as opposed to over the controller itself. By initially restricting ourselves to a finite time horizon, we are not burdened by the technical overhead associated with system stability, controller internal stability, or infinite dimensional optimization problems. In subsequent sections, we build upon the material presented here and extend these methods to infinite horizon problems, distributed control problems, and to the output-feedback setting.
Finite-horizon System Level Synthesis {#sec:time}
-------------------------------------
A new perspective on robustness {#sec:robust-time}
-------------------------------
Thus far, we have only shown that SLS can be used to cast standard optimal control problems as optimizations over system responses. As a preview of some of the additional benefits of working with system responses, we present here a novel robust control synthesis method that allows us to bound the performance degradation incurred by using approximate system responses that do not exactly satisfy the achievability constraints . In later sections, we extend these robustness results to the infinite horizon setting.
Preliminaries and Notation {#sec:LTIinfinite_horizon}
==========================
We now formally introduce the infinite horizon optimal control problems that we solve in this paper.
Notation
--------
We use lower and upper case Latin letters such as $x$ and $A$ to denote vectors and matrices, respectively, and lower and upper case boldface Latin letters such as $\tf x$ and $\tf G$ to denote signals and transfer matrices, respectively. We use calligraphic letters such as $\s$ to denote sets. In the interest of clarity, we work with discrete time linear time invariant systems, but unless stated otherwise, all results extend naturally to the continuous time setting. We use standard definitions of the Hardy spaces $\mathcal{H}_2$ and $\mathcal{H}_\infty$, and denote their restriction to the set of real-rational proper transfer matrices by $\mathcal{RH}_2$ and $\RHinf$. We use $G(i)$ to denote the $i$th spectral component of a transfer function $\tf G$, i.e., $\tf G(z) = \sum_{i=0}^{\infty} \frac{1}{z^i} G(i)$ for $| z | > 1$. We use $\FT$ to denote the space of finite impulse response (FIR) transfer matrices with horizon $T$, i.e., $\FT := \{ \tf G \in \RHinf \, | \, \tf G = \sum_{i=0}^T\frac{1}{z^i}G(i)\}$. We frequently use the notation $\tf G \in \frac{1}{z}\RHinf$ to denote that $\tf G$ is strictly proper. Informally this can be parsed as $z\tf G \in \RHinf$. Finally, we use $\tf G(T_1:T_2)$ to denote the projection of $\tf G$ onto $\mathcal{F}_{T_1}^\perp \cap \mathcal{F}_{T_2}$, i.e., $\tf G(T_1:T_2) = \sum_{i=T_1}^{T_2}\frac{1}{z^i}G(i)$.
State-Feedback System Level Synthesis {#sec:state_feedback_SLS}
=====================================
In this section, we extend the results of Section \[sec:sys\_resp\_intro\] to the infinite horizon setting, and propose a novel parameterization of internally stabilizing state-feedback controllers centered around *system responses*, which are defined by the closed loop maps from process disturbances to state and control action. We show that for a given system, the set of stable is an affine subspace of $\RHinf$, and that the corresponding internally stabilizing controller achieving the desired system response admits a particularly simple and transparent realization.
Distributed Control, Locality, and Scalability
----------------------------------------------
Localized LQR Optimal Control {#sec:llqr}
-----------------------------
Robustness {#sec:robust}
----------
The previous sections highlighted the benefits of spatiotemporal locality: imposing FIR constraints trivially lead to finite-dimensional optimization problems (and simple filter-bank based controller implementations), and spatial locality leads to huge wins in terms of controller synthesis and implementation complexity. However, both of these constraints can often be fairly restrictive in practice: indeed, if a system is stabilizable, but not controllable, we do not expect any such constraints to hold exactly. This section extends the robustness results presented in Section \[sec:robust-time\] to the infinite horizon setting, allowing us to circumvent these issues. Further, we show that this allows us to formalize the folk theorem that “good controllers are easy to compute.”
### A robustness result
Summary
-------
In this section, we extended the state-feedback finite-time horizon results from Section \[sec:sys\_resp\_intro\] to the infinite horizon settings. We further showed that the SLS framework allows us to impose convex locality constraints on distributed controllers, greatly improving the scalability of synthesis methods via a decomposition and dimensionality reduction based method. We also showed how robust counterparts to SLS problems can be used to transparently accommodate modeling errors, as well as how they can be used to derive finite dimensional approximations to infinite horizon problems with provable performance guarantees. In addition to the results presented here, there has also been interesting work looking at linking the SLS state-feedback formulation to classical notions such as passivity [@Ran18] and infinite-dimensional system theory and the spatial invariance framework [@JenB18]. In [@AndMC19] the $\ttf \Delta$-block is used to model discretization errors when a sparse approximate model is constructed from a structured continuous-time dynamics.
Output Feedback {#sec:output}
===============
This section extends many of the ideas presented in the previous section to the output feedback setting. As we will see, many of the core concepts developed in the state-feedback setting (working directly with system responses, affine achievability constraints) extend in a natural way to the output feedback setting. However, these extensions often come at the expense of more complex formulae and computational procedures.
Distributed Control, Locality, and Scalability {#sec:output-locality}
----------------------------------------------
Case Studies {#sec:simu}
------------
Summary
-------
In this section, we extended many of the state-feedback results from Section \[sec:state\_feedback\_SLS\] to the output feedback setting. In particular, we showed that by exploiting the partial separability of many SLS problems of interest, output feedback problems also enjoy improved the scalability via a decomposition and dimensionality reduction based method combined with distributed optimization methods such as ADMM. Lacking in the output feedback setting however is an analogous family of robustness results to those presented in Section \[sec:robust\]. Although preliminary results exist [@boczar2018finite], this remains in important direction for future work.
Conclusions {#sec:conclusion}
===========
This article reviewed the System Level Synthesis framework. Our aim was to provide a self-contained and accessible resource that summarizes progress in developing this novel approach to controller synthesis. We highlighted the benefits that SLS provides in the context of distributed control and in robust control. As we hope we were able to show, by working directly with system responses, there is a transparency in how system constraints, structure, and uncertainty affect controller synthesis, implementation, and performance, and it is this transparency that we exploit throughout to improve upon the state-of-the-art. As lengthy as this article is, we believe that we have just begun to scratch the surface of what SLS can do. Exciting current directions of research focus upon integrating SLS into model predictive control algorithms, further understanding the algebraic structure underlying localized controllers and their state-space realizations, further developing robust SLS theory (especially in the output feedback setting), and applying these tools to application areas spanning power-systems, computer networking, and machine learning/AI.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We employ a new method for studying compositionally disordered ferroelectric oxides. This method is based on the virtual crystal approximation (VCA), in which two or more component potentials are averaged into a composite atomic potential. In our method, we construct a virtual atom with the correctly averaged atomic size and atomic eigenvalues. We have used our new method to study the composition dependent phase transition in Pb(Zr$_{1-x}$Ti$_x$)O$_3$ lying between $x=0.5$ and $x=0.4$. We correctly predict the experimentally determined phase transition from the tetragonal phase to a low-temperature rhombohedral phase between these two compositions.'
address: |
$^*$Department of Chemistry, Long Island University - C. W. Post Campus, Brookville, NY 11548\
$^\S$Department of Chemistry and Laboratory for Research on the Structure of Matter,\
University of Pennsylvania, Philadelphia, PA 19104
author:
- 'Nicholas J. Ramer$^*$ and Andrew M. Rappe$^{\S}$'
title: Determination of ferroelectric compositional phase transition using novel virtual crystal approach
---
Introduction {#introduction .unnumbered}
============
Our recent work [@njr:Williamsburg99] has demonstrated the utility of the virtual crystal approximation (VCA) [@njr:VCA1; @njr:VCA2] to study compositionally disordered materials. Following our initial VCA study of a stress-induced phase transition in Pb(Zr$_{1-x}$Ti$_x$)O$_3$, we and other authors have extended the VCA to studies of compositional phase transitions [@njr:PRBVCA], temperature-dependent phase transitions [@njr:DHVVCA], and dynamical properties. [@njr:Rabe] Prior to this the VCA has had some success in providing qualitative, and in some instances quantitative agreement with large-scale solid-state calculations.
Previous implementations of the VCA have focused on the averaging of two or more component potentials in separable form at the solid-state level. [@njr:Papa; @njr:Slavenburg] By averaging the potentials, it is difficult to assess the quality of the resulting VCA potential. In some studies, these traditional VCA potentials have yielded unphysical results. [@njr:Chen; @njr:Bellaiche] Speculation has been made in the literature that it is the inability of these traditional VCA potentials to capture the important differences in chemical bonding and ionicity accurately that has given the poor results. [@njr:Chen]
Methodology {#methodology .unnumbered}
===========
Due to the inconsistencies in the literature regarding the quality of VCA results, we have recently formulated a new approach for the construction of VCA potentials. Our method represents a simple and intuitive way to construct VCA potentials and assess their quality. It is a severe departure from the more traditional methods for VCA potential construction. We have previously presented a thorough description of our method for the construction of VCA atoms. [@njr:Williamsburg99; @njr:PRBVCA] For brevity, we will only present here the salient points of our method.
- Our method averages the component atoms at the all-electron level, such that all-electron eigenvalues, potentials, wave functions and charge density are all computed for the virtual atom. Since the solution of the all-electron atom is common to all pseudopotential procedures, our method is general enough to be used in all types of pseudopotential construction.
Operationally we find the all-electron wave functions and eigenvalues for the atoms we want to average. We then determine the properly averaged eigenvalues for the VCA atom valence states from the eigenvalues of the all-electron atoms. We then construct a bare Coulombic potential from the properly averaged bare nuclear potentials of the component atoms and a properly averaged core charge density. In doing so, we have insured that the resulting VCA potential will have the correct averaged size. Using this averaged nuclear potential and frozen-core charge density, we construct new self-consistent wave functions for the VCA atom valence states. We require that these new wave functions solve the Kohn-Sham equations in the valence region and give the properly averaged eigenvalues.
- [From this new set of potentials and wave functions, we construct an optimized pseudopotential. [@njr:RappePS] This method enforces the exact agreement of the VCA eigenvalues with the averaged all-electron eigenvalues for the reference state [*only*]{}. For any other electronic configuration, we have made no requirements on the quality of the potential. Recently, we have formulated the designed nonlocal pseudopotential method, [@njr:RRPS] in which we exploit the inherent arbitrary separation of the local potential and semilocal correction terms required for the Kleinman-Bylander separable form. [@njr:KB] By adjusting the form of the local potential and therefore the correction terms, we may dramatically improve the transferability of the potential without affecting the exact agreement at the reference configuration. This transferability improvement over a wide range of electronic configurations insures correct ionicity and polarizability of the virtual atom.]{}
It is our assertion that the preservation of the proper averaged size of the VCA potential and a high level of transferability will provide accurate VCA atoms which yield excellent agreement with superlattice calculations. We have previously demonstrated the effectiveness of our new VCA approach. [@njr:Williamsburg99; @njr:PRBVCA] The remainder of this paper will focus on the construction and use of two VCA potentials to study the compositional phase transition in Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$.
Results and Discussion {#results-and-discussion .unnumbered}
======================
[lcdcd]{} & & &Step &Step\
Atom&$r_{c}$&$q_{c}$&Range&Height\
& & & &\
VCA ($s^{2}p^{6}d^{0}$)&1.38,1.51,1.40&7.07&0.00–0.56 & 1.18\
$x=0.5$& & &0.56–0.79 & 1.34\
& & & &\
VCA ($s^{2}p^{6}d^{0}$)&1.27,1.38,1.61&7.07&0.00–1.25 & 10.32\
$x=0.4$& & & &\
& & & &\
----------------------- --------- --------- --------- ---------
State
$s^2$ -7.5701 0.0000 -7.4266 0.0000
$p^6$ -5.9530 0.0000 -5.8303 0.0000
$s^0$ -2.5581 -0.0038 -2.5280 -0.0029
$d^0$ -3.4488 0.0000 -3.3629 0.0000
$\Delta E_{\rm{tot}}$ 0.0000 0.0000 0.0000 0.0000
$s^2$ -6.7808 0.0016 -6.6490 0.0019
$p^6$ -5.1720 0.0015 -5.0611 0.0017
$s^1$ -2.0304 -0.0009 -2.0061 -0.0013
$d^0$ -2.7034 0.0041 -2.6316 0.0050
$\Delta E_{\rm{tot}}$ -2.3067 -0.0019 -2.2790 -0.0014
$s^2$ -6.3652 -0.0031 -6.2570 -0.0027
$p^6$ -4.7744 -0.0049 -4.6868 -0.0038
$s^0$ -1.8399 0.0000 -1.8234 0.0004
$d^1$ -2.3571 -0.0033 -2.3087 -0.0047
$\Delta E_{\rm{tot}}$ -2.8936 -0.0012 -2.8270 -0.0007
$s^2$ -5.6586 -0.0029 -5.5599 -0.0023
$p^6$ -4.0734 -0.0047 -3.9953 -0.0040
$s^1$ -1.3453 0.0011 -1.3327 0.0019
$d^1$ -1.6889 -0.0018 -1.6517 -0.0010
$\Delta E_{\rm{tot}}$ -4.4923 -0.0004 -4.4107 0.0005
$s^2$ -4.1730 -0.0077 -4.1060 -0.0061
$p^6$ -2.6064 -0.0106 -2.5597 -0.0091
$s^2$ -0.3293 0.0018 -0.3285 0.0026
$d^2$ -0.3239 -0.0072 -0.3193 -0.0058
$\Delta E_{\rm{tot}}$ -6.2696 -0.0037 -6.1628 -0.0022
----------------------- --------- --------- --------- ---------
: Configuration testing for Zr$_{1-x}$Ti$_x$ virtual crystal (VCA) atoms generated with the method described in text. Averaged eigenvalues and total energy differences are given for Zr and Ti pseudopotentials (PS) for two values of $x$. The PS construction parameters have been presented elsewhere. Errors are computed as a difference between VCA and averaged PS results. All energies are in Ry.
We have applied our new VCA method to the Zr$_{1-x}$Ti$_{x}$ virtual atom for $x=0.5$ and $x=0.4$. All atomic energy calculations were done with the local density approximation and optimized [@njr:RappePS] and designed nonlocal pseudopotential [@njr:RRPS] methods were employed. The generation parameters for the two VCA potentials are included in Table 1. For all atoms, semi-core states were included as valence. It is important to note that although we have included multiple $s$-channel states, only one $s$ nonlocal projector is used. For both VCA atoms, we have used the $s$-potential with the addition of one or two square-step potentials as the local potential.
The transferability testing results for the two VCA potentials are contained in Table 2. From the table, it is evident that our new VCA pseudopotentials are highly transferable at either value of $x$.
We use both VCA potentials in solid-state calculations for three structural phases of Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$. The electronic wave functions are expanded in a plane-wave basis using a cutoff energy of 50 Ry. We included the 5$d$ shell as valence for the Pb atom as well as including scalar relativistic effects. Brillouin zone integrations are approximated accurately as sums on a 4$\times$4$\times$4 Monkhorst-Pack $k$-point mesh. [@njr:MNKPCK]
We have included three structurally distinct phases of Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$ in the current studies of compositions between $x=0.5$ and $x=0.4$: a tetragonal Ti-rich $P4mm$ and two rhombohedral Zr-rich ($R3c$ and $R3m$) phases. [@njr:Jaffe; @njr:PZTJaffe] Around 400-500K, the rhombohedral region exhibits a boundary between $R3c$ (low-temperature) and $R3m$ (high-temperature) phases, which depends weakly on composition. [@njr:Michel; @njr:Glazer] The $R3c$ phase shows complex oxygen octahedral tilting, which doubles the primitive unit cell to ten atoms. [@njr:Clarke]
We have completed full electronic and structural relaxations for five-atom unit cells for the tetragonal and high-temperature rhombohedral phases. Due to the complex oxygen octahedral distortions, we have used a 10-atom unit cell for the low-temperature rhombohedral phase. For all rhombohedral calculations, we have neglected the small shear relaxations.
In Figure 1, we show the equations of state for the three phases of Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$ at $x=0.5$. We find the tetragonal phase is the ground-state structure for this composition lying approximately 0.05 eV lower in energy than both rhombohedral phases. This is correct energy ordering found experimentally. [@njr:Jaffe; @njr:PZTJaffe]
=3.5in
In Figure 2, we show the equations of state for the three phases of Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$ at $x=0.4$. We find both rhombohedral phases have shifted down in energy relative to the tetragonal phase. In fact, the low-temperature rhombohedral phase has shifted by approximately 0.09 eV in energy and therefore becomes the ground-state structure for this composition of Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$. This result is in direct agreement with the experimental findings [@njr:Jaffe; @njr:PZTJaffe] and demonstrates the ability of our new VCA method to locate and predict compositional phase transitions.
=3.5in
Conclusions {#conclusions .unnumbered}
===========
In this paper, we have applied our new method for constructing a virtual crystal pseudopotential to the Zr and Ti atoms. Potentials constructed with our method not only possess the properly averaged atomic size but also a high level of accuracy in describing energetic differences due to changes in both ionicity and polarizability. This method is based on averaging all-electron information and therefore makes the method applicable to all types of pseudopotential construction algorithms. We have used our new method to construct two Zr$_{1-x}$Ti$_{x}$ VCA potentials at $x=0.5$ and $x=0.4$. With these potentials, we have studied three structural phase of Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$, finding that the tetragonal phase is the ground-state structure for Pb(Zr$_{1-x}$Ti$_{x}$)O$_3$ at $x=0.5$. When moving to $x=0.4$, we find that the low-temperature rhombohedral phase becomes the ground-state. This finding is in excellent agreement with experiment and represents the first $ab$ $initio$ determination of a compositionally-dependent phase transition in a ferroelectric oxide.
Recently yet another phase (monoclinic $Cm$) has been experimentally found at low temperatures for compositions between $x=0.45$ and $x=0.5$. [@njr:Cross] We plan to extend the current research by studying the monoclinic phases with our VCA atoms.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Ilya Grinberg for his help with programming the virtual crystal method. This work was supported by NSF grant DMR 97-02514 and the Laboratory for Research on the Structure of Matter at the University of Pennsylvania. AMR acknowledges the Alfred P. Sloan Foundation. Computational support was provided by the San Diego Supercomputer Center and the National Center for Supercomputing Applications.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The recent breakthroughs of deep reinforcement learning (DRL) technique in Alpha Go and playing Atari have set a good example in handling large state and actions spaces of complicated control problems. The DRL technique is comprised of (i) an offline deep neural network (DNN) construction phase, which derives the correlation between each state-action pair of the system and its value function, and (ii) an online deep Q-learning phase, which adaptively derives the optimal action and updates value estimates.
In this paper, we first present the general DRL framework, which can be widely utilized in many applications with different optimization objectives. This is followed by the introduction of three specific applications: the cloud computing resource allocation problem, the residential smart grid task scheduling problem, and building HVAC system optimal control problem. The effectiveness of the DRL technique in these three cyber-physical applications have been validated. Finally, this paper investigates the stochastic computing-based hardware implementations of the DRL framework, which consumes a significant improvement in area efficiency and power consumption compared with binary-based implementation counterparts.
author:
- |
Hongjia Li$^1$, Tianshu Wei$^2$, Ao Ren$^1$, Qi Zhu$^2$, and Yanzhi Wang$^1$\
\
\
, $^2${[email protected], [email protected]}
bibliography:
- 'references.bib'
title: |
Deep Reinforcement Learning: Framework, Applications, and Embedded Implementations\
[Invited Paper]{}
---
Deep reinforcement learning, optimal control, cyber-physical systems, stochastic computing.
Introduction
============
Reinforcement learning provides us a mathematical framework for learning or deriving strategies or policies that map situations (i.e., states) into actions with the goal of maximizing an accumulative reward [@sutton1998reinforcement]. It has been widely applied for solving problems in different fields, such as manufacturing, finance sector, and robotic control systems. Along with the resurgence of deep learning techniques, reinforcement learning has now evolved towards deep reinforcement learning (DRL), where deep neural networks (DNNs) are utilitzed in the policy-deriving process [@mnih2013playing; @mnih2015human; @silver2016mastering]. With offline-constructed and online-updated DNNs, DRL techniques demonstrate capabilities in handling complicated problems with high-dimensional state and action spaces and even enabling continuous action spaces [@lillicrap2015continuous]. These features make DRL distinguished from reinforcement learning. And recent breakthroughs in Alpha Go [@silver2016mastering] and playing Atari [@mnih2013playing] indicate the great success of DRL.
One major application scenario of DRL is the embedded computing environment, such as in unmanned aerial vehicles, autonomous driving, robotics, wearable devices and mobile computing systems. However, DNNs involved in the DRL can be both compute and memory intensive. Therefore, it is desirable to have dedicated hardware implementations (e.g., FPGA, ASIC) for DNNs in the DRL for the embedded computing platforms, in order to utilize the distributed-computing and parallelism of hardware resources for enhanced computing speed, energy efficiency, and resiliency. Stochastic computing (SC) [@alaghi2013survey; @gaines1969stochastic] as a low-cost substitute to the binary-based computing radically simplifies the hardware implementation of arithmetic units and has the potential to satisfy the low power and small hardware footprint requirements of DNNs in the embedded computing environment.
In this paper, we first present the general DRL framework, which can be widely utilized in many applications with different optimization objectives, such as resource allocation, residential smart grid, embedded system power management, and autonomous control. Followed by the introduction of three applications of the DRL framework, one for the cloud computing resource allocation problem, one for the residential smart grid user-end task scheduling problem and one for building HVAC system. The cloud computing resource allocation problem automatically and dynamically distributes resources (virtual machines or tasks) to servers by establishing efficient strategy. Through extensive experimental simulations using Google cluster traces [@clusterdata:Reiss2011], the DRL framework for cloud computing resource allocation achieves up to 54.1% energy saving compared with the baseline approach. The residential smart grid task scheduling problem determines the task scheduling and resource allocation with the goal of simultaneously maximizing the utilization of photovoltaic (PV) power generation and minimizing user’s electricity cost. Through extensive experimental simulations with realistic task modelings, the DRL framework for residential smart grid task scheduling achieves up to 22.77% total energy cost reduction compared with the baseline algorithm. The building HVAC system is designed for controlling a desired temperature within each zone with the factors of current zone temperature and outside environment disturbances. The proposed DRL control algorithm can achieve 20%-70% cost reduction compared with the rule-based baseline control strategy, while maintaining the temperature violation rate below 1.0%.
Additionally, as mentioned above, this paper investigates the stochatic computing (SC)-based hardware implementations of DNNs used in DRL using stochastic computing technique. To further enhance the performance (computing speed) and energy efficiency, pipelining techniques is employed in the SC-based hardware design. The stochastic computing-based ultra-low-power implementation consumes only 58771.53 $\mu m^{2}$ area and 7.73 $mW$ power with 261.12 $ns$ delay.
The rest of this paper is organized as as follows. Section 2 presents the related works on DRL. In Section 3, the general DRL basics and framework are introduced. Section 4 introduces three representative applications of DRL, along with simulation results. In the following Section 5, the hardware implementation of DRL using the stochastic computing technique is presented. The corresponding experimental results are showed in Section 6. The conclusion of this paper is presented in Section 7.
Related Works
=============
A lot of research efforts have been made recently on the development and applications of DRL. Mnih et al. are the first introducing deep learning model into the reinforcement learning and have succeeded in handling high-dimensional sensory input when playing Atari [@mnih2013playing].
In 2015, Mnih et al. further generalized DRL by developing the first artificial agent, called deep Q-network (DQN), capable of learning policies directly from high-dimensional sensory inputs and agent-environment interactions [@mnih2015human], in which convolutional neural networks with hierarchical layers of tiled convolutional filters were adopted. Lillicrap et al. proposed an actor-critic, model-free algorithm based on the deterministic policy gradient. Combined with DQN, the actor-critic approach can operate over continuous action spaces [@lillicrap2015continuous]. In 2016, Silver et al. combined supervised learning from games of human experts and reinforcement learning from self-play games to master the game of Go with DNN and tree search [@silver2016mastering]. In [@van2016deep] a specific adaptation to the DQN algorithm with double Q-learning was proposed, which is able to reduce the observed overestimations of the original DQN algorithm, and also lead to much better performance on several games including the Atari 2600 domain.
There are also extensive research works on enhancing the performance and energy efficiency of hardware implementations of DNNs. In order to effectively implement the deep convolutional neural networks onto embedded/portable systems, Ren et al. developed the first comprehensive design and optimization framework of stochastic computing-based deep convolutional neural networks [@ren2016sc]. In order to handle the challenges brought by stochastic computing including random error fluctuation, range limitation, and overhead in accumulation, Kim et al. adopted the approach of removing near-zero weights, applying weight-scaling, and integrating the activation function with the accumulator when designing an efficient DNN with stochastic computing [@kim2016dynamic]. In [@shafiee2016isaac], a pipelined architecture was employed for a convolutional neural network accelerator, with memristor crossbars dedicated for each neural network layer and eDRAM data buffers between pipeline stages. Ardakani et al. implemented the DNN using integer stochastic stream which is a sequence of integer numbers that are represented by either two’s complement or sign-magnitude [@ardakani2017vlsi] to solve the precision loss issue of conventional scaled adder, meanwhile reducing the latency.
DRL Framework
=============
Deep reinforcement learning shares the same basic concepts with reinforcement learning in that it is also an agent-environment interaction process. The learner and decision-maker is called the *agent*. The thing it interacts with, comprising everyting outside the agent, is called the *environment*. Specifically, the agent and environment interact at a sequence of decision epochs. At a decision epoch, the agent receives some representation of the environment’s *state* i.e., $s$, and on that basis selects an *action* i.e., $a$. In part as a consequence of its action, the agent receives a numerical *reward* and finds itself in a new state of the environment i.e., $s'$. A policy, denoted by $\pi$, of the agent is a mapping from each state to an action that specifies the action $a=\pi(s)$ that the agent will choose when the environment is in state $s$. The ultimate goal of an agent is to find the optimal policy, such that $$\label{eqn_1}
V^{\pi}(s) = \mathbf{E} \Big[\sum_{k=0}^{\infty }\gamma^kr(k) \Big| s\Big]$$ or $$\label{eqn_2}
V^{\pi}(s) = \mathbf{E} \Big[\int_{t_0}^{\infty}e^{-\beta(t-t_0)}r(t) dt \big| s\Big]$$ is maximized for each state $s$, where $r$ is the reward rate, and $\gamma$ and $\beta$ are the discount rates. The *value function* $V^{\pi}(s)$ is the expected return when the environment starts in state $s$ and follows policy $\pi$ thereafter. Eqn. (\[eqn\_1\]) is for a discrete-time system, while Eqn. (\[eqn\_2\]) is for a continuous-time system.
In order to derive the optimal policy, a $Q$ value, denoted by $Q(s,a)$, is associated with each state-action pair $(s,a)$, which approximates the expected discounted cummulative reward (i.e., the value function) of taking action $a$ at state $s$. The reinforcement learning algorithm has a convergence time proportional to $O(|A| \cdot |S|)$, where $|A|$ represents the total number of actions and $|S|$ represents the total number of states. And its computation complexity is $O(|A|+M)$ at each decision epoch, in which $M$ is the already known state-action pairs kept in the memory. Therefore, reinforcement learning becomes less effective when dealing with actual complicated problems with high-dimensional state and action spaces.
To overcome the drawbacks of reinforcement learning, DRL is comprised of an offline deep neural network (DNN) construction phase and an online deep Q-learning phase showed in Algorithm \[algorithm1\]. In the offline phase, we construct a DNN, which can infer for each state-action pair its $Q$ value to be used for the online phase. Sufficient training data is needed for the offline DNN construction. In [@rao2009vconf] a model-based procedure is adopted to accumulate the training samples, while in [@silver2016mastering] training data is obtained from actual measurement. To obtain the training data, we use an arbitrary but gradually refined policy to simulate the control process. An experience memory $D$ with capacity $N_{D}$ is used to store the state transition profiles and $Q$ values while smoothing out learning to avoid oscillations and divergence in the parameters [@mnih2013playing]. Then, a DNN with weight set $\theta$ can be trained using the state transition profile and $Q$ values.
**Offline DNN construction:**\
Simulate the control process using an arbitrary but gradually refined policy for enough long time Obtain the state transition profile and $Q(s,a)$ value estimates during the process simulation Store the state transition profile and $Q(s,a)$ value estimates in experience memory $D$ with capacity $N_D$ Train a DNN with features $(s,a)$ and outcomes $Q(s,a)$ **Online deep Q-learning:**\
In the online phase, deep Q-learning is adopted for action selection (i.e., the $\varepsilon$-greedy policy) and $Q$ value update. Specifically, suppose at decision epoch $t_k$, the system under control is in state $s_k$. The DRL agent enumerates all actions and obtains the corresponing $Q(s_k,a)$ value estimates using the offline-constructed DNN. According to the $\varepsilon$-greedy policy, the agent selects the action resulting in the maximum $Q(s_k,a)$ value estimate with probability 1 - $\varepsilon$, and selects a random action with probability $\varepsilon$. After the selected action $a_k$ is taken, the observed total reward $r_k(s_k,a_k)$ during $[t_k,t_{k+1})$ is used for $Q$ value update. In order to mitigate the potential oscillation in the DNN inference results, we adopt the duplicate $Q$ method from [@hasselt2010double], which maintains two $Q$ value estimates for each state-action pair and updates the two $Q$ value estimates interactively. At the end of an execution sequence of decision epochs, the DNN is then updated using the lately observed $Q$ values in a mini-batch manner, and will be employed in the next execution sequence.
From the above procedure, the DRL can now handle extremely large state space (even infinite continuous state space) by using offline-trained and online-updated DNN. For the action space, it should be kept within a reasonable size, due to the necessity to enumerate the action space for action selection at a decision epoch.
Representative Applications of Deep Reinforcement Learning
==========================================================
DRL Framework for Cloud Computing Resource Allocation
-----------------------------------------------------
In the cloud computing resource allocation problem, a server cluster consists of $M$ physical servers that can provide $P$ types of resources is considered. A first-come-first-served manner is deployed to process assigned jobs for the servers. A job will wait in the queue until sufficient resource is released in the server. We define the latency of a job as the actual duration from its arrival time to its complete time. A server has two working modes: active and sleep for energy saving. $T_{on}$ is the time needed by a server to transit from sleep mode to active mode. $T_{off}$ is the time needed by a server to transit from active mode to sleep mode when no job is pending or running. All the mode transitions are considered as uninterruptible. We assume the power consumption of a server in the sleep mode is zero. Based on an empirical non-linear model in [@fan2007power], the power consumption of a server in active mode is a function of CPU utilization as follows: $$P(u_t) = P(0\%)+(P(100\%)-P(0\%))(2u_t-u_t^{1.4})$$ where $u_t$ denotes the CPU utilization of the server at time $t$. In order to significantly reduce the action space, we adopt a continuous-time and event-driven decision making mechanism [@duff1995reinforcement] in which each decision epoch coincides with the arrival time of a new job. In the offline phase, we harness the power of *representation learning* and *weight sharing* for DNN construction. Specifically, we first employ an autoencoder to extract a lower-dimensional high-level representation of server group state for each possible server. The dimension difference reflects the relative importance of the targeting server group compared with other groups and results in reduction in the state space. Next, for estimating the Q-value of the action of allocating a job to servers in this group the neural network $\textbf{Sub-Q}$ takes the server group state, job’s state, all lower-dimensional high-level representations, and actions as input features. In addition, we introduce weight sharing among all autoencoders, as well as all $\textbf{Sub-Q}$’s to reduce the total number of parameters and the training time. For the online phase, at the beginning of each decision epoch, the Q value estimates are derived for each state-action pair by inference based on the offline trained DNN. An action is then selected for the current state using the $\epsilon$-greedy policy. At the next decision epoch, Q-value estimates are updated. After the execution of a whole control procedure, the DNN is updated in a mini-batch manner with the newly observed Q-value estimates.
In the simulation setup, we assume a homogeneous server cluster without loss of generality. The idle power consumption is $P(0\%) = 87$W, and the peak power consumption is $P(100\%) = 145$W [@fan2007power]. We set the server power mode transition times $T_{on}=30$s and $T_{off}=30$s. Based on the Google cluster traces [@clusterdata:Reiss2011], we simulate five different one-week job traces into the proposed online deep Q-learning framework and compare the average results against the baseline. Under the circumstances of $M$ = 20, 30 and 40, the proposed DRL-based framework on average can achieve 20.3%, 47.4% and 54.1% of power consumption saving while the accumulated latency only increases by 9.5%, 16.1% and 18.7%. The proposed framework effectively generates policies to decrease accumulated latency when the weight increases because of the more evenly jobs distributing. All tested cases can achieve at least 47.8% power consumption saving with only a slight increase in job latency. These results prove that weights of the reward function can take a effective control of the trade-off between power, latency, and resiliency.
Residential Smart Grid Task Scheduling
--------------------------------------
The present research focuses on task scheduling of residential appliance operations to minimize an individual electricity user’s cost in the Smart Grid factoring in photovoltaic (PV) power generation, due to the worldwide trend of transition to the Smart Grid and PV power usage in residential, industrial, and commercial sectors. In this work, we reduce users’ electricity cost by applying the deep reinforcement learning framework for the user-end task scheduling in the Smart Grid equipped with distributed PV power generation devices under dynamic pricing.
We employ a slotted time model i.e., the task scheduling frame (one day) is divided into $T = 24$ time slots each with duration of one hour. The tasks are non-interruptible, i.e., tasks need to be operated in continuous time slots. An inconvenience price is determined by the user to represent the penalty when scheduling task outside its desired operating window. We assume that the residential user is equipped with a distributed PV system. The power generation of the PV system in time slot $t$ is denoted by $P_{pv}(t)$. The power provided from the grid in time slot $t$ is denoted as $P_{grid}(t)$, which depends on $P_{pv}(t)$ and $P_{load}(t)$ according to the following: $$P_{grid}(t) =
\begin{cases}
0, & \text{when}\ P_{pv}(t) \geq P_{load}(t)\\
P_{load}(t)-P_{pv}(t), & \text{otherwise}
\end{cases}$$ We consider a dynamic price model $C(t,P_{grid}(t))$ consisting of a time-of-use (TOU) price component and a power consumption price component.
We simulate the control process using generated task sets and following a preliminary control policy. The state transition profile and $Q(s,a)$ value estimates are obtained through the simulation and used as the training data for offline DNN construction. We construct a three-layer artificial neural network with 26 hidden neurons, which is trained using the previously obtained training data. In the online phase, for each decision epoch $k$, according to the current system state $s_k$, the action resulting in the maximum $Q(s_k,a)$ estimate is selected using the $\epsilon$-greedy policy. And $Q(s_k,a)$ estimates are obtained by performing inference on the offline-trained neural network. Based on the selected actions and observed rewards, Q-value estimates are updated before the next decision epoch. At the end of one execution sequence, the neural network is updated for use in the next execution sequence. The PV power generation profiles are provide by [@PVpower], which are measured at Duffield, VA, in 2007. We adopt an approach using the negotiation-based task scheduling algorithm [@li2014negotiation] as our baseline system. We compare the total electric cost for the residential smart grid user using the DRL framework and the baseline algorithm on the following test cases: 100, 300 and 500 tasks for scheduling. According to the results, the DRL framework can schedule tasks to maximize the coverage of the PV power and avoid the peak of TOU price in a more effective manner compared to the baseline method. Correspondingly, the DRL framework can achieve $22.77\%$, $12.54\%$ and $12.45\%$ total energy cost reductions when the number of tasks are 100, 300, and 500, respectively.
DRL for Building HVAC Control
-----------------------------
The building HVAC system should be operated to maintain a desired temperature within each zone, based on current zone temperature and outside environment disturbances (e.g., ambient temperature and solar irradiance). The zone temperature at next time step is determined by the current system states, the environment disturbances, and the conditioned air input from the HVAC system. We have developed a DRL control algorithm to intelligently determine the optimal conditioned air flow input for each zone, for maintaining desired temperature while minimizing the total energy cost of the building HVAC system [@Wei_DAC17].
More specifically, we consider a building that is equipped with a VAV (variable air flow volume) HVAC system to maintain desired temperature for $z$ zones. The VAV terminal box in each zone provides conditioned air (typically at a constant temperature) with an air flow rate that can be chosen from multiple discrete levels (denoted as $F=\{f^1,f^2,...,f^m\}$). At each control time step, the optimal control action for each zone is determined based on the observation of the current system states, which include current physical time, zones’ temperature in the building and environment disturbances (i.e. ambient temperature and solar irradiance intensity). For environment disturbances, we also take into account a multi-step forecast of weather data in the system states. This enables our DRL algorithm to capture the trend of the weather condition and perform proactive control for time-variant systems.
We separately train a neural network for each zone by following the DRL Algorithm \[algorithm1\]. Each neural network is only responsible for approximating the Q-value in one zone. At each control time step, all neural networks will receive the entire system states of buildings and then determine the control action for each zone separately. This heuristic can greatly improve the training efficiency by reducing the number of output units in the neural network. $$\begin{aligned}
&r^i_{t} = -\lambda ( [{T_{t}^i} - {\overline{T}_{t}^i}]_+ + [{\underline{T}_{t}^i} - {T_{t}^i}]_+ ) -\hspace{30mm}\nonumber\\
&\hspace{8mm}cost(\sum_i a^i_{t-1}, s_{t-1})\cdot\frac{a^i_t}{\sum_i a^i_{t-1}},\hspace{3mm}a^i_{t-1}\in F\label{rewards}\end{aligned}$$
During the training process, our DRL algorithm will try to maximum the reward function for each zone. The first term measures the temperature violation in each zone, while the second term heuristically estimates the energy consumption cost contributed by each zone (which is assumed to be proportional to the air flow demand in each zone based on the total HVAC system energy cost in the building $cost(\cdot)$).
To calculate the Q-value estimates, we adopt a similar neural network structure as in [@mnih2013playing]. Each output unit in the neural network corresponds to the Q-value estimate of each available control action. By using this structure, the Q-value estimates for all control actions can be calculated by performing one forward pass. We calculate the optimal Q-value for the action in the current system state by following the Bellman Equation . $$\begin{aligned}
Q^*(s_{t-1},a_{t-1}) &= r_{t}+\gamma\max_{a_{t}}Q(s_{t},a_{t})\label{target}\\
&\Leftarrow\max[\frac{r_{t}}{\rho}+\gamma\underset{a_{t}}{\max}Q(s_{t},a_{t}), -1]\label{clipping}\end{aligned}$$ As shown in Equation , in practice we squash the original target Q-value to the range $[-1,0]$ by first shrinking the original reward with a factor $\rho$ and then clipping it if the target Q-value estimate is smaller than $-1$. This can help speedup the training process by reducing the variance of Q-value estimates.
We train the DRL algorithms on two different weather profiles in summer days. The first set of weather data has intensive solar radiation and large variance in temperature, while the second one has a milder weather profile. We calculate buildings’ energy cost by using the practical time-of-use price from the Southern California Edison, and demonstrate the effectiveness of our DRL algorithm by comparing with a rule-based HVAC control strategy (similarly as the one in [@Urieli:AAMAS2013]) and the conventional RL method. We evaluate the performance of our DRL algorithm with three building models, which have 1 zone, 4 zones and 5 zones, respectively. Our experiment results show that our DRL control algorithm is superior to the conventional RL method and is able to achieve $20\%-70\%$ cost reduction compared with the rule-based baseline control strategy, while maintaining the temperature violation rate below $1.0\%$ [@Wei_DAC17].
SC-Based DRL Implementation
===========================
Compared with conventional implementations in CMOS circuits, stochastic computing (SC) enables low-power and small-hardware-footprint implementations of arithmetic units using standard logic elements [@gaines1967stochastic]. The SC paradigm significantly simplifies the hardware implementation and thereby allowing very high clock rates. In addition, it can provide a high degree of fault tolerance and an opportunity for trade-off between computating speed and accuracy even without changing the hardware implementation.
In stochastic computing (SC), bit-streams are used for representing numbers. First, the occurance rate of 1’s i.e., $P(X=1)$ in a bit-stream is calculated. Next, according to unipolar encoding the number $x$ presented by the bit-stream is just $x=P(X=1)$, or according to bipolar encoding the number $x$ presented by the bit-stream is $x=2P(X=1)-1$ [@brown2001stochastic]. A bit-stream can represnt a number in the range of $[0, 1]$ in unipolar encoding or $[-1,1]$ in bipolar encoding. For representing a number beyond the range, a pre-scaling operation [@yuan2016design] is needed. In this paper, we choose bipolar encoding to cover both negtive and positive numbers in DNN related calculations. For instance, a bit-stream 1101001011 represents the number $0.2$.
SC Arithmetic Units
-------------------
The major arithmetic operations in DNNs are multiplication, addition, and activation function. These operations can be implemented with extremely small arithmetic units as follows.
![SC arithmetic units used in this work. (a) XNOR gate-based mulipication unit, (b) APC-based addition unit performing addition of 30 bit-streams, and (c) $K$-state FSM-based activation unit.[]{data-label="fig:AU"}](7.PNG){width="3.2in"}
**Multiplication Unit:** The multiplication of two numbers represented by bit-streams (in bipolar encoding) can be calculated as logic XNOR operation of the two bit-streams, as shown in Figure \[fig:AU\] (a). A brief derivation can be $a\times b = [2P(A=1)-1]\times[2P(B=1)-1]=2P(A=1)P(B=1)+2P(A=0)P(B=0)-1=2P(A=1)\odot P(B=1)-1=2P(Z=1)-1=z$. Regardless of the length of bit-streams (i.e., precision), the multiplication unit is simply an XNOR gate with two 1-bit inputs and one 1-bit output [@gaines1969stochastic].
![Addition units: (a) OR gate, (b) MUX, and (c) APC.[]{data-label="fig:add"}](2.PNG){width="3.2in"}
**Addition Unit:** The addition of $n$ numbers can be performed as logic OR operation of the $n$ bit-streams, or by an $n$-to-1 multiplexer where $n$ inputs take the bit-streams respectively and the output bit-stream equals to $1/n$ of the sum, or by an approximate parallel counter (APC) [@kim2015approximate], as shown in Figure \[fig:add\], where each of the inputs is a bit-stream. The APC counts the number of ones from its $n$ inputs, that is to say, it adds the $i$-th bit of each of the bit-streams into a $\log n$-bit binary number with the value approximately equivalent to the sum. In summary, OR gate is the most area efficient but the accuracy is too low, MUX is area efficient with limited accuracy, and APC achieves the highest accuracy at the cost of a larger footprint. We adopt APC for addition considering accuracy, power consumption, and footprint according to [@ren2016sc].
An APC employs two parts, an approximate unit (AU), consisting of AND and OR gates for accumulating approximation, and an adder tree consisting of adders to calculate the binary summation of all input bits, each coming from an input bit-stream. We propose an improved APC design as shown in Figure \[fig:AU\] (b), where the last pair of inputs are feeded to a half adder directly instead of an AND or OR gate. For an APC with 30 inputs as in Figure \[fig:AU\] (b), the output should be 5-bit binary numbers. In order to further reduce the hardware footprint, we employ inverse mirror full adders as proposed in [@kim2015approximate] for the adder tree in an APC. Inverse mirror full adders are smaller and more responsive adders that output inverse logic of true summation and carry-out bits. The internal results in the even layers correspond to the number of ones in the primary input, while the internal results in the odd layers represent the number of zeros. We compare inaccuracy rates of our improved APC design to those of the original APC. As shown in Table \[table1:Inaccuracies\], our improved designs significantly reduce inaccuracy rate to less than 0.7% and at the same time with more than 40% reduction of gate count.
----------------------- --------- --------- ----------
**256** **512** **1024**
**26-input** 2.56% 2.12% 1.71%
**30-input** 2.34% 2.03% 1.56%
**26-input improved** 0.63% 0.61% 0.57%
**30-input improved** 0.61% 0.58% 0.55%
----------------------- --------- --------- ----------
: Inaccuracy rates of the improved and orginal APC designs.[]{data-label="table1:Inaccuracies"}
**Activation Unit:** The most popular activation functions used for deep neural networks are sigmoid, tanh, and Rectified Linear Unit (ReLU). In this work, we select tanh due to its convenience for SC implementation and comparable effectiveness as ReLU and sigmoid [@krizhevsky2012imagenet]. The tanh function can be easily implemented with a $K$-state finite-state-machine (FSM) in the SC domain with significantly reduced hardware footprint compared to its conventional computing counterpart [@brown2001stochastic]. Figure \[fig:AU\] (c) includes a $K$-state FSM design of the tanh function in SC domain for use in the activation unit. It outputs a zero if the current state is on the left half of the states, and a one otherwise. By this design, we have $$Stanh(K,X) \cong tanh(\frac{K}{2}x)$$ where $Stanh$ stands for the tanh function in SC domain. The $K$ value represents the precision of $Stanh$, and therefore higher accuracy can be achieved with a larger $K$ value. We use a $K$ value in the range of $[-\frac{K}{2}x,\frac{K}{2}x]$ in our experiments. $Stanh(K,X)$ takes bit-streams as input, while inner products calculated from an APC are in the binary format. Therefore, we use a saturated up/down counter [@kim2016dynamic] to convert the binary format input from APC to a bit-stream. The whole design of the activation unit is shown in Figure \[fig:AU\] (c).
System Design
-------------
Figure \[fig:overview\] shows the whole system diagram of the proposed DRL implementation for embedded computing platforms. It consists of an SC-based hardware DNN, a software controller in an embedded processor, and a B/S conversion block in between, which converts data in binary format for software controller to/from bit-streams for SC-based hardware DNN. For the SC-based hardware DNN, the previously discussed SC arithmetic units including multiplication units, addition units and activation units are utilized to perform DNN calculations. More specifically, the DNN consists of $M$ layers, each with $N_i$ $(1\leq i \leq M)$ neurons. The inputs($x_i$) and its corresponding weights($w_i$) are operated by the multiplication and addition units. In order to insure the next layer’s input are within \[-1,1\] range, the outputs are transformed by an activation function.
![Whole system diagram of the DRL implementation including the SC-based hardware DNN and interface with the software controller in an embedded processor. []{data-label="fig:overview"}](4.PNG){width="3.2in"}
The software controller performs both online control for each decision epoch and offline control for a sequence of decision epochs. The offline control first constructs a DNN using previously collected data and the resultant weights of the DNN are sent to the hardware DNN as parameters for online inference. The online control at each decision epoch $k$ performs action selection and $Q$ value update, during which state-action pairs $(s_k,a)$ for each action $a$ are sent to the hardware DNN for the calculation of $Q$ values $Q(s_k,a)$ (i.e., DNN inference). $Q$ values calculated from the hardware DNN are then sent back to the software controller for use in action selection and $Q$ value update. After the online execution at a sequence of decision epochs, the offline control takes charge again to update DNN weights with training based on the newly updated $Q$ values.
Design Optimization
-------------------
![Deep pipelining technique in the SC-based hardware DNN.[]{data-label="fig:pipeline"}](6.PNG){width="3.2in"}
Different from [@ren2016sc], we use the “deep” pipelining technique in the SC-based hardware DNN, where the pipeline stages can be within the DNN layers, while in [@ren2016sc] only inter-layer pipelining is considered. The clock rate of a pipelined architecture is in general increased with deeper pipelining, but is also clamped by the slowest pipeline stage. In order to increase clock rate while balancing each pipeline stage, we implement two pipeline stages within each DNN layer i.e., registers are inserted between addition units and activation units as shown in Figure \[fig:pipeline\].
In conventional CMOS circuits performing binary computing, a higher data precision will slow down the clock rate. However, in SC circuits the clock rate is now independent of data precision. In SC, a higher data precision is achieved by longer bit-streams, while the clock rate should be set to cover the operations in each pipeline stage on just **1-bit** of data. To measure the performance of the SC pipelined architecture, we define *delay* as the bit-stream length times the clock cycle. In this way, the inverse of the delay is equivalent to the throughput of the pipelined architecture of the SC-based hardware DNN.
Experimental Results
====================
This section demonstrates the effectiveness of our optimized hardware implementation. We adopted one DRL network for the residential smart grid with one 26-neuron input layer, one 30-neuron hidden layer and one single-neuron output layer to implement the hardware application. Therefore the input layer is consisted of 30 XNOR gates for processing the inputs and weight, 30 26-input APCs and Btanh as the activation function. The hidden layer mainly includes a 30-input APC. Converters between stochastic and binary numbers are employed when processing the inputs and generating the outputs. Table \[table:binary\] presents the hardware implementation of the fixed network using conventional binary computing with the bit size ranging from 8 bits to 32 bits. It can be observed that the SC-based implementation can achieve a much smaller power and area cost compared with the binary-based hardware implementations. Table \[table: result\] shows the result of our proposed DRL hardware implementation based on SC with the impact of pipelining. The bit stream length ranges from 256 to 1024. As showed in the table, the pipelined optimization can significantly reduce the delay, i.e. increase the system throughput, while maintaining small power and area cost.
---- ----------------- ----------------- -----------------------
**Delay($ns$)** **Power($mW$)** **Area($\mu m^{2}$)**
8 7.60 63.31 1056958.13
16 10.53 217.79 1080106.41
32 14.76 880.25 3450187.80
---- ----------------- ----------------- -----------------------
: Performance of Binary-based Hardware Implementation of the DRL Framework[]{data-label="table:binary"}
[max width=,center]{}
-- ---------------- --------- ------- ----------
Pipelining 261.12 7.73 58771.53
Non-pipelining 412.47 6.30 57941.61
Pipelining 522.24 7.73 58820.74
Non-pipelining 824.63 6.30 57990.82
Pipelining 1044.48 7.73 58919.16
Non-pipelining 1648.95 10.76 58089.24
-- ---------------- --------- ------- ----------
: Performance of Optimized Hardware Implementation of the DRL Framework[]{data-label="table: result"}
Conclusion
==========
In this paper, we first present the general DRL framework, which can be widely utilized in many applications with different optimization objectives. This is followed by the introduction of three specific applications: the cloud computing resource allocation problem, the residential smart grid task scheduling problem, and building HVAC system optimal control problem. The effectiveness of the DRL technique in these three cyber-physical applications have been validated. Finally, this paper investigates the stochastic computing-based hardware implementations of the DRL framework, which consumes a significant improvement in area efficiency and power consumption compared with binary-based implementation counterparts.
Acknowledgements
================
This work was supported in part by the National Science Foundation under grants CCF-1553757, CCF-1646381, CNS-1739748 and CNS-1704662, CASE Center at Syracuse University, and Riverside Public Utilities.
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
[^1]\
Florida State University\
E-mail:
title: Experimental Overview of Light Mesons
---
Introduction
============
The mesons are bound states primarily consisting of a quark and anti-quark, and are the simplest strongly interacting systems. We have a theory of the strong interaction: Quantum Chromodynamics (QCD). However, the development of a detailed understanding of the properties of QCD in the strongly coupled regime has so far been incomplete, particularly with regards to its physical states. The study of the properties and decays of these states, particularly of the mesons, is expected to shed light on some of the open questions in QCD. These questions include which color-singlet quark states are allowed by nature, and what is the detailed nature of the interaction responsible for quark confinement. Recent advances in experiment and theoretical understanding have allowed to continue to move forward towards answers to these questions.
The light mesons composed of $uds$ quarks are particularly interesting for two reasons. Due to their light mass, they probe relativistic effects and the confinement regime of the quark-antiquark interaction more strongly than those containing heavier quarks. We also have the ability to experimentally generate them in copious amounts. In this paper, I will only discuss light-quark mesons that are “non-exotic”, i.e., appear to be primarily composed of a quark-antiquark pair. Heavier quarks and more exotic states are covered by other papers from this conference. Also, even this more restricted topic is incredibly rich, so I will only focus on discussing a few of the most prominent latest experimental results in the spectroscopy of these light mesons.
Mesons are most simply understood in terms of the constituent quark model, as a valence quark and anti-quark pair. They are classified by their quantum numbers, total spin $J$, parity $P$, and charge conjugation $C$. The light mesons are found to come in “octets” of mesons with the same $J^{PC}$. The spectrum of these states can be calculated using phenomenological models or numerical calculations of QCD on a lattice (LQCD) [@lattice], and are broadly found to be in good agreement with states that have been observed so far. In the latest edition of the PDG, over 80 mesons have been identified [@pdg]. In the most general sense, we can say that most lowest state octets are well-established (with some notable exceptions, such as the the $0^{++}$ scalar mesons), while the excited states in the range $M>2$ GeV are less understood. It is fair to say that we will not truly understand the spectrum of light mesons unless we can understand the radial excitations of the lowest mass mesons. Also, since several types of exotic states are expected to lie in this higher mass range, the firm establishment of the spectrum of light mesons has become important not just for the intrinsic understanding of these states, but as a prerequisite for the study of these exotic states.
However, the reality of observed states is more complicated than the simple constituent quark model, as more complex states with the same quantum numbers as normal mesons can quantum-mechanically mix with them to produce the states that are experimentally observed. These more complex states include multiquark states and those with significant gluonic content. There are then two important experimental challenges. Most mesons have large intrinsic widths, and given the large number of meson states, multiple states will overlap with each other in the mass spectra for a given final state. To identify individual states, amplitude analyses that can disentangle their separate contributions are required. Then, to further identify the meson states and determine the contribution of different components to their wavefunctions, it is necessary to measure their decays to different final state particles, and to analyze these multiple decay channels in a consistent framework. This program of analysis is making fresh progress in recent years thanks to large, high-quality data collected by modern, large-acceptance detectors, and improvements in theoretical models needed to understand these data. In the following, I will discuss results from three of these new sets of data collected by different experiments using different reactions, which are illustrated in Fig. \[fig:production\].
![Examples of light meson production processes: (left) Production in hadron decay, in association with another particle, which can be selected to tune the properties of the produced light meson X; (middle) Diffractive hadroproduction from a high-energy pion beam, for the example of the production of a light meson $\mathrm{X}^-$ decaying to $\pi^-\pi^+\pi^-$, which is dominated by Pomeron exchange; (right) Photoproduction off a proton target, where the quantum numbers of the light meson X depend on the quantum numbers of the quanta exchanged between the proton and the photon.[]{data-label="fig:production"}](meson_production.pdf "fig:"){width="2.1in"} ![Examples of light meson production processes: (left) Production in hadron decay, in association with another particle, which can be selected to tune the properties of the produced light meson X; (middle) Diffractive hadroproduction from a high-energy pion beam, for the example of the production of a light meson $\mathrm{X}^-$ decaying to $\pi^-\pi^+\pi^-$, which is dominated by Pomeron exchange; (right) Photoproduction off a proton target, where the quantum numbers of the light meson X depend on the quantum numbers of the quanta exchanged between the proton and the photon.[]{data-label="fig:production"}](compass_production.pdf "fig:"){width="2.4in"}
Light Mesons at BES-III
=======================
Studying mesons produced in the decay of other hadrons has several advantages. It is often possible to select a clean sample of parent hadrons, yielding a sample of events with little background and a well defined initial state. For two-body hadron decays, such as illustrated in Fig. 1(a), by selecting one of the particles produced in the decay to be of a particular type, one can select for the $J^{PC}$ and primary quark content of the other particle produced in the decay. A classic example is the charmonium decay $J/\psi~(1^{--}) \to \gamma~(1^{--}) + \mathrm{X}~(0^{++}~\mathrm{and}~2^{++})$. Instead of a photon, one could select an $\omega$ or $\phi$ meson to enhance the associated production of mesons with $ud$ and $s$ quark content, respectively. This process has been utilized by many experiments, from CDF and DØin high-energy $p\bar{p}$ collisions, and ATLAS, CMS, and LHCb in higher-energy $pp$ collisions, to BaBar, Belle, BES, CLEO, KLOE, and others in $e^+e^-$ annihilations.
BES-III is an experiment at the BEPC II $e^+e^-$ collider located at the Institute of High Energy Physics (IHEP) in Beijing, China. The BES-III detector [@besiii] is cylindrically symmetric and has reconstruction and particle identification capabilities to detect all particles produced in the $e^+e^-$ annihilations generated at $\sqrt{s} \sim 2-5$ GeV. BES-III has collected the world’s largest data samples in the charmonium region, consisting of $\sim 1.3\times10^9~J/\psi$, $\sim 5.0 \times 10^8 ~ \psi(2S)$, and $>6$ fb$^{-1}$ collected above $D\overline{D}$ thresholds. These data have allowed for the study of light mesons with $M\lesssim2.5$ GeV in a wide variety of charmonium decays.
![Summary of the properties of the states identified by BES-III in the mass region $1790-1890$ MeV. The axes give the Breit-Wigner masses and widths, and the $J^{PC}$ are given, when known.[@besiiippbar; @besiiietappipi; @besiii6pi; @besiiietaksks; @besiiiomegaeta; @besiiiomegaetapipi]. Taken from Ref. [@zhanghadron].[]{data-label="fig:x1835"}](bes1800_summary.pdf){width="5.in"}
One of the biggest current mysteries raised by BES-III is the nature of the states that have been observed in the region near the $p\bar{p}$ threshold, $M\sim1800 - 1880$ MeV. The first of the states was named X(1835), and was seen as a threshold enhancement in the $M(p\bar{p})$ distribution in the decay $J/\psi \to \gamma p\bar{p}$ by BES-II, and interpreted as a sub-threshold resonance [@besiippbar]. A partial wave analysis (PWA) of this reaction with a larger data sample by BES-III identified its $J^{PC}$ as $0^{-+}$ [@besiiippbar]. Subsequent studies of the decays of $J/\psi$ and $\psi(2S)$ to $(\gamma,\omega,\pi^0,\eta) p\bar{p}$ yielded little additional evidence for such a state [@otherpp]. However, in X, a study of the decay $J/\psi \to \gamma \eta' \pi^+\pi^-$ found an enhancement in the $M(\eta' \pi^+\pi^-)$ spectrum with mass $\sim1835$ MeV, but a width of $\sim200$ MeV [@besiietappipi; @besiiietappipi], much larger than the $<50$ MeV found for the X(1835) in the $p\bar{p}$ channel.
Since then, several other states with various $J^{PC}$ have been found in this mass region. Several of their mass spectra are shown in Fig. 2, and the current status of their resonance parameters and $J^{PC}$ is summarized in Fig. \[fig:x1835\]. The main questions are now: how many states actually exist in this region and what is the relation of these different enhancements to each other? The first step must be to firmly assign $J^{PC}$ values to all of these enhancements, either through PWA or other angular analysis. Other complications to the interpretation of these states are the large backgrounds in some of the decay channels, which could affect the mass and width determinations, and the closeness of these states to the $p\bar{p}$ threshold. These factors point towards the need for a coupled-channel analysis in order to accurately determine the lineshapes and resonance parameters of a possible state in these different decay channels.
![Fits to the most recent $\eta'\pi^+\pi^-$ mass spectrum from BES-III, using different parameterizations for the peak at $M\sim1.8$ GeV. (Left) A single Breit-Wigner shape, as used in the previous publication; (middle) a Flatte shape that is strongly coupled to the nearby $p\bar{p}$ threshold along with a narrow Breit-Wigner of mass $M\sim1.92$ GeV; (right) a coherent sum of two Breit-Wigners, a wide X(1835) and a narrow X(1870). From Ref. [@besiiietappipimore].[]{data-label="fig:prl_interfere"}](fit_simpleBW_new.pdf "fig:"){width="1.9in"} ![Fits to the most recent $\eta'\pi^+\pi^-$ mass spectrum from BES-III, using different parameterizations for the peak at $M\sim1.8$ GeV. (Left) A single Breit-Wigner shape, as used in the previous publication; (middle) a Flatte shape that is strongly coupled to the nearby $p\bar{p}$ threshold along with a narrow Breit-Wigner of mass $M\sim1.92$ GeV; (right) a coherent sum of two Breit-Wigners, a wide X(1835) and a narrow X(1870). From Ref. [@besiiietappipimore].[]{data-label="fig:prl_interfere"}](fit_flatte_new.pdf "fig:"){width="1.9in"} ![Fits to the most recent $\eta'\pi^+\pi^-$ mass spectrum from BES-III, using different parameterizations for the peak at $M\sim1.8$ GeV. (Left) A single Breit-Wigner shape, as used in the previous publication; (middle) a Flatte shape that is strongly coupled to the nearby $p\bar{p}$ threshold along with a narrow Breit-Wigner of mass $M\sim1.92$ GeV; (right) a coherent sum of two Breit-Wigners, a wide X(1835) and a narrow X(1870). From Ref. [@besiiietappipimore].[]{data-label="fig:prl_interfere"}](fit_interference_new.pdf "fig:"){width="1.9in"}
A first step towards a more sophisticated analysis was taken in the analysis of a larger data set for the reaction $J/\psi \to \gamma \eta' \pi^+\pi^-$ [@besiiietappipimore]. Three fits to the $M(\eta' \pi^+\pi^-)$ spectrum are shown in Fig. \[fig:prl\_interfere\]. As opposed to the previous measurement, the peak at $M\sim1.8$ GeV is no longer described well by a simple Breit-Wigner function. Results from two other models are shown: one with a Flatte shape that is strongly coupled to the nearby $p\bar{p}$ threshold along with a narrow Breit-Wigner of mass $M\sim1.92$ GeV, and a coherent sum of two Breit-Wigners, a wide X(1835) and a narrow X(1870). Both models have a similar fit quality, and more information is needed to accurate describe the spectrum, again pointing in the direction of a coupled channel analysis.
![Summary of results from the PWA of the reaction $J/\psi \to \gamma \phi\phi$ from BES-III. The left panel shows the mass spectrum and contributions from the different partial waves. The right gives the results for the different resonant contributions in tabular form. From Ref. [@besiiiphiphi].[]{data-label="fig:phiphi"}](phiphi_all_binfit){width="2.5in"}
Several PWAs of radiative $J/\psi$ decays have also been performed. The most recently published result was of the PWA of the $J/\psi \to \gamma \phi\phi$ decay [@besiiiphiphi]. This reaction allowed for the study of scalar, pseudoscalar, and tensor mesons in the little-studied region of $M>2.0$ GeV. The results are summarized in Fig. \[fig:phiphi\]. This analysis confirmed the contribution of several $f_2$ states and the $\eta(2225)$, and identified two new states, the $\eta(2100)$ and X(2500). Several other PWA’s are ongoing. The results of the PWA of the radiative production two pseudoscalar mesons is particularly interesting for the information it gives on the spectrum of scalar mesons and the potential contribution of glueballs. The results for $J/\psi \to \gamma \pi^0\pi^0$ [@besiiipi0pi0] and $J/\psi \to \gamma \eta\eta$ [@besiiietaeta] have been already published, and the results of the analysis of the $\pi^+\pi^-$, $K^+K^-$, and $K_SK_S$ final states are eagerly awaited.
Light Mesons at COMPASS
=======================
The study of mesons produced using beams of hadrons (primarily charged $\pi$ and $K$ mesons) has several advantages, including the ability to tune the beam particle type in order to preferentially create different types of hadrons, and the ability to collect large sets of data, since hadron beam experiments are almost always fixed-target experiments. The primary downside of such hadroproduction experiments, compared to the meson production in decays discussed in the previous section, is that their analysis is more complicated. Instead of starting from an initial state with well-defined quantum numbers, the final states observed in hadroproduction experiments can be produced through several different processes. Untangling their contributions for the study of mesons requires the application of appropriate models. Due to the ease of production of pion beams, meson pionproduction has been studied in many experiments, including E852 at Brookhaven, VES, and COMPASS.
![Summary of preliminary results of the analysis of $\sim50\times10^6$ exclusive $\pi^- + p \to \pi^-\pi^+\pi^- + p_\mathrm{recoil}$ events. Center of the boxes represents the mass, height of the boxes the width of the states. The different colors show ground and excited states. The circles represent the latest measurements according to PDG 2014, the triangles the results of this analysis.[]{data-label="fig:compasssummary"}](compass_results.pdf){width="5.in"}
The COMPASS experiment [@compass] is a fixed target experiment located at CERN which can support a variety of muon and hadron beams. Data were taken in several periods with a 190 GeV/$c$ $\pi^-$ beam incident on a liquid hydrogen target in order to study the light meson spectrum with $M\lesssim2$ GeV using diffractive pion-proton scattering (see Fig. 1(middle)). Several different final states including $\pi$, $\eta$, and $\eta'$ mesons have been investigated. Notably, COMPASS has a collected a large sample of $\sim50\times10^6$ exclusive $\pi^- + p \to \pi^-\pi^+\pi^- + p_\mathrm{recoil}$ events. A $t-$resolved analysis in bins of momentum transfer has been performed, using an impressive 88 partial waves, the largest set to date. The results of this partial wave analysis have been already published [@compasspwa].
![Preliminary fits to COMPASS data showing contributions from the $2^{++}$ $a_2$ mesons.[]{data-label="fig:compassa2"}](compass_a2.pdf){width="4.5in"}
![Preliminary fits to COMPASS data showing contributions from $1^{++}$ $a_1$ mesons.[]{data-label="fig:compassa1"}](compass_a1.pdf){width="4.5in"}
![Preliminary fits to COMPASS data showing contributions from $2^{-+}$ $\pi_2$ mesons.[]{data-label="fig:compasspi2"}](compass_pi2.pdf){width="5.5in"}
The next step in the analysis of this reaction is to perform a resonance model fit to this data in order to determine the contributions of different intermediate states in this reaction and their properties. The current status of this model fit is discussed in detail elsewhere in these proceedings [@compassnew], but briefly it models the $M(\pi^-\pi^+\pi^-)$ dependence using resonant and non-resonant contributions from 11 ground and excited states in a simultaneous fit to 14 partial waves. This is the largest model used in such an analysis so far, and extensive systematic studies have been done with this fit.
The preliminary results from this fit are summarized in Fig \[fig:compasssummary\], and results from individual partial waves are shown in Figs. \[fig:compassa2\], \[fig:compassa1\], and \[fig:compasspi2\]. In the $2^{++}$ channels, a robust signal for the well-known $a_2(1320)$ is found, as illustrated in Fig. \[fig:compassa2\], while evidence for the lesser-known $a_2(1700)$ is seen as destructive interference at low $t'$, particularly in the $f_2(1270)\,\pi$ P-wave channel. In the $1^{++}$ channels, evidence for a substantial non-resonant contribution is seen, for example the primary contributions to the $\rho(770)\,\pi$ S-wave are found to be the well-known $a_1(1260)$ and a non-resonant contribution of similar size, as illustrated in Figs. \[fig:compassa1\]. A potential signal for the $a_1(1640)$ is also seen, with the strongest evidence in the $f_2(1270)\,\pi$ P-wave channel. Figs. \[fig:compasspi2\] illustrates the evidence for the well-known $\pi_2(1670)$, the lesser-known $\pi_2(1880)$, and a new $\pi_2(2005)$. To summarize, COMPASS has made robust measurements of the properties of the ground states for several $J^{PC}$ mesons, and has provided valuable measurements for several excited states, for which few measurements currently exist.
The further analysis of this COMPASS data is expected to continue to provide a wealth of knowledge on light quark states. Ongoing projects related to the $\pi^-\pi^+\pi^-$ final state include the extraction of resonance contributions to the $\pi^+\pi^-$ subsystem, and analysis of models and partial waves using “semi-automatic” algorithmical methods [@compassfuture]. Other non-strange final states are also being studied, and collaborations with other groups are leading to analysis that move beyond the standard isobar model. One such example is the collaboration with JPAC, which yielded an analysis including analyticity and unitarity constraints [@compassjpac]. There is also the possibility of studying strange meson final states using data collected with a charged kaon beam.
Light Mesons at GlueX
=====================
The study of light mesons in photoproduction has gained new interest with the availability of multi-GeV photon beams at Jefferson Lab. In meson photoproduction, as illustrated in the right panel of Fig. 1, the photon couples through vector meson dominance (VMD) with quanta exchanged from the target (usually a proton) which are generally modeled as the exchange of a virtual meson. The wide variety of possible couplings allows for the production of a wide variety of mesons of different quark content and quantum number. Conversely, this flexibility means that more complicated theoretical models are generally needed to describe these interactions than for hadroproduction.
The GlueX experiment [@gluex] is the flagship experiment for the newly constructed Hall D in Jefferson Lab located in Newport News, Virginia. GlueX takes the highest-energy electrons extracted from the upgrade 12 GeV CEBAF electron beam accelerator and scatters them off of a thin diamond radiator, to create a broadband photon beam peaked at 9 GeV, with a high degree of linear polarization. The photon beam is incident on a liquid hydrogen target, and is surrounded by a spectrometer with good capabilities to detect both charged and neutral particles. The experiment had a commissioning run in 2016, and has started a multi-year program of data taking in early 2017. The primary goal of GlueX is the identification and study of the spectrum of hybrid mesons, which are mesons where the confining gluonic field contributions directly to the properties of the meson. The study of the spectrum of light mesons is a prerequisite to these exotic searches, and the data taken from GlueX can be used for many studies of hadronic physics. The GlueX detector and physics program are discussed in more detail elsewhere in these proceedings [@gluexhadron], but I will discuss a few highlights below.
![Beam asymmetry $\Sigma$ from 2016 GlueX data as a function of Mandelstam$-t$ for the production of $\pi^0$ (middle) and $\eta$ (right), along with several model predictions. From Ref. [@gluexpi0].[]{data-label="fig:gluexsigma"}](fig5_Sigma-eta.pdf){width="2.5in"}
Photoproduction near 9 GeV has been little studied in many years, with no new experimental data since the SLAC experiments of the 1970s and early 1980s. The first step towards the GlueX program of studying the light meson spectrum and searching for hybrid mesons is necessarily to study the processes through which they are produced. These studies will yield important inputs into the amplitude analyses necessary for meson spectroscopy, and to do so requires working closely with theoretical colleagues, notably the JPAC Collaboration. Such collaborations are already yielding new models for interpreting the data coming out of GlueX, which bodes well for the future program of photoproduction studies.
To begin the study of photoproduction processes, a program of studying the beam asymmetries ($\Sigma$) for the photoproduction of single psuedoscalar mesons ($\pi^0,\pi^-,\eta,\eta',...$) and the spin-density matrix elements for the photoproduction of single vector mesons ($\rho,\omega,\phi,...$) has begun. Preliminary results for many of these channels have been recently presented at conferences, and the first measurements of the beam asymmetry in $\pi^0$ and $\eta$ photoproduction using the 2016 data set has resulted in the first GlueX publication [@gluexpi0]. These results are illustrated in Fig. \[fig:gluexsigma\]. The value of $\Sigma$ of near unity indicates the dominance of vector meson exchange in this process. The $\eta$ measurement is the first in this energy range, and these results illustrate the first steps towards a detailed understanding of the production mechanisms of mesons in this energy range.
![Mass spectra from exclusive $\gamma p \to p + (4,5,6)\gamma$ events in early GlueX data. Contributions likely due to well-known states are labeled.[]{data-label="fig:gluexsigma"}](gluex_mass_spectra.pdf){width="6.in"}
![(Left) Invariant mass spectrum of $\pi^+\pi^-$ from exclusive $\gamma p \to \pi^+\pi^-p$ events measured at SLAC, illustrating claimed $\rho(1600)$ contribution [@slacrho]. (Middle) Timelike pion form factors determined by BaBar in $e^+e^- \to \gamma_{\mathrm{ISR}} \pi^+\pi^-$ [@babarpipi]. The solid line illustrates the fit to the data with various $\rho$ resonances. (Right) Preliminary $\pi^+\pi^-$ invariant mass spectrum from early GlueX data illustrating various potential resonance contributions.[]{data-label="fig:gluexsigma"}](pipi_spectroscopy.pdf){width="6.in"}
Final states containing neutral particles are almost unexplored in this energy range. As an illustration of the prospects for the analysis of such reactions with GlueX, example mass spectra for the reactions $\gamma p \to p + (4,5,6)\gamma$ from initial GlueX data are shown in Fig. X. Many well-known mesons are clearly seen in this data. With at least an order of magnitude more data expected, the prospects for spectroscopy from amplitude analyses of these reactions look promising..
An example of the prospects for studying charged particle final states can be seen in $\gamma p \to p + \pi^+\pi^-$. A basic question for understanding the light meson spectrum is, what is the spectrum of excited $\rho$ mesons? Several candidates for excited $\rho$’s have been seen over the years, including the observation of a state called the $\rho(1600)$ in photoproduction in SLAC in 1984, shown in Fig. X(left) [@slacrho]. However, the analysis of other reactions and other final states (notably the $4\pi$ final state) have led to the conclusion recorded in the PDG that there are most likely two different states in this mass region, the $\rho(1450)$ and $\rho(1700)$. Many of these data have limited statistical precision, however, one high-statistics analysis is illustrated in Fig. X(middle). The BaBar data for $e^+e^- \to \gamma_\mathrm{ISR} \pi^+\pi^-$ around 1.5 GeV is found to be best described by the interference of the $\rho(1450)$ and $\rho(1700)$ [@babarpipi].
These studies then beg the question: what is going on in photoproduction? Are different states being produced, or are the $\rho(1450)$ and $\rho(1700)$ simply manifesting differently in the photoproduced $\pi^+\pi^-$ spectrum due to the different production processes that contribute, compared to other reactions. GlueX can already take another look at this reaction, with two orders of magnitude more data than available at SLAC. In a first look at the $\pi^+\pi^-$ mass spectrum, shown in Fig. X(right), several enhancements are indeed seen with $M(\pi^+\pi^-) > 1$ GeV. Moment and amplitude analyses are underway to determine their nature.
Summary
=======
Although the study of light mesons has a venerable history, their study has been reinvigorated in recent years. We are entering the era of large, high-quality data sets, and confronting the challenge of understanding these precise data is leading the way towards a better understanding of the light meson spectrum. We can expect more data and refined analyses from BES-III and COMPASS in the coming years. Besides the start of GlueX data taking, we can also look forward to data from the CLAS12 experiment at Jefferson Lab, which will use both electron and photon beams, and the PANDA $p\bar{p}$ annihilation experiment at GSI/FAIR, currently slated to start running no earlier than 2022. With renewed efforts from both experiment and theory working in close collaboration, the future of this field looks bright indeed!
[99]{} For a general review of the subject, see, e.g., S. Godfrey and J. Napolitano, Rev. Mod. Phys. **71**, 1411 (1999). C. Patrignani et al. \[Particle Data Group\], Chin. Phys. C, **40**, 100001 (2016) and 2017 update. For example, J. J. Dudek, R. G. Edwards, P. Guo, and C. E. Thomas \[Hadron Spectrum Collaboration\], Phys. Rev. D **88**, 094505 (2013). M. Ablikim et al., \[BES-III Collaboration\], Nucl. Instr. Meth. Phys. A **614**, 345 (2010). J. Z. Bai et al. \[BES-II Collaboration\], Phys. Rev. Lett. **91**, 022001 (2003). M. Ablikim et al. \[BES-III Collaboration\], Phys. Rev. Lett. **108**, 112003 (2012). M. Ablikim et al. \[BES-II Collaboration\], Phys. Rev. Lett. **95**, 262001 (2005). M. Ablikim et al. \[BES-III Collaboration\], Phys. Rev. Lett. **106**, 072002 (2011). M. Ablikim et al. \[BES-III Collaboration\], Phys. Rev. D **88**, 091502(R) (2013). M. Ablikim et al. \[BES-III Collaboration\], Phys. Rev. Lett. **115**, 091803 (2015). M. Ablikim et al. \[BES-III Collaboration\], Phys. Rev. D **87**, 032008 (2013). M. Ablikim et al. \[BES-III Collaboration\], Phys. Rev. Lett. **107**, 182001 (2011). M. Ablikim et al. \[BES-II Collaboration\], Eur. Phys. J. C **53**, 15 (2008); M. Ablikim et al. \[BES-II Collaboration\], Phys. Rev. D **80**, 052004 (2009); J. P. Alexander, et al. \[CLEO Collaboration\], Phys. Rev. D **82**, 092002 (2010); M. Ablikim et al. \[BES-III Collaboration\], Phys. Rev. Lett. 110, 022001 (2013). M. Ablikim et al., \[BES-III Collaboration\], Phys. Rev. Lett. **117**, 042002 (2016). J. Zhang, elsewhere in these proceedings. M. Ablikim et al., \[BES-III Collaboration\], Phys. Rev. D **93**, 112011 (2016). M. Ablikim et al., \[BES-III Collaboration\], Phys. Rev. D **92**, 052003 (2015). M. Ablikim et al. \[BESIII Collaboration\], Phys. Rev. D **87**, 092009 (2013).
P. Abbon, et al. \[COMPASS Collaboration\], Nucl. Instrum. and Meth. A **577**, 455 (2007). C. Adolph et al. \[COMPASS Collaboration\], Phys. Rev. D **95**, 032004 (2017). S, Wallner, elsewhere in these proceedings. F. Krinner and B. Grube, elsewhere in these proceedings. A. Jackura et al. \[COMPASS and JPAC Collaborations\], `arXiv:1707.02848 [hep-ph]`.
H. Al Ghoul et al. \[GlueX Collaboration\], AIP Conf. Proc. **1735** 020001 (2016). S. Dobbs, elsewhere in these proceedings. H. Al Ghoul et al. \[GlueX Collaboration\], Phys. Rev. C **95**, 042201 (2017). K. Abe et al. \[SLAC Hybrid Facility Photon Collaboration\], Phys. Rev. Lett. **53**, 751 (1984). J. P. Lees et al. \[BaBar Collaboration\], Phys. Rev. D **86** 032013 (2012).
[^1]: Previous affiliation: Northwestern University.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We show, in the context of single photon detection, that an atomic three-level model for a transmon in a transmission line does not support the predictions of the nonlinear polarisability model known as the cross-Kerr effect. We show that the induced displacement of a probe in the presence or absence of a single photon in the signal field, cannot be resolved above the quantum noise in the probe. This strongly suggests that cross-Kerr media are not suitable for photon counting or related single photon applications. Our results are presented in the context of a transmon in a one dimensional microwave waveguide, but the conclusions also apply to optical systems.'
author:
- 'Bixuan Fan$^{1}$, Anton F. Kockum$^{2}$, Joshua Combes$^{3}$, Göran Johansson$^{2}$, Io-chun Hoi$^2$, Christopher Wilson$^2$, Per Delsing$^2$, G. J. Milburn$^{1\dag}$ and Thomas M. Stace$^{1\ast}$'
bibliography:
- 'xuan\_ref.bib'
title: ' Breakdown of the cross-Kerr scheme for Photon Counting'
---
The cross-Kerr effect, whereby the phase of one field is changed proportional to the intensity of another, arises from the nonlinear response of an atomic medium to applied fields. It is usually described phenomenologically in terms of a third order term in the nonlinear polarisability, a description that is valid when the applied fields are strong and absorption is weak [@Boyd]. A derivation of the nonlinear polarisability description of the Kerr effect based on an underlying microscopic atomic model has been given by many authors including [@DrumWalls; @Hilico; @Sinclair].
Many proposed applications of the cross-Kerr effect however suppose that at least one of the fields is very weak — perhaps only a single photon — including non-demolition measurements [@imoto; @grangier1998; @munro; @roos], quantum state preparation [@ham; @gerry; @wubiao], quantum teleportation [@vitali] and quantum logic gates build-up [@milburn1989; @chuang; @nemoto; @munroJOP; @munro05NJP]. All these schemes require strong Kerr nonlinearities at the level of a single photon. It is not clear that the standard model of a cross-Kerr effect, based on a third-order nonlinear polarisability, should be valid for fields with only a few photons.
Doubts regarding the utility of the Kerr effect in single photon applications have been raised before. Shapiro and Razevi [@shapiro06; @shapiro07] took the multimode nature of the single photon pulse into consideration and found that there is extra phase noise compared to simple single mode calculations, leading to constraints on the achievable phase shifts. Gea-Banacloche [@gea10] pointed out that it is impossible to obtain large phase shifts via the Kerr effect with single photon wave-packets. None of this prior work has addressed in detail the question of the cross-Kerr phase shift on a coherent probe field in the presence or absence of a single photon in the control field.
Recently, superconducting circuits have become important test-beds for microwave quantum optics, demonstrating quantised fields, artificial “atoms” (i.e. with well-resolved energy levels), and strong “atom”-field interactions. The transmon [@koch] is one of most promising superconducting artificial atoms due to its insensitivity to $1/f$ noise, strong anharmonicity, and large dipole moment. Indeed, the typical size of the transmon is comparable to the dielectric gap in an on-chip microwave waveguide, and so the dipole moment is within an order of magnitude of the maximum that it can possibly be, given the geometrical constraints of the dielectric gap [@devoret2007circuit]. This fact leads to the possibility of very large cross-Kerr nonlinearities, where the transmon provides the non-linear polarisability. Recent experiments using a superconducting transmon in a 1D microwave transmission line have demonstrated gigantic cross-Kerr nonlinearities: a control field with [*on average*]{} 1 photon induces a phase shift in the probe field of 11 degrees [@iochun]. Importantly, in this experiment, the microwave fields were freely propagating; no cavity was involved.
This large cross-Kerr phase shift immediately suggests the possibility of constructing a broadband, number-resolving, microwave-photon counter, as long as the cross-Kerr induced displacement of the probe exceeds the intrinsic quantum noise in the probe. Indeed, broadband microwave photon counting is a crucial missing piece of the experimental quantum microwave toolbox, although there are several proposals for detecting microwave photons [@Romero; @RomeroPhysica; @Johnson; @Peropadre; @ChenHover].
In fact, the cross-Kerr interaction is strictly an effective interaction based on weak field–dipole coupling approximations. Ultimately it is mediated by the strong nonlinearities inherent in an anharmonic oscillator (e.g. an atom), so it must eventually break down as a useful description of the physics. It is therefore important to understand the contribution of the transmon (or atomic) dynamics to the effective nonlinearity in the limit of very strong coupling, which was achieved in [@iochun]. In this work we investigate the coupled field–transmon dynamics in this limit, using proposals for microwave–photon counting as a technical objective to evaluate the validity of the cross-Kerr approximation.
We consider two fields, a probe and a control, incident on a superconducting transmon qubit, which is treated as a three-level, $\Xi$-type system in a one-dimensional transmission line. Such three-level systems are prototypes for analysing cross-Kerr nonlinearities [@grangier1998]. We do not eliminate the transmon, but instead treat its dynamics exactly, including quantum noise in the incident fields. The probe is assumed to be a coherent field (or possibly squeezed), while the control field is in a Fock state, whose photon number, $n$, we are trying to measure. For our purposes, we restrict to $n=0$ or 1.
We show that in spite of the very large cross-Kerr nonlinearity, the induced probe displacement (i.e. the signal) in the presence of a single control photon is limited by saturation effects in the transmon, and is always less than the probe’s own quantum noise. That is, the signal-to-noise ratio (SNR) is always below unity. Moreover, our conclusion also extends to the -type four-level atomic level configuration, with which cross-Kerr media are often modelled [@schmidt; @kang; @chen; @hu]. These conclusions have rather profound implications for the exploitation of cross-Kerr phenomena in quantum technologies.
The transmon levels are $\{{|a\rangle},{|b\rangle},{|c\rangle}\}$, with corresponding energy levels, $\omega_i$, and decay rates, $\gamma_i$, as shown in Fig. \[schematic\]. Relaxation between transmon energy levels is relatively fast compared to dephasing rates, which we neglect. The probe field, $\hat{b}$, is in a coherent state ${|\beta\rangle}$, and is nearly resonant with the ${|b\rangle}\leftrightarrow{|c\rangle}$ transition, whilst the control field is in a Fock state of $n=0$ or 1 photons, at a frequency $\omega_{con}$ close to the ${|a\rangle}\leftrightarrow{|b\rangle}$ transition. Qualitatively, the control field induces a transient population transfer into the state ${|b\rangle}$, and the probe field induces a coherence, $\sigma_{bc}$, between states ${|b\rangle}$ and ${|c\rangle}$. This polarisation couples back to the probe field, so that the probe field is modified from its input state according to the standard input-output relation $$\begin{aligned}
\label{inout}
\hat{b}_{out}=\hat{b}_{in}+\sqrt{\gamma_c}\hat{\sigma}_{bc}.\label{inout}\end{aligned}$$ The homodyne detector monitoring the output probe field yields a photocurrent given by $$\begin{aligned}
J^{hom}_n(t)=\left\langle \hat{y}\right\rangle+\xi.\label{photo}\end{aligned}$$ where $\hat{y}=-i\sqrt{\gamma_c}(\hat{\sigma}_{bc}-\hat{\sigma}_{cb})$ is the transmon polarisation, $\hat{\sigma}_{ij}={|i\rangle}{\langlej|}$ and $\xi dt=dW(t)$ is a Weiner process satisfying $E[dW]=0$, $E[d^2W]=dt$. Finally, the useful signal is the integral of the homodyne current over the lifetime, $T$, of the photon wave packet $$S_{n}=\int_0^T dt\,J^{hom}_n(t)$$ If $n=0$ the transmon dynamics are trivial, and $E[ S_0]=0$. For $n=1$, $E[ S_1]\neq0$, and so $S_1$ represents the useful signal associated with a single photon in the control field. However, in any given measurement, the homodyne current includes quantum noise, characterised by the variance $(\sigma_{S_n})^{2}=E[ S_n^{2}]-E[S_n]^{2}$. To a good approximation, $\sigma_{S_n}$ is independent of the photon number, $n$, and so we define the signal-to-noise ratio, $\textrm{SNR}=E[ S_1]/(\sqrt{2}\sigma_{S})$. Note that we assume that the homodyne current will also include technical noise sources. We ignore these, so that $SNR$ represents the quantum limit for the proposed scheme.
To study quantitatively the system consisting of a transmon interacting with propagating microwave fields, we adopt two different (but consistent) formulations, yielding both numerical and analytic results.
In the first formulation we suppose the control photon is generated by a fictitious cavity which is initially in a Fock state. The field in the cavity decays into the 1D waveguide, and propagates to the transmon, which mediates the interaction between the control and the probe. We emphasize that the cavity is included simply as a model photon source; the transmon is not contained within the cavity. To analyse this system, we use the stochastic cascaded master equation method [@gardiner04; @wiseman11]. The stochastic master equation describing the conditional dynamics of the cascaded cavity field–transmon density matrix, $\rho$, is given by $$\begin{aligned}
\label{ME}
d\rho&=& (-i[H_s,\rho]+\gamma_{con}\mathcal{D}[\hat{a}_{con}]\rho +
\mathcal{D}[\hat{L}_b]\rho+\mathcal{D}[\hat{L}_c]\rho)dt \nonumber\\
&&{}+\sqrt{\gamma_{con}}( [\hat{L}_b,\rho\hat{a}^\dag_{con}]+
[\hat{a}_{con}\rho,\hat{L}_b^\dagger])dt
\nonumber \\
&&{}+\mathcal{H}[\hat{L}_ce^{-i\pi/2}]\rho \,dW\label{me1}\end{aligned}$$ where $\hat{L}_b=\sqrt{\gamma_b}\hat{\sigma}_{ab}$, $\hat{L}_c=\sqrt{\gamma_c}\hat{\sigma}_{bc}$ and $$\begin{aligned}
H_{s}&=&\Delta _{c}\hat{\sigma}_{cc}+\Delta _{b}\hat{\sigma}_{bb}+\Omega _{p}(\hat{\sigma}_{bc}+\hat{\sigma}_{cb}),\\
\mathcal{D}[\hat{r}]\rho &=&\frac{1}{2}(2\hat{r}\rho \hat{r}^{\dag
}-\rho \hat{r}^{\dag }\hat{r}-\hat{r}^{\dag }\hat{r}\rho ),\\
\mathcal{H}[\hat{r}]\rho &=&\hat{r}\rho+\rho\hat{r}^\dag-\rm Tr[\hat{r}\rho+\rho\hat{r}^\dag]\rho,\end{aligned}$$ $\Delta _{b}=\omega _{ba}-\omega _{con}$, $\Delta _{c}=\Delta
_{p}+\Delta _{b}$ $(\Delta _{p}=\omega _{bc}-\omega _{p})$ and $\Omega _{p}=\sqrt{\gamma _{con}}\beta $. We solve [Eq. (\[me1\])]{} for the conditional state of the field–transmon system, from which we compute the conditional homodyne photocurrent, using [Eq. (\[photo\])]{}. This approach allows us to generate a simulated measurement record for ensembles of events in which $n=0$ or 1. From these simulated measurement records, we obtain a histogram of homodyne currents, from which we estimate the SNR.
The second formulation uses the Fock state master equation [@josh; @zoller98]. Instead of simulating the free space photon as the output of a fictitious cavity, the propagating photon wave packet drives the transmon directly. The transmon density matrix acquires indices $m,n$ representing coherences between the transmon and photon Fock subspaces $m$ and $n$. The corresponding master equation for the hierarchy of transmon density matrices $\rho_{m,n}$ is $$\begin{aligned}
\dot{\rho}_{m,n}(t) &=&-i[H_s,\rho _{m,n}]+\mathcal{D}[\hat{L}_{b}]\rho _{m,n}+\mathcal{D}[\hat{L}_{c}]\rho _{m,n} \\\nonumber
&&+\sqrt{n}f^\ast(t)[\hat{L}_{b},\rho _{m,n-1}]+\sqrt{m}f(t)[\rho _{m-1,n},\hat{L}_{b}^{\dagger }]
\notag\end{aligned}$$where $f(t)$ is a complex valued probability amplitude that determines the photon counting rate as proportional to $|f(t)|^2$. We first solve the dynamics for $\rho_{0,0}(t)$, which drives $\rho_{0,1}(t)$ and $\rho_{1,0}(t)$, which in turn drives $\rho_{1,1}(t)$. We solve these analytically and numerically, and use the quantum regression theorem [@lax63] to calculate the SNR (see part A of the supplementary information). The great advantage of this approach is that we obtain the SNR without resorting to stochastic simulations.
If the photon is derived from the exponential (E) decay of a cavity mode, then $f(t)=\sqrt{\gamma _{con}}\exp (-\gamma _{con}t/2)$. Further, this method can handle arbitrary photon wave packets, and we include Gaussian (G) and rectangular (R), as shown in Fig. \[pulses\](top). All pulses contains exactly one photon, that is $\int^T_0 |f(t)|^2 dt=1$ and their common width is $1/\gamma^2_{con}$. The photon induces a polarisation in the transmon, shown in Fig. \[pulses\](bottom). We see different pulse shapes yield modest differences in the transmon polarisation $\left\langle\hat{y}(t)\right\rangle$.
.\[pulses\]
In Fig. \[SNR\], the SNR is shown as a function of the probe amplitude with other parameters (the detunings and the single photon pulse width) optimised. The points are calculated by averaging 5000 trajectories of the stochastic master equation, whilst the solid line is computed from the Fock state master equation. There is good agreement between the two approaches. The inset shows a histogram of the results of the stochastic simulation at the value of $\beta$ that optimises the SNR. Clearly the SNR is everywhere less than unity, and so it is impossible to reliably distinguish between zero and one photon in a single shot. The histogram confirms that the distribution of integrated homodyne current is much broader than the separation of the means.
As an example of the influence of the various parameters on the SNR, we plot the SNR as a function of the detunings $\Delta_b$ and $\Delta_c$, as illustrated in Fig. \[detunings\]. Clearly, the optimal SNR is located at $\Delta_b=\Delta_c=0$.
The reason that SNR$<1$ can be understood in the following way: a single control photon induces a variation in the transmon polarisation $\hat \sigma_{bc}$, which manifests as a fluctuation in the homodyne current according to [Eq. (\[photo\])]{}. However the polarisation of the transmon is a bounded operator: $||\hat y||\leq \sqrt\gamma_b$. The optimal photon wave packet width is $T\sim\gamma_b^{-1}$ (any shorter and the transmon cannot respond to the field; any longer and vacuum noise in the homodyne signal grows), so the integrated polarisation can be no larger than $|E[S_1]|\leq\int_0^Tdt\,||\hat y||\leq\gamma_b^{-1/2}$. Quantum noise in [Eq. (\[photo\])]{} gives $\sigma_{S}^2\geq\textrm{var}[\int_0^Tdt\,\xi]=\gamma_b^{-1}$ we see that the signal to noise ratio is necessarily less than unity. Fig. \[SNR\] bears out this analysis: for small probe field amplitudes, the SNR increases quickly, however the transmon dynamics quickly saturates at higher probe amplitudes.
This argument suggests that the fundamental problem is the saturation of the transmon transition. It may be thought that this can addressed by increasing the number of transmons. We therefore briefly consider a system of $N$ transmons, arranged such that the spacing between adjacent transmons is much smaller than the wavelength, the transmons are described by the collective atomic spin operators $$\begin{aligned}
\hat{S}_{ij}=\frac{1}{\sqrt{N}}\sum_k\sigma^k_{ij}\end{aligned}$$ The stochastic master equation describing the $n$-transmon system is given by $$\begin{aligned}
d\rho&=& -i[H_s,\rho]dt+\gamma_{con}\mathcal{D}[\hat{a}_{con}]\rho dt+N\gamma_b\mathcal{D}[\hat{S}_{ab}]\rho dt\nonumber \\
&+&N\gamma_c\mathcal{D}[\hat{S}_{bc}]\rho dt-\sqrt{N\gamma_{con}\gamma_b}([\hat{S}_{ba},\hat{a}_c\rho]+[\rho\hat{a}^\dag_c,\hat{S}_{ab}])dt \nonumber \\
&+&\sqrt{N\gamma_c}\mathcal{H}[\hat{S}_{bc}e^{-i\pi/2}]\rho dW \label{ntrans}\end{aligned}$$ where $$\begin{aligned}
H_s=N(\Delta_{c}\hat{S}_{cc}+\Delta_{b}\hat{S}_{b})+\sqrt{N\gamma_c}\beta(\hat{S}_{bc}+\hat{S}_{cb})\end{aligned}$$ We can see that the ensemble master equation (\[ntrans\]) is the same form as the single-transmon master equation, albeit with decay rates and energies scaled by a factor of $N$, and its dynamics must therefore be correspondingly faster. However, this cannot change the optimised SNR, so the SNR for the $N$ transmon case will be the same as for the single transmon case. It is worth commenting on a number of other avenues that we have explored, but which yield similar negative results. Firstly, it may be thought that squeezing the probe field in an appropriate quadrature would reduce the noise, and therefore improve the SNR. Since we are monitoring the displacement of the probe field, we should squeeze the phase quadrature. However this will enhance noise in the conjugate, amplitude quadrature. The additional noise in the probe amplitude adds noise to the transmon dynamics arising from fluctuations in $\Omega_p$. We find numerically that these tradeoffs yield no net improvement in the SNR. (see part B of the supplementary information)
Secondly, there is another multi-transmon limit, in which the transmons are sufficiently separated that they may be considered as a series of cascaded systems. It might be hoped that the control field will interact sequentially with each transmon. A local probe at each transmon would then yield an independent estimate of the probe displacement. After $M$ such independent probes, the SNR would be improved by a factor of $M^{1/2}$, and the SNR could be made arbitrarily large by increasing $M$. However, the Kramers-Kronig relations impose a tradeoff between the phase shift at each transmon and the probability that the control field is reflected: a large phase shift necessarily implies a large reflection probability. Again, we find numerically that the tradeoff yields no net improvement (see part C of the supplementary information).
Thirdly, a number of proposals for inducing cross-Kerr nonlinearities in optical systems use an -type four-level system [@schmidt; @kang], with a strong classical field addressing the intermediate transition. As we show in part D of the supplementary information, in the limit of strong driving, this maps onto the same three-level structure we consider in this work, so the conclusions we have reached here also apply to such -type systems.
Fourthly, We also numerically investigated the effect of varying the ratio $\gamma_c /\gamma_b$ and for $1 < \gamma_c/ \gamma_b < 100$ find that the SNR is still much less than unity (see part E of the supplementary information).
A number of proposals suggest using weak Kerr media to build controlled phase and C-NOT gates with fewer resources than linear optical schemes [@nemoto; @munro05NJP]. In these schemes the cross-Kerr phase shift per photon is much less than $\pi$, so a strong coherent bus compensates for the weak nonlinearity, such that the small cross-Kerr phase shift manifests as a large displacement of the strong coherent field. However the saturation of the cross-Kerr effect described above indicates that once the displacement of the strong coherent field approaches its own quantum noise, saturation effects lead to the breakdown of the effective cross-Kerr description, rendering such protocols ineffective.
In summary, we have investigated the feasibility of microwave photon-counting based on an induced cross-Kerr nonlinearity arising from coupling to a large anharmonic dipole. We find that saturation of the transmon transition limits the SNR to less than unity. As such, it is not possible to use strong, atom-induced cross-Kerr nonlinearities to perform single photon detection. This conclusion applies to a number of extensions of the basic model, including multiple transmons, cascaded transmons and $\textsf{N}$-type, four-level system. Further, it limits the applicability of any proposal that requires a cross-Kerr nonlinearity to produce a displacement of a coherent field by an amount greater than the intrinsic quantum noise in the coherent field: it is precisely this condition where the effective cross-Kerr description breaks down, and saturation effects become dominant.
This work is sponsored by ARC Center of Excellence for Engineered Quantum Systems and the China Scholarship council. Bixuan Fan would like to thank Dr.Zhenglu Duan and Dr. Matt Woolley for helpful discussions.
SUPPLEMENTARY INFORMATION {#supplementary-information .unnumbered}
=========================
The analytical solution for a three-level system on resonance
-------------------------------------------------------------
Here we provide the details of the analytical solution for a three level system. As presented in the main text, we consider a three-level transmon coupling with a coherent field at the ${|b\rangle}\leftrightarrow{|c\rangle}$ transition and a single photon at the ${|a\rangle}\leftrightarrow{|b\rangle}$ transition. The one-photon Fock state master equations [@josh] are given by $$\begin{aligned}
\dot{\rho}_{0,0}(t) &=&-i[H_s,\rho _{0,0}]+\mathcal{D}[\hat{L}_{b}]\rho _{0,0}+\mathcal{D}[\hat{L}_{c}]\rho _{0,0} \\
\dot{\rho}_{0,1}(t) &=&-i[H_s,\rho _{0,1}]+\mathcal{D}[\hat{L}_{b}]\rho _{0,1}+\mathcal{D}[\hat{L}_{c}]\rho _{0,1}\\\nonumber
&+&f ^{\ast }(t)[\hat{L}_{b},\rho _{0,0}] \\
\dot{\rho}_{1,1}(t) &=&-i[H_s,\rho _{1,1}]+\mathcal{D}[\hat{L}_{b}]\rho _{1,1}+\mathcal{D}[\hat{L}_{c}]\rho _{1,1}\\\nonumber
&+&f^\ast (t)[\hat{L}_{b},\rho _{1,0}]+f (t)[\rho _{0,1},\hat{L}_{b}^{\dagger }]\end{aligned}$$where the temporal profile function $f(t)$, system operators $\hat{L}_b$, $\hat{L}_c$ and the system Hamiltonian $H_s$ have been defined in the main text. Here $\rho_{m,n}$ is a matrix and the subscripts $m$ and $n$ denote the photon number basis.
Initially the transmon is prepared at the ground state. The lowest equation for $\rho _{0,0}(t)$ can be easily solved as $\rho _{0,0}(t)=\rho _{0,0}(0)$. Then it is substituted to the next equation for $\rho_{0,1}(t)$, which is traceless. For our system and an arbitrary input Fock state, the generalized density matrices $\rho_{m,n}$ in the Bloch-like representation can be parameterized as $$\rho_{m,n} =\frac{1}{3}\mathrm{I} \delta(m,n)+\frac{1}{2}\bar{a}_{m,n}\bar{\lambda}$$where I is a 3 by 3 identity matrix and vectors $\bar{a}_{m,n}=(a_{1mn},a_{2mn},a_{3mn},a_{4mn},a_{5mn},a_{6mn},a_{7mn},a_{8mn})$ and $\bar{\lambda}=(\lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4},\lambda
_{5},\lambda _{6},\lambda _{7},\lambda _{8})$. The Gell-Mann matrices [@hioe] for qutrit are $$\lambda _{1}=\begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0\end{pmatrix}\lambda _{2}=\begin{pmatrix}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0\end{pmatrix}\lambda _{3}=\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0\end{pmatrix}$$$$\lambda _{4}=\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0\end{pmatrix}\lambda _{5}=\begin{pmatrix}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0\end{pmatrix}\lambda _{6}=\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0\end{pmatrix}$$ $$\lambda _{7}=\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0\end{pmatrix}\lambda _{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -2\end{pmatrix}$$These matrices are traceless, Hermitian, and obey the relation $\mathrm{tr}(\lambda _{i}\lambda _{j})=2\delta _{ij}$.
Substituting $\rho_{0,1}$ in terms of $\bar{a}_{0,1}$ into the Fock master equation, we can have the coefficient equations for $a_{i01}(t) (i=1,2,...8)$. With initial conditions $a_{i01}(0)(i=1,2...8)=0$, we have the solutions: $$\begin{aligned}
a_{101}(t) &=&a_{201}(t)=a_{301}(t)=a_{801}(t)=0 \\
a_{501}(t) &=&C_{1}C_{5}\exp (-\theta _{1}t)+C_{1}C_{6}\exp (-\theta
_{1}t)\\\nonumber
&+&C_{1}C_{7}\exp (-\gamma _{con}t/2) \\
a_{601}(t) &=&C_{1}C_{2}\exp (-\theta _{1}t)+C_{1}C_{3}\exp (-\theta
_{2}t)\\\nonumber
&+&C_{1}C_{4}\exp (-\gamma _{con}t/2)] \\
a_{401}(t) &=&ia_{501}(t) \\
a_{701}(t) &=&-ia_{601}(t)\end{aligned}$$ where $$\begin{aligned}
\theta _{1} &=&3\gamma_b /4+\sqrt{\gamma_b }\theta /4,\theta _{2}=3\gamma_b /4-\sqrt{\gamma_b }\theta /4 \\
\theta &=&\sqrt{-32\beta ^{2}+\gamma_b } \\
C_{1} &=&\frac{\sqrt{\gamma_b \gamma _{con}}}{\theta \lbrack 2\gamma_b (4\beta
^{2}+\gamma_b )-3\gamma_b \gamma _{con}+\gamma _{con}^{2}]} \\
C_{2} &=&4\sqrt{\gamma_b }(8\beta ^{2}-\gamma_b +\sqrt{\gamma_b }\theta )+\gamma_{con}(\sqrt{\gamma_b }-\theta ) \\
C_{3} &=&4\sqrt{\gamma_b }(-8\beta ^{2}+\gamma_b +\sqrt{\gamma_b }\theta -\gamma_{con}(\sqrt{\gamma_b }+\theta ) \\
C_{4} &=&-4\gamma_b \theta +2\gamma _{con}\theta \\
C_{5} &=&2\sqrt{2}\beta (-3\gamma_b +2\gamma _{con}+\sqrt{\gamma_b }\theta ) \\
C_{6} &=&2\sqrt{2}\beta (3\gamma_b -2\gamma _{con}+\sqrt{\gamma_b }\theta ) \\
C_{7} &=&4\sqrt{2\gamma_b }\beta \theta\end{aligned}$$ and the density matrix $\rho _{0,1}$ at time $t$ can be presented as $$\rho _{0,1}(t)=\frac{1}{2}\bar{a}_{0,1}\bar{\lambda}=\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
a_{401}(t) & a_{601}(t) & 0\end{array}\right)$$ where we used $\gamma _{c}=2\gamma _{b} $ (for a transmon).
The top level equation $\rho _{1,1}(t)$ which represents the actual system evolution is not traceless. In the Bloch representation the equation is $\rho _{1,1}=\frac{1}{3}\mathrm{I}+\frac{1}{2}\bar{a}_{1,1}\bar{\lambda}$ with initial conditions: $a_{811}(t=0)=-2/\sqrt{3}$ and $a_{i11}(t=0)=0$ for $i\neq 8$. Substituting this expression and the solution for $\rho_{0,1}$ into the master equation we have the motion equation for $\bar{a}_{1,1}$.
Only motion equations for $a_{211}$, $a_{311}$ and $a_{811}$ are coupled and non-zero. We define a vector: $$x=(a_{211},a_{311,}a_{811})^{T}$$with initial condition $$x(0)=(0,0,-\frac{2}{\sqrt{3}})^{T}$$ $$\frac{dx}{dt}=Ax+B(t)$$with coefficient matrices: $$A=\left(
\begin{array}{ccc}
-3\gamma_b /2 & -2\sqrt{2\gamma_b }\beta & 0 \\
2\sqrt{2\gamma_b }\beta & -5\gamma_b /2 & -\sqrt{3}\gamma_b /2 \\
0 & \sqrt{3}\gamma_b /2 & -\gamma_b /2\end{array}\right)$$ $$B(t)=\left(
\begin{array}{c}
-2\sqrt{\gamma_b }\xi a_{501} \\
-\gamma_b +2\sqrt{\gamma_b }\xi a_{601} \\
-\frac{\gamma_b }{\sqrt{3}}-2\sqrt{3\gamma_b }\xi a_{601}\end{array}\right)$$By diagonalization and integration, we obtain the solutions as: $$\begin{aligned}
&&x[i](t) \\\nonumber
&=&V_{i1}C_{11}(t)[\int [C_{11}(-t^{\prime })Q_{11}B_{1}(t^{\prime
})+C_{11}(-t^{\prime })Q_{12}B_{2}(t^{\prime })\\\nonumber
&+&C_{11}(-t^{\prime
})Q_{13}B_{3}(t^{\prime })]dt^{\prime }-\frac{2}{\sqrt{3}}Q_{13}] \notag \\\nonumber
&+&V_{12}C_{22}(t)[\int [C_{12}(-t^{\prime })Q_{21}B_{1}(t^{\prime
})+C_{22}(-t^{\prime })Q_{22}B_{2}(t^{\prime })\\\nonumber
&+& C_{22}(-t^{\prime
})Q_{23}B_{3}(t^{\prime })]dt^{\prime }-\frac{2}{\sqrt{3}}Q_{23}] \notag \\\nonumber
&+&V_{13}C_{33}(t)[\int [C_{13}(-t^{\prime })Q_{31}B_{1}(t^{\prime
})+C_{33}(-t^{\prime })Q_{32}B_{2}(t^{\prime })\\\nonumber
&+&C_{33}(-t^{\prime
})Q_{33}B_{3}(t^{\prime })]dt^{\prime }-\frac{2}{\sqrt{3}}Q_{33}] \notag \\\nonumber
&=&V_{i1}S_{1}+V_{i2}S_{2}+V_{i3}S_{3}-\frac{2}{\sqrt{3}}V_{i1}C_{11}(t)Q_{13}\\\nonumber
&-&\frac{2}{\sqrt{3}}V_{i2}C_{22}(t)Q_{23}-\frac{2}{\sqrt{3}}V_{i3}C_{33}(t)Q_{33}\end{aligned}$$
where$$\begin{aligned}
S_{i} &=&C_{1i}(t)\int [C_{1i}(-t^{\prime })Q_{i1}B_{1}(t^{\prime
})+C_{1i}(-t^{\prime })Q_{i2}B_{2}(t^{\prime })\\\nonumber
&+&C_{1i}(-t^{\prime
})Q_{i3}B_{3}(t^{\prime })]dt^{\prime } \\\nonumber
&=&\frac{\gamma_b }{\lambda _{i}}\left( Q_{i2}+\frac{1}{\sqrt{3}}Q_{i3}\right)
\left( 1-\exp \left( \lambda _{i}t\right) \right) \\\nonumber
&+&\frac{2\sqrt{\gamma_b \gamma _{con}}C_{1}C_{4}}{\gamma _{con}+\lambda _{i}}( Q_{i2}-\sqrt{3}Q_{i3}) ( \exp (-(\gamma_b +\gamma_{con})t)\\\nonumber
&-&\exp ( \lambda _{i}t)) \notag \\\nonumber
&-&\frac{2\sqrt{\gamma_b \gamma _{con}}C_{1}C_{2}}{\Delta _{1}+\lambda
_{i}+\frac{\gamma _{con}}{2}}\left( Q_{i2}-\sqrt{3}Q_{i3}\right)( \exp (-(\Delta
_{1}+\frac{\gamma _{con}}{2})t)\\\nonumber
&-&\exp( \lambda _{i}t)) \notag \\\nonumber
&-&\frac{2\sqrt{\gamma_b \gamma _{con}}C_{1}C_{3}}{\Delta _{2}+\lambda
_{i}+\frac{\gamma _{con}}{2}}\left( Q_{i2}-\sqrt{3}Q_{i3}\right)( \exp (-(\Delta
_{2}+\frac{\gamma _{con}}{2})t)\\\nonumber
&-&\exp ( \lambda _{i}t)) \notag\end{aligned}$$ with matrices $V$, $Q$ and $C$ being the eigen-vector matrix, inverse eigen-vector matrix and the diagonalized matrix of the coefficient matrix $A$.
Then we have the system density matrix at time t: $$\begin{aligned}
&&\rho _{1,1}(t)=\frac{1}{3}\mathrm{I}\\\nonumber
&+&\frac{1}{2}\left(
\begin{array}{ccc}
a_{311}(t)+\frac{a_{811}(t)}{\sqrt{3}} & -ia_{211}(t) & 0 \\
ia_{211}(t) & \frac{a_{811}(t)}{\sqrt{3}}-a_{311}(t) & 0 \\
0 & 0 & -2\frac{a_{811}(t)}{\sqrt{3}}\end{array}\right) ,\end{aligned}$$ The transmon polarisation for the homodyne detection is $$\left\langle \hat{y}(t)\right\rangle=\rm Tr[-i(\hat{L}_{c}-\hat{L}_{c}^{\dag })\rho _{1,1}](t)=\rm Tr\left[ \lambda _{2}\rho _{1,1}\right](t) =a_{211}(t)$$
In Fig. \[y\_AN\], we compare the results for $\left\langle \hat{y}(t)\right\rangle$ by analytical and numerical methods. It is a perfect agreement between the results by the two methods.
The noise or the variance of the detected signal, can be calculated by the quantum regression theorem [@lax63]: $$\begin{aligned}
&&(\Delta S)^{2}=E\left[ S^{2}-\bar{S}^{2}\right] \\\nonumber
&&=\int^{T}_0 dt\int^{T}_0 dt' u(t'-t)\rm Tr[\hat{y}(t)e^{\mathcal{L}(t'-t)}(-i\hat{L}_c\rho(t)\\\nonumber
&&+i\rho(t)\hat{L}^\dag_c)]+u(t-t')\rm Tr[\hat{y}e^{\mathcal(t-t')}(-i\hat{L}_c\rho(t')\\\nonumber
&&+i\rho(t')\hat{L}^\dag_c)]+T-\bar{S}^{2}\end{aligned}$$where $S=\int^T_0 dtJ_{hom}(t)$, $\bar{S}=\int^T_0 dt y_{uc}(t)$ and the function $u(t)=1 (t>0); u(t)=0 (t<0)$. The subscripts $uc$ means unconditional results.
Squeezed probe {#squeezing}
--------------
In this subsection we replace the coherent probe field of a phase-squeezed state and the corresponding stochastic master equation is
$$\begin{aligned}
&&d\rho=
(-i[H_s,\rho]+\gamma_{con}\mathcal{D}[\hat{a}_{con}]\rho+\gamma_b\mathcal{D}[\hat{\sigma}_{ab}]\rho
\\\nonumber
&-&\sqrt{\gamma_{con}\gamma_b}([\hat{\sigma}_{ba},\hat{a}_{con}\rho]+[\rho\hat{a}^\dag_{con},\hat{\sigma}_{ab}])+\gamma_c (N+1)\mathcal{D}[\hat{\sigma}_{bc}]\\\nonumber
&+&\gamma_c
N\mathcal{D}[\hat{\sigma}_{cb}]+\gamma_c
M\hat{\sigma}_{bc}\rho\hat{\sigma}_{bc}-\gamma_c
M^*\hat{\sigma}_{cb}\rho\hat{\sigma}_{cb})dt\\\nonumber
&+&\sqrt{\frac{\gamma_c}{L}}\mathcal{H}[(N+1+M)\hat{\sigma}_{bc}e^{-i\pi/2}-(N+M^*)\hat{\sigma}_{cb}e^{i\pi/2}]\rho
dW \label{SME}\end{aligned}$$
where $$\begin{aligned}
H_s &=&
\Delta_{c}\hat{\sigma}_{cc}+\Delta_{b}\hat{\sigma}_{bb}+\sqrt{\gamma_c}\int d\nu
\beta_\nu(\hat{\sigma}_{bc}+\hat{\sigma}_{cb})\end{aligned}$$ with $M=\sinh(r)\cosh(r)e^{i\theta}$, $N=\sinh^2(r)$ and $L=1+2N+M+M^*$. The instantaneous photocurrent is $$\begin{aligned}
I^{hom}_c(t)=\left\langle
\hat{y}\right\rangle_c(t)+\sqrt{L}\xi(t)\end{aligned}$$ This equation indicates that the noise term $\xi$ is multiplied by a factor of $\sqrt{L}$. Notice that when $\theta$ is chosen as zero, $L=e^{-2r}$ and the noise is reduced while when $\theta=\pi$, the noise is amplified.
In Fig. \[SNR\_db\] we show that the squeezing in the phase quadrature can only help to improve the SNR slightly. According to uncertainty relation the phase squeezing indicates an amplification of the amplitude noise, which results in larger dynamical noise in the atomic response.
Cascaded n transmons
--------------------
In the main text we have discussed the single transmon and a transmon ensemble for single microwave detection. In this subsection, we investigate the feasibility of cascading multiple transmons, each with a probe field.
First we evaluate the transmission rate of the signal field after passing one transmon. As is known that in low dimensional systems photons will be reflected completely by qubits on resonance [@SHFan05][@zumofen]. On the other hand, EIT-like effects appear when there are more energy levels [@SHFan09]. In this three-level structure, two-pathway interference forms and the familiar EIT-like transparency window appears, shown in Fig. \[T\_D1\]. However, how wide is the window and whether the parameters for this transparency window is consistent with those at best SNR are questions. In the following we calculate the transmission rate by the similar procedure in [@SHFan05].
The cascaded system is equivalent to the model of transmons interacting with a single photon pulse centered at $\omega_c$ and a coherent field in the transmission line directly. After linearizing the dispersion in the vicinity of $\omega_c$ and formally adding the non-Hermitian damping terms (which come from the Markovian approximation by tracing out the bath operators), the Hamiltonian in the real space can be written as $$\begin{aligned}
H &=& \int
dx[\hat{a}^\dag_R(x)(\omega_c-iv_g\frac{\partial}{\partial
x})\hat{a}_R(x)\\\nonumber
&+&\hat{a}^\dag_L(x)(\omega_c+iv_g\frac{\partial}{\partial
x})\hat{a}_L(x)]+(\omega_{c}-\omega_p-i\gamma_c/2)\hat{\sigma}_{cc}\\\nonumber
&+&(\omega_b-i\gamma_b/2)\hat{\sigma}_{bb}+\int
dx\sqrt{\gamma_b}\delta(x)[\hat{a}^\dag_R\hat{\sigma}_{ab}+\hat{\sigma}_{ba}\hat{a}_R\\\nonumber
&+&\hat{a}^\dag_L\hat{\sigma}_{ab}+\hat{\sigma}_{ba}\hat{a}_L]+\sqrt{\gamma_c}\alpha(\hat{\sigma}_{bc}+\hat{\sigma}_{cb})\end{aligned}$$ where $v_g$ is the group velocity of the signal photon, which depends on the geometry and material of the waveguide. For a typical coplanar waveguide of quantum circuit system, $v_g=\frac{1}{\sqrt{C'L'}}=\frac{1}{C'
Z_0}=c/\sqrt{\epsilon_{eff}}$ with the effective permittivity $\epsilon_{eff}$ around 5.9. The time-independent eigen-equation is $$\begin{aligned}
H\left\vert E_k\right\rangle=E_k\left\vert E_k\right\rangle\end{aligned}$$ with $E_k=\omega=\omega_c+v_g k_R$ and $$\begin{aligned}
\left\vert E_k\right\rangle&=&\int dx\phi_R\hat{a}^\dag_R(x)\left\vert 0_R,0_L,a\right\rangle\\\nonumber
&+&\int dx \phi_L\hat{a}^\dag_L\left\vert0_R,0_L,a\right\rangle\\\nonumber
&+&c_1\left\vert0_R,0_L,b\right\rangle+c_2\left\vert 0_R,0_L,c\right\rangle\end{aligned}$$
Then we have the equations for the coefficients: $$\begin{aligned}
(\omega_{con}-iv_g)\frac{\partial}{\partial x}\phi_R+g_1\delta(x)c_1=\omega\phi_R\end{aligned}$$ $$\begin{aligned}
(\omega_{con}+iv_g)\frac{\partial}{\partial
x}\phi_L+\alpha\delta(x)c_1=\omega\phi_L\end{aligned}$$ $$\begin{aligned}
(\omega_b-i\frac{\gamma_c}{2})c_1+\sqrt{\gamma_b}(\phi_R(0)+\phi_L(0))+\sqrt{\gamma_c}c_2\alpha=\omega c_1\end{aligned}$$ $$\begin{aligned}
(\omega_c-i\gamma_c/2-\omega_p)c_2+\sqrt{\gamma_c}c_1\alpha=\omega c_2\end{aligned}$$ where $$\begin{aligned}
\phi_R=exp(ikx)\theta(-x)+t exp(ikx)\theta(x)\end{aligned}$$ $$\begin{aligned}
\phi_L=rexp(-ikx)\theta(-x)\end{aligned}$$ with $\theta(x)$ being the Heaviside step function.
By solving the equations above, the transmission amplitude can be obtained;
$$\begin{aligned}
t=\frac{(\Delta'_b+i\sqrt{\gamma_c}/2)(\Delta'_c+i\gamma_c/2)-\gamma_c\alpha^2}{(\Delta'_b+i\gamma_c/2+i
\gamma_b/vg)(\Delta'_c+i\gamma_c/2)-\gamma_c\alpha^2}\end{aligned}$$
where $\Delta_b'=\omega_b-\omega$ and $\Delta_c'=\omega_c-\omega_p-\omega$. Here the frequency width of the signal photon is much smaller than its central frequency, that is to say, it is a relatively narrow pulse, therefore in the the following we use $\Delta_b$ and $\Delta_c$ to replace $\Delta_b'$ and $\Delta_c'$.
From Fig. \[T\_D1\], an induced transparency window appears in our system as expected, which is caused by the two-channel interference between transmon transitions. The width of window is twice of the coupling $\sqrt{\gamma_c}\alpha$. One can find that to achieve a high transmission rate, either large $\alpha$ or large $\Delta_b$ is required. This indicates that a large transmission rate corresponds a low SNR.
Assuming we cascade n transmons with separate probes and detectors, seen in Fig. \[scheme\_n\] ideally it is equivalent to average over n trajectories. However, after including the reflection, the effective $SNR_n$ becomes (high order terms $o(R)$ are omitted here): $$\begin{aligned}
SNR_{n}=SNR_{1}(\sqrt{n}T^{n-1}+\sum^{n-1}_{j=1}j/\sqrt{n}T^{j-1}R)\end{aligned}$$ where $SNR_1$ is the SNR in one transmon and one probe case. After optimizing parameters, we obtain the SNR as a function of probe number as shown in Fig. \[n\_probe\]. Obviously, the signal can not win the noise in this method.
![(Color online) The schematic for microwave photon counting using cascaded n tranmons.[]{data-label="scheme_n"}](Ntransmon1.eps){width="45.00000%"}
![The plot of SNR as a function of transmon number n.[]{data-label="n_probe"}](n_probe.eps){width="45.00000%"}
Conversion from N-type Four-level structure to ladder-type three-level structure
--------------------------------------------------------------------------------
The N-type four-level structure has been suggested to be a promising candidate for implementing cross-Kerr nonlinearity. Here in this subsection, we show that it can be approximately mapped to a three-level ladder system.
The Hamiltonian for a four-level system coupling with a signal field $\beta$ at transition 0-1, a control field $\beta$ at transition 1-2 and a probe $\alpha$ at transition 2-3 in a rotating frame is given by $$\begin{aligned}
H&=&(\Delta_{10}-\Delta_{12})\hat{\sigma}_{00}+\Delta_{12}\hat{\sigma}_{11}+\Delta_{32}\hat{\sigma}_{33}\\\nonumber
&+& \sqrt{\gamma_{01}}\beta(\hat{\sigma}_{10}+\hat{\sigma}_{01})+\frac{\Omega}{2}(\hat{\sigma}_{12}+\hat{\sigma}_{21})\\\nonumber
&+&\sqrt{\gamma}_{32}\alpha(\hat{\sigma}_{23}+\hat{\sigma}_{32})\end{aligned}$$ When the transition 1-2 is strongly driven by $\Omega$, the subsystem of the transition 1-2 and the control field can be diagonalized independently. We define a unitary transformation U as $$\left(
\begin{array}{ccc}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{array}
\right)$$
By diagonalizing $U^\dag H_{12,\Omega} U$, we have: $$\begin{aligned}
\theta = \frac{1}{2}\arctan(\Omega/\Delta_{12})\end{aligned}$$and the states ${\left\vert 1\right\rangle, \left\vert 2\right\rangle}$ can be represented in the dressed states $\left\vert -\right\rangle$ and $\left\vert +\right\rangle$[@Cohen]: $$\begin{aligned}
{\left\vert 1\right\rangle}=\cos\theta\left\vert - \right\rangle-\sin\theta\left\vert + \right\rangle\\\nonumber
{\left\vert 2 \right\rangle}=\cos\theta\left\vert +\right\rangle+\sin\theta\left\vert- \right\rangle\end{aligned}$$ Then the Hamiltonian can be rewritten as $$\begin{aligned}
H&=&-\Delta_{10}\hat{\sigma}_{00}+\Delta_{32}\hat{\sigma}_{33}+\lambda_+\hat{\sigma}_{++}+\lambda_-\hat{\sigma}_{--}\\\nonumber
&+& \sqrt{\gamma_{01}}\beta(\cos\theta(\hat{\sigma}_{-0}+\hat{\sigma}_{0-})-\sin\theta(\hat{\sigma}_{+0}+\hat{\sigma}_{0+}))\\\nonumber
&+&\sqrt{\gamma}_{32}\alpha(\cos\theta(\hat{\sigma}_{+3}+\hat{\sigma}_{3+})+\sin\theta(\hat{\sigma}_{-3}+\hat{\sigma}_{3-}))\end{aligned}$$ where $\lambda_{\pm}=\frac{1}{2}\Delta_{12}\pm \frac{1}{2}\sqrt{\Delta^2_{12}+\Omega^2}$.
![(Color online) The comparison of the population of the excited state in the two configurations. The parameters are $\Delta_{12}=0$, $\Delta_{10}=\Delta_{32}=\lambda_-$, $\Omega=10\gamma_{10}$, and $\alpha=\beta=\gamma_{12}=\gamma_{32}=\gamma_{10}$.[]{data-label="4L_3L"}](N4L_ladder3L.eps){width="45.00000%"}
We assume the signal field and probe field are tuned to be resonant with the state $\left\vert -\right\rangle$, that is, $\Delta_{10}=\lambda_-$ and $\Delta_{32}=\lambda_-$. We perform a rotating and ignore the fast-varying terms like $e^{i\sqrt{\Delta^2_{12}+\Omega^2}t}$, then we have an approximate three level system: $$\begin{aligned}
H= \cos\theta\sqrt{\gamma_{01}}\beta(\hat{\sigma}_{-0}+\hat{\sigma}_{0-})+\sin\theta\sqrt{\gamma_{32}}\alpha(\hat{\sigma}_{-3}+\hat{\sigma}_{3-})\end{aligned}$$ As shown in Fig. \[4L\_3L\], it is a good approximation when there is a strong driving ($\Omega\gg \gamma_{01}, \alpha, \beta$) at the dressed transition.
Effect of different relaxation rate ratio $\gamma_c/\gamma_b$
-------------------------------------------------------------
For a transmon, the relaxation rate of the upper transition is twice that of the lower transition but in other three-level system we may have different ratio of the relaxation rates. Here we investigate the effect of different relative relaxation rates on the SNR numerically. Fig. \[rc2rb\] shows the SNR as a function of the ratio $\gamma_c/\gamma_b$. It is clear that no matter lower ratio or higher ratio the best SNR can not be improved much and still far below unity.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- François Ménard
- Xavier Delfosse
- 'Jean-Louis Monin'
date: 'Received: / Accepted: '
title: 'Optical Linear Polarimetry of Ultra Cool Dwarfs[^1]'
---
Introduction
============
Recent sky surveys have uncovered large populations of objects cooler than M dwarfs (e.g., Delfosse et al. 1997; Kirkpatrick et al. 1999). Among them, the L dwarfs were the first to be identified (Martín et al. 1997). They cover a range of effective temperature between $\sim$2200 K and $\sim$1400 K and are characterized by the presence of condensates (i.e., grains) in their atmospheres. Of particular interest, (spectro-)photometric monitoring revealed variability (e.g., Gelino et al. 2002; Bailer-Jones 2002) that was attributed to the presence of rapidly evolving clouds of particles covering the photospheres not uniformly. Depending on the geometry, light scattering by these clouds may yield net disk-integrated polarisation.
However, there are various ways by which the light of a star can be polarised. The presence of a magnetic field can induce polarisation by Zeeman effect. Light scattering by inhomogeneous clouds or rapid rotation leading to an elliptical photospheric disk can also, in principle, yield net disk-integrated polarisation. In this paper we explore, from an observational point of view, the linear polarisation properties of field brown dwarfs.
We present the linear polarisation measurements of eight ultra cool dwarfs (i.e., 1 very late M and 7 L dwarfs) obtained in the red, at 768nm, in a first step to constrain the dust distribution across the photospheres of cool objects. The observations and results are presented in §2. In §3, we present arguments regarding the origin of the detected polarisation. In §4 we discuss realistic photospheric scattering geometries and propose observational tests. The behaviour of the polarisation as a function of T$_{\rm eff}$ is presented in §5.
Observations and results
========================
All the polarimetric data were obtained with the imaging polarimetry mode of FORS1 attached to Melipal, UT3 of ESO’s VLT facility located atop Cerro Paranal, Chile. FORS1 is mounted at the cassegrain focus and provides a classical set-up for accurate dual beam imaging polarimetry[^2].
-------------------------- ---------- --------- --------- ---------- ----------- ------------------ --------------- -------------
name I I-J J-K sp.ty. $\pi$ v$\sin{i}$ EW(H$\alpha$) phot. var.
mas mag
LHS 102B 17.0(3) 3.70(3) 1.90(3) L5(3) 104.7(7) 32.5$\pm$2.5(11) $<$4.0(4)
2MASSW J001544.7+351603 17.3(8) 3.50(8) 1.58(2) L2(2) 54.2(10) 2(2) 0.02?(13)
2MASSW J003615.9+182110 16.11(6) 3.67(6) 1.41(2) L3.5(5) 114.2(6) 15(12) $<$0.5(2) $<$0.01(13)
DENIS-P J0255.0-4700 17.14(4) 3.66(4) 1.62(4) L8(4,2) 159.2(10) 40$\pm$10(11) $<$2(4)
DENIS-P J200048.4-752306 15.88(1) 3.23(1) 1.19(1) M9(8) 58.5(9)
DENIS-P J203608.6-130638 18.24(1) 3.54(1) 1.17(1) L1-L2(8) 31.3(9)
DENIS-P J205754.1-025229 16.61(1) 3.42(1) 1.56(1) L0-L1(8) 55.7(9)
2MASSW J222443.8-015852 18.02(6) 3.97(6) 2.03(2) L4.5(2) 88.1(6) 1(2) 0.08(13)
-------------------------- ---------- --------- --------- ---------- ----------- ------------------ --------------- -------------
\[tab:generaldata\] Note: (1)From DENIS survey, Delfosse et al. (2002), in preparation; (2) Kirkpatrick et al. (2000); (3) Goldman et al. (1999); (4) Martín et al. (1999); (5) Reid et al. (2000); (6) Dahn et al. (2002); (7) Proper motion of LHS 102A, from Van Altena et al. (1995); (8) Following the (I-J) vs. spectral type relations of Dahn et al. (2002); (9) Following the (I-J) vs. M$_I$ relation of Dahn et al. (2002); (10) Following the M$_J$ vs. spectral type relation of Dahn et al. (2002); (11) Basri et al. (2000); (12) Schweitzer et al. (2001); (13) Gelino et al. (2002)
The observations were carried out during the periods 9-11 December 2001 and 16-19 May 2002. The Moon was set at the time of each observation. All data were collected with a broadband I$_{\rm
Bessel}$ filter centered on 768nm and 138nm wide (FWHM). The efficiency and stability of the instrument was checked and confirmed on 6 occasions by measuring the highly polarised star Vela 1-95. All our measurements fell well within 1-$\sigma$ (i.e., $<$0.08%) of the catalog value and no night-to-night efficiency corrections were applied. We also checked for instrumental polarisation by measuring 3 nearby non-magnetic white dwarfs. With a typical 1-$\sigma$ error bar of $\sigma_{\rm P}$=0.02% all three measurements are compatible with zero polarisation. In the following we will assume the imaging set-up to be free of instrumental polarisation.
The data were detrended in a standard way with [noao/iraf]{}. We applied bias subtraction, cosmetic correction for deviant pixels, and division by twilight flatfields. The flatfield frames were obtained without the polarising optics. Due to a slight but systematic variation of the bias level from night to night, i.e., an increase by 1-2 ADU every day, we obtained and used new sets of calibrations frames every night for safety.
Except for LHS 102B (Goldman et al. 1999), we selected our targets from the 2MASS and DENIS near-infrared sky surveys. They are listed in Table\[tab:generaldata\] together with photometric and spectroscopic information. Columns 2, 3, and 4 list the I-band magnitudes and I-J and J-K colors respectively. Column 5 lists the spectral types. A range is given when the spectral type is estimated from near-infrared colors (except for the M9 DENIS-P J200048.4-752306), all others are confirmed by spectroscopy. When there is ambiguity, the spectral types are given in the Kirkpatrick et al. (1999) system. The last four columns list the annual parallax, $\pi$, the projected rotational velocities, v$\sin{i}$, the H$\alpha$ equivalent widths and the photometric variability, when available. References are listed in parenthesis next to the data and refer to the notes at the bottom of the table.
Our results are presented in Table \[tab:poldata\]. In order, the columns list the abreviated target name, the date of observation, the modified Julian date of the middle of the observation, and the polarisation data, P, its associated error $\sigma_{\rm P}$, and the position angle of the plane of vibration of the E-vector in the equatorial coordinate system, when P/$\sigma_{\rm P}\sim$ 3.0 or more. The modified Julian date (MJD) is related to the julian date (JD) by MJD = JD - 2400000.5.
---------- ----------- ----------- ------- ------------------ ----------
name obs. date MJD P $\sigma_{\rm P}$ $\theta$
d/m/y (%) (%) $\deg$
LHS 102B 10/12/01 52253.050 0.105 0.036 70.1
2M J0015 9/12/01 52252.047 0.065 0.032 —
2M J0036 9/12/01 52252.089 0.199 0.028 17.6
D J0255 11/12/01 52254.048 0.167 0.040 80.6
D J2000 17/05/02 52411.394 0.083 0.017 122.9
D J2036 19/05/02 52413.391 0.122 0.042 170.4
D J2057 18/05/02 52412.398 0.044 0.023 —
2M J2224 8/05/01 52251.046 0.095 0.046 —
---------- ----------- ----------- ------- ------------------ ----------
: I-band linear polarisation data
\[tab:poldata\]
The origin of the polarisation
==============================
An interstellar origin?
-----------------------
The annual parallaxes listed in Tab. \[tab:generaldata\] place all the objects between 6pc and 32pc from the Sun. Leroy (1993, 1999) measured a sample of 1000 stars within $\sim$150pc from the Sun. Out to a distance of 50pc no significant interstellar polarisation is found and only 18 stars have P$\ge 0.1$% in the distance range between 60pc and 90pc of the Sun, from Hipparcos distances. Furthermore, they are found in small and well defined regions of the sky, away from our targets. An interstellar origin for the linear polarisation presented here is therefore extremely unlikely and we rule it out. [*A consequence of this result is that the frequency of intrinsically polarised brown dwarfs in our sample (i.e., from M9 to L8 dwarfs) appears extremely high, $\sim$ 50%*]{} (i.e., 3/8 (37%), or 5/8 (62%) if we include the marginal detections, see §4)[^3]. For comparison, in nearby FGKM stars (dwarfs and giants), the fractions are 2.5%, 7%, 5.5%, and 11% respectively in the distance range 60-90pc from the Sun (Leroy 1999). Those fractions go to zero for smaller distances.
Possible mechanisms for intrinsic polarisation
----------------------------------------------
A possibility to produce intrinsic linear polarisation is via the presence of magnetic field, either from Zeeman splitting of atomic or molecular lines or synchrotron emission. The evidence for magnetic field in L dwarfs is not clear yet. Observations show that the H$_{\alpha}$ activity rapidly declines from mid-M to L dwarfs (Gizis et al. 2000). This may result from the high electrical resistivities of their cool, hence mostly neutral, atmospheres (Mohanty et al. 2002). On the other hand, Berger (2002) detected high persistent levels of radio emission in 3 late M and L dwarfs, including 2MASSW J003615.9+182110, the most highly polarised source in our sample. Magnetic fields in the range 10-1000G are deduced assuming that the radio emission is coronal and that it peaks sharply at 8.5 GHz. Therefore, the (gyro-)synchrotron processes invoked will not lead to significant polarisation in the optical, especially linear polarisation.
Zeeman splitting of atomic lines is also unlikely the source of the linear polarisation. For comparison, a sample of Ap stars with a $\sim1$kG dipolar field at the surface show a maximum net linear polarisation of order of a few times 0.01% only (Leroy 1995). This polarisation, produced by Zeeman splitting in saturated atomic lines, is maximum where a large number of atomic lines are present in the spectrum, i.e., in the blue for these stars. For ultra cool dwarfs, atomic absorption does not dominate in the red where we made our measurements. Wider molecular bands do, but these are complex, made of numerous molecular lines side by side (e.g. Valenti et al. 1998) and their global Zeeman polarisation, especially linear, is almost always much lower than that of atomic lines.
Pending definitive measurements of the magnetic fields, we will assume that they are not powerful enough at the surface of late-M and L type brown dwarfs to induce a detectable linear polarisation in the I-band. Scattering by photospheric dust grains therefore remains the most likely mechanism for the polarisation.
The scattering geometry: oblate photospheres or inhomogeneous dust clouds?
==========================================================================
The polarised sources
---------------------
In a recent paper, Sengupta & Krishan (2001, hereafter SK01) argued that the photosphere of a brown dwarf will in general be oblate due to fast rotation. The fast rotation is suggested by the large v$\sin{i}$ values measured in late-M and L dwarfs (Basri et al. 2000). This oblateness will result in an asymmetric scattering geometry where cancellation of the contribution of each point on the photosphere is not perfect (as it would on a sphere) and a net disk-integrated linear polarisation results. They calculated the linear polarisation expected from single and multiple scattering by uniformly distributed dust in an oblate photosphere seen edge-on (see their Figs. 1 & 2 respectively).
In our sample, 3 targets have a significant polarisation: 2MASSW J003615.9+182110, DENIS-P J0255.0-4700, and DENIS-P J200048.4-752306. The maximum polarisation we detect is $\sim$0.2% at 768nm.
From these data, the single scattering case of SK01 for highly eccentric ($e>0.3$) photospheres and most grain sizes except the very smallest ($\alpha << 1.0$) can be excluded, because they would produce too much polarisation, unless all targets in our sample are seen pole-on and their projected photospheric disk is circular. This is unlikely, and not compatible with the v$\sin{i}$ values presented in Tab.\[tab:generaldata\] for 2MASSW J003615.9 and DENIS-P J0255.
Multiple scattering usually lowers the polarisation because the planes of the scattering events are randomly oriented and average each other’s contribution out from the final polarisation. At 768nm, the predictions made by SK01 reflect that fact but the curves for small (0.1$\mu$m) and large (1.0$\mu$m) grains are too close to the polarisations we measured to choose between the two cases. Measurements at longer wavelengths are needed to decide.
On the other hand, the models of SK01 do not consider the possibility that the dust is distributed non uniformly in the photosphere. This configuration may also be relevant to produce net linear polarisation and in principle does not require projected photospheric oblateness. Schubert & Zhang (2000) argue that the dust in L dwarfs should be organized in one of two states: in bands as in the giant planets of our Solar System or in chaotic clouds.
Although numerical simulations are needed to assess the polarising power of these configurations, predictions can be made on geometrical arguments. An oblate and uniformly dusty photosphere will always produce stable polarisation, with both the position angle and the polarisation level fixed, as the scattering geometry is constant. Dust bands in the atmospheres will also produce a stable polarisation. On the other hand, a photosphere covered with randomly distributed clouds is likely to see its polarisation change because the scattering geometry will change with time as the object rotates and the clouds form, move and disappear. Therefore, the detection of variable linear polarisation, especially variations of the polarisation position angle, would clearly point toward cloud covered photospheres rather than homogeneous or banded dust distributions in ultra cool dwarfs.
The unpolarised sources
-----------------------
In our sample, there are also three objects whose polarisation is not detected, and two more which are just marginally detected, i.e., with P/$\sigma_P$ at 2.9, just below 3. Many possibilities exist to explain their non detection. Obviously they could be devoid of dust, but this is unlikely in view of the models and observations. For example, 2MASSW J222443.8-015852 is the only target in our list with a confirmed photometric variability, presumably caused by inhomogeneous dust clouds, but its polarisation is not detected.
Also, the objects that do not have v$\sin{i}$ measurements could be slow rotators, or could be seen close to pole-on, hence showing a low ellipticity to the observer and/or symmetric dust band structure. Although statistically improbable, this possibility cannot be ruled out yet.
To yield a low net polarisation in the optical, the scattering grains may also be much larger than 1$\mu$m, which would agree with recent theoretical predictions for dust size in L dwarf atmospheres. Also, it is possible that the photospheres are covered by a very large number of small randomly distributed dust clouds. Such a configuration should not produce a large detectable polarisation. A few large clouds probably being more favorable.
Linear polarisation vs. spectral type
=====================================
Models predict that the amount and vertical location of dust is a function of T$_{\rm eff}$ (e.g., Allard et al. 2001). In order to match the rapid blueing of mid-L dwarfs, Ackerman & Marley (2001) suggest that the horizontal distribution of the dust is also a function of T$_{\rm eff}$.
Fig.\[fig:polvsspt\] is a plot of the measured linear polarisation as a function of spectral type. Different symbols are used for detections and non-detections (see the figure caption for details). A slight trend for larger polarisation in cooler objects may be present, but more data are clearly needed before we dare claim of anything real. It would be interesting to extend the sample to include cooler objects as their dust is expected to settle below the photosphere and no polarisation from scattering should result.
Conclusions
===========
We have measured the linear polarisation of one very late-M and seven L dwarfs at a wavelength of 768nm. We have 3 detections, 2 marginal cases, and 3 unpolarised targets. In all cases the polarisations remain low, below P=0.2%. The linear polarisation is intrinsic to the objects and not of interstellar origin. The fraction of polarised nearby brown dwarfs in our sample is high, $\sim$50%. It appears much higher than for nearby FGKM stars.
Our small sample does not allow to identify the mechanism responsible for the linear polarisation. However, fast spinning dwarfs with oblate photospheres and uniform, or banded, dust clouds are expected to produce a polarisation constant in time. On the other hand, large randomly distributed dust clouds may produce more erratic polarisations. Searching for polarimetric variability is needed to solve this issue.
For now, we find no definite correlation of polarisation with spectral type although we note a potential trend upward for cooler objects, up to spectral type mid-L. More data are clearly needed, as well as the extension of the sample to cooler T dwarfs. These objects are not expected to have significant amounts of dust in their photospheres and should therefore not show detectable linear polarisation.
It is a pleasure to thank Emanuela Pompei and Michael Dahlem from ESO for their dedicated and efficient support during the observations. The expertise of Thomas Szeifert with FORS1 polarimetry data is also gratefully acknowledged. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. We thank the [*Programme National de Physique Stellaire*]{} (PNPS) for partial financial support.
Ackerman, A.S., Marley, M.S. 2001, ApJ, 556, 872
Allard, F., Hauschildt, P.H., Alexander, D.R., et al. 2001, ApJ, 556, 357
Bailer-Jones, C.A.L., 2002, A&A, 389, 963
Basri, G., Mohanty, S., Allard, F., et al. 2000, ApJ, 538, 363
Berger, E. 2002, ApJ, 572, 503
Dahn, C.C., et al. 2002, AJ, in press
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Gelino, C.R., Marley, M.S., Holtzman, J.A., Ackerman, A.S., Lodders, K. 2002, ApJ, 577, 433
Gizis, J.E., Monet, D.G., Reid, I.N., et al. 2000, AJ, 120, 1085
Goldman, B., Delfosse, X., Forveille, T., et al. 1999, A&A, 351, L5
Kirkpatrick, J.D., et al. 1999, ApJ, 519, 802
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Leroy, J.-L. 1993, A&A, 274, 203
Leroy, J.-L. 1995, A&AS, 114, 79
Leroy, J.-L. 1999, A&A, 346, 955
Martín, E.L., Basri, G., Delfosse, X., Forveille, T., 1997, A&A, 327, L29
Martín, E.L., Delfosse, X., Basri, G., et al. 1999, AJ, 118, 2466
Mohanty, S., Basri, G., Shu, F., Allard, F., Chabrier, G., 2002, ApJ, 571, 469
Reid, I.N., Kirkpatrick, J.D., Gizis, J.E., et al. 2000 AJ, 119, 369
Schubert, G., Zhang, K., 2000, in ASP Conf. Ser. 212, [*From Giant Planets to Cool Stars*]{}, ed. C.A. Griffith & M.S. Marley, (San Francisco: ASP), 210
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Sengupta, S., Krishan, V. 2001, ApJ, 561, L123
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van Altena, W.F., Lee, J.T., Hoffleit, E.D. 1995, [*The General Catalog of Trigonometric paralaxes*]{}, 4$^{th}$ Edition, Yale University Observatory.
[^1]: Based on data collected at ESO/VLT with the FORS1 instrument during observing programs 68-C.0171 and 69-C.0679.
[^2]: Details of the imaging polarimetry mode of FORS1 can be found at [http://www.eso.org/instruments/fors1]{}.
[^3]: We choose to quote 50% by taking the average of the two, hence 4/8.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
A process of the decay of the anomalously low lying nuclear isomer $^{229m}$Th$(3/2^+,8.28 \pm 0.17$ eV) in the Thorium anion (Th$^-$) via the internal conversion (IC) channel is studied. We show that the half life of the nuclear isomer in the $6d_{3/2}^37s_{1/2}^2$ ground state and in the $6d_{3/2}^2
7s_{1/2}^2 7p_{1/2}^1$ excited state of Th$^-$ is $\approx1.5$ and $\approx1.1$ times bigger than in the $6d_{3/2}^2 7s_{1/2}^2$ ground state of the Th atom. The IC probabilities in the anion decreases despite the decay via the additional $6d_{3/2}$ or $7p_{1/2}$ electrons. This counterintuitive result is a consequence: a) of a decrease in the amplitudes of the $6d_{3/2}$ and $7s_{1/2}$ wave functions near the nucleus due to an increase in their diffuseness of upon the addition of extra electron, b) of mutual compensation in the IC probability due to a kinematic factor, which depends on the energy of the conversion electron in the continuum as $E_c^{-1/2}$, and the $E_c^{1/4}$ growth of the amplitudes of the electron wave functions.
author:
- 'E. V. Tkalya'
- 'R. Si'
title: 'Internal conversion of the low energy $^{229m}$Th isomer in the Thorium anion'
---
Introduction {#sec:Introduction}
============
The $^{229}$Th nucleus has a unique low-lying isomeric state $^{229m}$Th($3/2^+,E_{\text{is}}=8.28 \pm 0.17$ eV) [@Seiferle-19]. The dramatic and controversial story of the experimental studies of this state — the discovering of the level [@Kroger-76; @Reich-90; @Burke-90; @Irwin-97; @Richardson-98; @Utter-99; @Tkalya-99-JETPL; @Shaw-99; @Browne-01; @Zhao-12; @Peik-13], the measuring its energy [@Kroger-76; @Reich-90; @Helmer-94; @Beck-07; @Wense-16; @Seiferle-17; @Masuda-19; @Seiferle-19], magnetic and quadrupole moments [@Thielking-18], charge radius [@Safronova-18], and half-life [@Wense-16; @Seiferle-17] is still far from complete. Increasing the accuracy of the measurements is important for the creation of ultra precise clock at the nuclear transition of the optical range [@Peik-03; @Rellergert-10; @Campbell-12; @Peik-15], which in turn can be used to study the relative effects of the variation of the fine structure constant and the strong interaction parameter [@Flambaum-06; @Litvinova-09; @Berengut-09]. Investigations of this nuclear state are important for the design the laser at the nuclear transition [@Tkalya-11; @Tkalya-13], to control the isomeric level $\gamma$ decay via boundary conditions [@Tkalya-18-PRL], to detect the decay of the ground state sublevel of the nucleus into the isomeric state sublevel in the muonic atom $^{229}$Th [@Tkalya-16-PRA], to check the exponentiality of the decay law of at long times [@Dykhne-98] and others.
We know now five possible decay channels of the $^{229m}$Th isomer — four processes where the electron shell is involved, and the alpha decay. It is natural to systematize the first four channels in the framework of the perturbation theory for the quantum electrodynamics using the order of relevant Feynman diagrams [@Tkalya-04].
In the first order of the perturbation theory, this is the process of the emission of a photon by the nucleus, which for low-energy nuclear levels is practically unobservable. However, the photon observation is possible if the $^{229m}$Th isomer is put in the dielectrics with a large band gap, where the thorium atom becomes effectively “ionized” by the chemical environment. For the first time, this possibility was indicated in [@Tkalya-00-JETPL; @Tkalya-00-PRC].
Internal conversion (IC) is a second-order process. IC is the main decay channel of the isomer $^{229m}$Th on the valence shells of the ground state of the Th atom [@Strizhov-91], on the excited atomic states of Th [@Strizhov-91; @Bilous-17], and on the Rydberg states [@Tkalya-19-PRC-IC_Rydb]. Experimentally, IC was observed in Refs. [@Wense-16; @Seiferle-17; @Shigekawa-19]. Another second-order process, namely, the decay of the $^{229m}$Th isomer during inelastic scattering by metal conduction electrons, was considered in [@Tkalya-99-JETPL]. The nuclear excitation by electron transition (NEET) [@Morita-73] is also of the second order process. A detailed theory of NEET is given in [@Tkalya-92; @Tkalya-07]. This process can play an important role as an integral part of the electron bridge.
Electron bridge, a third-order process suggested in [@Krutov-68], was considered for the decay and excitation of $^{229m}$Th in Refs. [@Strizhov-91; @Tkalya-92-JETPL; @Tkalya-92-SJNP]. Later, this process was thoroughly studied theoretically (see in [@Kalman-94; @Tkalya-96; @Porsev-10-PRL; @Porsev-10-PRA-1+; @Karpeshin-17; @Muller-19; @Borisyuk-19-PRC]), and there are high hopes for the effective excitation of the $^{229}$Th nuclei in ion traps.
The $\alpha$ decay of $^{229m}$Th considered in [@Dykhne-96; @Varlamov-97] is an important decay channel. The detection of this process or accompanying bremsstrahlung [@Tkalya-99-PRC] could be the most reliable proof of both the existence of the $^{229m}$Th isomer and its excitation by the laser radiation [@Tkalya-96].
In this paper, we investigate the decay of $^{229m}$Th in the thorium anion, Th${^-}$. Anions are negative ions created when an atom gains one or more electrons. Particularly often the anions are formed from the chemical elements in the groups 17 and 16 of the Periodic table (F$^-$, O$^{2-}$ and so on). These elements lack, respectively, one or two electrons with respect to the complete electronic configuration of a noble gas. However, anions of the other chemical elements can be formed in various physicochemical processes, too. Laser plasma is a well-studied universal source of the negative ions [@Sil'nov-07]. The laser ablation method allows one to get the anions of any chemical element in the Periodic table. In the laser plasma, anions are produced at a certain stage of the expansion during cooling of the plasma bunch after the end of the laser pulse. Typical values of the negative ion currents are tens to hundreds of microamperes. In Ref. [@Tang-19], the Th$^-$ anions were produced via the pulsed Nd:Y-Al-garnet laser ablation of a thorium metal disk. Further, the anions were accumulated and cooled via buffer gas cooling in the ion trap. After that, anions were photodetached by a tunable dye laser and the outgoing photoelectrons were detected.
It turned out that Th${^-}$ is a stable system with the ionization potential of about 0.6 eV [@Malley-09; @Tang-19]. It has at least a couple of levels connected by a strong electric dipole transition suitable for laser cooling [@Tang-19]. Since the cooled ions in traps manipulated by laser and the laser ablation (which produces plasma contained as well as positive and negatively charged ions), as a method of loading the ion trap, are currently considered as a promising system for studying of $^{229m}$Th, the knowledge of the decay channels and the lifetime of the $^{229m}$Th isomer in anions can be very useful for future experimental research.
Internal conversion in Th$^-$ {#sec:IC}
=============================
Until recently, the $6d_
{3/2}^2{}7s_{1/2}^2{}7p_{1/2}^1$ $^4G_{5/2}^{\circ}$ state with the binding energy of 0.368 eV was considered as the ground state of the thorium anion [@Malley-09]. However, as has been shown in [@Tang-19], $6d_{3/2}^2{}7s_{1/2}^2{}7p_{1/2}^1$ $^4G_{5/2}^{\circ}$ is an excited state, and the true ground state of Th${^-}$ is the configuration $6d_{3/2}^3{}7s_{1/2}^2$ $^4F_{3/2}$ with the binding energy of 0.6 eV. In addition, there are several strong electric dipole transitions between the bound levels arising from configurations $6d_{3/2}^3{}7s_{1/2}^2$ and $6d_{3/2}^2{}7s_{1/2}^2{}7p_{1/2}^1$. These conclusions have been reached on the basis of measurements of the ionization potential for the Thorium anion, and large-scale numerical multiconfiguration Dirac-Hartree-Fock calculations [@Si-18; @Tang-19].
In the following, we present the analysis of the internal conversion, which uses the bound state wave functions obtained in Ref. [@Tang-19]: the $6d_{3/2}^2{}7s_{1/2}^2$ wave function for the ground state of the Thorium atom (Th), the $6d_{3/2}^2{}7s_{1/2}^2$ wave function for the ground state of the anion (Th$^-$), and the $6d_{3/2}^2{}7s_{1/2}^2{}7p_{1/2}^1$ wave function for the anion excited state (Th$^{-^*}$). Thus, we are able to compare the internal conversion coefficients (ICC) for the Th atom and anions, obtained within the same approach. This is important because different codes can give significantly different results when one calculates ICC from valence shells at ultra-low energy nuclear transition (see below Table \[tab:ICC\_TBSB\]).
The internal conversion coefficients per one electron for the $E(M)L$ nuclear transition with the energy $\omega_N=E_{\text{is}}$ were calculated using the formulas $$\begin{aligned}
\alpha_{E/ML} = \frac{\omega_N}{m} \frac{E+m}{p} \frac{L}{L+1}
\sum_{f} \left( C^{j_f 1/2}_{j_i 1/2 L 0}\right)^2
|{\cal{M}}^{E/ML}_{if}|^2,
\label{eq:ICC_EML}\end{aligned}$$ where $m$ is the mass of the electron, $E$ and $p$ are the energy and momentum of the conversion electron satisfying the condition $E^2 = m^2 + p^2$ (the system of units is $\hbar=c=1$), $j$ is the total angular momentum of the electron, $C^{j_f 1/2}_{j_i 1/2 L
0}$ is the Clebsch-Gordan coefficient. The electron matrix elements in Eq. (\[eq:ICC\_EML\]) are $$\begin{split}
{\cal{M}}^{EL}_{if} &= \int_0^{\infty} h_L^{(1)}(\omega_N a_B x)
[g_i(x)g_f(x)+ f_i(x)f_f(x)]x^2 dx,\\
{\cal{M}}^{ML}_{if} &=\cfrac{\kappa_i+\kappa_f}{L}
\int_0^{\infty} h_L^{(1)}(\omega_N a_B x)[g_i(x)f_f(x)+ \\
&\qquad\qquad {}f_i(x)g_f(x)]x^2 dx.
\end{split}
\label{eq:ME}$$ Here $x=r/a_B$, $a_B$ is the Bohr radius, $h_L^{(1)}(\omega_N a_B
x)$ is the Hankel function of the first kind [@Abramowitz-64], $\kappa = l(l+1)-j(j+1)-1/4$, where $l$ is the orbital angular momentum of the electron.
One can use the well-known representation for the Clebsch-Gordan coefficient through the $6j$ symbol [@Bohr-98-I] $$\left(C^{j_f 1/2}_{j_i 1/2 L 0}\right)^2 =(2l_i+1)(2j_f+1)
\left(C^{l_f 0}_{l_i 0 L 0} \right)^2
\left\{
\begin{array}{ccc}
l_i & 1/2 & j_i \\
j_f & L & l_f
\end{array}
\right\}^2
\label{eq:CGC}$$ in order for the selection rules for the parity and the orbital angular momentum in Eq. (\[eq:ICC\_EML\]) to be satisfied automatically. In this case, the $l_i\rightarrow l_i' = 2j_i-l_i$ substitution should be made in Eq. (\[eq:CGC\]) for the magnetic type nuclear transitions.
In the case we are considering here, the nuclear transition energy is small and the following conditions are fulfilled: $E_b
<\omega_N \ll 1/a_B$, where $E_b$ is the electron binding energy in the initial state. Therefore, Eqs. (\[eq:ICC\_EML\]–\[eq:ME\]) take the form $$\begin{aligned}
\alpha_{E/ML} &=& e^2 \sqrt{\frac{2m}{\omega_N-E_b}}
\frac{[(2L-1)!!]^2}{(\omega_Na_B)^{2L+1}}\frac{L}{L+1} \nonumber
\\
&&\sum_{f} \left( C^{j_f 1/2}_{j_i 1/2 L 0}\right)^2
|{\text{\textsl{m}}}^{E/ML}_{if}|^2, \label{eq:ICC_EML_2}\end{aligned}$$ where the new electron matrix elements are $$\begin{split}
{\text{\textsl{m}}}^{EL}_{if} &= \int_0^{\infty}
[g_i(x)g_f(x)+ f_i(x)f_f(x)]x^{2-L-1} dx,\\
{\text{\textsl{m}}}^{ML}_{if} &= \cfrac{\kappa_i+\kappa_f}{L}
\int_0^{\infty}[g_i(x)f_f(x)+f_i(x)g_f(x)]x^{2-L-1} dx.
\end{split}
\label{eq:ME_2}$$ The energy of the nonrelativistic conversion electron in the continuum is $E_c=mv^2/2=\omega_N-E_b$, where $v$ is the electron speed. Thus, the factor $\sqrt{2m/(\omega_N-E_b)}$ in Eq. \[eq:ICC\_EML\_2\] is equal to $2/v$. We will see below that this factor significantly “increases” the internal conversion coefficient in the Th atom (whose valence shells have a binding energy of about 6–7 eV), because it compensates for the small amplitudes of the electronic wave functions in the continuum.
The matrix elements (\[eq:ME\]) and (\[eq:ME\_2\]) were calculated by numerical integration. We used the wave functions from the work [@Tang-19] for the initial states, and the wave functions of the continuum for the final state.
The wave functions of the initial state are shown partly in Fig. \[fig:WF\] (these regions give main contributions to the electronic matrix elements).
![Wave functions of the $7s_{1/2}$ and $6d_{3/2}$ electron states in the Th atom and Thorium anion in the ground and excited states: (a) and (c) — the large $g_i(x)$ components, (b) and (d) — the small $f_i(x)$ components of the Dirac bispinor.[]{data-label="fig:WF"}](Tkalya-Si_Fig1.eps){width="0.98\hsize"}
Extra electron contributes to an additional nuclear screening for other valence electrons. As a result, the electron shell becomes more diffuse. This is clearly seen in Fig. \[fig:<x>\] — the average orbital radius of the $6d_{3/2}$ and $7s_{1/2}$ states increases when an electron is added to the $7p_{1/2}$ and $6d_{3/2}$ states of the Thorium atom. (Note also, in the $6d_{3/2}^2{}7s_{/2}^2{}7p_{1/2}^1$ anion excited state, the average radii $\langle{}6d_{3/2}|x|6d_{3/2}\rangle$ and $\langle{}7s_{1/2}|x|7s_{1/2}\rangle$ are smaller than the corresponding radii in the $6d_{3/2}^3{}7s_{1/2}^2$ ground state. It can be easily explained — the $7p_{1/2}$ shell shields the nuclear charge less effective than the $6d_{3/2}$ shell because $\langle{}7p_{1/2}|x|7p_{1/2}\rangle{} >
\langle{}6d_{3/2}|x|6d_{3/2}\rangle$, Fig. \[fig:<x>\].)
![Averaged radii of the $7p_{1/2}$, $7s_{1/2}$ and $6d_{3/2}$ orbitals in various electron configurations.[]{data-label="fig:<x>"}](Tkalya-Si_Fig2.eps){width="0.98\hsize"}
As a result of the indicated “swelling”, on the one hand, and the conservation of the normalization volume of the wave functions, on the other hand, the amplitudes of the $7s_{1/2}$ and $6d_{3/2}$ wave functions decrease in the region near the nucleus, Fig. \[fig:WF\].
The wave functions of the continuum spectrum are the numerical solutions of the Dirac equations with the electron energies $E>m$ ($E\approx{}E_c+m$) $$\left.
\begin{array}{ll}
g'(x)+\cfrac{1+\kappa}{x}g(x)-
\cfrac{1}{e^2}\left(\cfrac{E}{m}+1-\cfrac{V(x)}{m}\right)f(x) =0,\\
f'(x)+\cfrac{1-\kappa}{x}f(x)+
\cfrac{1}{e^2}\left(\cfrac{E}{m}-1-\cfrac{V(x)}{m}\right)g(x) =0,
\end{array}
\right.
\label{eq:EqDirac}$$ normalized at $x\rightarrow\infty$ with the condition $g_f(x)=\sin(pa_Bx+\varphi_{lj})/x$, where $\varphi_{lj}$ is a phase. In Eq. (\[eq:EqDirac\]), $e$ is the electron charge.
As an example, two wave functions of the final state are shown in Fig. \[fig:WF-continuum\] for the IC transitions $7s_{1/2}\rightarrow{}S_{1/2}$ and $6d_{3/2}\rightarrow{}D_{1/2}$. The energies of conversion electrons are: $E_c(S_{1/2})=1.79$ eV and $E_c(D_{3/2})=1.03$ eV for IC in the Th atom, $E_c(S_{1/2})=6.68$ eV and $E_c(D_{3/2})=7.65$ eV for IC in the Th anion in the ground state, and $E_c(S_{1/2})=5.63$ eV and $E_c(D_{3/2})=5.47$ eV for IC in the Th anion in the excited state. The solutions $g_f(x)$ (and $f_f(x)$) of Eq. (\[eq:EqDirac\]) reliably reach the asymptotic behavior $xg_f(x)={\text{Const}}\times{}\sin(pa_Bx)$ at $x\approx300$ in the Th$^+$ potential (for IC in the Th atom) and at $x\approx30$ in the potential of the Th atom (for IC in Th$^-$ and Th$^{-^*}$). Further, the obtained wave functions $g_f(x)$ and $f_f(x)$ are renormalized by dividing by the constant “Const”.
![Wave functions of the $S_{1/2}$ and $D_{3/2}$ electron states in the continuum after IC on the $7s_{1/2}$ and $6d_{3/2}$ electron states in Th, Th$^-$ and Th$^{-^*}$: (a) and (c) — the large $g_f(x)$ components, (b) and (d) — the small $f_f(x)$ components of the Dirac bispinor.[]{data-label="fig:WF-continuum"}](Tkalya-Si_Fig3.eps){width="0.98\hsize"}
The potential energy $V(x)$ of the electron in Eq. \[eq:EqDirac\] is $ V(x)= V_{\text{nucl}}(x) +
V_{\text{shell}}(x)$, where $V_{\text{shell}}(x)$ is the potential energy of the electron in the shell electron potential, and $V_{\text{nucl}}(x)$ is the potential energy of electron in potential of the unscreened nucleus. That is, the positive charge, $Z$, has been uniformly distributed within a sphere of the radius $x_{R_0} = R_0/a_B$ ($R_0 = 1.2A^{1/3}$ fm is the radius of the nucleus with the atomic number $A$): $V_{\text{nucl}}(x) =
-{\cal{E}}_0 (Z/2x_{R_0}) [3-(x/x_{R_0})^2]$ for $0\leq{}x\leq{}x_{R_0}$, and $V_{\text{nucl}}(x) =-{\cal{E}}_0
Z/x$ for $x\geq{}x_{R_0}$ where ${\cal{E}}_0=me^4$ is the atomic unit of energy.
The electron shell potential has been found by solving the Poisson equation with the given electron density. The electron density in the Th$^+$ ion and in the Th atom for the IC calculations in the Th atom and in the Th$^-$ anion respectively has been obtained within the DFT theory [@Nikolaev-15; @Nikolaev-16] through the self-consistent procedure taking into account the exchange and correlation effects. Moreover, for the internal conversion in the neutral Th atom we consider two various electron densities for Th$^+$, corresponding to the $6d_{3/2}^2{}7s_{1/2}^1$ and $6d_{3/2}^1{}7s_{1/2}^2$ configurations.
Results and Discussion {#sec:Discussion}
======================
Calculated internal conversion coefficients are presented in Table \[tab:ICC\]. We estimated the half-life of the $^{229m}$Th isomer in the anion for the two sets of reduced nuclear probabilities given in Table \[tab:Bwu\]. The first set (see in Ref. [@Tkalya-15-PRC]) was obtained with Alaga rules from the available experimental data [@Bemis-88; @Gulda-02; @Barci-03; @Ruchowska-06] for the $M1$ and $E2$ transitions between the rotation bands $3/2^+[631]$ and $5/2^+[633]$ in the $^{229}$Th nucleus [@Dykhne-98; @Tkalya-15-PRC]). The second set (taken from Ref. [@Minkov-17]) is based on a detailed computer calculation using modern concepts of nuclear interactions. The corresponding probabilities of radiative transitions in the $^{229}$Th nucleus from the isomeric to the ground state are also given in Table \[tab:Bwu\].
Th $7s_{1/2}$(-6.49 eV) $6d_{3/2}$(-7.25 eV)
--------------- ---------------------- ---------------------- ----------------------
$\alpha_{M1}$ $7.93\times10^8$ $2.31\times10^6$
$\alpha_{E2}$ $1.06\times10^{15}$ $4.80\times10^{15}$
Th$^-$ $7s_{1/2}$(-1.60 eV) $6d_{3/2}$(-0.63 eV)
$\alpha_{M1}$ $5.40\times10^8$ $1.45\times10^6$
$\alpha_{E2}$ $6.59\times10^{14}$ $3.08\times10^{15}$
Th$^{-^*}$ $7s_{1/2}$(-2.65 eV) $6d_{3/2}$(-2.81 eV) $7p_{1/2}$(-0.61 eV)
$\alpha_{M1}$ $6.76\times10^8$ $1.21\times10^6$ $3.37\times10^7$
$\alpha_{E2}$ $6.85\times10^{14}$ $2.76\times10^{15}$ $5.44\times10^{16}$
: Internal conversion coefficients per one electron for nuclear transition with the energy $\omega_N=8.27$ eV for the Thorium atom in the $6d_{3/2}^2{}7s_{1/2}^2$ ground state (Th), for the Thorium anion in the $6d_{3/2}^3{}7s_{1/2}^2$ ground state (Th$^-$) and in the $6d_{3/2}^2{}7s_{1/2}^2{}7p_{1/2}^1$ excited state (Th$^{-^*}$). (The binding energies on the shells are given in parentheses).[]{data-label="tab:ICC"}
Set Mult. $B_{\text{W.u.}}$ $\Gamma^{\text{rad}}$ (eV)
----- ------- --------------------- ----------------------------
1 $M1$ $3.1\times 10^{-2}$ $3.65\times10^{-19}$
$E2$ $11.7$ $3.06\times10^{-29}$
2 $M1$ $0.76\times10^{-2}$ $8.94\times10^{-20}$
$E2$ $27$ $7.05\times10^{-29}$
: Reduced matrix elements ($B_{\text{W.u.}}$) of the $3/2^+[631](8.27\,\,\text{eV})\rightarrow 5/2^+[633](0.0)$ nuclear transition in the $^{229}$Th nucleus and corresponding radiation widths ($\Gamma^{\text{rad}}$).[]{data-label="tab:Bwu"}
With data presented in Tables \[tab:ICC\] and \[tab:Bwu\] we calculate the half-life $T_{1/2}$ of the isomer $^{229m}$Th in the atom and anion. The results are summarized in Table \[tab:T\_1/2\]. They must be treated with some caution. The accuracy of calculating the IIC is relatively small for the nuclear transitions with ultralow energies (see below in Table \[tab:ICC\_TBSB\]). This is mainly due to the accuracy of the calculation of the wave functions of valence states and their binding energies. That is why we have calculated the internal conversion probabilities not only for the thorium anion, but also for the thorium atom. Since the calculations have been performed in a unified approach, we consider these results as reliable for the description of relative changes in the conversion probabilities and half-lives of $^{229m}$Th in going from atom to anion.
As can be seen from Table \[tab:T\_1/2\], the lifetime of the isomer in the $6d_{3/2}^3{}7s_{1/2}^2$ ground state of the anion is approximately 1.4–1.5 times longer than the isomer lifetime in the atom. For the Thorium anion in the $6d_{3/2}^2{}7s_{1/2}^2{}7p_{1/2}^1$ excited state this excess is insignificant, only $\approx$10%.
Set Th Th$^-$ Th$^{-^*}$
----- ------------------------------- ---------------------- ---------------------- ---------------------
1 $T_{1/2}$ $7.86\times 10^{-7}$ $1.15\times 10^{-6}$ $8.98\times10^{-7}$
$T_{1/2}/T_{1/2}^{\text{Th}}$ 1 1.47 1.14
2 $T_{1/2}$ $3.19\times 10^{-6}$ $4.67\times10^{-6}$ $3.55\times10^{-6}$
$T_{1/2}/T_{1/2}^{\text{Th}}$ 1 1.47 1.11
: $^{229m}$Th isomer half life (in s) in the Th atom and in the Thorium anion in the ground (Th$^-$) and in the excited (Th$^{-^*}$) states.[]{data-label="tab:T_1/2"}
This result at first glance looks counterintuitive. First, the internal conversion proceeds on four valence electrons in the thorium atom, and on five valence electrons in the anion. Second – less obvious – reason is that the electron matrix elements in the anion are larger than in the Th atom (see in Fig. \[fig:ME2\]). (Note that the latter effect is nontrivial, since, as we have seen, the WF amplitudes of the bound $6d_{3/2}$ and $7s_{1/2}$ states decrease upon transition from the Th atom to the thorium anion. The matrix elements growth in Fig. \[fig:ME2\] is explained by a faster increase in the amplitudes of the electron wave functions in the continuous spectrum with an increase of its energy (see in Fig. \[fig:WF-continuum\]). Due to diffusion of the electron shell, the binding energies of the electrons in the Thorium anion are smaller than in the atom, and the kinetic energy of the conversion electrons in the continuum is greater. In the nonrelativistic case, the amplitudes of the Coulomb wave functions (see in [@Abramowitz-64]) in the continuum increase with the energy of the conversion electron as $E_c^{1/4}$, i.e. significantly faster than the decrease of the amplitudes of the wave functions in the discrete spectrum. This explains somewhat unexpected form of the plots in Fig. \[fig:ME2\].)
![Matrix Elements Eq. (\[eq:ME\_2\]) for the main IC transitions. (Matrix elements for the $M1$ $6d_{3/2}$-$D_{3/2}$ electronic transitions are given in the units of $10^{-1}$.)[]{data-label="fig:ME2"}](Tkalya-Si_Fig4.eps){width="0.98\hsize"}
There is also a third factor that places an important role in the process. This is the kinetic energy of the conversion electron in the denominator in Eq. (\[eq:ICC\_EML\_2\]). It gives the factor $1/v$ in the expression for the IC probability and increases it near the threshold of the Th atom. In the thorium anion, this factor is 2-3 times smaller. And such a decrease turns out to be the most significant reason, which compensates for the increase in amplitudes of the electron wave functions in the continuum, and leads to a decrease in the IC probability of the anion in both the ground and excited states.
It is necessary to emphasize one more feature of the internal conversion in the anion. According to Fig. \[fig:<x>\], $\langle{}6d_{3/2}|x|6d_{3/2}\rangle$ is the largest in the ground state of the anion. Nevertheless, the partial internal conversion coefficients on the $6d_{3/2}$ shell in Th$^-$ exceed ICC in Th$^{-^*}$ (see Table \[tab:ICC\]). This is caused by the lack of the cancellation of two effects: the $E_c^{1/4}$ increase in the amplitudes of WF of the conversion electron in the continuum and the decrease of the factor $1/v$ for the electron promoted during IC from the $6d_{3/2}$ shell of Th$^-$. We recall that we use the energies for the bound states of the electron orbitals from Ref. [@Tang-19], where the multi-configuration electron terms were taken into account. In this case, the binding energies of the terms shift up or down, while the radial wave function remains the same. As a result, one gets the indicated inconsistency between the amplitude of the radial wave function and the binding energy of the orbital. Note that this effect practically does not affect the main results of the work.
In conclusion, it will be useful to compare the total IC coefficients for the $7s_{1/2}$ and $6d_{3/2}$ shells of the Th atom obtained in different works and using different codes. The relevant data are given in Table IV. We see that on average all the data correspond to each other with the accuracy of the factor of two. This is sufficient for preliminary estimates of the isomer lifetime and planning of experiments. For more delicate effects, it is necessary to perform calculations within the same code.
------------------------ -------------------- -------------------- -------------------- --------------------
This
work
$\alpha_{M1}$ $1.6\times10^9$ $4.6\times10^6$
$\alpha_{E2}$ $2.1\times10^{15}$ $9.6\times10^{15}$
Ref. [@Bilous-18]
$\alpha_{M1}$ $1.1\times10^9$ $2.0\times10^6$
$\alpha_{E2}$ $4.8\times10^{15}$ $4.3\times10^{15}$
Ref. [@Tkalya-15-PRC]
code[@Soldatov-79]
$\alpha_{M1}$ $1.6\times10^9$ $1.9\times10^9$ $4.1\times10^6$ $4.9\times10^6$
$\alpha_{E2}$ $2.1\times10^{15}$ $2.8\times10^{15}$ $9.4\times10^{15}$ $1.3\times10^{16}$
Ref. [@Borisyuk-18-QE]
code[@Band-79]
$\alpha_{M1}$ $0.96\times10^9$ $1.1\times10^9$ $2.8\times10^6$ $3.3\times10^6$
$\alpha_{E2}$ $1.2\times10^{15}$ $1.6\times10^{15}$ $6.0\times10^{15}$ $8.0\times10^{15}$
------------------------ -------------------- -------------------- -------------------- --------------------
: Total IC coefficients for the $7s_{1/2}$ and $6d_{3/2}$ shells of the Th atom obtained by different codes for the two values of the isomeric level energy: $E_{\text{is}} = 8.28$ eV, and 7.8 eV. (In the parentheses, there are the binding energies on the shells.)[]{data-label="tab:ICC_TBSB"}
Conclusion {#sec:Conclusion}
==========
In the paper, for the first time, we have studied the decay of the low lying isomer $3/2^+(8.28\pm0.17$ eV) of the $^{229}$Th nucleus in the Thorium anion. It has been found that the half life of the isomer in the $6d_{3/2}^3{}7s_{1/2}^2$ ground state of the anion is approximately 40-50[%]{} above the value in the $6d_{3/2}^2{}7s_{1/2}^2$ ground state of the Th atom and $\approx$10[%]{} larger in comparison with the $6d_{3/2}^2{}7s_{1/2}^2{}7p_{1/2}^1$ excited state of the anion. The IC decay probability is correspondingly reduced, despite “extra” fifth electron involved in the internal conversion process. The reason is as follows. Extra electron contributes to an additional nuclear screening for other valence electrons. As a result, the valence electron shells become more diffuse and amplitudes of the $6d_{3/2}$ and $7s_{1/2}$ wave functions near the nucleus decrease. Following the amplitudes, the probability of the internal conversion decreases too.
This research was supported by a grant of the Russian Science Foundation (Project No 19-72-30014).
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---
abstract: 'In this paper, using the method of moving frames, we generalise some of Terracini’s results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with “small” higher fundamental forms as contained in scrolls.'
address:
- 'Dipartimento di Matematica e Informatica, Università degli Studi di Udine, via delle Scienze, 206, 33100 Udine, Italy'
- 'Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, 80126 Napoli, Italy'
author:
- Pietro De Poi
- Roberta Di Gennaro
- Giovanna Ilardi
title: On varieties with higher osculating defect
---
[^1]
Introduction {#introduction .unnumbered}
============
The starting point of this paper is given by the classical papers of Terracini [@T], [@T1], [@T2], [@T3] on the description of the $k$-dimensional varieties $V$ of ${\mathbb{P}}^N(\mathbb C)$, $(N>2k)$, such that the embedded tangent variety $\operatorname{Tan}(V)$ is defective, i.e. it has dimension less than $2k$ ($2k-\ell$ with $\ell>0$). In [@T1], Terracini links this problem to the determination of the linear systems of quadrics such that the Jacobian matrix has rank $k-\ell$. After Terracini, there have been further classical papers on this subject: here we cite only as an example [@De1], [@M1], [@M2], [@M3].
Terracini proved results bounding the tangent defect of $V$ and on the structure of the varieties satisfying a certain number of Laplace equations. To state the results, we will say that $V$ *satisfies $\delta_s$ Laplace equations of order $s$* if—given a local parametrisation ${\mathbf{x}}(t_1,\dots,t_k)=(x_1(t_1,\dots,t_k),\dots,x_N(t_1,\dots,t_k))$ and denoted by ${\mathbf{x}}^I=\frac{\partial^{\vert I\vert}{\mathbf{x}}(t_1,\dots,t_k)}
{\partial t_1^{i_1}\dots\partial t_k^{i_k}}$ the partial derivatives of $\mathbf x$ —it satisfies the following partial differential equations: $$\sum_{0\leq\vert I\vert \leq s} E_I^{(h)}{\mathbf{x}}^{I} =0, \hspace{1cm} h=1,\dotsc, \delta_s$$ where at least a $E_I^{(h)}\neq 0$ with $|I|=s$ and these equations are linearly independent.
In this paper, we apply the method of moving frames, developed by Darboux, Cartan and others, to understand the relationship between the algebraic geometry of subvarieties of the projective space and their local projective differential geometry. This was a project of the classical geometers, revived by Akivis and Goldberg (see [@AG] and references therein) and Griffiths and Harris (see [@G-H]) and more recently by Landsberg (see for example [@L], [@L1] and with other authors, see for example [@IL], [@LR]).
We generalise Terracini’s Theorem to varieties with defect of higher order studying linear system of hypersurfaces (the fundamental forms) instead of Laplace equations of every order satisfied by the variety. We prove the following.
Let $V \subseteq {\mathbb{P}}^N$ be a $k$-dimensional irreducible variety whose $t$-th fundamental form has dimension $k-\ell-1$, with $\ell >0$; then $V$ has $(t-1)$-osculating defect $\geq \ell$ and moreover it holds
1. $V$ is contained in a $d$-dimensional scroll $S(\Sigma^h_r)$ in ${\mathbb{P}}^r$, with $0 \leq h \leq k-\ell$ and $k-h\le r$.
2. The tangent ${\mathbb{P}}^{d}$’s to $S(\Sigma^h_r)$ at the smooth points of a generic ${\mathbb{P}}^r$ of $S(\Sigma^h_r)$ are contained in a linear space of dimension $d_t-h=d_{t-1}+ k-\ell-h$—where $d_t$ is the dimension of the $t$-th osculating space to $V$ at its general point. In particular, $r\le d\le d_{t-1}+ k-\ell-h$.
See Theorem \[mainthm\].\
Moreover, we have obtained some classifications for the extremal cases of the preceding theorem; for example, we show that, if $\ell=k-1$ and $t=2$, then $V$ is either a hypersurface or a developable ${\mathbb{P}}^{k-1}$-bundle.
Successively, Terracini studied again varieties with tangent defect, but satisfying a number of Laplace equations less than $\binom{k}{2} + l$ in [@T1].
We generalise also this result as follows, in terms of fundamental forms:
Let $V \subseteq {\mathbb{P}}^N$ be a $k$-dimensional irreducible variety. $V$ has $t$-th osculating defect $o_t=\ell>0$ and the $(t+1)$-th fundamental form has dimension at least $k-\ell$ if and only if the Jacobian matrix of the $(t+1)$-th fundamental form of $V$ has rank $k-\ell$.
See Theorem \[thm terr gen\].
The article is structured as follows. In Section \[notation\_sec\] we give the basic notations and preliminaries, and we show some results that we need. Many of them either are natural generalisations of known results (mainly from [@G-H]) or are not very surprising; nevertheless, we think that including them can be useful by lack of adequate references. More precisely, after fixing some notations and recalling the basic definitions such as Laplace equations, Darboux frames, the second fundamental form and apolarity, we prove the relation between the dimension of the second fundamental form and the number of Laplace equations of order two for a $k$-dimensional projective variety $V\subset{\mathbb{P}}^N$: more precisely, if $V$ satisfies $\delta_2$ Laplace equations, then the second fundamental form has dimension $\binom{k+1}{2}-1-\delta_2$.
Then, after recalling the definition of the osculating spaces of every order, we link them to the higher fundamental forms proving in particular that the Jacobian system of the $t$-th fundamental form is contained in the $(t-1)$-th fundamental form. We also prove the equivalence between the dimension of the $t$-th fundamental form and the number of Laplace equations of order $t$, extending the above result for the second fundamental form.
We recall the definition of the $t$-th Gauss map and we show that its differential can be interpreted as the $t$-th fundamental form. Finally, we introduce the definition of $t$-th dual variety of $V$ and we prove some lemmas about it.
In Section \[sec:terr\] we state and prove the main theorems of the article, i. e. Theorems \[mainthm\] and \[thm terr gen\]. In order to do so, we also prove a lemma on the tangent space of the higher osculating variety of $V$.
Notation and preliminaries {#notation_sec}
==========================
We will follow notation as in [@G-H], [@H]. Let $V \subset {\mathbb{P}}^N$ be a projective variety of dimension $k$ over $\mathbb C$ that will be always irreducible. For any point $P\in V$ we use the following notation: $\widetilde {T}_P(V)\subset {\mathbb{P}}^N$ is the embedded tangent projective space to $V$ in $P$ and $T_P(V)$ is the Zariski tangent space.
With abuse of notation, we agree with [@G-H] by identifying the embedded tangent space in ${\mathbb{P}}^N$ with its affine cone in $\mathbb C^{N+1}$ , in order to avoid any complicating of writing. With this interpretation $T_P(V)\cong \frac{\widetilde T_P(V)}{\mathbb C}$. By $\mathbb G(N,t)$ we denote the Grassmannian of $t$-planes of ${\mathbb{P}}^N$.
We define $\operatorname{Tan}(V) :=\overline{\bigcup_{P \in V_0}\tilde {T}_P(V) }$ where $V_0\subset V$ is the smooth locus of $V$. $\operatorname{Tan}(V)$ has expected dimension $2k$, and the case in which $\operatorname{Tan}(V)$ is less than expected was studied by many algebraic geometers: classically Terracini (see [@T1]) linked the dimension $2k-\ell$ of $\operatorname{Tan}(V)$ with the number of Laplace equations that the variety $V$ satisfies, and more recently Griffiths and Harris [@G-H] analysed the same dimension in terms of second fundamental form $II$.
Actually, for studying Laplace equations, it is usual to consider a parametric representation of $V$; instead in [@G-H] and [@IL] the authors use the language of the Darboux frames. So, our first step is to understand in this language what means that $V$ satisfies a Laplace equation.
We begin to expose the definition of Laplace equations. Let $V \subseteq {\mathbb{P}}^N$ and let $ {\mathbf x} = {\mathbf x} (t_1 , \dotsc , t_k) = {\mathbf x} (\mathbf {t})$ be a local affine parametrisation of $V$ centred at the smooth point $P=[p_0:p_1:\dotsb:p_N]$, with—for example—$p_0\neq 0$ and ${\mathbf{x}}(\mathbf 0)=P$. Let $I=(i_1,\dotsc,i_k)$ be a multiindex, that is a $k$-tuple of non negative integers. We shall denote by $\vert I\vert$ the sum of the components of $I$, i.e. $\vert I\vert=i_1+\dotsb+i_k$. If ${\mathbf{x}}(t_1,\dots,t_k)=(x_1(t_1,\dots,t_k),\dots,x_N(t_1,\dots,t_k))$ is the above vector function, we shall denote by ${\mathbf{x}}^I$ the partial derivatives of $\mathbf x$: $${\mathbf {x}}^I=\frac{\partial^{\vert I\vert}{\mathbf{x}}(t_1,\dots,t_k)}
{\partial t_1^{i_1}\dots\partial t_k^{i_k}}.$$
\[de:le\] By saying that $V$ *satisfies $\delta_s$ Laplace equations of order $s$* we mean that—with the above local parametrisation $\mathbf{x}$ of $V$—$\mathbf{x}$ satisfies the following system of partial differential equations $$\begin{aligned}
\label{laplaceequations}
\sum_{0\leq\vert I\vert \leq s} E_I^{(h)}{\mathbf{x}}^{I} &=0, &E_I^{(h)}&\in\mathbb C,\ h =1,\dotsc, \delta_s\end{aligned}$$ such that at least a $E_I^{(h)}\neq 0$ with $|I|=s$ and that these equations are linearly independent. We say equivalently that $V$ represents the system of differential equations or that $V$ is an integral variety for it.
It is not restrictive to suppose that $P=[1:0:\dotsb:0]$, ant therefore we have ${\mathbf{x}}(\mathbf 0)=P= \mathbf 0\in \mathbb A^N$, and that $$\begin{aligned}
x_i(t_1,\dots,t_k)&=t_i & \forall i &\le k,\end{aligned}$$ i.e. $x_{k+1}=\dotsb=x_N$ defines $\widetilde {T}_P(V)\subset {\mathbb{P}}^N$. In these hypotheses, Equations become $$\begin{aligned}
\label{laplacecorto}
\sum_{2\leq\vert I\vert \leq s} E_I^{(h)}{\mathbf{x}}^{I} &=0 & h &=1,\dotsc, \delta_s.\end{aligned}$$ In what follows, we will make this assumption.
At same time, to study the behaviour of $V$ in $P$, following [@G-H] (and references \[2\], \[6\], \[7\] and \[10\] therein) and [@L], we consider the frame manifold on $V$, $\mathcal F (V)$; an element of it is a Darboux frame centred in $P$, that means an $(N+1)$-tuple $$\left\{A_0;A_1,\ldots,A_k;\dotsc,A_N\right\}$$ which is a basis of $\mathbb C^{N+1}$ such that, if $\pi:\mathbb C^{N+1}\setminus\{0\}\rightarrow {\mathbb{P}}^N$, $$\pi(A_0)=P$$ and $$\pi(A_0), \pi(A_1),\ldots,\pi(A_k) \textup{ span } \widetilde{T}_P(V).$$ Let us make this frame to move in $\mathcal F (V)$; then we can give structure equations (with Maurer-Cartan $1$-forms $\omega_i$, $\omega_{i,j}$ on $\mathcal F({\mathbb{P}}^N)$ restricted on $V$) for the exterior derivatives of this moving frame
$$\label{derivate_frame_eq}
\begin{cases}
\omega_\mu=0 &\forall \mu>k\\
d A_0=\sum_{i=0}^k \omega_i A_i \\
d A_i=\sum_{j=0}^N \omega_{ij}A_j &i=1,\dotsc,N\\
d\omega_j= \sum_{h=0}^k \omega_h \wedge \omega_{h,j} & j=0,\dotsc,k\\
d\omega_{ij}= \sum_{h=0}^N \omega_{i,h} \wedge \omega_{h,j} &i=1,\dotsc,N,\ j=0,\dotsc,N.
\end{cases}$$
Geometrically, the frame $\{A_i\}$ defines a coordinate simplex in ${\mathbb{P}}^N$. The $1$-forms $\omega_i, \omega_{ij}$ give the rotation matrix when the coordinate simplex is infinitesimally displaced; in particular, modulo $A_0$, as $d A_0\in T^*_P({\mathbb{P}}^N)$ (the cotangent space), the $1$-forms $\omega_1,\dotsc,\omega_k$ give a basis for the cotangent space $T^*_P(V)$, the corresponding $\pi(A_i)=v_i\in T_P(V)$ give a basis for $T_P(V)$ such that $v_i$ is tangent to the line $\overline{A_0A_i}$, and $\omega_{k+1} =\cdots =\omega_N=0$ on $T_P(V)$.
In such notation, we can define locally the second fundamental form:
\[II\] The *second fundamental form of $V$ in $P$* is the linear system $|II|$ in the projective space ${\mathbb{P}}(T_P(V))\cong {\mathbb{P}}^{k-1}$ of the quadrics defined by the equations: $$\begin{aligned}
\sum_{i,j=1}^k q_{ij\mu} \omega_i\omega_j&=0, & \mu&=k+1,\ldots,N\end{aligned}$$ where the coefficients $q_{ij\mu}$ are defined by the relations $$\begin{aligned}
\label{eq:second}
\omega_{i\mu}&=\sum_{j=1}^k q_{ij\mu}\omega_j, & q_{ij\mu}&=q_{ji\mu}\end{aligned}$$ obtained from $d\omega_\mu=0$, $\forall \mu>k$, via the Cartan lemma (see [@G-H (1.17)]).
We may symbolically write the second fundamental form as in [@G-H (1.20)] as $$\begin{aligned}
\label{eq_II}
d^2 A_0 \equiv \sum_{\substack{
0\le i,j\le k\\
k+1\le \mu \le N}
}q_{ij\mu}\omega_i \omega_j A_\mu& \mod \tilde T(V)\end{aligned}$$ or, more intrinsically the following (global) map $$\begin{aligned}
\label{sym2}
II\colon \operatorname{Sym}^{(2)} T(V) &\rightarrow N(V)\end{aligned}$$ where $N(V)$ is the normal bundle ($N_P(V):=\dfrac{\mathbb C^{N+1}}{\tilde T_P(V)}$ as in [@G-H]) which in coordinates is $$II(\sum_{i,j} a_{ij}v_i v_j)=\sum_{\substack{
0\le i,j\le k\\
k+1\le \mu \le N}
}
q_{ij}a_{ij}A_\mu.$$
To relate the second fundamental form to the Laplace equations , for ease our exposition, we consider the case $s=2$ . If there are $\delta_2$ independent relations of type: $$\begin{aligned}
\sum_{i, j=1}^k a_{i j}^{(\alpha)} {\mathbf x}^{(i j)} + \sum_{i=1}^k b_i ^{(\alpha)} {\mathbf x}^{(i)} + c^{(\alpha)}{\mathbf x} &= 0, & \alpha&=1,\dotsc,\delta_2,\end{aligned}$$ that, with our assumption on the coordinate become $$\begin{aligned}
\label{2nd}
\sum_{i, j=1}^k a_{i j}^{(\alpha)} {\mathbf x}^{(i j)} &= 0, & \alpha&=1,\dotsc,\delta_2;\end{aligned}$$ we can consider the linear system of quadrics of ${\mathbb{P}}(T_P(V)^*)$ of dimension $\delta_2 -1$, generated by the quadrics of equations: $$\begin{aligned}
\label{eq_quadriche}
\sum_{i, j=1}^k a_{i j}^{(\alpha)} v_i v_j &= 0 &\alpha &= 1, \dotsc,\delta_2;\end{aligned}$$ it defines the linear system of quadrics *associated* to the system of Laplace equations.
We recall now some notions of apolarity. Since our definitions are base dependent, for ease our exposition, we can say that two forms $f\in\mathbb C[x_0,\dotsc,x_N]$ and $g\in \mathbb C[y_0,\dotsc,y_N]= C[x_0,\dotsc,x_N]^*$ of the same degree $n$, are *apolar* if $$\sum_{I=(i_0,\dotsc,i_N)} a_I b_I=0,$$ where $f=\sum_I a_I\mathbf x^I$ and $g=\sum_I b_I\mathbf y^I$. Since $f$ and $g$ define hypersurfaces $F:=V(f)\subset {\mathbb{P}}^N=\operatorname{Proj}(\mathbb C[x_0,\dotsc,x_N])$ and $G:=V(g)\subset {\mathbb{P}}^{N*}=\operatorname{Proj}(\mathbb C[y_0,\dotsc,y_N])$, we will say also that $F$ and $G$ are *apolar* if $f$ and $g$ are apolar.
Given a system of hypersurface $H$ in ${\mathbb{P}}^N$ we say that the linear system $K$ in ${\mathbb{P}}^{N*}$ given by the hypersurfaces which are apolar to all the ones in $H$ is the *apolar system* of $H$.
The following result is classical:
\[prop\_IIapolar\] $\vert II\vert$ is the apolar system to the system of quadrics ; so, if $V$ satisfies $ \delta_2$ independent Laplace equations, then $ \dim |II| =\binom{k+1}{2}-1-\delta_2$.
Since we can identify the parametrisation ${\mathbf{x}}$ around $P$ with $\pi(A_0)$, then, by $$\begin{aligned}
d^2 A_0 (\sum_{i, j=1}^k a_{i j}^{(\alpha)} v_i v_j) &= \sum_{1\le i,j\le k}q_{ij\mu} a_{i j}^{(\alpha)} & \alpha &= 1, \dotsc,\delta_2 & \mu &=k+1,\dotsc,N;\end{aligned}$$ for our choice of the coordinates. On the other hand, $$\begin{aligned}
d^2 A_0 (\sum_{i, j=1}^k a_{i j}^{(\alpha)} v_i v_j) &=\sum_{i, j=1}^k a_{i j}^{(\alpha)} \frac{d^2A_0}{d v_i d v_j} =\sum_{i, j=1}^k a_{i j}^{(\alpha)} {\mathbf x}^{(i j)}
& \alpha &= 1, \dotsc,\delta_2.\end{aligned}$$
The second fundamental form can be related also with the second osculating space that we define as follows
\[T2\] Let $P\in V$, the *second osculating space* to $V$ at $P$ is the subspace $\tilde T^{(2)}_P(V)\subset {\mathbb{P}}^N$ spanned by $A_0$ and by all the derivatives $\dfrac{d A_0}{d v_\alpha}=A_{\alpha}$ and $\dfrac{d A_\alpha}{d v_\beta}= \dfrac{d A_\beta}{d v_\alpha} $ for $1\leq\alpha,\beta\leq k$.
So from now on we can consider the Darboux frame $$\{A_0;A_1,\dotsc,A_k;A_{k+1},\dotsc,A_{k+r};A_{k+r+1},\dotsc,A_{N}\}$$ so that $A_0;A_1,\dotsc,A_k;A_{k+1},\dotsc,A_{k+r}$ in $P$ span $\tilde T^{(2)}_P(V)$. It is straightforward to see—for example from the proof of Proposition \[prop\_IIapolar\]—that $$\dim |II|=r-1 \Longleftrightarrow \dim \tilde T^{(2)}_P(V)=k+r.$$
Generalising Definition \[II\], we can define the $t$-th fundamental form and the $t$-th osculating space at $P\in V$, for $t\geq 3$, and relate them with .
\[t\^t def\] Let $P\in V$, let $t\geq 3$ be an integer and $I=(i_1,\ldots,i_k)$ such that $|I|\leq t$. The *$t-$th osculating space* to $V$ at $P$ is the subspace $\tilde T^{(t)}_P(V)\subset {\mathbb{P}}^N$ spanned by $A_0$ and by all the derivatives $\dfrac{d^{|I|}A_0}{d v_1^{i1}\dotsm d v_k^{i_k}}$, where $v_1,\dotsc, v_k$ span $T_P(V)$.
We will put $$\begin{aligned}
d_t&:= \dim(\tilde T^{(t)}_P(V)),\\
e_t&:= \operatorname{expdim}(\tilde T^{(t)}_P(V))=\min(N, d_{t-1}+\binom{k-1+t}{t}).\end{aligned}$$
If $V$ satisfies $\delta_t$ Laplace equations of order $t$, we have $d_t = e_t - \delta_t$. Moreover, since a Laplace equation of order $t$ contains at least one of the $\binom{k-1+t}{t}$ partial derivatives of order $t$, we have $ \delta_t \le \binom{k-1+t}{t}$.
Put $k_t:=\binom{k+t}{t}-1$. Obviously, $d_t\le \min(k_t,N)$. If $N<k_t$, then $V\subseteq{\mathbb{P}}^N$ represents at least $k_t-N$ Laplace equations of order $t$. These Laplace equations are called *trivial*.
Let $t\geq 2$ and $V_0\subseteq V$ be the quasi projective variety of points where $\tilde T ^{(t)}_P (V)$ has maximal dimension. The variety $${\operatorname{Tan}}^t(V) := { \overline{\bigcup_{P \in V_0} \tilde T^{(t)}_P(V)}}$$ is called the *variety of osculating $t$-spaces to $V$*. Its expected dimension is $$\operatorname{expdim}{\operatorname{Tan}}^t(V):=\min(k+ d_t,N)$$ The *$t$-th osculating defect of $V$* is the integer $$o _t:=\operatorname{expdim}{\operatorname{Tan}}^t(V) -\dim {\operatorname{Tan}}^t(V).$$ If $t=1$, we call $o_1$ the *tangent defect*.
Obviously we have $$d_t\le d_{t-1}+ \binom{k-1+t}{t}\le \dotsm \le \sum_{i=1}^t\binom{k+i-1}{i}= k_t.$$
We will study the osculating defects related to the fundamental forms. Following [@G-H] and recalling we give
\[ff\_def\] The *$t-$th fundamental form* of $ V$ in $P$ is the linear system $|I^t|$ in the projective space ${\mathbb{P}}(T_P(V))\cong {\mathbb{P}}^{k-1}$ of hypersurfaces of degree $t$ defined symbolically by the equations: $$d^t A_0=0;
$$ more intrinsically, we write $I^t$ as the map $$I^t\colon \operatorname{Sym}^{(t)} T(V) \rightarrow N^t(V)$$ where $N^t(V)$ is the bundle defined locally as $N^t_P(V):=\displaystyle{\frac{\mathbb C^{N+1}}{{\tilde T}^{(t-1)}_P(V)} }$ and the map $I^t$ is defined locally on each $v\in T_P(V)$ as $$v^t \mapsto \displaystyle{\frac{d^t A_0}{d v^t} }\mod {\tilde T}^{(t-1)}_P(V).$$
Choose a Darboux frame $$\label{eq:darbu}
\{A_0;A_1,\dotsc,A_k;A_{k+1},\dotsc,A_{d_2};A_{d_2+1},\dotsc,A_{d_s};\dotsc,A_{d_t};\dotsc, A_N\}$$ such that $A_0,A_1,\dotsc,A_{d_s}$ span ${\tilde T}^{(s)}_P(V)$ $\forall s=1,\dotsc, t$, with $d_1:=k$. By the definition of ${\tilde T}^{(s)}_P(V)$, we have that $$\begin{aligned}
\label{eq:dsup}
d A_{\alpha_{s-1}}& \equiv 0 \mod \tilde T^{(s)}_P(V) & \alpha_{s-1}&=d_{s-2}+1,\dotsc, d_{s-1}, & s&=2,\dotsc, t-1,\end{aligned}$$ where we put $d_0=0$, from we have $$\begin{aligned}
\label{eq:omega}
\omega_{\alpha_{s-1},\mu_s}&=0 & \alpha_{s-1}&=d_{s-2}+1,\dotsc, d_{s-1},\ \mu_s>d_s & s&=2,\dotsc t-1,\end{aligned}$$ from which we infer, after some computations $$\label{eq:tfond}
d^ t A_0 \equiv
\sum_{\substack{d_{s-1}+1\le \alpha_s \le d_{s}\\
s=1,\dotsc, t-1\\
d_{t-1}+1\le \alpha_{t}\le N}}
\omega_{\alpha_1}\omega_{\alpha_1,\alpha_2}\dotsm \omega_{\alpha_s,\alpha_{s+1}}\dotsm \omega_{\alpha_{t-1},\alpha_{t}}A_{\alpha_{t}} \mod {\tilde T}^{(t-1)}_P(V)$$ or, using Cartan lemma, $$\label{eq:tfond1}
\begin{array}{ll}d^ t A_0 & \equiv
\displaystyle{\sum_{\substack{
1\le i_1,\dotsc,i_t\le k\\
d_{t-1}+1\le \alpha_t\le N
}} q_{i_1,\dotsc,i_t,\alpha_t}\omega_{i_1}\dotsm\omega_{i_t} A_{\alpha_t}=}\\ &
=\displaystyle{\sum_{\substack{
|I|=t\\
d_{t-1}+1\le \alpha_t\le N
}} q_{I,\alpha_t}\omega_I A_{\alpha_t}
\ \mod {\tilde T}^{(t-1)}_P(V)},
\end{array}$$ with the natural symmetries for the indices $i_1,\dotsc,i_t$ of $q_{i_1,\dotsc,i_t,\alpha_t}$ which can be expressed as $$\begin{aligned}
\label{eq:eulero}
\frac{d^{t-1}A_i}{d v_j}&\equiv \frac{d^{t-1}A_j}{d v_i} &\mod &{\tilde T}^{(t-1)}_P(V), & i,j=1,\dotsc,k.\end{aligned}$$
From $$\begin{aligned}
0&=d\omega_{\alpha_{s-1},\mu_s} =\sum_{h_s=d_{s-1}+1}^{d_s}\omega_{\alpha_{s-1},h_s}\wedge\omega_{h_s,\mu_s} \\ \alpha_{s-1}&=d_{s-2}+1,\dotsc, d_{s-1},\ \ \ \ \mu_s>d_s , \ \ \ \ s=2,\dotsc t-1.\end{aligned}$$ Now, $\omega_{\alpha_{s-1},h_s}$ and $\omega_{h_s,\mu_s}$ are horizontal for the fibration $\tilde T^{(t-1)}_P(V)\to V$, and therefore—by induction on $s$, since the case $s=2$ is in [@G-H page 374]—they are a linear combination of $\omega_1,\dotsc,\omega_k$; then, we have $$\begin{gathered}
0=d\omega_{\alpha_{s-1},\mu_s}(\frac{\partial}{\partial \omega_\gamma})=\sum_{h_s=d_{s-1}+1}^{d_s}(\frac{\partial \omega_{\alpha_{s-1},h_s}}{\partial \omega_\gamma}
\omega_{h_s,\mu_s}-
\omega_{\alpha_{s-1},h_s}\frac{\partial \omega_{h_s,\mu_s}}{\partial \omega_\gamma}) \\
\alpha_{s-1}=d_{s-2}+1,\dotsc, d_{s-1},\ \mu_s>d_s\\
\gamma=1,\dotsc,k,\ s=2,\dotsc t-1.\end{gathered}$$ which means $$\begin{gathered}
\label{eq:tanti}
\sum_{h_s=d_{s-1}+1}^{d_s}(\frac{\partial \omega_{\alpha_{s-1},h_s}}{\partial \omega_\gamma} \omega_{h_s,\mu_s})
=\sum_{h_s=d_{s-1}+1}^{d_s}(\frac{\partial \omega_{h_s,\mu_s}}{\partial \omega_\gamma}\omega_{\alpha_{s-1},h_s}) \\
\alpha_{s-1}=d_{s-2}+1,\dotsc, d_{s-1},\ \mu_s>d_s\\
\gamma=1,\dotsc,k,\ s=2,\dotsc t-1.\end{gathered}$$ Since the linear system $|I^t|$ is generated, from Relation , by the following polynomials of degree $t$ $$\begin{aligned}
V_{\alpha_t}&:=\sum_{\substack{d_{s-1}+1\le \alpha_s \le d_{s}\\
s=1,\dotsc, t-1}}
\omega_{\alpha_1}\omega_{\alpha_1,\alpha_2}\dotsm \omega_{\alpha_s,\alpha_{s+1}}\dotsm \omega_{\alpha_{t-1},\alpha_t}, & d_{t-1}+1&\le \alpha_{t}\le N\end{aligned}$$ then, we can prove Theorem \[thm:I\]; in order to do so, we recall that
Let $\Sigma$ be the linear system of dimension $d$ of hypersurfaces of degree $n$ ($n>1$) in ${\mathbb{P}}^N$ ($N>1$), generated by the $d+1$ hypersurfaces $f_0=0, \dotsc, f_d=0$.
The Jacobian matrix of the forms $f_0, \dotsc, f_d$, $$J(\Sigma):=(\partial f_i/\partial x_j)_{i=0, \dotsc,d; j=0,\dotsc,r}$$ is said the *Jacobian matrix* of the system $\Sigma$.
The *Jacobian system* of $\Sigma$ is the linear system of the minors of maximum order of $J(\Sigma)$. Obviously, the Jacobian system does not depend on the choice of $f_0, \dotsc, f_d$, but only on $\Sigma$.
\[thm:I\] Given a $k$-dimensional projective variety $V\subset{\mathbb{P}}^N$, its $t$-th fundamental form $|I^t|$ is a linear system of polynomials of degree $t$ whose Jacobian system is contained in the $(t-1)$-th fundamental form $|I^{t-1}|$.
With notation as above, we start considering, with $d_{t-1}+1\le \alpha_{t}\le N$, $$\begin{gathered}
\frac{\partial V_{\alpha_t}}{\partial \omega_\gamma}=\sum_{\substack{d_{s-1}+1\le \alpha_s \le d_{s}\\
s=2,\dotsc, t-1}}
\omega_{\gamma,\alpha_2}\dotsm \omega_{\alpha_s,\alpha_{s+1}}\dotsm
\omega_{\alpha_{t-1},\alpha_t}+\dotsb\\
\dotsb +\sum_{\substack{d_{s-1}+1\le \alpha_s \le d_{s}\\
s=1,\dotsc, t-1}}
(\omega_{\alpha_1}\dotsm \frac{\partial \omega_{\alpha_s,\alpha_{s+1}}}{\partial \omega_\gamma}\dotsm
\omega_{\alpha_{t-1},\alpha_t}+\dotsb\\
\dotsb +\omega_{\alpha_1}\dotsm \omega_{\alpha_s,\alpha_{s+1}}\dotsm
\frac{\partial\omega_{\alpha_{t-1},\alpha_t}}{\partial \omega_\gamma});\end{gathered}$$ then from , we deduce $$\begin{gathered}
\frac{\partial V_{\alpha_t}}{\partial \omega_\gamma}=\sum_{\substack{d_{s-1}+1\le \alpha_s \le d_{s}\\
s=2,\dotsc, t-1}}
\omega_{\gamma,\alpha_2}\dotsm \omega_{\alpha_s,\alpha_{s+1}}\dotsm
\omega_{\alpha_{t-1},\alpha_t}+\\
+(t-1)\sum_{\substack{d_{s-1}+1\le \alpha_s \le d_{s}\\
s=1,\dotsc, t-1}}
\omega_{\alpha_1}\dotsm \omega_{\alpha_s,\alpha_{s+1}}\dotsm
\frac{\partial\omega_{\alpha_{t-1},\alpha_t}}{\partial \omega_\gamma};\end{gathered}$$ then, for example from $$\omega_{\gamma,\alpha_2}=\sum_{\alpha_1=1}^k q_{\gamma,\alpha_1,\alpha_2}\omega_{\alpha_1}= \sum_{\alpha_1=1}^k q_{\alpha_1,\gamma,\alpha_2}\omega_{\alpha_1}=
\sum_{\alpha_1=1}^k\frac{\partial \omega_{\alpha_1,\alpha_2}}{\partial \omega_\gamma}\omega_{\alpha_1}$$ and again from , $$\frac{\partial V_{\alpha_t}}{\partial \omega_\gamma}=t\sum_{\substack{d_{s-1}+1\le \alpha_s \le d_{s}\\
s=1,\dotsc, t-1}}
\omega_{\alpha_1}\dotsm \omega_{\alpha_s,\alpha_{s+1}}\dotsm
\frac{\partial\omega_{\alpha_{t-1},\alpha_t}}{\partial \omega_\gamma}.$$
Actually, as for the second fundamental form, Proposition \[prop\_apolarity\] holds, with an adapted proof from the one as in Proposition \[prop\_IIapolar\]. In order to do so, we suppose to fix a Darboux frame as ; then, with this choice, if we have a system of $\delta_t$ Laplace equations of order $t$ as in , they can be expressed as $$\begin{aligned}
\label{laplaceequationsgen}
\sum_{|I|= t} E_I^{(h)}{\mathbf{x}}^{I} &=0 & h &=1,\dotsc, \delta_t;\end{aligned}$$ then, as in we can define the linear systems of homogeneous polynomials of degree $t$ *associated* to : $$\begin{aligned}
\label{asso}
\sum_{|I|= t} E_I^{(h)}{\mathbf{v}}_{I} &=0 & h &=1,\dotsc, \delta_t,\end{aligned}$$ where $\mathbf v_I=\prod_{\substack{i=1,\dotsc,k\\ i_1+\dotsm+ i_k=t}}
v_i^{i_j}.$
\[prop\_apolarity\] If $V$ satisfies $\delta_t$ Laplace equations of order $t$ like in , the $t$-th fundamental form is the apolar system to the system of the hypersurfaces of degree $t$ associated to the system of Laplace equations (i. e. the hypersurfaces in ), and vice versa.
It is enough to repeat the proof of Proposition \[prop\_IIapolar\] with an adapted local coordinate system. More precisely, we can choose a Darboux frame as in . Since we can identify the parametrisation ${\mathbf{x}}$ around $P$ with $\pi(A_0)$, then, by our hypothesis, Laplace equations of order $t$ become $$\begin{aligned}
\sum_{|I| =t} E_I^{(h)}{\mathbf{x}}^{I} &=0 & h &=1,\dotsc, \delta_t.\end{aligned}$$ By , we have $$\begin{aligned}
d^ t A_0 (\sum_{|I| =t} E_I^{(h)}{\mathbf{v}}_{I})&= \sum_{|I| =t} q_{I,\beta} E_I^{(h)} & h &=1,\dotsc, \delta_t & \beta&=d_{t-1}+1,\dotsc, N,\end{aligned}$$ and, on the other hand, $$\begin{aligned}
d^ t A_0 (\sum_{|I| =t} E_I^{(h)}{\mathbf{v}}_{I})&= \sum_{|I| =t} E_I^{(h)}\frac{d^t A_0}{(d\mathbf{v})^I}=\sum_{|I| =t} E_I^{(h)}{\mathbf{x}}^{I} & h &=1,\dotsc, \delta_t.\end{aligned}$$
From Proposition \[prop\_apolarity\] we recover immediately the following
\[cor\_dimensione\] If $V$ satisfies $\delta_t$ Laplace equations of order $t$ like in , the $t$-th fundamental form has dimension $\binom{k-1+t}{t}-1-\delta_t$, and vice versa.
We will denote the dimension of the $t$-th fundamental form as $\Delta_t$: $$\Delta_t:=\dim(|I^t|).$$
\[cor\_dimensione ff\] If $N\ge k_t$, we have that $$d_t= d_{t-1}+\Delta_t+1,$$ and vice versa, if $d_t= d_{t-1}+\Delta+1$, then the $t$-th fundamental form has dimension $\Delta$.
From now on, we will suppose that our Darboux frame is as in .
In order to prove the results of the following section, we recall also the following notation and definitions.
Let $\Sigma^h_t \subset \mathbb{G}(N,t)$ be a subvariety of pure dimension $h$. Let $I_{\Sigma^h_t}\subset \Sigma^h_t \times {\mathbb{P}}^N$ be the incidence variety of the pairs $(\sigma, q)$ such that $q\in \sigma$ and let $p_1\colon I_{\Sigma^h_t} \to \Sigma^h_t$ and $p_2\colon I_{\Sigma^h_t} \to {\mathbb{P}}^N$ be the maps induced by restricting to $I_{\Sigma^h_t}$ the canonical projections of $\Sigma^h_t \times {\mathbb{P}}^N$ to its factors.
The morphism $p_1\colon I_{\Sigma^h_t} \to \Sigma^h_t$ is said to be a *family of $t$-dimensional linear subvarieties of ${\mathbb{P}}^N$*. $\Sigma^h_t$ is the parameter space of the family, but for brevity we will often refer to it as to the family itself. Obviously, $$\dim(I_{\Sigma^h_t})= t + \dim(\Sigma^h_t).$$ Let us suppose that $\Sigma^h_t$ is irreducible. We will denote by $S(\Sigma^h_t)$ the image of $I_{\Sigma^h_t}$ under $p_2$. $S(\Sigma^h_t)$ is—by definition—a *scroll in ${\mathbb{P}}^r$* of ${\mathbb{P}}^N$. The previous notation will be useful to study osculating variety.
Let $t\geq 1$, the *$t$-th projective Gauss map* is the rational map $$\begin{aligned}
\gamma^t\colon &V \dashrightarrow \mathbb G({\mathbb{P}}^N,d_t)\\
& P \mapsto {\tilde T}^{(t)}_P(V).\end{aligned}$$
\[rem\_dimension\] The *$t$-th osculating variety* is $\widetilde{{\operatorname{Tan}}^t}(V) = {\overline{\bigcup_{P \in V_0}\gamma^t(P)}}\subset \mathbb G(k_t,{\mathbb{P}}^N)$ where, as before, $\ V_0$ denotes the open subset of $\ V$ of the points for which $\ \dim {\tilde T}^{(t)}_P(V)=d_t$ and then ${\operatorname{Tan}}^t(V)$ is the scroll $S(\widetilde{{\operatorname{Tan}}^t}(V))$ of dimension $$\dim{\operatorname{Tan}}^t(V)\leq\dim \operatorname{Im}\gamma^t+d_t =k+d_t-\dim((\gamma^t)^{-1}(\Pi)),$$ where $\Pi$ is a general element of $\widetilde{{\operatorname{Tan}}^t}(V)$.
We prove now
\[13\] The first differential of $\gamma^t$ at $P$ is the $(t+1)$-th fundamental form at $P$.
We have, by the definition of $\gamma^t$, that $$d\gamma^t_P\colon T_P V \dashrightarrow T_{\tilde T^{(t)}_P V} \mathbb G({\mathbb{P}}^N,d_t),$$ and we recall that $T_{\tilde T^{(t)}_P V} \mathbb G({\mathbb{P}}^N,d_t)\cong \operatorname{Hom}(\tilde T^{(t)}_P V, N^{t+1}_P(V))$; moreover if we choose a Darboux frame as in , we have that $d A_0\in \tilde T_P V \subset \tilde T^{(t)}_P V$ and $$\frac{\tilde T^{(t)}_P V}{\mathbb C A_0}=T^{(t)}_P V$$ and therefore $d\gamma^t_P\in \operatorname{Hom}(T_P V\otimes T^{(t)}_P V, N^{t+1}_P(V)) $.
Now, we remark that, in our Darboux frame, we can interpret $\gamma^t$ as $$\gamma^t(P)=A_0\wedge\dotsb \wedge A_{d_t},$$ and therefore by , $$\begin{aligned}
d\gamma^t_P&\equiv\sum_{\substack{
1\le i \le d_t\\
d_t+1\le j \le N
}}(-1)^{d_t-i+1} \omega _{i,j} A_0\wedge\dotsb\wedge \hat {A_i}\wedge\dotsb \wedge A_{d_t}\wedge A_{j}, & \mod \tilde T^{(t)}_P V;\end{aligned}$$ now, a basis for $T_P V\otimes T^{(t)}_P V$ can be expressed by $(A_\alpha\otimes A_\mu)_{\substack{\alpha=1,\dotsc,k\\ \mu=1,\dotsc, d_t}}$, and $$\begin{aligned}
d\gamma^t_P(A_\alpha\otimes A_\mu) &=\sum_{
d_t+1\le j \le N} \omega _{\mu,j}(A_\alpha) A_{j}\in N^{t+1}_P(V)\end{aligned}$$ on the other hand, for the $(t+1)$-th fundamental form we have $$\begin{aligned}
\frac{d A_\mu}{d v_\alpha} &\equiv \sum_{\substack{
d_t+1\le j \le N
}} \omega _{\mu,j}(A_\alpha) A_{j} & \mod \tilde T^{(t)}_P(V).\end{aligned}$$
We recall now the definition of higher order dual varieties (see [@pi]), which is the natural extension of the definition of the dual variety:
Let $V\subset{\mathbb{P}}^N$ be a projective variety; for *$t$-th dual variety* of $V$, $\check V^{(t)}$, we mean $$\label{tan}
\check V^{(t)}=\overline{\bigcup_{P\in V_0} C_P^{(t)}(V)}$$ where—as before—$V_0\subset V$ is the set of the points for which $\dim T^{(t)}_P(V)=d_t$ and $C_p^{(t)}(V)$ is $$C_P^{(t)}(V):=\bigcap_{K\in T_P^{(t)}(V)} K= \{H\in{{\mathbb{P}}^N}^* \mid H\supset \tilde T^{(t)}_P(V)\} \subset {{\mathbb{P}}^N}^*.$$ and it is classically called the *$t$-th characteristic space* of $V$ in $P$.
We can now make observation similar to the ones in [@G-H §3(a)]: the elements of $C_P^{(t)}(V)$ can be naturally identified with the hyperplanes in ${\mathbb{P}}(N_P^{t+1}(V))$ and therefore $\check V^{(t)}$ is just the image of the map $$\delta^{t}\colon {\mathbb{P}}(N^{t+1}(V)^*)\to {{\mathbb{P}}^N}^*,$$ analogous to the one of [@G-H (3.1)]. In term of the frames, a hyperplane $\xi$ of ${\mathbb{P}}(N^{t+1}_P(V))$ can be given by choosing $A_{d_t+1}, \dotsc, A_{N-1}$ such that their projection in $N_P^{t+1}(V)=\mathbb C^{N+1}/\tilde T^{(t)}_P(V)$ spans $\xi$. Therefore, in term of coordinates, $\delta^t$ can be expressed as $$\delta^t(P,\xi)=A_0\wedge A_1\wedge \dotsb \wedge A_{N-1},$$ (see [@G-H (3.2)]) or, if we choose dual coordinates $$A_i^*:=(-1)^{N-i}A_0\wedge \dotsb \wedge A_{i-1}\wedge A_{i+1}\wedge \dotsb \wedge A_N,$$ $\delta^t(P,\xi)=A_N^*$. From relations we deduce $$d A_j^*= \sum_{i\neq j} (-\omega_{i,j}A_i^*+ \omega_{i,i}A_j^*),$$ and in particular $$d A_N^*= \sum_{i=0}^{N-1} (-\omega_{i,N}A_i^*+ \omega_{i,i}A_N^*)=\sum_{i=1}^{N-1} (-\omega_{i,N}A_i^*)+(\omega_0 +\sum_{i=1}^{N-1} \omega_{i,i})A_N^*.$$ By the definition of $\check V^{(t)}$, for its dimension we have $$N-d_t-1\le \dim\check V^{(t)}=:d_{t,1}\le N-d_t-1+k.$$
If we choose a Darboux frame as in , these formulas become, thank to $$\begin{aligned}
d A_j^* &= \sum_{\substack{i\neq j\\ i>u-2 }}(-\omega_{i,j}A_i^*+ \omega_{i,i}A_j^*),
&\begin{cases}
j=d_{u-1}+1,\dotsc, d_{u}&\textup{if $u=0,\dotsc, t-1$,}\\
j=d_{t-1}+1,\dotsc,N& \textup{if $u=t$},
\end{cases}\end{aligned}$$ where we put $d_{-1}:=-1$ when we vary $j$; in particular, $$d A_N^* = \sum_{i= t-1}^{N-1}(-\omega_{i,N}A_i^*+ \omega_{i,i}A_N^*))=\sum_{i=t-1}^{N-1} (-\omega_{i,N}A_i^*)+(\omega_0 +\sum_{i=t-1}^{N-1} \omega_{i,i})A_N^*,$$ and therefore $$\begin{aligned}
\label{eq:deg}
d A_N^* &\equiv \sum_{i=t-1}^{N-1} (-\omega_{i,N}A_i^*)&\mod A_N^*.\end{aligned}$$
We say that $\check V^{(t)}$ is *degenerate*, if it has dimension less than expected: $d_{t,1}< N-1-d_{t}+k$.
In Relation the last $N-d_t-1$ forms $\omega_{i,N}$, $i=d_t+1,\dotsc,N-1$, restrict to a basis for the forms of the fibres ${\mathbb{P}}(N^{t+1}_P)^*={\mathbb{P}}^{N-1-d_t}$; in fact, they describe the variation of $\xi$ when $P$ is held fixed. The first $\omega_{i,N}$, with $i\le d_t$ are horizontal for the fibreing ${\mathbb{P}}(N^{t+1}(V)^*)\to V$, and therefore $\check V^{(t)}$ is degenerate if and only if $$\begin{aligned}
\omega_{i_1,N}\wedge\dotsb \wedge \omega_{i_k,N} &=0& \forall i_1, \dotsc i_k\ \textup{with}\ t-1\le i_1 < \dotsb < i_k \le d_t.\end{aligned}$$
Now, if we put $d_{t,s}:=\dim \tilde T^{(s)}_\xi(\check V^{(t)})$, if $N-d_{t,s}\ge d_{t-1}+1$, i.e. $d_{t,s}\le N-d_{t-1}-1$ (otherwise $\tilde T^{(s)}_\xi(\check V^{(t)})={\mathbb{P}}^{N*}$), we can choose a Darboux frame such that $T^{(s)}_\xi(\check V^{(t)})$ is generated by $A_N^*$ and $A_{N-1}^*,\dotsc,A_{N-d_{t,s}}^*$.
Let us now define the characteristic varieties of a projective variety $V\subset{\mathbb{P}}^N$.
The *variety of the $s$-th characteristic spaces of the $t$-th osculating spaces* of $V$ is the $s$-th dual of the $t$-th dual variety of $V$, that is $$\operatorname{Car}^s_t(V):=\overline{\bigcup_{\xi \in \check V^{(t)}_0} C_\xi^{(s)}(\check V^{(t)})}\subset{\mathbb{P}}^N$$ where $\check V^{(t)}_0$ is the open subset of $\check V^{(t)}$ of the points $\xi$ are such that $\xi\supset {\tilde T}^{(t)}_P(V)$ and $\dim {\tilde T}^{(t)}_P(V)=d_t$; we will denote in the following this $s$-th characteristic space of $\check V^{(t)}$ in a general $\xi\supset \tilde T_P^{(t)}(V)$ as $C^{(s)}_{t,P}(V):= C_\xi^{(s)}(\check V^{(t)})$; then, using the above notation, $\dim (C^{(s)}_{t,P}(V))=N-1-d_{t,s}$.
\[lemma\] With notations as above, if $P\in V$, we have
1. \[a\] $\tilde T^{(t-1)}_P(V) \subset C^{(1)}_{t,P}(V)$;
2. \[b\] $P\in C^{(s)}_{t,P}(V)$;
3. \[c\] if $\xi\in C^{(t)}_P(V)$, $\tilde T_\xi(\check V^{(t)}) \subset C^{(t-1)}_P(V)$.
: using the above notations, we have that, since $\tilde T_\xi(\check V^{(t)})$ is generated by $A_{N}^*$ and $A_{N-1}^*,\dotsc,A_{t-1}^*$, and we can choose a frame such that we have that the first $d_{t,1}$, $A_{N-1}^*,\dotsc,A_{N-d_{t,1}}^*$ are a base of $\tilde T_\xi(\check V^{(t)})$. Then, $C_\xi^{(1)}(\check V^{(t)})$ contains $A_0$ and $A_1,\dotsc,A_{N-d_{t,1}-1}$, and since $d_{t-1}\le d_t-k\le N-d_{t,1}-1$, we have the assertion.
: since $\ \tilde T^{(s)}_\xi(\check V^{(t)})\ $ is generated, in an appropriate frame, by $\ A_N^*\ $ and $\ A_{N-1}^*,\dotsc,A_{{N-d_{t,s}}}^*$, we have that $C_\xi^{(s)}(\check V^{(t)})$ contains $A_0$.
: it is just in the dual space.
\[cor:lem\] With notations as above, if $P\in V$, $\xi \in \check V^{(t)}$ and $Q\in C^{(s)}_{t,P}(V)$ are general points, then $\tilde T_\xi (\operatorname{Car}^s_t(V))\subset C^{(s-1)}_{t,P}(V)$.
It is simply the dual of Lemma \[lemma\], .
Terracini’s theorems and generalisations {#sec:terr}
========================================
In this section we generalise classical Terracini’s results in terms of osculating defect and higher fundamental forms instead of Laplace equations, so that we forget the parametrisation of $V$. First of all, by Corollary \[cor\_dimensione\] we rewrite the results of [@T] and [@T1 Section 3] as follows:
\[1ca\] Let $V \subseteq {\mathbb{P}}^N$ be a $k$-dimensional irreducible variety whose second fundamental form has dimension $k-\ell-1$, with $\ell >0$. Then $V$ has tangent defect at least $\ell$ and it is contained in a scroll $S(\Sigma^h_t)$ in ${\mathbb{P}}^t$ such that $T_{{\mathbb{P}}^t_v}(S(\Sigma^h_t)) \subset {\mathbb{P}}^{2k-h-\ell}$ with $0 \leq h \leq k-\ell$, where $v\in \Sigma^h_t$ is a general point, and ${\mathbb{P}}^t_v$ is the corresponding fibre of the scroll.
\[2ca\] Let $V \subseteq {\mathbb{P}}^N$ be a $k$-dimensional irreducible variety. $V$ has tangent defect $o_1=\ell>0$ and the second fundamental form has dimension at least $k-\ell$ if and only if the Jacobian matrix of the second fundamental form of $V$ has rank $k-\ell$.
We will prove Theorems \[mainthm\] and \[thm terr gen\]; Theorems \[1ca\] and \[2ca\] are just corollaries of them.
\[lemsbagliato\] Let $V \subseteq {\mathbb{P}}^N$ be a $k$-dimensional variety and let $P\in V$. Then, the tangent cone to $\operatorname{Tan}^{t-1} (V)$ in $P$ is contained in ${\tilde T}^{(t)}_P(V)$, and therefore $\tilde T_P(\operatorname{Tan}^{t-1} (V)) \subset {\tilde T}^{(t)}_P(V)$.
Let us take a frame on $V$ as above, i.e. such that $\{A_0;A_1,\dotsc,A_{k_t};\dotsc, A_N\}$ the first $k$-elements $A_1,\dotsc, A_k$ generate $T_P(V)$, and so on, and therefore $A_1,\dotsc, A_{k_t}$ generate $T_P^{(t)}(V)$.
Let us take also a frame on $\operatorname{Tan}^{t-1} (V)$ centred at $P$, $\{B_0;B_1,\dotsc,B_{\ell};\dotsc, B_N\}$, such that $B_0$ represents $P\in\operatorname{Tan}^{t-1} (V)$ and $B_1,\dotsc,B_{\ell}$ generate $T_P(\operatorname{Tan}^{t-1} (V))$. By definition, we have $$B_0=C_0A_0+\sum_{i=1}^{k_{t-1}}C_i A_i;$$ taking the exterior derivative $$dB_0=d C_0 A_0+C_0 d A_0+ \sum_{i=1}^{k_{t-1}}(d C_i A_i+C_i d A_i),$$ from which we infer that the tangent cone to $\operatorname{Tan}^{t-1} (V)$ in $P$ is contained in ${\tilde T}^{(t)}_P(V)$; since the tangent cone spans the tangent space, we conclude that $\tilde T_P(\operatorname{Tan}^{t-1} (V)) \subset {\tilde T}^{(t)}_P(V)$.
\[mainthm\] Let $V \subseteq {\mathbb{P}}^N$ be a $k$-dimensional irreducible variety whose $t$-th fundamental form has dimension $k-\ell-1$, with $\ell >0$. Then:
1. \[primo\] $V$ has $(t-1)$-osculating defect $o_{t-1}\geq \ell$.
2. \[secondo\] $V$ is contained in a $d$-dimensional scroll $S(\Sigma^h_r)$, $(d\le h+r)$, in linear spaces of dimension $r$, with $0 \leq h \leq k-\ell$ and $k-h\le r$.
3. \[terzo\] Let $\ {\mathbb{P}}^r\subset \ S(\Sigma^h_r)\ $ be a general $\ r\ $- dimensional space of the scroll; then $\ \langle \cup_{A\in {\mathbb{P}}^r}\tilde T_A( S(\Sigma^h_r))\rangle$ is contained in a linear space of dimension $d_t-h=d_{t-1}+ k-\ell-h
(\le\binom{k+t-1}{t-1}-1+k-\ell-h)$. In particular, $r\le d\le d_{t-1}+ k-\ell-h$.
\[primo\]: By hypothesis, Lemma \[lemsbagliato\] and Corollary \[cor\_dimensione ff\] (and with the above notations) $$\begin{gathered}
\dim \operatorname{Tan}^{t-1} (V)\leq \dim T_P( \operatorname{Tan}^{t-1} (V))\leq \\
\leq\dim(\tilde T^{(t)}_P(V))=d_{t-1}+\Delta_t+1\le \operatorname{expdim}\operatorname{Tan}^{t-1} (V)-\ell.\end{gathered}$$ \[secondo\]: Let—as above—$\gamma^t\colon V\dasharrow \mathbb G (N,d_t)$ the $t$-th Gauss map. Let $h:=\dim \operatorname{Im}(\gamma^t)$, so that $k-h$ is the dimension of the general fibre of $\gamma^t$. Let $\Phi_{k-h}(\Pi):=(\gamma^t)^{-1}(\Pi)$ be a general fibre; this is just the set of points $Q\in V$ for which $\Pi={\tilde T}^{(t)}_Q(V)$. $\Phi_{k-h}(\Pi)$ generates a linear space ${\mathbb{P}}^r$, $k-h\le r \le d_t$. Let us consider the scroll over $\operatorname{Im}(\gamma^t)=:\Sigma^h_r$ of these spaces, $S(\Sigma^h_r)$. Just by definition, $V\subset S(\Sigma^h_r)$.
Let $\check V^{(t)}\subset {{\mathbb{P}}^N}^*$ be the $t$-th dual variety variety; we have $$\dim(\check V^{(t)})=h+N-1-\dim(\tilde T^{(t)}_P(V)).$$
Moreover, by Lemma \[lemma\], , $\tilde T^{(t-1)}_P(V))\subset C^{(1)}_{t,P}(V)$, so that (by Corollary \[cor\_dimensione ff\]) $$\label{eq:equazione}
d_{t-1}\leq N-1-d_{t,1}=d_t-h=d_{t-1}+\delta_t+1-h=d_{t-1}+k-\ell-h,$$ and therefore $h\leq k-\ell$.
\[terzo\]: We have, by Lemma \[lemma\], , $Q\in C_{t,P}^{(t)}(V)$ if $Q\in \Phi_{k-h}(\Pi)$, $\Pi=\tilde T^{(t)}_P(V)$. Since $C_{t,P}^{(t)}(V)$ is a linear space, we have that $\langle\Phi_{k-h}(\Pi)\rangle={\mathbb{P}}^r \subset C_{t,P}^{(t)}(V)$, and therefore $S(\Sigma^h_r)\subset \operatorname{Car}^t_t(V)$. Finally, apply Corollary \[cor:lem\], to get that, if $R\in {\mathbb{P}}^r$ is a general point, $\tilde T _R S(\Sigma^h_r)\subset C^{(t-1)}_{t,P}(V)$ and moreover, since $\dim (\check V^{(t)})=N-1-d_t+h$, we have that $$\dim C^{(t-1)}_{t,P}(V)\le d_t-h=d_{t-1}+ k-\ell-h\le \binom{k+t-1}{t-1}-1+k-\ell-h.$$
Let us see some applications of this theorem.
\[ex:1\] Clearly, when $h=0$ we have that $V$ is contained in a ${\mathbb{P}}^{d_{t}}$. For example, this is the only possibility when $k=1$ i.e the case of the curves; but in this case we can say even more: we have $\ell=1$, $k-\ell=0=h$ and from we deduce that the curve is contained in a ${\mathbb{P}}^{d_{t-1}}$. So, if the theorem holds for $k=1$ and $t=2$, $V={\mathbb{P}}^1$ and for $k=1$ and $t=3$, $V$ is a plane curve, etc.
More generally, if $\ell=k$, $h=0=k-\ell$, and also in this case, thanks to , we deduce that $V$ is contained in a ${\mathbb{P}}^{d_{t-1}}$. In particular, if the theorem holds for $t=2$, we deduce $V={\mathbb{P}}^k$.
Let us pass to the next case $\ell=k-1$; in this case $h=0,1$. If $h=0<1=k-\ell$, thanks to , we infer that $d_{t}=d_{t-1}+1$. Hence $V\subset {\mathbb{P}}^{d_{t-1}+1}$ by Example \[ex:1\]. For $t=2$, we deduce that $V$ is a hypersurface in a ${\mathbb{P}}^{k+1}$.
If $h=1=k-\ell$, again by , we infer that $d_{t}=d_{t-1}+1$. Since, $k-1\le r\le d_t-1$ for $t=2$, we have that $k-1\le r\le d\le k$, but we cannot have $r=k$, since otherwise we would have that $V={\mathbb{P}}^r= S(\Sigma^h_r)$ and for it we would have $h=0$. Therefore, $r=k-1$, $\Phi_{k-h}(\Pi)={\mathbb{P}}^{k-1}$ and $V$ is a developable ${\mathbb{P}}^{k-1}$-bundle.
Our result generalising Theorem \[2ca\] is the following.
\[thm terr gen\] Let $V \subseteq {\mathbb{P}}^N$ be a $k$-dimensional irreducible variety. $V$ has $t$-th osculating defect $o_t=\ell>0$ and the $(t+1)$-th fundamental form has dimension at least $k-\ell$ if and only if the Jacobian matrix of the $(t+1)$-th fundamental form of $V$ has rank $k-\ell$.
Let us fix as usual a Darboux frame for $V$ as in ; if $P \in V$ is a general point, then, by Definition \[t\^t def\], $$\tilde T^t_P(V)=\langle (\frac{d^{|I|}A_0}{d v_1^{i_1}\dotsc d v_{k}^{i_{k}}})_{|I|\le t}\rangle,$$ with the convention that $d^0A_0=A_0$, therefore, we can fix a Darboux frame $\{B_0;B_1,\dotsc,B_d;B_{d+1},\dotsc,B_N\}$ ($d:=\dim \operatorname{Tan}^{(t)} (V) = \operatorname{expdim}\operatorname{Tan}^{(t)}(V)-\ell$) for $\operatorname{Tan}^{(t)}(V)$ centred at $Q \in \tilde T^t_P(V)$, where $B_1,\dotsc,B_d$ span $T_Q(\operatorname{Tan}^{(t)}(V))$, and so $$\label{eq:b}
B_0=A_o\sum_{|I|=t}\lambda^{(I)}\frac{d^{|I|}A_0}{d v_1^{i_1}\dotsc d v_{k}^{i_{k}}}.$$
Saying $\dim \operatorname{Tan}^{(t)} (V) = \operatorname{expdim}\operatorname{Tan}^{(t)}(V)-\ell$ means that there are $\ell$ linearly independent linear homogeneous relations between the (first) partial derivatives of $B_0$ with respect to the $v_j$’s and the $\lambda^{(I)}$’s: $$\begin{aligned}
\sum_{j=1}^ka_{\alpha,j}\frac{\partial B_0}{\partial v_j}+\sum_{|I|= t}a_{\alpha,I}\frac{\partial B_0}{\partial \lambda^{(I)}}&=0 & \alpha=1,\dotsc,\ell;\end{aligned}$$ then, by , $$\begin{aligned}
\sum_{j=1}^ka_{\alpha,j}(\sum_{|I|= t}\lambda^{(I)}\frac{d^{|I|+1}A_0}{d v_j d \mathbf v^I})&\equiv 0 & \alpha=1,\dotsc,\ell, & \mod T^{(t)}_P(V)\end{aligned}$$ i.e. these relations are indeed relations between the partial derivatives up to order $t+1$ of $A_0$, and we can think of it as a system of Laplace equations of order $t+1$: $$\begin{aligned}
\sum_{j=1}^ka_{\alpha,j}(\sum_{|I|= t}\lambda^{(I)}\mathbf x^{I+j}) &=0 & \alpha=1,\dotsc,\ell,\end{aligned}$$ and their associated polynomials are all reducible $$\begin{aligned}
\label{eq:apose}
(\sum_{j=1}^ka_{\alpha,j}v_j)(\sum_{|I|= t}\lambda^{(I)}\mathbf v_I)&=0 & \alpha=1,\dotsc,\ell,\end{aligned}$$ with the same factor of degree $t$, $\sum_{|I|= t}\lambda^{(I)}\mathbf v_I$. Since these homogeneous polynomials are independent, the $\ell$ linear forms $\sum_{j=1}^ka_{\alpha,j}v_j$, $\alpha=1,\dotsc,\ell$, are independent. In particular, up to a change of coordinates, it is not restrictive to suppose that these forms are $v_1,\dotsc,v_\ell$.
By Proposition \[prop\_apolarity\], we have that the $(t+1)$-fundamental form is the apolar system associated to ; in particular, we have that all the partial derivatives of the $(t+1)$-fundamental form with respect to $v_1,\dotsc,v_\ell$, are zero, from which we get that the rank of the Jacobian is $k-\ell$.
Since all the above can be reverted, the vice versa easily follows.
From this Theorem starts our next goal, i.e. to classify varieties with tangent or—more generally—higher osculating defect, which we will pursue in a subsequent paper. This goal relies on the study of linear systems of quadrics, or, more generally, of higher degree hypersurfaces, with Jacobian matrix of rank less than expected.
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[^1]: The first author was partially supported by MiUR, project “Geometria delle varietà algebriche e dei loro spazi di moduli” and by the University of Naples “Federico II”, project “F.A.R.O. 2010: Algebre di Hopf, differenziali e di vertice in geometria, topologia e teorie di campo classiche e quantistiche”.
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---
abstract: 'N-body simulations are used to model the tidal disruption of the Sagittarius (Sgr) dwarf galaxy with constraints set by the positions and velocities of M giants in the Sgr tidal arms recently revealed by the Two Micron All-Sky Survey (2MASS). The simulated Sgr dwarf is placed on a variety of orbits within a Milky Way potential parameterized by variable circular velocities, halo flattenings and radial profiles. Two hundred separate test particle orbits have been used to explore a wide range of model Milky Way potentials and dwarf galaxy characteristics. The family of models is delimited by the data to a relatively narrow allowed range of parameters, and then input into N-body simulations. We present our best-fitting model, and discuss the orbital period, apoGalacticon distance, current space velocity, mass-to-light ratio, and other characteristics of the Sgr dwarf. In addition, we discuss the implications of this model for the flattening of the Galactic halo.'
author:
- 'David R. Law, Steven R. Majewski, Michael F. Skrutskie'
- 'Kathryn V. Johnston'
title: Modeling the Tidal Tails of the Sagittarius Dwarf Galaxy
---
Introduction
============
Since the discovery of the Sgr dwarf by Ibata et al. (1994) many groups (e.g., Johnston, Hernquist, & Bolte 1996, Ibata et al. 1997, Ibata & Lewis 1998, G[' o]{}mez-Flechoso, Fux, & Martinet 1999, Johnston et al. 1999, Helmi & White 2001) have sought to model the Sgr - Milky Way interaction with respect to a modest patchwork of observational constraints. Recently, Majewski et al. (2003a, hereafter “Paper I”) have shown that the extensive length of the Sgr tidal tails can be traced by M giant stars visible in the all-sky view of the system provided by the 2MASS database. Spectroscopy of Sgr candidate stars has allowed determination of radial velocities throughout the trailing tail (Majewski et al. 2003b, hereafter “Paper II”), and these substantial new constraints can be used to develop more refined models of the Sgr system.
In this contribution, we briefly describe some of the major results of such modeling. A comprehensive description of this new Sgr disruption model can be found in Law, Johnston, & Majewski (2003, hereafter “Paper III”).
Modeling the Sgr System
=======================
Following previous work by Johnston et al. (1996, 1999) the Milky Way potential is represented numerically by a Miyamoto-Nagai (1975) disk, Hernquist spheroid, and a logarithmic halo. The total mass and radial profile are fixed by requiring that the rotation curve of this model Galaxy be consistent with HI & CO tangent point observations (e.g., Honma & Sofue 1997).
The Sgr dwarf itself is represented by $10^5$ self-gravitating particles (representing both the dark and light matter components of the satellite), which are initially distributed according to a Plummer (1911) model. This satellite is evolved through the simulated Galactic potential for five orbital periods using a self-consistent field code (Hernquist & Ostriker 1992). The present-day simulated dwarf is constrained to be located at $(l,b) = (5.6^{\circ},-14.2^{\circ})$ at a solar distance of $D_{\rm Sgr} = 24$ kpc (Paper I, Ibata et al. 1995) and have a radial velocity of $v_{\rm LOS,Sgr} = 171$ km s$^{-1}$ (Ibata et al. 1997). The direction of the dwarf’s space velocity vector is determined by requiring that the dwarf orbit in the orbital plane observed in Paper I.
Subject to these requirements, test-particle orbits (i.e. orbits calculated for a test particle with the observed kinematical characteristics of Sgr) and N-body simulations are performed for simulated satellites with a variety of orbital speeds. These simulations can be additionally constrained using the 2MASS M giant distance and radial velocity data presented in Papers I and II. Fig. 1 compares the M giant data (Panels a-b, filled squares) with the model Sgr dwarf whose tidal tails best reproduce the observations (Panels c-d). Note the close agreement between model and observed debris distances and radial velocities along the trailing debris tail ($\Lambda_{\odot} = 0^{\circ}$ - $100^{\circ}$)[^1]. This best-fit model is characterized by a period of 0.75 Gyr with apoGalacticon 52 kpc, periGalacticon 14 kpc, and a present space velocity of $(U,V,W) = (237.2, -43.4, 218.9)$ km s$^{-1}$.
Although we do not attempt to model the Sgr core in detail, it is nonetheless possible to use the width of the Sgr debris stream to estimate such global characteristics as the bound mass of the dwarf. The simulated dwarf which appears to best fit the width of streams shown in Fig. 1 has a present mass of $M_{\rm Sgr} = 3 \times 10^8 M_{\odot}$ and a mass-to-light ratio $M_{\rm Sgr}/L_{\rm Sgr} = 21$.
Discussion
==========
As demonstrated in the previous section, the tidal tails of this model provide a good fit to the all-sky view of M giants presented in Papers I and II. It is therefore possible to use this model to determine what range of Milky Way models permit simulated satellites to reproduce observations. Particularly, N-body simulations can be used to constrain the flattening of the Galactic halo (e.g., Ibata et al. 2001). Fitting an orbital plane to leading and trailing M giant debris separately, we determine that the orbital pole of Sgr debris has precessed by $1.7^{\circ} \pm 2.4^{\circ}$ over about $300^{\circ}$ of orbital longitude. Repeating this calculation for N-body simulations in model dark halos with a variety of flattenings, we calculate pole precessions of $2.2^{\circ} \pm 1.6^{\circ}$, $3.5^{\circ} \pm 1.7^{\circ}$, and $5.6^{\circ} \pm 1.4^{\circ}$ for flattenings in the halo potential of $q = 1, 0.95$, and $0.90$ respectively. It therefore appears likely that the halo of the Milky Way can be described by an almost spherical potential.
Although this model provides a good match to the distances and velocities of trailing Sgr debris given in Papers I and II, it does not fit recent data obtained by Majewski et al. (2003c, hereafter “Paper IV”) in the region of the Sgr leading arm. Fig. 1 (Panel a, filled triangles) plots these new data, which has velocities slower than that of the model by up to 200 km s$^{-1}$ in the range $\Lambda_{\odot} = 300^{\circ}$ - $200^{\circ}$. There is no simple modification of the velocity of the model satellite that serves to reproduce this new trend, and this may be an indication of such other effects as dynamical friction. However, simulations suggest that including corrections from Chandrasekhar’s formulation of dynamical friction should not have a substantial effect on the observed velocities of leading tidal debris for model satellites with mass $M_{\rm Sgr, 0} \leq 10^{10} M_{\odot}$, and we find that accurately reproducing the observed trend is difficult even for satellites with initial masses greater than this.
This inconsistency and implications of the best-fit model for the size and shape of the Milky Way are discussed at greater length in Paper III.
DRL acknowledges support from the Local Organizing Committee, a U.Va. Small Research Fellowship, and the U.Va. Echols Program. The authors also acknowledge NASA/JPL contracts 1228235 (SRM/KVJ) and 1234021 (MFS).
G[' o]{}mez-Flechoso, M. A., Fux, R., & Martinet, L. 1999, , 347, 77
Helmi, A. & White, S. D. M. 2001, , 323, 529
Hernquist, L. & Ostriker, J. P. 1992, , 386, 375
Honma, M. & Sofue, Y. 1997, , 49, 453
Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nature, 370, 194
Ibata, R. A., Gilmore, G. & Irwin, M. J. 1995, , 277, 781
Ibata, R. A. & Lewis, G. F. 1998, , 500, 575
Ibata, R.A., Lewis, G. F., Irwin, M.J., Totten, E., & Quinn, T. 2001, , 551, 294
Ibata, R. A., Wyse, R. F. G., Gilmore, G., Irwin, M. J., & Suntzeff, N. B. 1997, , 113, 634
Johnston, K. V., Hernquist, L., & Bolte, M. 1996, , 465, 278
Johnston, K. V., Majewski, S. R., Siegel, M. H., Reid, I. N., & Kunkel, W. E. 1999, , 118, 1719
Law, D. R., Johnston, K. V., & Majewski, S. R. 2003, in prep. (“Paper III”)
Majewski, S. R., Skrutskie, M. F., Weinberg, M. D. & Ostheimer, J. C. 2003a, ApJ submitted (“Paper I”)
Majewski, S. R. et al. 2003b, ApJ submitted (“Paper II”)
Majewski, S. R. et al. 2003c, in prep. (“Paper IV”)
Miyamoto, M. & Nagai, R. 1975, , 27, 533
Plummer, H. C. 1911, , 71, 460
[^1]: We use the orbital longitude coordinate system in the Sgr orbital plane defined in Paper I.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal steps in a Schröder path. As an application we obtain a new combinatorial description of the large and small Schröder numbers and their q-analogues.'
address:
- 'Giovanni Cerulli Irelli:Mathematisches Institut, Universität Bonn, Bonn, Germany 53115'
- 'Evgeny Feigin:Department of Mathematics,National Research University Higher School of Economics,Russia, 117312, Moscow, Vavilova str. 7Tamm Department of Theoretical Physics, Lebedev Physics Institute '
- 'Markus Reineke: Fachbereich C - Mathematik, Bergische Universität Wuppertal, D - 42097 Wuppertal'
author:
- 'Giovanni Cerulli Irelli, Evgeny Feigin, Markus Reineke'
title: 'Degenerate flag varieties: moment graphs and Schröder numbers'
---
Introduction
============
For $n\ge 1$, let ${\EuScript{F}}^a_{n+1}$ be the degenerate flag variety attached to the Lie algebra $\msl_{n+1}$ (see [@Fe1], [@Fe2]). This is a flat degeneration of the classical flag variety, defined using the PBW filtration on irreducible representations of $\msl_{n+1}$ (see [@FFoL]). By construction, the ${\EuScript{F}}^a_{n+1}$ is acted upon by the degenerate Lie group $SL_{n+1}^a$, which is the semi-direct product of the Borel subgroup $B$ and the abelian group ${{\mathbb G}}_a^N$, where ${{\mathbb G}}_a$ is the additive group of the field. In particular, ${{\mathbb G}}_a^N$ acts on ${\EuScript{F}}^a_{n+1}$ with an open dense orbit. The degenerate flag varieties are singular normal projective algebraic varieties, sharing many nice properties with their classical analogues. In particular, they enjoy a description in terms of linear algebra as subvarieties inside a product of Grassmann varieties.
It has been observed in [@CFR] that the degenerate flag varieties can be identified with certain quiver Grassmannians of the equioriented quiver of type $A_n$. More precisely, ${\EuScript{F}}^a_{n+1}$ is isomorphic to the quiver Grassmannian ${\rm Gr}_{{{\bf dim}}A}(A\oplus A^*)$, where $A$ and $A^*$ are the path algebra of the equioriented $A_n$ quiver, resp. its dual. This observation was used in two different ways: first, to get a deeper understanding of the geometry and combinatorics of the degenerate flag varieties, and, second, to generalize the results and constructions to a wider class of quiver Grassmannians. In this paper we continue the study of the varieties ${\EuScript{F}}^a_{n+1}$ using the techniques from the theory of quiver Grassmannians. More concretely, we achieve two things: first, we describe the combinatorial structure of the moment graph of ${\EuScript{F}}^a_{n+1}$. Second, we describe explicitly the smooth and singular loci of the degenerate flag varieties. Let us give a brief description of our results.
Recall that the notion of the moment graph attached to an algebraic variety $X$ acted upon by an algebraic torus was introduced in [@GKM], [@BM]. This combinatorial object captures the structure of zero- and one-dimensional orbits of $T$. It turns out to be very useful for describing various geometric properties of $X$, such as cohomology and intersection cohomology. Our first task is to describe the moment graph $\Gamma$ of ${\EuScript{F}}^a_{n+1}$. We note that the automorphism group ${\rm Aut}(A\oplus A^*)$ acts on ${\EuScript{F}}^a_{n+1}$. The maximal torus $T$ of the automorphism group acts with a finite number of fixed points (this number is equal to the normalized median Genocchi number, see [@CFR],[@Fe2],[@Fe3]). It is proved in [@CFR] that there exists a codimension one subgroup ${\mathfrak A}\subset {\rm Aut}(A\oplus A^*)$ containing the torus $T$ such that ${\mathfrak A}$-orbits through $T$-fixed points are affine cells that provide a cellular decomposition of ${\EuScript{F}}^a_{n+1}$. We describe ${\mathfrak A}$ as a quotient of the Borel subgroup of $SL_{2n}$. Using this description, we prove the following theorem (for a more precise formulation see section \[MG\]):
The number of one-dimensional $T$-orbits in ${\EuScript{F}}^a_{n+1}$ is finite. The edges of $\Gamma$ correspond to the one-parameter subgroups of ${\mathfrak A}$.
We note that the structure of $\Gamma$ has many common features with its classical analogue (see [@C], [@GHZ], [@T]).
Our next goal is to describe the smooth locus of the degenerate flag varieties. Since ${\EuScript{F}}^a_{n+1}$ has a cellular decomposition by ${\mathfrak A}$-orbits of $T$-fixed points, it suffices to decide which $T$-fixed points are smooth. We recall that the $T$-fixed points are labeled by collections ${{\bf S}}=(S_1,\dots,S_n)$ of subsets of $\{1,\dots,n+1\}$ such that $\#S_i=i$ and $S_i\subset S_{i+1}\cup\{i+1\}$. We denote the corresponding $T$-fixed point by $p_{{{\bf S}}}$.
\[I2\] A point $p_{{{\bf S}}}$ is smooth if and only if for all $1\leq j<i\leq n$, the condition $i\in S_j$ implies $j+1\in S_i$. The number of smooth $T$-fixed points is given by the large Schröder number $r_n$.
We recall (see [@St], [@G]) that the large Schröder number $r_n$ is equal to the number of Schröder paths, i.e. subdiagonal lattice paths starting at $(0,0)$ and ending at $(n,n)$ with the following steps allowed: $(1,0)$, $(0,1)$ and $(1,1)$. In particular, Theorem \[I2\] implies that the Euler characteristic of the smooth locus of ${\EuScript{F}}^a_{n+1}$ is equal to $r_n$. Moreover we prove the following theorem:
The Poincaré polynomial of the smooth locus of ${\EuScript{F}}^a_{n+1}$ is equal to the (scaled) $q$-Schröder number $q^{n(n-1)/2}r_n(q)$, where $r_n(q)$ is defined via the statistics on Schröder paths, counting the number of $(1,1)$ steps in a path.
As an application, we obtain a new proof of the statement that $r_n(q)$ is divisible by $1+q$. The ratio is known to give a $q$-analogue of the small Schröder numbers.
Let us mention two more results of the paper. First, we prove that, for a general Dynkin type quiver $Q$ and a projective $Q$-module $P$ and an injective $Q$-module $I$, the quiver Grassmannian ${\rm Gr}_{{{\bf dim}}P} (P\oplus I)$ is smooth in codimension $2$. Second, we prove that the smooth locus of ${\EuScript{F}}^a_{n+1}$ can be described as the subvariety of points where the desingularization map $R_{n+1}\to {\EuScript{F}}^a_{n+1}$ (see [@FF]) is one-to-one.
Finally, we note that all the results of the paper can be generalized to the case of the degenerate partial (parabolic) flag varieties.
Our paper is organized as follows:\
In Section $1$ we introduce the main objects and recall the main definitions and results needed in the rest of the paper.\
In Section $2$ we describe the moment graph of the degenerate flag varieties.\
In Section $3$ we prove a criterion for smoothness of a $T$-fixed point and compute the Euler characteristics and Poincaré polynomials.\
In Appendix $A$ we prove the regularity in codimension $2$ of certain quiver Grassmannians.\
In Appendix $B$ we describe the smooth locus in terms of the desingularization.\
In Appendix $C$ we compute the moment graph for the degenerate flag variety ${\EuScript{F}}^a_4$.\
Quiver Grassmannians and degenerate flag varieties
==================================================
In this section we recall definitions and results on the degenerate flag varieties and quiver Grassmannians to be used in the main body of the paper.
Degenerate flag varieties
-------------------------
Let ${\EuScript{F}}_{n+1}$ be the complete flag variety for the group $SL_{n+1}$, i.e. the quotient $SL_{n+1}/B$ by the Borel subgroup $B$. This variety has an explicit realization as the subvariety of the product of Grassmannians $\prod_{k=1}^n {\rm Gr}_k({{\mathbb C}}^{n+1})$ consisting of collections $(V_1,\dots,V_n)$ such that $V_i\subset V_{i+1}$ for all $i$. In [@Fe1],[@Fe2] flat degenerations ${\EuScript{F}}^a_{n+1}$ of the classical flag varieties were introduced. The degenerate flag varieties ${\EuScript{F}}^a_{n+1}$ are (typically singular) irreducible normal projective algebraic varieties, sharing many nice properties with their classical analogues. In particular, they also have a very explicit description in linear algebra terms. Namely, let $W$ be an $(n+1)$-dimensional vector space with a basis $w_1,\dots,w_{n+1}$. Let $pr_k:W\to W$ be the projection operators defined by $pr_k w_k=0$ and $pr_kw_i=w_i$ if $i\ne k$. The following lemma is proved in [@Fe2], Theorem $2.1$.
\[Lemma:DegFlagGrass\] The degenerate flag variety ${\EuScript{F}}^a_{n+1}$ is a subvariety of the product of Grassmannians $\prod_{k=1}^n {\rm Gr}_k(W)$, consisting of collections $(V_k)_{k=1}^n$ such that $$pr_{k+1} V_k\subset V_{k+1}\ \text{ for all }\ k=1,\dots,n-1.$$
Another important property of the varieties ${\EuScript{F}}^a_{n+1}$ is that they admit a cellular decomposition into a disjoint union of complex cells. Moreover, there exists an algebraic group ${\mathfrak A}$ and a torus $T\subset {\mathfrak A}$ acting on ${\EuScript{F}}^a_{n+1}$ such that each cell contains exactly one $T$-fixed point and the ${\mathfrak A}$-orbit through this point coincides with the cell. Let us describe the combinatorics of the cells, postponing the description of the group action to the next subsection. So let ${{\bf S}}=(S_1,\dots,S_n)$ be a collection of subsets of the set $\{1,\dots,n+1\}$ such that each $S_i$ contains $i$ elements. Then the cells in ${\EuScript{F}}^a_{n+1}$ are labeled by the collections satisfying the following property $$\label{mG}
S_k\subset S_{k+1}\cup \{k+1\},\quad k=1,\dots,n-1.$$ We call such collections admissible. The number of admissible collections (and hence the Euler characteristic of ${\EuScript{F}}^a_{n+1}$) is equal to the normalized median Genocchi number $h_{n+1}$ (see [@Fe2],[@Fe3], [@CFR]). We note that the correspondence between the admissible collections and $T$-fixed points is very explicit. Namely, for a collection ${{\bf S}}$ we denote by $p_{{{\bf S}}}\in {\EuScript{F}}^a_{n+1}$ a point defined by $$p_{{{\bf S}}}=(V_1,\dots,V_n),\qquad V_k=\mathrm{span}(w_i,\ i\in S_k).$$ Clearly, such a point belongs to ${\EuScript{F}}^a_{n+1}$ if and only if the collection ${{\bf S}}$ is admissible.
Quiver Grassmannians. {#QGr}
---------------------
The construction above can be reformulated in the language of quiver Grassmannians (see e.g. [@Sc], [@CR]). Let $Q$ be the equioriented type $A_n$ quiver with vertices labeled by numbers from $1$ to $n$ and arrows $i\to i+1$, $i=1,\dots,n-1$: $$Q:\qquad \bullet \to \bullet \to \dots\to\bullet$$ For a representation $M$ of $Q$ we denote by $M_k$ the subspace of $M$ attached to the vertex $k$. For a pair $1\le i\le j\le n$ let $R_{i,j}$ be an indecomposable representation of $Q$ supported on the vertices $i,\dots,j$ (i.e. $(R_{i,j})_k={{\mathbb C}}$ for $i\le k\le j$ and is trivial otherwise). We have the following immediate lemma.
\[HE\] $$\dim {\rm Hom} (R_{i,j}, R_{k,l})=\begin{cases} 1, \text{ if } k\le i\le l\le j,\\
0, \text{ otherwise}\end{cases};$$ $$\dim {\rm Ext}^1 (R_{i,j}, R_{k,l})=\begin{cases} 1, \text{ if } i+1\leq k\leq j+1\leq l,\\
0, \text{ otherwise}\end{cases}.$$
We note that the representations $R_{1,j}$ are injective and the $R_{i,n}$ are projective (note that these are all indecomposable injective and projective representations of $Q$). We set $$I_k=R_{1,k},\quad P_k=R_{k,n}, \quad P=\bigoplus_{k=1}^n P_k,\quad I=\bigoplus_{k=1}^n I_k.$$ Hence, $P$ is isomorphic to the path algebra of $Q$ and $I$ is isomorphic to its linear dual. For a dimension vector ${{\bf e}}=(e_1,\dots,e_n)$ and a representation $M$ of $Q$, we denote by ${\rm Gr}_{{\bf e}}(M)$ the quiver Grassmannian of ${{\bf e}}$-dimensional subrepresentations of $M$. Then by definition one gets $$\label{QG}
{\EuScript{F}}^a_{n+1}\simeq {\rm Gr}_{{{\bf dim}}P} (P\oplus I).$$
\[Rem:CoeffQuiver\] The representation $P\oplus I$ can be visualized by the following picture (here $n=4$). Each fat dot corresponds to a basis vector and two dots corresponding to the vectors $u$ and $v$ are connected by an arrow $u\to v$ if $u$ is mapped to $v$. The quiver obtained in this way is called the coefficient–quiver of $P\oplus I$. $$\label{pic}
\xymatrix@R=6pt@C=8pt
{
\bullet\ar[r]&\bullet\ar[r]&\bullet\ar[r]&\bullet&\\
\bullet\ar[r]&\bullet\ar[r]&\bullet&\bullet&\\
\bullet\ar[r]&\bullet&\bullet\ar[r]&\bullet&\\
\bullet&\bullet\ar[r]&\bullet\ar[r]&\bullet&\\
\bullet\ar[r]&\bullet\ar[r]&\bullet\ar[r]&\bullet&\\
}$$
The isomorphism has many important consequences. In particular the automorphism group of the $Q$-module $P\oplus I$ acts on ${\EuScript{F}}^a_{n+1}$. The group ${\rm Aut} (P\oplus I)$ is of the following form $
{\rm Aut} (P\oplus I)=
\begin{pmatrix}
{\rm Aut} P & {\rm Hom} (I,P)\\
{\rm Hom} (P,I) & {\rm Aut} I
\end{pmatrix}.
$ The part ${\rm Hom} (I,P)$ is one-dimensional (${\rm Hom} (I,P)={\rm Hom} (I_n,P_1)$). We denote by ${\mathfrak A}\subset {\rm Aut} (P\oplus I)$ the following subgroup $${\mathfrak A}=\begin{pmatrix}
{\rm Aut} P & 0\\
{\rm Hom} (P,I) & {\rm Aut} I
\end{pmatrix}.$$ The group ${\mathfrak A}$ contains a torus $T$ isomorphic to $({{\mathbb C}}^*)^{2n}$, where each factor scales the corresponding indecomposable summand in $P\oplus I$. The importance of the group ${\mathfrak A}$ comes from the following lemma, proved in [@CFR].
The group ${\mathfrak A}$ acts on ${\EuScript{F}}^a_{n+1}$ with a finite number of orbits. Each orbit is a complex affine cell, containing exactly one $T$-fixed point. The orbits are labeled by admissible collections.
For an admissible collection ${{\bf S}}$ we denote by $C_{{{\bf S}}}$ the cell containing the $T$-fixed point $p_{{{\bf S}}}$.
\[torus\] We note that $T$ contains the identity automorphism which acts trivially on the degenerate flag variety. Hence one gets a $(2n-1)$-dimensional torus acting effectively on ${\EuScript{F}}^a_{n+1}$, while the maximal torus $T^c$ acting on the classical flag variety ${\EuScript{F}}_{n+1}$ is $n$-dimensional. We note that there is a natural embedding $T^c\subset T$. In fact recall that any point of ${\EuScript{F}}^a_{n+1}$ is of the form $(V_k)_{k=1}^n$, $V_k\subset W\simeq {{\mathbb C}}^{n+1}$. Hence any diagonal (in the basis $w_i$) matrix in $SL(W)$ induces an automorphism of the degenerate flag variety. Hence we obtain the embedding $T^c\subset T$.
Finally, we note that the torus $T$ contains a one-dimensional subtorus $T_0$ with the following properties: the set of $T$-fixed points coincides with the set of the $T_0$-fixed points and the attracting set of a fixed point $p$ coincides the the orbit ${\mathfrak A}p$ (which is an affine cell) [@CFR Theorem 5.1]. The action of the one-dimensional torus can be illustrated as follows ($n=4$, the scalar $\lambda\in{{\mathbb C}}^*$ is the parameter of the torus and the power of $\la$ corresponds to the scaling factor of the $T_0$ action): $$\label{la}
\xymatrix@R=6pt@C=8pt
{
1&&&&&\bullet&\\
\la&&&&\bullet\ar[r]&\bullet&\\
\la^2&&&\bullet\ar[r]&\bullet\ar[r]&\bullet&\\
\la^3&&\bullet\ar[r]&\bullet\ar[r]&\bullet\ar[r]&\bullet&\\
\la^4&&\bullet\ar[r]&\bullet\ar[r]&\bullet\ar[r]&\bullet&\\
\la^5&&\bullet\ar[r]&\bullet\ar[r]&\bullet&&\\
\la^6&&\bullet\ar[r]&\bullet&&&\\
\la^7&&\bullet&&&&\\
}$$ This picture is obtained from the picture by putting the $P$-part on top of the $I$-part.
We conclude this section by describing the action of the torus $T$ on the tangent space at a $T$–fixed point $p_{{\bf S}}$. Recall that the tangent space at $p_\mathbf{S}$ is isomorphic to ${\rm Hom}(p_\mathbf{S},M/p_\mathbf{S})$ where $M=P\oplus I$ ([@CFR Lemma 2.3], [@CR], [@Sc]). Let $\theta_M$ be the coefficient quiver of M (see Remark \[Rem:CoeffQuiver\]) and let $\pi: \theta_M\rightarrow Q$ be the natural projection onto the $A_n$ quiver $Q$. The coefficient quiver of $M/p_\mathbf{S}$ is $\theta_M\setminus \mathbf{S}$. The vector space ${\rm Hom}(p_\mathbf{S},M/p_\mathbf{S})$ has a distinguished basis, denoted by $\mathcal{B}$, parameterized by triples $(A,f,B)$ where A is a predecessor–closed connected sub quiver of $\mathbf{S}$, $B$ is a successor–closed connected sub quiver of $\theta_M\setminus\mathbf{S}$ and $f:A\rightarrow B$ is a quiver isomorphism compatible with $\pi$ (see [@CrawleyTree]). For example, in the left–hand side of the picture below $$\label{Eq:ExTangSpaceBasis}
\begin{array}{ccc}
\xymatrix@R=6pt@C=8pt{
\cdot\ar[r]&\cdot\ar[r]&\bullet&\\
\cdot\ar[r]&\bullet&\cdot&\\
\bullet&*+[F]{\bullet}\ar[r]&\bullet &A\ar^f[d]\\
\cdot\ar[r]&*+[F]{\cdot}\ar[r]&\bullet& B
}
&&
\xymatrix@R=6pt@C=8pt{
\lambda_3\ar[r]&\lambda_3\ar[r]&\lambda_3\\
\lambda_4\ar[r]&\lambda_4&1\\
\lambda_5&\lambda_1\ar[r]&\lambda_1\\
\lambda_2\ar[r]&\lambda_2\ar[r]&\lambda_2
}
\end{array}$$ the fat dots highlight the coefficient–quiver $\mathbf{S}$ of a T–fixed point $p_\mathbf{S}$ of ${\EuScript{F}}_4^a$ and the frames highlight a distinguished basis vector of the tangent space at $p_\mathbf{S}$.
\[Prop:TangentSpaceFicedPoint\] Given a $T$–fixed point $p_{{\bf S}}$ of ${\EuScript{F}}_{n+1}^a$, the torus $T$ acts on the tangent space at $p_{{\bf S}}$ diagonally in the basis $\mathcal{B}$. Moreover the eigenvalues are (generically) distinct.
Given $\lambda\in T$ and $f\in {\rm Hom}(p_\mathbf{S}, M/p_\mathbf{S})$, $(\lambda.f)(v)=\lambda.f(\lambda^{-1}.v)$. Now, by definition of $T$, each connected component $R$ of $\theta_M$ has a weight $wt(R)$ and hence a basis vector $(A,f,B)$ receives the weight $wt(B)/wt(A)$.
To illustrate the previous proposition, let us consider ${\EuScript{F}}_{4}^a$ and the action of $T$ depicted in the right–hand side of . The tangent space at $p_\mathbf{S}$ has dimension $7$ and the torus acts in the standard basis $\mathcal{B}$ as the diagonal matrix $diag(\frac{1}{\lambda_3}, \frac{\lambda_3}{\lambda_4},\frac{\lambda_2}{\lambda_4}, \frac{\lambda_3}{\lambda_1}, \frac{\lambda_2}{\lambda_1}, \frac{1}{\lambda_2}, \frac{\lambda_4}{\lambda_5})$. The one–dimensional torus $T_0$ is given by putting $\lambda_i:=\lambda^i$. In particular its action on the tangent space at $p_{{\bf S}}$ is given by the diagonal matrix $diag(\lambda^{-3}, \lambda^{-1},\lambda^{-2}, \lambda^{2}, \lambda^{1}, \lambda^{-2}, \lambda^{-1})$. Notice that the eigenvalues of the $T_0$ action are *not* distinct.
\[Cor:T-actionTangent\] The $T$–fixed one–dimensional vector subspaces of ${\rm Hom}(p_\mathbf{S}, M/p_\mathbf{S})$ are precisely the coordinate ones, i.e. those generated by standard basis vectors.
Partial flag varieties.
-----------------------
The whole picture described above has a straightforward generalization to the case of partial flag varieties. Namely, given a collection ${{\bf d}}=(d_1,\dots,d_k)$, where $1\le d_1<d_2<\dots <d_k\le n$, let ${\EuScript{F}}_{{\bf d}}$ be the corresponding partial flag variety for $SL_{n+1}$ (${\EuScript{F}}_{{\bf d}}$ is a quotient of $SL_{n+1}$ by a parabolic subgroup). Explicitly, ${\EuScript{F}}_{{\bf d}}$ consists of collections $(V_{d_1},\dots,V_{d_k})$ of subspaces of an $(n+1)$-dimensional vector space $W$ such that $\dim V_m=m$ and $V_{d_i}\subset V_{d_{i+1}}$. These varieties can be degenerated in the same way as the complete flag variety (see [@Fe1],[@Fe2]). As a result one gets a variety ${\EuScript{F}}^a_{{\bf d}}$, consisting of collections of subspaces $(V_{d_1},\dots,V_{d_k})$ of $W$ such that $\dim V_m=m$ and $$pr_{d_i+1}\dots pr_{d_{i+1}}V_{d_i}\subset V_{d_{i+1}},\ i=1,\dots,k-1.$$ These varieties are also certain quiver Grassmannians (see [@CFR]). Namely, consider the equioriented quiver of type $A_k$. Then the degenerate partial flag variety ${\EuScript{F}}^a_{{\bf d}}$ is isomorphic to $$\label{part}
{\rm Gr}_{(d_1,\dots,d_k)}
(P_1^{d_1}\oplus P_2^{d_2-d_1}\oplus\dots \oplus P_k^{d_k-d_{k-1}}
\oplus I_1^{d_2-d_1}\oplus\dots\oplus I_{k-1}^{d_k-d_{k-1}}\oplus I_k^{n+1-d_k}),$$ where $P_i$ and $I_j$ are projective and injective modules of the $A_k$ quiver. There is a natural surjection ${\EuScript{F}}^a_{n+1}\to {\EuScript{F}}^a_{{\bf d}}$, sending $(V_i)_{i=1}^n$ to $(V_{d_j})_{j=1}^k$. The group $\fA$ thus acts on ${\EuScript{F}}^a_{{\bf d}}$; the orbits are affine cells containing exactly one $T$-fixed point. These $T$-fixed points are parametrized by collections ${{\bf S}}=(S_{d_1},\dots,S_{d_k})$ of subsets of $\{1,\dots,n+1\}$ subject to the conditions $\#S_{d_i}=d_i$ and $$\label{dadm}
S_{d_i}\subset S_{d_{i+1}}\cup\{d_i+1,\dots,d_{i+1}\},\ i=1,\dots,k-1.$$ We call such collections ${{\bf d}}$-admissible. As for the complete flags, the corresponding $T$-fixed point $p_{{\bf S}}=(V_{d_1},\dots,V_{d_k})$ is given by $V_{d_i}={\mathrm span}(w_j,\ j\in S_{d_i})$.
The moment graph {#MG}
================
In this section we study the combinatorics and geometry of the cellular decomposition of the degenerate flag varieties.
The group action.
-----------------
Recall the group ${\mathfrak A}$ acting on ${\EuScript{F}}^a_{n+1}$. The following lemma is simple, but important for us. Let $B\subset GL_{2n}$ be the Borel subgroup of lower-triangular matrices and $N\subset B$ be the subgroup of matrices $(a_{i,j})_{i\ge j}$ such that $a_{i,i}=1$ and $a_{i,j}=0$ unless $i-j>n$. For example, for $n=5$ the froup $N$ looks as follows: $$\left(
\begin{array}{cccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
* & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
* & * & 0 & 0 & 0& 0 & 0 & 1 & 0 & 0\\
* & * & * & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
* & * & * & * & 0 & 0 & 0 & 0 & 0 & 1
\end{array}\right).$$
\[B\] The group ${\mathfrak A}$ is isomorphic to the quotient group $B/N$.
Consider the isomorphism ${\rm Aut} (P\oplus I)\simeq {\rm Aut} (\bigoplus_{i=1}^n P_i
\oplus \bigoplus_{k=1}^n I_k)$. We note that for any pair of indecomposable summands of $P\oplus I$ the space of homomorphisms between them is either one-dimensional or trivial. More precisely, let us introduce the following notation for the indecomposable summands of $P\oplus I$: $$\label{phi}
R_1=P_n, R_2=P_{n-1},\dots, R_n=P_1,\ R_{n+1}=I_n, R_{n+2}=I_{n-1},\dots, R_{2n}=I_1.$$ Then for two indecomposable summands $R_i$ and $R_j$ one has $\dim {\rm Hom} (R_i,R_j)=1$ if and only if $i\le j$ and $j-i\le n$ (see Lemma \[HE\]). Hence we obtain a surjection of groups $B\to \fA$ and the kernel coincides with $N$.
Let us fix a non zero element $\gamma_{i,j}\in {\rm Hom} (R_i,R_j)$ for each pair $i,j$ with $i\le j$, $j-i\le n$. Then any element $g\in\fA$ can be uniquely written as a sum $\sum g_{i,j}\gamma_{i,j}$, defining a matrix in $B$. This produces a section $\fA\to B$.
\[Rem:Ri\] We note that the direct summands $R_i$ in type $A_4$ are visualized in . Namely, $R_1$ is represented by the only fat dot in the upper line, $R_2$ is represented by the two dots in the next to the upper line, and so on up to $R_8$. In general, if $i\le n$, then the dimension vector of $R_i$ is $(0,\dots,0,1,\dots,1)$ with $i$ units and each non zero $(R_i)_k$ is spanned by $w_{n+1-i}$. If $i>n$, then the dimension vector of $R_i$ is $(1,\dots,1,0,\dots,0)$ with $2n-i+1$ units and each non-zero $(R_i)_k$ is spanned by $w_{2n-i+2}$.
Recall that the $T$ fixed points in ${\EuScript{F}}^a_{n+1}$ are labeled by the admissible collections. For an admissible collection ${{\bf S}}$ let $p_{{{\bf S}}}$ be the corresponding $T$-fixed point and $C_{{{\bf S}}}$ be the cell containing $p_{{{\bf S}}}$. We know that $C_{{{\bf S}}}={\mathfrak A} p_{{{\bf S}}}$. Our goal now is to describe a unipotent subgroup $U_{{\bf S}}\subset {\mathfrak A}$ such that the map $U_{{\bf S}}\to C_{{{\bf S}}}$ is one-to-one. Let ${\mathfrak a}$ be the Lie algebra of the group ${\mathfrak A}$. Then $${{\mathfrak a}}={\rm Hom} (P,P)\oplus {\rm Hom} (I,I)\oplus {\rm Hom} (P,I).$$ The Lie algebra ${{\mathfrak a}}$ is the quotient of the Borel subalgebra ${{\mathfrak b}}\subset\gl_{2n}$ of lower triangular matrices by the ideal ${{\mathfrak n}}$ consisting of matrices $(a_{j,i})_{j\ge i}$ such that $a_{i,j}=0$ unless $j-i>n$ (this is exactly the Lie algebra of $N$). In particular, the one-dimensional hom-spaces ${\rm Hom}(R_i,R_j)$, $i\le j$, $j-i\le n$ between two indecomposable summands of $P\oplus I$ correspond to the root vectors of the form $E_{j,i}\in{{\mathfrak b}}$ ($E_{j,i}$ are matrix units). We have $${{\mathfrak a}}={{\mathfrak t}}\oplus\bigoplus_{\genfrac{}{}{0pt}{}{1\le i<j\le 2n}{j-i\le n}} {{\mathfrak a}}_{i,j},$$ where ${{\mathfrak t}}$ is the Lie algebra of the torus $T$ and ${{\mathfrak a}}_{i,j}={\rm Hom}(R_i,R_j)$.
Consider a pair $R_i,R_j$ of direct summands of $P\oplus I$ such that $\dim {\rm Hom}(R_i,R_j)=1$ and fix a non-zero $\gamma\in {\rm Hom}(R_i,R_j)$.
A pair of indices $(i,j)$ (a pair of representations $R_i,R_j$) is called ${{\bf S}}$-effective, if $p_{{\bf S}}\cap R_i\ne 0$ and $\gamma(p_{{\bf S}}\cap R_i)$ does not sit inside $p_{{\bf S}}$.
\[Remark:S-effectiveTorus\] $\mathbf{S}$–effective pairs have the following geometric interpretation: they are in bijection with standard basis vectors of the tangent space at $p_{{\bf S}}$ on which $T_0$ acts with *positive* weight (see the end of subsection \[QGr\]). Let us prove this statement. In notation , we denote by $\mathbf{R}_k$ the coefficient–quiver of $R_k$. Given an ${{\bf S}}$–effective pair $(i,j)$ a non–zero $\gamma\in{\rm Hom}(R_i,R_j)$ is determined (up to scalar multiplication) by a (unique) triple $(A,f,B)$. So $A\subset \mathbf{R}_i$ is predecessor–closed, $B\subset \mathbf{R}_j$ is successor closed and $f:A\rightarrow B$ is a quiver isomorphism compatible with $\pi$ (see subsection \[QGr\]). The sub representation $\gamma(p_{{\bf S}}\cap R_i)\subset R_j$ determines the successor–closed sub quiver $f({{\bf S}}\cap A)$ of $B$. Since by definition $\gamma(p_{{\bf S}}\cap R_i)$ does not sit inside $p_{{\bf S}}$, $f({{\bf S}}\cap A)$ strictly contains ${{\bf S}}\cap B$ and the difference $f({{\bf S}}\cap A)\setminus ({{\bf S}}\cap B)$ is the coefficient quiver of the non trivial quotient $\gamma(p_{{\bf S}}\cap R_i)/(\gamma(p_{{\bf S}}\cap R_i)\cap p_{{\bf S}})$. The map $$\gamma\mapsto b_\gamma:=({{\bf S}}\cap A\setminus f^{-1}({{\bf S}}\cap B), f|_{{{\bf S}}\cap A\setminus f^{-1}({{\bf S}}\cap B)}, f({{\bf S}}\cap A)\setminus ({{\bf S}}\cap B))$$ gives a bijection between ${{\bf S}}$–effective pairs and standard basis vectors of the tangent space $T_{p_{{\bf S}}}({\EuScript{F}}_{n+1}^a)={\rm Hom}(p_\mathbf{S},M/p_\mathbf{S})$ on which $T_0$ acts with a positive weight. To see this we notice that ${{\bf S}}\cap A$ is predecessor–closed in ${{\bf S}}$ and ${{\bf S}}\cap B$ is successor closed in $B$. Then $f^{-1}({{\bf S}}\cap B)$ is successor closed in ${{\bf S}}\cap A$ and hence ${{\bf S}}\cap A\setminus f^{-1}({{\bf S}}\cap B)$ is predecessor closed in ${{\bf S}}\cap A$ and hence in ${{\bf S}}$. We notice that ${{\bf S}}\cap B$ coincides with ${{\bf S}}\cap \mathbf{R}_j$ (otherwise ${{\bf S}}\cap B$ would not be strictly contained in $f({{\bf S}}\cap A)$). Since $f({{\bf S}}\cap A)$ is successor closed in $\mathbf{R}_j$ and ${{\bf S}}\cap B={{\bf S}}\cap \mathbf{R}_j$, it follows that $f({{\bf S}}\cap A)\setminus ({{\bf S}}\cap B)$ is successor closed in $\mathbf{R}_j\setminus ({{\bf S}}\cap \mathbf{R}_j)$ and hence in $\theta_M\setminus {{\bf S}}$. The quiver morphsim $f|_{{{\bf S}}\cap A\setminus f^{-1}({{\bf S}}\cap B)}$ is a quiver isomorphism between ${{\bf S}}\cap A\setminus f^{-1}({{\bf S}}\cap B)$ and $f({{\bf S}}\cap A)\setminus ({{\bf S}}\cap B)$ compatible with $\pi$, since $f$ is so. The image $b_\gamma$ of $\gamma$ is hence a standard basis vector of ${\rm Hom}(p_\bs,M/p_{{\bf S}})$. The action of $T_0$ on $b_\gamma$ is given by $\lambda. b_\gamma=\lambda^{j-i} b_\gamma$. Since $\gamma\neq 0$, then $i\leq j$ and hence $b_\gamma$ has positive weight. The map is hence well–defined and injective. Let us show that it is surjective. Let $b=(A',f',B')$ be a standard basis vector of ${\rm Hom}(p_{{\bf S}},M/p_{{\bf S}})$ on which $T_0$ acts with a positive weight. Then there are indices $i$ and $j$ such that $A'$ is a predecessor–closed sub quiver of ${{\bf S}}\cap \mathbf{R}_i$, and $B'$ is a successor–closed sub quiver of $\mathbf{R}_j\setminus (\mathbf{R}_j\cap{{\bf S}})$. The torus $T_0$ acts on $b$ as $\lambda.b=\lambda^{j-i}b$ and hence $j>i$. We claim that $j-i\leq n$. Indeed if $j-i>n$ then $\pi(\mathbf{R}_j)$ and $\pi(\mathbf{R}_i)$ are disjoint in $Q$ (otherwise ${\rm Hom}(R_i,R_j)\neq 0$ against the hypothesis $j-i>n$) and hence the quiver isomorphism $f':A'\rightarrow B'$ could not exist. In view of Lemma \[HE\] and the proof of Lemma \[B\], it follows that there is a non–zero standard basis vector $\gamma\in{\rm Hom}(R_i, R_j)$ defined by a triple $(A, f, B)$. Notice that $\pi(A)=\pi(B)=\pi(R_i)\cap\pi(R_j)\supset \pi(A')=\pi(B')$. It follows that $A'\subset A$, $B'\subset B$ and $f'=f|_{A'}$. From this we conclude that $p_{{\bf S}}\cap R_i\neq0$ and $\gamma(p_{{\bf S}}\cap R_i)$ does not sit inside $p_{{\bf S}}$ and hence $(i,j)$ is an ${{\bf S}}$–effective pair.
Let $U_{i,j}\subset \fA$ be the one-parameter subgroup with the Lie algebra ${{\mathfrak a}}_{i,j}$. The importance of effective pairs is explained by the following lemma:
If a pair $(i,j)$ is not ${{\bf S}}$-effective then $U_{i,j} p_{{{\bf S}}}=p_{{{\bf S}}}$. Otherwise, the map $U_{i,j}\to{\EuScript{F}}^a_{n+1}$, $g\mapsto gp_{{{\bf S}}}$ is injective.
Assume that a pair $R_i,R_j$ is not ${{\bf S}}$-effective and take a non trivial $\gamma\in {\rm Hom}(R_i,R_j)$. By definition, $\gamma p_{{{\bf S}}}\subset p_{{{\bf S}}}$ and hence the exponent of the (scaled) operator $\gamma$ fixes $p_{{{\bf S}}}$. To prove the second claim we note that $$\exp(c\gamma) p_{{{\bf S}}}=({\mathrm Id} + c\gamma) p_{{{\bf S}}}.$$ Hence, if $\gamma p_{{{\bf S}}}$ does not sit inside $p_{{{\bf S}}}$, then all the points $\exp(c\gamma) p_{{{\bf S}}}$, $c\in{{\mathbb C}}$ are different.
For an admissible ${{\bf S}}$ let ${{\mathfrak a}}_{{\bf S}}\subset{{\mathfrak a}}$ be the subspace defined as the direct sum of one-dimensional spaces ${\rm Hom}(R_i,R_j)$ for all ${{\bf S}}$-effective pairs $R_i,R_j$.
The subspace ${{\mathfrak a}}_{{\bf S}}$ is a Lie subalgebra of ${{\mathfrak a}}$.
Let $\gamma_1\in{{\mathfrak a}}_{i,j}$ and $\gamma_2\in{{\mathfrak a}}_{k,l}$, $i>j$, $k>l$ be two elements such that $[\gamma_1,\gamma_2]\ne 0$. Then either $j=k$ or $i=l$. We work out the first case (the second is very similar). We have $[\gamma_1,\gamma_2]=\gamma_1\gamma_2\in{{\mathfrak a}}_{i,l}$. Since $\gamma_2$ is ${{\bf S}}$-effective, we have $$\gamma_2(p_{{{\bf S}}}\cap R_l)\supsetneq p_{{{\bf S}}}\cap R_k.$$ Now, since $$\gamma_1(p_{{{\bf S}}}\cap R_j)\supsetneq p_{{{\bf S}}}\cap R_i$$ and $j=k$, we obtain that $$\gamma_1\gamma_2(p_{{{\bf S}}}\cap R_l)\supsetneq p_{{{\bf S}}}\cap R_i$$ and hence $\gamma_1\gamma_2$ is ${{\bf S}}$-effective.
Let $U_{{\bf S}}$ be the Lie group of ${{\mathfrak a}}_{{\bf S}}$, i.e. $U_{{\bf S}}$ is generated by all $U_{i,j}$ with ${{\bf S}}$-effective $(i,j)$. We note that $U_{{\bf S}}$ is invariant with respect to the torus $T$ action by conjugation.
\[U\] The map $U_{{\bf S}}\to C_{{\bf S}}$, $g\mapsto gp_{{{\bf S}}}$ is bijective and $T$-equivariant.
First, we note that $T$-equivariance follows from $Tp_{{\bf S}}=p_{{\bf S}}$. Now let us prove that the map $U_{{\bf S}}\to C_{{\bf S}}$ is surjective. Let us write an element $\g\in\fA$ as $g=g_{{\bf S}}g_1h$, where $h\in T$, $g_{{\bf S}}\in U_{{\bf S}}$ and $g_1$ belongs to the subgroup of of $\fA$, generated by $U_{i,j}$ with non ${{\bf S}}$-effective $(i,j)$. Then $g p_{{\bf S}}=g_{{\bf S}}p_{{\bf S}}$ and hence we are done. Finally, let us prove the injectivity. Assume that there exists $g\in U_{{\bf S}}$ such that $gp_{{\bf S}}=p_{{\bf S}}$. We identify $g$ with the corresponding lower triangular matrix in $GL_{2n}$ with enries $g_{i,j}$ satisfying $g_{i,i}=1$ and $g_{i,j}=0$ if $i-j>n$. Our goal is to prove that $g_{i,j}=0$ for all $i>j$. Let $p({{\bf S}})=(V_1,\dots,V_n)$ and assume that $g_{i,j}\ne 0$ for $i>j$. Since $g\in U_{{\bf S}}$, the pair $(i,j)$ is ${{\bf S}}$-effective. Consider a non-zero element $\gamma\in {{\mathfrak a}}_{i,j}$ (so $\gamma\in{\mathrm Hom}(R_i,R_j)$). Let $t=1,\dots,n$ be a number such that $V_t\cap R_i\ne 0$ and $\gamma V_t\cap V_t=0$. Choose a non-zero vector $w\in V_t\cap R_i$. Then $gw\notin V_t$ and hence $gp_{{\bf S}}\ne p_{{\bf S}}$.
We note that Theorem \[U\] is analogous to the corresponding theorem for classical flag varieties, see e.g. [@T], Lemma $3.2$.
\[3\] The number of ${{\bf S}}$-effective pairs $(i,j)$ is equal to the sum $N_{PI}({{\bf S}})+N_{PP}({{\bf S}})+N_{II}({{\bf S}})$ of three numbers defined by:
- $N_{PI}({{\bf S}})$ is the number of pairs $1\le k<l\le n+1$ such that there exists $t$ with $k\le t<l$ such that $k\in S_t$, $l\notin S_t$.
- $N_{PP}({{\bf S}})$ is the number of pairs $1\le k<l\le n$ such that there exists $t\ge l$ such that $l\in S_t$, $k\notin S_t$.
- $N_{II}({{\bf S}})$ is the number of pairs $2\le k<l\le n+1$ such that there exists $t<k$ such that $l\in S_t$, $k\notin S_t$.
We divide ${{\bf S}}$-effective pairs into three parts $R_i,R_j\subset P$, $R_i,R_j\subset I$ and $R_i\subset P, R_j\subset I$. We claim that the number of ${{\bf S}}$-effective pairs from the first (second, third) part is equal to $N_{PP}({{\bf S}})$ ($N_{II}({{\bf S}})$, $N_{PI}({{\bf S}})$).
1. The case $R_i\subset P$, $R_j\subset I$. Then $1\leq i\leq n<j\leq 2n$. Since $(i,j)$ is ${{\bf S}}$–effective, there exists an index $t:\, n+1-i\leq t\leq 2n+1-j$ such that $n+1-i\in S_t$ and $2(n+1)-j\notin S_t$. Put $k=n+1-i$ and $l=2(n+1)-j$.
2. The case $R_i,R_j\subset P$. Since $(i,j)$ is ${{\bf S}}$–effective then $1\leq i< j\leq n$ and there is an index $t:\, t\geq n+1-i> n+1-j$ such that $n+1-i\in S_t$ and $n+1-j\notin S_t$. Put $l=n+1-i$ and $k=n+1-j$.
3. The case $R_i,R_j\subset I$. Since $(i,j)$ is ${{\bf S}}$–effective then $n+1\leq i< j\leq 2n$ and there is an index $t:\, t\leq 2n+1-j< 2n+1-i$ such that $2(n+1)-i\in S_t$ and $2(n+1)-j\notin S_t$. Put $l=2(n+1)-i$ and $k=2(n+1)-j$.
\[Cor:CellDim\] The dimension of $C_{{{\bf S}}}$ is equal to the sum $N_{PI}({{\bf S}})+N_{PP}({{\bf S}})+N_{II}({{\bf S}})$.
Thanks to Theorem \[U\] the dimension of the cell $C_{{{\bf S}}}$ is equal to the number of ${{\bf S}}$-effective pairs $R_i$, $R_j$. Now Proposition \[3\] implies the corollary.
\[Cor:PoincPoly\] The Poincaré polynomial of ${\EuScript{F}}^a_{n+1}$ is equal to the sum of the terms $q^{N_{PI}({{\bf S}})+N_{PP}({{\bf S}})+N_{II}({{\bf S}})}$, where the sum runs over the set of admissible collections.
In [@CFR Theorem 5.1] it is shown that although ${\EuScript{F}}_{n+1}^a$ is not smooth, the one–dimensional sub torus $T_0$ of $T$ still produces a Białynicki–Birula type cell decomposition ([@BB], [@CG Theorem 2.4.3]). In other words, the attracting set of a $T_0$–fixed point $p_{{\bf S}}$ is a cell and it has dimension equal to the dimension of the positive part of the tangent space at $p_{{\bf S}}$ (the positive part is the vector subspace generated by vectors on which $T_0$ acts with positive weight). In view of Remark \[Remark:S-effectiveTorus\], this dimension is precisely the number of $\mathbf{S}$–effective pairs. Theorem \[U\] provides another and more explicit proof of this fact.
\[Rem:CellDimDiagram\] From the discussion above (see Corollary \[Cor:CellDim\] and Remark \[Remark:S-effectiveTorus\]), the dimension of the cell with center $p_{{\bf S}}$ can be easily read off from ${{\bf S}}$, viewed inside the coefficient quiver of $P\oplus I$ written in the form . Indeed in this diagram let us color a vertex black if it belongs to ${{\bf S}}$ and white otherwise. In the $i$–th columns (counting from left to right) there are precisely $i$ black vertices. Some of them are sources of ${{\bf S}}$. For every such source $t\in S_i$ let us count the number $w_t$ of white vertices below it. Let $c_i$ be the sum of the $w_t$’s. Then the dimension of the cell with center $p_{{\bf S}}$ equals the sum $c_1+c_2+\cdots+c_n$. For example let us consider the following $T$–fixed point of ${\EuScript{F}}_5^a$: $$\xymatrix@R=6pt@C=8pt
{
&&&&\bullet&\\
&&&\circ\ar[r]&\bullet&\\
&&\circ\ar[r]&\bullet\ar[r]&\bullet&\\
&\circ\ar[r]&\circ\ar[r]&\bullet\ar[r]&\bullet&\\
&\circ\ar[r]&\circ\ar[r]&\circ\ar[r]&\circ&\\
&\bullet\ar[r]&\bullet\ar[r]&\bullet&&\\
&\circ\ar[r]&\bullet&&&\\
&\circ&&&&\\
}$$ then $c_1=2$, $c_2=0$, $c_3=2$ and $c_4=2$. The cell has hence dimension $6$.
Moment graph.
-------------
We briefly recall the definition of a moment graph (see [@BM],[@GKM]). Let $X$ be a projective algebraic variety acted upon by a torus $T=({{\mathbb C}}^*)^d$ with a fixed one-dimensional subtorus $\imath:{{\mathbb C}}^*\subset T$. Assume that the $T$ action on $X$ has finitely many fixed points and one-dimensional orbits and any ${{\mathbb C}}^*$ fixed point is $T$-fixed ($X^T=X^{{{\mathbb C}}^*}$). Assume further that $X$ has a decomposition as a disjoint union of $T$-invariant affine cells in such a way that each cell $C$ contains exactly one ${{\mathbb C}}^*$-fixed point $p$ and $C=\{x\in X:\ \lim_{\lambda\to 0} \imath(\la)x=p\}$ (i.e. the cell consists of all points attracted by $p$, see [@BB]). We denote this cell by $C_p$. The moment graph $\Gamma$ has its set of vertices labeled by the $T$-fixed points. Two points $p_1$ and $p_2$ are connected by an edge in $\Gamma$ if there exists a one-dimensional $T$-orbit $L$ such that $\bar L=L\sqcup p_1\sqcup p_2$ (i.e. $p_1$ and $p_2$ are the $T$-fixed points in the closure of $L$). Thus the edges of $\Gamma$ are labeled by the one-dimensional $T$-orbits. We orient $\Gamma$ by the following rule: for two vertices $p_1$ and $p_2$ we say $p_1\ge p_2$ if $C_{p_2}\subset \bar C_{p_1}$. If there is an edge connecting $p_1$ and $p_2$ in $\Gamma$ then we put an arrow $p_1\to p_2$. Finally, one defines a labeling $\al_L$ of the edges $L$ of $\Gamma$ by the elements $\al_L\in {{\mathfrak t}}^*$, where ${{\mathfrak t}}$ is the Lie algebra of the torus $T$. Namely, for an edge $L$ let $T_x\subset T$ be the stabilizer of a point $x\in L$ (obviously, $T_x$ is independent of $x\in L$). Then the Lie algebra ${{\mathfrak t}}_x\subset {{\mathfrak t}}$ is a hyperplane. We define $\al_L$ as a non-zero element in the annihilator of ${{\mathfrak t}}_x$.
Here we give an example of the moment graph for the classical flag variety ${\EuScript{F}}_3=SL_3/B$. The torus $T$ has $6$ fixed points labeled by pairs $(S_1,S_2)$ of subsets of $\{1,2,3\}$ such that $\#S_1=1$, $\#S_2=2$ and $S_1\subset S_2$. The moment graph of ${\EuScript{F}}_3$ looks as follows: $$\xymatrix{
& & (1,12)\ar[dll]\ar[dr]\ar[ddd] &\\
(2,12) \ar[d] \ar[drrr]& & & (1,13)\ar[d] \ar[dlll]\\
(2,23) \ar[drr] & & & (3,13)\ar[dl] \\
& & (3,23) & \\
}$$ We note that usually the arrows in the moment graph direct from bottom to top. However for our purposes it is more convenient to draw the vertices from top to bottom, since in the degenerate situation the dense cell corresponds to the point $(1,12)$, see Example \[A2\]. This is not important in the classical situation due to the Chevalley involution, but crucial in the degenerate case.
Our goal is to describe the moment graph of the degenerate flag varieties.
We note that the moment graphs turn out to be a powerful tool for computing various cohomology groups of algebraic varieties (see [@BM], [@GKM], [@T], [@Fi], [@FW]). A crucial role is played by the notion of sheaves on moment graphs. In this paper we do not discuss $\Gamma$-sheaves, but only describe the combinatorial structure of the graphs. Computation of the (equivariant) cohomology as well as the (equivariant) intersection cohomology of the degenerate flag varieties is an interesting open problem.
\[A2\] Here we give a picture of the moment graph for the degenerate flag variety ${\EuScript{F}}^a_3$. Recall that the $T$-fixed points are labeled by pairs $(S_1,S_2)$ of subsets of the set $\{1,2,3\}$ such that $\#S_1=1$, $\#S_2=2$ and $S_1\subset S_2\cup\{2\}$. $$\xymatrix{
& (1,12)\ar[dl]\ar[dr]\ar[dd] &\\
(2,12)\ar[dd]\ar[dddr] & & (1,13)\ar[dd]\ar[dddl]\\
& (3,23)\ar[dr]\ar[dl] &\\
(2,23)\ar[dr] & & (3,13)\ar[dl]\\
& (2,13)&
}$$
The moment graph for the degenerate flag variety ${\EuScript{F}}^a_4$ is computed in Appendix C.
We now give an explicit combinatorial description of the moment graph. We identify the Lie algebra ${{\mathfrak t}}$ of $T$ with the diagonal traceless $2n\times 2n$ matrices. For a pair of indices $i,j$, $1\le i<j\le 2n$, we denote by $\al_{i,j}\in{{\mathfrak t}}^*$ the element $\al_{i,j}({\mathrm diag}(x_1,\dots,x_{2n}))=x_i-x_j$.
\[Thm:OneDimOrbits\] The number of one-dimensional $T$-orbits in ${\EuScript{F}}^a_{n+1}$ is finite. The orbits are of the form $U_{i,j} p_{{{\bf S}}}\setminus p_{{{\bf S}}}$, where ${{\bf S}}$ is admissible and $(i,j)$ is ${{\bf S}}$-effective. The edge in the moment graph, which corresponds to $U_{i,j} p_{{{\bf S}}}\setminus p_{{{\bf S}}}$ is labeled by $\al_{i,j}$.
Thanks to Theorem \[U\], we only need to describe the one-dimensional $T$-orbits in $U_{{\bf S}}$. It is easy to see that these are non-identity elements in $U_{i,j}$.
Theorem \[Thm:OneDimOrbits\] also follows from Corollary \[Cor:T-actionTangent\] and Remark \[Remark:S-effectiveTorus\]. Indeed in view of Corollary \[Cor:T-actionTangent\], the directions around $p_{{\bf S}}$ of the one–dimensional $T$–orbits containing $p_{{\bf S}}$ are precisely the standard basis vectors of the tangent space $T_{p_{{\bf S}}}({\EuScript{F}}_{n+1}^a)$ at $p_{{\bf S}}$. In particular the number of such $T$–orbits is bigger or equal than ${\rm dim} {\EuScript{F}}_{n+1}^a$ and it is equal if and only if $p_\mathbf{S}$ is smooth. Any such curve $\ell$ consists of three T–orbits $\ell=\{p_\mathbf{S}\}\cup\{\ell'\}\cup\{p_\mathbf{R}\}$. The direction of $\ell$ is fixed also by the one–dimensional torus $T_0$. In particular this standard basis vector of $T_{p_\mathbf{S}}({\EuScript{F}}_{n+1}^a)$ has either positive or negative $T_0$ weight. If the weight is positive then $\{p_\mathbf{S}\}\cup\{\ell'\}$ sits inside the attracting set of $p_\mathbf{S}$ which is the cell ${\mathfrak A}p_\mathbf{S}$ and hence $p_\mathbf{R}$ (and its attracting cell) is in the closure of this cell. It follows that in the moment graph there is an arrow $p_\mathbf{S}\rightarrow p_\mathbf{R}$. In particular the number of arrows starting from $p_{{\bf S}}$ in the moment graph, equals the number of standard basis vector of $T_{p_{{\bf S}}}({\EuScript{F}}_{n+1}^a)$ on which $T_0$ acts with positive weight. In view of Remark \[Remark:S-effectiveTorus\] this number equals the number of ${{\bf S}}$–effective pairs.
The dimension of a cell $C_{{{\bf S}}}$ is equal to the number of edges in the moment graph which are directed outwards the vertex $p_{{{\bf S}}}$.
The following theorem generalizes the results as above to the case of the degenerate partial flag varieties.
The number of one-dimensional $T$ orbits on ${\EuScript{F}}^a_{{\bf d}}$ is finite. Each of these orbits is covered by a one-dimensional $T$-orbit in ${\EuScript{F}}^a_{n+1}$ via the surjection ${\EuScript{F}}^a_{n+1}\to{\EuScript{F}}^a_{{\bf d}}$. All the orbits are of the form $U_{i,j}p\setminus p$ for some $i,j$ and a $T$-fixed $p\in {\EuScript{F}}^a_{{\bf d}}$.
Smooth locus and the Schröder numbers
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In this section we describe the smooth locus of the degenerate flag varieties ${\EuScript{F}}^a_{n+1}$ and compute Euler characteristics and Poincaré polynomials.
Smooth cells.
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Take a point $N\in {\rm Gr}_{{{\bf dim}}P}(P\oplus I)$. Then $N$ can be split as $N=N_P\oplus N_I$, where $N_P\subset P$ and $N_I\subset I$, such that $N_I$ and $P/N_P$ are of the same dimension vector (see [@CFR Theorem 1.3]).
\[T\] A point $N$ in a quiver Grassmannian ${\rm Gr}_{{{\bf dim}}P}(P\oplus I)$ is smooth if and only if ${\rm Ext}^1(N_I,P/N_P)=0$.
Let $\langle\cdot,\cdot\rangle$ be the Euler form of the quiver $Q$, given on a pair of dimension vectors ${\bf d}$, ${\bf e}$ by $\langle {\bf d},{\bf e}\rangle=\sum_{i=1}^n d_ie_i-\sum_{i=1}^{n-1} d_ie_{i+1}$. Then $\langle{{\bf dim}}X,{{\bf dim}}Y\rangle=\dim {\rm Hom}(X,Y)-\dim{\rm Ext}^1(X,Y)$ for arbitrary representations $X$ and $Y$ of $Q$. By [@CFR Theorem 1.1], we have $$\langle{{\bf dim}}P,{{\bf dim}}I\rangle=\dim {\rm Gr}_{{{\bf dim}}P}(P\oplus I).$$ By the formula [@CFR Lemma 2.3] for the dimension of the tangent space $T_N$ to the point $N\in{\rm Gr}_{{{\bf dim}}P}(P\oplus I)$, we then have $$\begin{gathered}
\dim T_N=\dim{\rm Hom} (N_I\oplus N_P, P/N_P\oplus I/N_I)=\\
\langle{{\bf dim}}P,{{\bf dim}}I\rangle - \dim {\rm Ext}^1 (N_I\oplus N_P, P/N_P\oplus I/N_I).\end{gathered}$$ Since $N_P$ is projective and $N/N_I$ is injective, we obtain $$\dim {\rm Ext}^1 (N_I\oplus N_P, P/N_P\oplus I/N_I)=\dim {\rm Ext}^1 (N_I, P/N_P).$$ Hence, the dimension of the tangent space at a point $N$ is equal to the dimension of the Grassmannian if and only if ${\rm Ext}^1 (N_I, P/N_P)$ vanishes.
Recall that the quiver Grassmannian ${\rm Gr}_{{{\bf dim}}P}(P\oplus I)$ can be decomposed into the disjoint union of ${\mathfrak A}$-orbits of the form ${\mathfrak A} p_{{{\bf S}}}$. Hence all the points of the orbit are smooth or singular together with $p_{{{\bf S}}}$. So it suffices to understand what are the conditions for an admissible collection ${{\bf S}}$ that guarantee the smoothness of $p_{{{\bf S}}}$. We use Lemma \[T\] above.
\[main\] A point $p_{{{\bf S}}}$ is smooth if and only if for all $1\leq j<i\leq n$, the condition $i\in S_{j}$ implies $j+1\in S_i$.
Given an admissible collection ${{\bf S}}=(S_i)_{i=1}^n$, we introduce the following numbers for all $i=1,\ldots,n+1$: $$k_i=\min\{1\leq k<i\, :\, i\in S_k\},\;\;\; l_j=\min\{j\leq l\leq n\, :\, j\in S_l\}.$$ Recall the indecomposable representations $R_{k,l}$ with the support on the interval $[k,l]$. A representation $p_{{{\bf S}}}$ is isomorphic to the direct sum $N_I\oplus N_P$, where $N_I\subset I$ and $N_P\subset P$. It is easy to see that $$N_I=\bigoplus_i R_{k_i,i-1},\;\;\; P/N_P=\bigoplus_j R_{j,l_j-1}.$$ The extension groups between the indecomposables are given by Lemma \[HE\]. Thus we obtain that $0\not={\rm Ext}^1(N_I,P/N_P)$ if and only if there exist indices $i$ and $j$ such that $k_i+1\leq j\leq i\leq l_j-1$. This holds (writing out the three inequalities) if and only if there exist indices $j\leq i$ such that $$\min\{1\leq k<i\, :\, i\in S_k\}<j,\;\;\; \min\{j\leq l\leq n\, :\, j\in S_l\}>i.$$ This translates into the condition that there exist $j\leq i$ such that $i\in S_{j-1}$, but $j\not\in S_i$. Conversely, this means that the orbit is smooth if and only if for all $1\leq j\leq i\leq n+1$, if $i\in S_{j-1}$, then $j\in S_i$. Note that this condition is void in case $j=1$ or $i=n+1$, so that we can replace $j$ by $j-1$, and obtain the assertion of theorem.
In what follows we call an admissible collection ${{\bf S}}$ smooth iff $p_{{{\bf S}}}$ is a smooth point.
The large Schröder numbers.
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Let $r_n$ be the $n$-th large Schröder number, defined as the number of Schröder paths, i.e. subdiagonal lattice paths from $(0,0)$ to $(n,n)$ consisting of the steps $(0,1)$, $(1,0)$ or $(1,1)$. The sequence $r_0,r_1,r_2,...$ starts with $1,2,6,22,90,394$. Here are the six Schröder paths for $n=2$: $$\begin{picture}(40,30)
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We note that (see e.g. [@BEK],[@BSS],[@St]) $$r_n=r_{n-1}+\sum_{k=0}^{n-1} r_kr_{n-1-k}.$$ The small Schröder numbers $s_n$ are defined as halves of the large ones.
Recall that a collection ${{\bf S}}=(S_a)_{a=1}^n$ of subsets of the set $ \{1,\dots,n+1\}$ is smooth if $\#S_a=a$, $S_a\subset S_{a+1}$ and for all $1\le a< b\le n$ the following condition holds (see Theorem \[main\]): $$\text{if } b\in S_a, \text{ then } a+1\in S_b.$$ Let $LS_n$ be the set of length $n$ smooth collections and $\bar r_n$ be the cardinality of $LS_n$.
The numbers $\bar r_n$ satisfy the recursion $$\bar r_n=\bar r_{n-1}+\sum_{k=0}^{n-1}\bar r_k\bar r_{n-1-k}.$$
We divide all smooth collections according to the values of $S_1$. So first, let $S_1=\{1\}$. Let us show that the number of such smooth collections is equal to $\bar r_{n-1}$. Note that all $S_a$ contain $1$. For $a=1,\dots,n-1$ we set $$S'_a=\{i:\ i+1\in S_{a+1}\}\subset \{1,\dots,n\}.$$ We claim that the collection $(S'_a)_{a=1}^{n-1}$ is smooth and all (length $n-1$) smooth collections arise in this way. First, obviously, $\#S'_a=a$ and $S'_a\subset S'_{a+1}$. Now the conditions ($b\in S_a$ implies $a+1\in S_b$), $2\le a<b\le n$ are equivalent to the conditions ($b\in S'_a$ implies $a+1\in S'_b$), $1\le a<b\le n-1$.
Let $LS_n^k\subset LS_n$ be the set of smooth collections satisfying $S_1=\{k\}$, $2\le k\le n+1$. We want to show that the cardinality of $LS^k_n$ is equal to $\bar r_{k-2}\bar r_{n+1-k}$. To this end we construct a bijection $F:LS_n^k\to LS_{k-2}\times LS_{n+1-k}$. For convenience, we write $F=(f,g)$, where $$f:LS_n^k\to LS_{k-2},\ g: LS_n^k\to LS_{n+1-k}.$$
First, since $S_1\subset S_a$ for any $a$, we have $k\in S_a$, $2\le a\le n$. Now the conditions $k\in S_a$ for $a=1,\dots,k-1$ imply $$\{2,\dots,k\}\subset S_a \text{ for all } a\ge k.$$ Given a collection ${{\bf S}}\in LS_n^k$, we define $$g({{\bf S}})=(g({{\bf S}})_1,\dots,g({{\bf S}})_{n+1-k})$$ as follows: $$\label{g}
g({{\bf S}})_a=
\begin{cases}
\{i:\ 2\le i\le n+1-k,\ i+k-1\in S_{a+k-1}\},\text{ if } 1\notin S_{a+k-1},\\
\{1\}\cup \{i:\ 2\le i\le n+1-k,\ i+k-1\in S_{a+k-1}\}\text{ otherwise}.
\end{cases}$$ We note that the image depends only on the sets $S_a$ with $a\ge k$.
Now, given a collection ${{\bf S}}\in LS_n^k$, we need to define $$f({{\bf S}})=(f({{\bf S}})_1,\dots,f({{\bf S}})_{k-2}).$$ Let $S_k=\{2,\dots,k\}\cup \{i\}$ for some $i=1,k+1,\dots,n+1$. We note that since $k\in S_a\subset S_k$ for $a<k$, each $S_a\setminus k$ for $2\le a\le k-1$ is an $(a-1)$-element subset of the fixed set of cardinality $k-1$ (this set is $\{2,\dots,k-1\}\cup\{i\}$). We now define the map $f$ as follows: $$\label{f}
f({{\bf S}})_a=
\begin{cases}
\{i:\ 1\le i\le k-2,\ i+1\in S_{a+1}\},\text{ if } S_{a+1}\subset \{2,\dots,k\},\\
\{i:\ 1\le i\le k-2,\ i+1\in S_{a+1}\}\cup \{k-1\}, \text{ otherwise}.
\end{cases}$$ We note (this is important) that $f_1({{\bf S}})$ depends only on $S_2,\dots,S_{k-1}$.
Our first goal is to show that $f({{\bf S}})\in LS_{k-2}$ and $g({{\bf S}})\in LS_{n-k+1}$ for any ${{\bf S}}\in LS_n^k$. By definition, $g({{\bf S}})_a\subset g({{\bf S}})_{a+1}$ for $1\le a\le n-k$ and $$g({{\bf S}})_a\in \{1,\dots,n-k+2\}, \ \#g({{\bf S}})_a=a \text{ for } 1\le a\le n-k+1.$$ Let us show that for $1\le a<b\le n-k+1$ the inclusion $b\in g({{\bf S}})_a$ implies $a+1\in g({{\bf S}})_b$. Since $b>1$, $b\in g({{\bf S}})_a$ implies $b+k-1\in S_{a+k-1}$. Since ${{\bf S}}$ is smooth, we obtain $a+k\in S_{b+k-1}$, which gives $a+1\in g({{\bf S}})_b$ and we are done. Similarly, one proves that $f({{\bf S}})\in LS_{k-2}$.
Finally, we have to prove that the map $F=(f,g):LS_n^k\to LS_{k-2}\times LS_{n-k+1}$ is one-to-one. Given an element $({{\bf S}}',{{\bf S}}'')\in LS_{k-2}\times LS_{n-k+1}$, we use formulas and to reconstruct ${{\bf S}}$ such that $F({{\bf S}})=({{\bf S}}',{{\bf S}}'')$.
The Euler characteristic of the smooth locus of ${\EuScript{F}}^a_{n+1}$ is equal to the $n$-th Schröder number $r_n$.
Finally, let us formulate the analogue of Theorem \[main\] for the degenerate partial flag varieties. We omit the proof since it is very close to the proof of Theorem \[main\]. Recall that the $T$-fixed points in ${\EuScript{F}}^a_{{\bf d}}$ are labeled by ${{\bf d}}$-admissible collections ${{\bf S}}=(S_{d_1},\dots,S_{d_k})$ (see ).
A $T$-fixed point $p_{{\bf S}}\in {\EuScript{F}}^a_{{\bf d}}$ is smooth if and only if the following conditions hold: if $b\in S_{d_i}$ and $d_{j+1}\ge b>d_j$ for some $j\ge i$, then $\{d_i+1,\dots,d_{i+1}\}\subset S_{d_{j+1}}$.
Poincaré polynomials
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There are several ways to define $q$-analogues of the Schröder numbers (see [@BDPP], [@BSS],[@BEK]). We will need the simplest one (see [@BSS], page $37$, polynomials $d_n(q)$). They are called Narayana polynomials there, but in other papers the same polynomials are also referred to as the Schröder polynomials, see e.g. [@G]). For a Schröder path $P$ let $diag(P)$ be the number of the diagonal steps in $P$. Define $r_n(q)$ as the sum of the terms $q^{diag(P)}$ over the set of Schröder paths $P$. Here are the first several polynomials $$\begin{gathered}
r_0(q)=1,\qquad r_1(q)=1+q,\qquad r_2(q)=2+3q+q^2,\\
r_3(q)=5+10q+6q^2+q^3,\qquad r_4(q)=14+35q+30q^2+10q^3+q^4.\end{gathered}$$ Clearly, $r_n(0)$ is the $n$-th Catalan number. Let $P_n^{sm}(q)$ be the Poincaré polynomial of the smooth locus of ${\EuScript{F}}^a_{n+1}$. Our goal here is to prove the following theorem:
\[Pnsm\] $P_n^{sm}(q)=q^{n(n-1)/2}r_n(q).$
Recall (see [@BEK], [@BSS]) that $$\label{r(q)}
r_n(q)=qr_{n-1}(q)+\sum_{k=0}^{n-1} r_k(q)r_{n-1-k}(q).$$
\[PP\] The Poincaré polynomials of the smooth locus satisfy the following recursion: $$\label{qS}
P_n^{sm}(q)=q^nP_{n-1}^{sm}(q) + \sum_{l=0}^{n-1} q^{(l+1)(n-l)-1} P_l^{sm}(q)P_{n-1-l}^{sm}(q).$$
First, let us consider a smooth collections $(S_1,\dots,S_n)$ with $S_1=1$. Then the cells labeling such collections are in one-to-one correspondence with smooth collections ${{\bf S}}'$ of length $n-1$: $S_i'=S_{i+1}\setminus \{1\}$. We claim that $$\label{k=1}
\dim C_{{{\bf S}}}= \dim C_{{{\bf S}}'} + n.$$ We use Proposition \[3\]. Clearly, the terms $N_{PP}$ and $N_{II}$ for $p_{{{\bf S}}}$ and $p({{\bf S}}')$ do coincide and the difference of the terms $N_{PI}$ is equal to $n$ (since $S_1=\{1\}$, in the definition of $N_{PI}$ we can take $t=1$, $i=1$, $j=2,\dots,n+1$). Now produces the first term of the right hand side of .
Recall the bijection $F=(f,g):LS_n^k\to LS_{k-2}\times LS_{n-k+1}$, $k\ge 2$, from the set of smooth collections with $S_1=\{k\}$ to the product $LS_{k-2}\times LS_{n-k+1}$. Our goal is to prove that $$\label{k}
\dim C_{{{\bf S}}}= \dim C_{f({{\bf S}})} + \dim C_{g({{\bf S}})} + (k-1)(n+2-k)-1$$ (after the shift $l=k-2$ one gets the corresponding term in ). Recall that since $k\in S_1$ we have $$\{2,\dots,k\}\subset S_m \text{ for all } m\ge k.$$ In particular, $S_k=\{2,\dots,k\}\cup \{r\}$ for some number $r=1,k+1,\dots,n+1$. We claim that $$\begin{gathered}
N_{PI}({{\bf S}})+ N_{PP}({{\bf S}})=\\
= N_{PI}(f({{\bf S}})) + N_{PI}(g({{\bf S}})) + N_{PP}(f({{\bf S}}))+ N_{PP}(g({{\bf S}})) + (k-1)(n+1-k),\end{gathered}$$ and $$N_{II}({{\bf S}})=N_{II}(f({{\bf S}})) + N_{II}(g({{\bf S}})) + k-2.$$ First, let us prove the first formula. Assume that $1=r=S_k\setminus\{2,\dots,k\}$. Then $$\begin{gathered}
N_{PI}({{\bf S}})= N_{PI}(f({{\bf S}})) + N_{PI}(g({{\bf S}}))+ (k-1)(n+1-k),\\
N_{PP}({{\bf S}})= N_{PP}(f({{\bf S}})) + N_{PP}(g({{\bf S}})).\end{gathered}$$ Here the term $(k-1)(n+1-k)$ comes from the fact that in the definition of $N_{PI}({{\bf S}})$ one can take $i=2,\dots,k$, $j=k+1,\dots,n+1$ and $t=k$. These possibilities are not counted in $N_{PI}(f({{\bf S}})) + N_{PI}(g({{\bf S}}))$. Now assume that $r>k$. Then one has $$\begin{gathered}
N_{PI}({{\bf S}})= N_{PI}(f({{\bf S}})) + N_{PI}(g({{\bf S}}))+ (k-1)(n-k),\\
N_{PP}({{\bf S}})= N_{PP}(f({{\bf S}})) + N_{PP}(g({{\bf S}})) + k-1.\end{gathered}$$ Here the term $(k-1)(n-k)$ comes from the fact that in the definition of $N_{PI}({{\bf S}})$ one can take $i=2,\dots,k$, $j\in\{k+1,\dots,n+1\}\setminus r$ and $t=k$. The term $k-1$ in the right hand side of the second equality comes from the fact that in the definition of $N_{PP}({{\bf S}})$ one can take $i=1$, $j=2,\dots,k$ and $t=k$. All these possibilities are lost when computing $N_{PI}(f({{\bf S}}))$, $N_{PI}(g({{\bf S}}))$, $N_{PP}(f({{\bf S}}))$ and $N_{PP}(g({{\bf S}}))$.
Now let us prove that $$N_{II}({{\bf S}})=N_{II}(f({{\bf S}})) + N_{II}(g({{\bf S}})) + k-2.$$ Here the argument is even simpler: the missing $k-2$ comes from the following possibilities for $N_{II}({{\bf S}})$ missing in $N_{II}(f({{\bf S}})) + N_{II}(g({{\bf S}}))$: $i=2,\dots,k-1$, $j=k$, $t=1$.
We thus obtain $$\dim C_{{{\bf S}}}= \dim C_{f({{\bf S}})} + \dim C_{g({{\bf S}})} + (k-1)(n+1-k) + (k-2),$$ which implies and as well as the proposition.
Theorem \[Pnsm\] holds.
We note that $P_1^{sm}(q)=1+q=r_1(q)$. Now the induction procedure combined with and Proposition \[PP\] gives the desired result.
It is natural to define a $q,t$-version $h_{n+1}(q,t)$ of the normalized median Genocchi numbers as the sum over admissible collections ${{\bf S}}$ of the terms $$q^{\dim C_{{{\bf S}}}}t^{\dim T_{p_{{{\bf S}}}} {\EuScript{F}}^a_{n+1}}t^{-n(n+1)/2}.$$ Then the value $h_n(1,1)$ is exactly the normalized median Genocchi number and $h_{n+1}(q,0)=q^{n(n-1)/2}r_n(q)$ is the (scaled) $n$-th Schröder polynomial. Here are first few $q,t$-Genocchi polynomials: $$\begin{gathered}
h_2(q,t)=1+q,\quad h_3(q,t)=2q+3q^2+q^3 + t,\\
h_4(q,t)=q^3(5+10q+6q^2+q^3)+tq(2q+7q^2+5q^3)+t^2(1+q).\end{gathered}$$
Schröder numbers: from large to small
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Recall the polynomials $P_n^{sm}(q)$, which are equal to $q^{n(n-1)/2}r_n(q)$, $r_n(q)$ being the $q$-Schröder polynomials. Recall (see [@G]) that the polynomials $r_n(q)$ are divisible by $1+q$. The ratios are denoted by $s_n(q)$ (thus $r_n(q)=s_n(q)(1+q)$). These are the small $q$-Schröder polynomials. (In particular, $s_n(1)$ are the small Schröder numbers). Our goal here is to show that the divisibility of $r_n(q)$ by $1+q$ has a very simple and concrete explanation within our approach.
There exists a fixed-point free involution $\sigma$ on the set of smooth collections. For any smooth collection ${{\bf S}}$ and the corresponding cell $C_{{{\bf S}}}$ one has $$\dim C_{{{\bf S}}} = \dim C_{\sigma{{\bf S}}} \pm 1.$$
Consider the map $w:\{1,\dots,n+1\}\to \{1,\dots,n+1\}$, which interchanges $1$ and $n+1$ and stabilizes all other elements. Define a map $\sigma$ by the formula $$\sigma(S_1,\dots,S_n)=(wS_1,\dots,w S_n).$$ First, we note that $\sigma$ maps each smooth ${{\bf S}}$ to a smooth collection. Second, since $w^2$ is the identity map, $\sigma^2=\mathrm{Id}$. Third, let us show that $\sigma$ is fixed-point free. In fact, a smooth ${{\bf S}}$ is fixed by $\sigma$ if and only if for all $k=1,\dots,n$ the set $S_k$ either contains both $1$ and $n+1$ or does not contain any of these elements. We note that $\# S_n=n$ and hence $S_n$ contains at least one of the elements $1$, $n+1$. If $\sigma {{\bf S}}={{\bf S}}$, then $S_n\supset \{1,n+1\}$. Now let $1\le k<n$ be a number such that $\{1,n+1\}$ is contained in $S_{k+1}$ but not in $S_{k+1}$ (since $\# S_1=1$ such $k$ does exist). If $\sigma{{\bf S}}={{\bf S}}$, then we have $1,n+1\notin S_k$. Since ${{\bf S}}$ is smooth, $S_k\subset S_{k+1}$ and therefore $S_{k+1}$ contains two non-intersecting sets $S_k$ and $\{1,n+1\}$. This contradicts with $\#S_{k+1}=k+1$.
Now let ${{\bf S}}$ be a smooth collection. Let $k$ be a number such that $1\in S_k\setminus S_{k-1}$ and, similarly, let $l$ be a number such that $n+1\in S_l\setminus S_{l-1}$. As we proved above, $k\ne l$. Assume that $k<l$. We claim that $$\dim C_{{{\bf S}}} = \dim C_{\sigma{{\bf S}}} + 1.$$ Recall that $\dim C_{{{\bf S}}}$ is the sum of three numbers $N_{PI}({{\bf S}})+N_{PP}({{\bf S}})+N_{II}({{\bf S}})$ (see Proposition \[3\]). First, we note that a pair $i=1,j=n+1$ adds one to $N_{PI}({{\bf S}})$, but not to $N_{PI}(\sigma{{\bf S}})$. Second, each pair $i,j$ with $1<i,j<n+1$, either shows up for both ${{\bf S}}$ and $\sigma{{\bf S}}$ in the dimension counting as in Proposition \[3\] or does not show up for both cells. Now let us look at other pairs and compute the difference between the dimensions of $C_{{{\bf S}}}$ and that of $C_{\sigma{{\bf S}}}$.
Take $m$ satisfying $k\le m<l$ and consider $j$ such that $j>m$, $j\notin S_m$. Then a pair $i=1,j$ adds one to $N_{PI}({{\bf S}})$, but not to $N_{PI}(\sigma{{\bf S}})$ (since $1\in S_m$, but $1\notin (\sigma S)_m$). However, let us look at a pair $i=m$, $j=n+1$. Since $n+1\in (\sigma S)_m\setminus S_m$, the pair $(m,n+1)$ adds one to $N_{II}(\sigma{{\bf S}})$, but not to $N_{II}({{\bf S}})$.
Now take $m$ satisfying $k\le m<l$ and consider $i$ such that $i\le m$, $i\in S_m$. Then a pair $i,j=n+1$ adds one to $N_{PI}({{\bf S}})$, but not to $N_{PI}(\sigma{{\bf S}})$ (since $n+1\notin S_m$, but $n+1\in (\sigma S)_m$). However, let us look at a pair $i=1$, $j=m$. Since $1\in S_m\setminus (\sigma S)_m$, the pair $(1,m)$ adds one to $N_{PP}(\sigma{{\bf S}})$, but not to $N_{PP}({{\bf S}})$.
Summarizing, the difference $$N_{PI}({{\bf S}})+N_{PP}({{\bf S}})+N_{II}({{\bf S}}) - N_{PI}(\sigma{{\bf S}}) - N_{PP}(\sigma{{\bf S}}) - N_{II}(\sigma{{\bf S}})$$ is equal to one (coming from the pair $i=1$, $j=n+1$). This implies the second statement of the theorem.
The polynomials $P_n^{sm}(q)$ and $r_n(q)$ are divisible by $1+q$. The ratio $P_n^{sm}(q)/(1+q)$ is equal to the sum of the terms $q^{\dim C_{{{\bf S}}}}$ taken over smooth ${{\bf S}}$ satisfying the following conditions for all $m=1,\dots,n$: if $1\in S_m$ then $n+1\in S_m$.
The Theorem above states that $P_n^{sm}(q)$ is equal to the sum over the orbits of the involution $\sigma$ of the terms $q^d(1+q)$, where $d$ is the minimum of the dimensions of the cells corresponding to the collections in the orbit. But we know that $\dim C_{{{\bf S}}}= \dim C_{\sigma{{\bf S}}} - 1$ if there exists $m$ such that $n+1\in S_m$, but $1\notin S_m$. This implies the corollary.
Let us relabel the smooth collections as follows. To a smooth collection ${{\bf S}}$ we attach a permutation $\pi\in S_{n+1}$ by the formula $\pi(m)=S_m\setminus S_{m-1}$. Then ${{\bf S}}$ is smooth if and only if the corresponding permutation satisfies the following conditions for all $1\le a<b\le n$: $$\text{if } \pi^{-1}(b)\le a\ \text{ then }\ \pi^{-1} (a+1)\le b.$$
The number of permutations, corresponding to smooth collections, is equal to the large Schröder number. The number of such permutations satisfying $\pi^{-1}(n+1)<\pi^{-1}(1)$ is equal to the small Schröder number.
Appendix A: Regularity in codimension $2$.
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We consider the Grassmannian ${\rm Gr}_{{{\bf dim}}P}(P\oplus I)$ for $P$ a projective and $I$ an injective representation over a Dynkin quiver $Q$. Recall that a variety $X$ is said to be regular in codimension $d$ if there exists a codimension $d+1$ subvariety $Y\subset X$ such that all points of $X\setminus Y$ are smooth. For example, normal varieties are regular in codimension one. In [@CFR] it is proved that all ${\rm Gr}_{{{\bf dim}}P}(P\oplus I)$ are normal. We now prove a stronger statement.
${\rm Gr}_{{{\bf dim}}P}(P\oplus I)$ is regular in codimension $2$.
Recall that the group $\fA\subset {\rm Aut} (P\oplus I)$ acts on ${\rm Gr}_{{{\bf dim}}P}(P\oplus I)$ with orbits parametrized by pairs of representations $N_I$, $Q_P$ of the same dimension vector such that $N_I$ is a subrepresentation of $I$ and $Q_P$ is a quotient of $P$. Assume that an orbit, parametrized by a pair $(N_I,Q_P)$ of dimension vector ${\bf f}$, and admitting exact sequences $$0\rightarrow N_I\rightarrow I\rightarrow Q_I\rightarrow 0,\;\;\; 0\rightarrow N_p\rightarrow P
\rightarrow Q_P\rightarrow 0,$$ is a singular codimension $2$ stratum. Using the codimension formula of the proof of [@CFR Theorem 4.5], this means that $$\langle {\bf f},{\bf f}\rangle+[N_I,N_I]^1+[Q_P,Q_P]^1=2\qquad \text{ and }\qquad [N_I,Q_P]^1\not=0$$ (we use the abbreviations $[X,Y]=\dim{\rm Hom}(X,Y)$ and $[X,Y]^1=\dim{\rm Ext}^1(X,Y)$). If $\langle{\bf f},{\bf f}\rangle=0$, then ${\bf f}=0$, thus $N_I=0=Q_P$, and all extension groups are zero, a contradiction. If $\langle{\bf f},{\bf f}\rangle=2$, then $[N_I,N_I]^1=0=[Q_P,Q_P]^1$, thus both $N_I$ and $Q_P$ are isomorphic to the unique exceptional representation $G$ of dimension vector ${\bf f}$. In particular, $[N_I,Q_P]^1=[G,G]^1=0$, a contradiction. Thus we have $\langle {\bf f},{\bf f}\rangle=1$ and (without loss of generality) $[N_I,N_I]^1=0$ and $[Q_P,Q_P]^1=1$. Thus ${\bf f}$ is a root and $N_I$ is the corresponding indecomposable. $Q_P$ is a minimal degeneration of $N_I$, thus by [@B Theorem 4.5]) there exists a non-split short exact sequence $$0\rightarrow U\rightarrow N_I\rightarrow V\rightarrow 0$$ such that both $U$ and $V$ are indecomposable, and $Q_P\simeq U\oplus V$. In particular, $[V,U]^1\ne 0$, thus $[U,V]^1=0$ since Dynkin quivers are representation-directed. We thus have $1=[Q_P,Q_P]^1=[U\oplus V,U\oplus V]^1=[V,U]^1$. From $[N_I,N_I]^1=0$ it follows that $[N_I,V]^1=0$ using the above exact sequence, thus $0\not=[N_I,Q_P]^1=[N_I,U\oplus V]^1=[N_I,U]^1$. Applying ${\rm Hom}(\_,U)$ to the above sequence yields $${\rm Hom}(U,U)\rightarrow {\rm Ext}^1(V,U)\rightarrow{\rm Ext}^1(N_I,U)\rightarrow{\rm Ext}^1(U,U)=0.$$ The first two terms in this sequence are both one-dimensional. The connecting map is non-zero since the above exact sequence is non-split, thus it is invertible. This implies that $[N_I,U]^1=0$, a contradiction.
Appendix B: desingularization and the smooth locus.
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Let $\pi_{n+1}:R_{n+1}\to {\EuScript{F}}^a_{n+1}$ be the desingularization of the degenerate flag variety of type $A_n$ of [@FF]. Our goal here is to prove the following theorem.
\[oto\] $\pi^{-1}_{n+1}(x)$ is a single point iff $x$ is a smooth point of ${\EuScript{F}}^a_{n+1}$.
Recall that $R_{n+1}$ can be explicitly realized as follows. Let $W$ be an $(n+1)$-dimensional space with a basis $(w_1,\dots,w_{n+1})$. For a pair $1\le i\le j\le n$, let $W^{n+1}_{i,j}=\mathrm{span}(w_1,\dots,w_i,w_{j+1},\dots,w_{n+1})$. Then $R_n$ is the variety of collections $(V_{i,j})_{1\le i\le j\le n}$ such that $V_{i,j}\in \mathrm{Gr}_i(W^{n+1}_{i,j})$ and $V_{i,j}\subset V_{i+1,j}$ and $pr_{j+1}V_{i,j}\subset V_{i,j+1}$.
$R_{n+1}$ can be embedded into ${\EuScript{F}}^a_{n+1}\times R_n$ in such a way that $\pi_{n+1}$ is simply the projection to the first factor.
We first note that the map $\pi_{n+1}:R_{n+1}\to {\EuScript{F}}^a_{n+1}$ is explicitly given by $(V_{i,j})_{i\le j}\mapsto (V_{i,i})_{i=1}^n$. Now consider the forgetful map $$(V_{i,j})_{1\le i\le j\le n}\to (V_{i,j})_{1\le i< j\le n}$$ (the diagonal terms $V_{i,i}$ are omitted). We claim that the image is isomorphic to $R_n$. Namely, for a pair $1\le i<j\le n$, we consider the “shift” map $sh_{i,j}: W_{i,j}^{n+1}\to W_{i,j-1}^n$ given by $$sh_{i,j} w_k=\begin{cases} w_k, \text{ if } k\le i,\\ w_{k-1}, \text{ if } k>j.\end{cases}$$ Then for a point $(V_{i,j})_{i\le j}\in R_{n+1}$, the collection $$(V'_{i,j})_{1\le i\le j\le n-1}=(sh_{i,j+1} V_{i,j+1})_{1\le i\le j\le n-1}$$ belongs to $R_n$. We denote the map $R_{n+1}\to R_n$ by $\psi_{n+1}$. Now the embedding $R_{n+1}\to {\EuScript{F}}^a_{n+1}\times R_n$ is given by the map $A=(\pi_{n+1},\psi_{n+1})$.
Let ${{\bf S}}$ be a length $n$ smooth collection. Then $$\pi_n\psi_{n+1} \pi_{n+1}^{-1} p_{{{\bf S}}}\subset{\EuScript{F}}^a_n$$ is a single point. Moreover, it is a smooth torus fixed point.
Recall that $$p_{{{\bf S}}}=((p_{{{\bf S}}})_i)_{i=1}^n,\ (p_{{{\bf S}}})_i=\mathrm{span}(w_a:\ a\in S_i).$$ Our first goal is to prove that there exists a unique way to define spaces $(V_{i,i+1})_{i=1}^{n-1}$ such that there exists a point in $R_{n+1}$ with the diagonal components being $(p_{{{\bf S}}})_i$ and the $(i,i+1)$-st components being $V_{i,i+1}$. In fact, fix some $i$ with $1\le i\le n-1$. We need $V_{i,i+1}$ such that $\dim V_{i,i+1}=i$ and $$pr_{i+1} (p_{{{\bf S}}})_i\subset V_{i,i+1}\subset W^{n+1}_{i,i+1}\cap (p_{{{\bf S}}})_{i+1}.$$ If $i+1\notin S_i$, then $\dim pr_{i+1} (p_{{{\bf S}}})_i=i$ and hence $V_{i,i+1}=pr_{i+1} (p_{{{\bf S}}})_i$. If $i+1\in S_i$, then since ${{\bf S}}$ is smooth, we have $i+1\in S_{i+1}$. Therefore the intersection $$W^{n+1}_{i,i+1}\cap (p_{{{\bf S}}})_{i+1}=\mathrm{span}(w_a: a\ne i+1)\cap \mathrm{span}(w_a:\ a\in S_{i+1})$$ is $i$-dimensional and hence $V_{i,i+1}$ is forced to coincide with this intersection. Note that in both cases $V_{i,i+1}$ is the linear span of some basis vectors. We denote by $S_{i,i+1}\subset \{1,\dots,i,i+2,\dots,n+1\}$ the set of indices of these vectors, i.e. $$V_{i,i+1}=\mathrm{span}(w_a:\ a\in S_{i,i+1}).$$ We note that $S_{i,i+1}\subset S_{i+1}$ and $S_i\subset S_{i,i+1}\cup\{i+1\}$.
We identify the collection of subspaces $(V_{i,i+1})_{i=1}^{n-1}$ constructed above with the point $(sh_{i,i+1}V_{i,i+1})_{i=1}^{n-1}\in{\EuScript{F}}^a_n$. As mentioned above, each component of this point is a linear span of basis vectors and thus $(sh_{i,i+1}V_{i,i+1})_{i=1}^{n-1}=p(\bar {{\bf S}})$ for some collection $\bar{{\bf S}}=(\bar S_1,\dots,\bar S_{n-1})$. Explicitly, $$\bar S_i=\{a:\ a\in S_{i,i+1}, a\le i\}\cup \{a-1:\ a\in S_{i,i+1}, a> i+1\}.$$ Our goal is to prove that this collection is smooth. In fact, assume $b\in \bar S_a$ for some $1\le a<b\le n-1$. Then since $b>a$ we have $b+1\in S_{a,a+1}$. We consider two cases: $b+1\in S_a$ and $b+1\notin S_a$. If $b+1\in S_a$, then $a+1\in S_{b+1}$ (${{\bf S}}$ is smooth). Since $S_a\subset S_{b+1}$, we have $b+1\in S_{b+1}$. Therefore, $S_{b,b+1}=S_{b+1}\setminus \{b+1\}$ and, in particular, $a+1\in S_{b,b+1}$. Since $a+1\le b$, this implies $a+1\in \bar S_b$. Now assume $b+1\notin S_a$. Then $S_{a,a+1}\ne S_a$ and hence $a+1\in S_a$. This implies $a+1\in S_b$ and so $a+1\in S_{b,b+1}$ (because $w_{a+1}=pr_{b+1} w_{a+1}\in V_{b,b+1}$). We thus arrive at $a+1\in\bar S_b$, which means that $\bar {{\bf S}}$ is smooth.
The map $\pi_{n+1}$ is one-to-one over the smooth locus of ${\EuScript{F}}^a_{n+1}$.
We note that since the fibers over any two points of a given cell in ${\EuScript{F}}^a_{n+1}$ are isomorphic, it suffices to prove that the fiber is a single point over a smooth torus fixed point. Let ${{\bf S}}$ be a smooth collection and $p(\bar{{\bf S}})=\pi_n\psi_{n+1} \pi_{n+1}^{-1}$. Since $\bar{{\bf S}}$ is smooth, our corollary follows by induction on $n$.
To complete the proof of Theorem \[oto\], we need to show that the fiber over a non-smooth point has positive dimension. It suffices to prove that if a collection ${{\bf S}}$ is not smooth, then the preimage of $p_{{{\bf S}}}$ has positive dimension. We first prove the following lemma.
Assume that $S_a$ is not a subset of $S_{a+1}$ for some $a$. Then the dimension of the fiber $\pi_{n+1}^{-1} p_{{{\bf S}}}$ is positive.
Assume that $p_{{{\bf S}}}$ is the image of $(V_{i,j})_{1\le i\le j\le n}$. Let us look at possible sets $V_{a,a+1}$. We know that $$\label{ii}
pr_{a+1} (p_{{{\bf S}}})_a\subset V_{a,a+1}\subset (p_{{{\bf S}}})_{a+1}\cap \mathrm{span}(w_i: i\ne a+1).$$ Since $S_a$ is not a subset of $S_{a+1}$ and $S_a\subset S_{a+1}\cup\{a+1\}$, we obtain $a+1\in S_a$, $a+1\notin S_{a+1}$. Therefore, $$\dim pr_{a+1} (p_{{{\bf S}}})_a=a-1,\ \dim (p_{{{\bf S}}})_{a+1}\cap \mathrm{span}(w_i: i\ne a+1)=a+1.$$ Thus the choice of $V_{i,i+1}$ as in is equivalent to the choice of a point in ${{\mathbb P}}^1$. Therefore the preimage $\pi^{-1}_{n+1} p_{{{\bf S}}}$ is at least one-dimensional.
If ${{\bf S}}$ is not smooth, then the dimension of the fiber $\pi^{-1}_{n+1} p_{{{\bf S}}}$ is positive.
Let $k\ge 1$ be a minimal number such that there exists a number $a$, $1\le a\le n-k$ such that $a+k\in S_a$, but $a+1\notin S_{a+k}$. We prove our corollary by induction on $k$. First, we note that the case $k=1$ means that $S_a\notin S_{a+1}$ and we are done by the lemma above. Now let $k>1$. Since $k>1$ the sets $S_{a,a+1}$ satisfying $$S_a\cup \{a+1\}\subset S_{a,a+1}\subset S_{a+1}$$ are defined uniquely. Now define a length $n-1$ collection $\bar {{\bf S}}$ as above: $$\bar S_i=\{l:\ l\in S_{i,i+1}, l\le i\}\cup \{l-1:\ l\in S_{i,i+1}, l> i+1\}.$$ Since $a+k\in S_a$ and $k>1$ we obtain $a+k-1\in \bar S_a$. Also, since $a+1\notin S_{a+k}$, we obtain $a+1\notin \bar S_{a+k}$ and hence $a+1\notin \bar S_{a+k-1}$ (since $k>1$ we have $S_{a+k-1}\subset S_{a+k}$). This proves that $k$ becomes $k-1$ for $\bar{{\bf S}}$. By the inductive assumption we know that the preimage $\pi^{-1}_n p(\bar{{\bf S}})$ is positive-dimensional. But $\pi^{-1}_{n+1} p_{{{\bf S}}}=\pi^{-1}_n p(\bar{{\bf S}})$ and we are done.
Appendix C
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In this appendix we compute the moment graph of ${\EuScript{F}}_{4}^a$. The T–fixed points of ${\EuScript{F}}_4^a$ are listed in figure \[Fig:TFixedA3\]. Recall that such points are parameterized by successor–closed subquivers of the following quiver $$\label{Eq:DiagBasis}
\xymatrix@R=3pt@C=10pt{3&&&\cdot\\2&&\cdot\ar[r]&\cdot\\1&\cdot\ar[r]&\cdot\ar[r]&\cdot\\4&\cdot\ar[r]&\cdot\ar[r]&\cdot\\3&\cdot\ar[r]&\cdot&\\2&\cdot&&}$$ having one vertex in the first column, two in the second and three vertices in the third column.
$$\begin{array}{|c|c|c|c|c|c|c|}
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35\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*[F]{\cdot}&&}\\\hline
\end{array}$$ $$\begin{array}{|c|c|c|}
\hline
36\xymatrix@R=3pt@C=3pt{&&*[F]{\cdot}\\&*{\cdot}\ar@{-}[r]&*{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*[F]{\cdot}&&}&
37\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}&\\*[F]{\cdot}&&}&
38\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*[F]{\cdot}&&}\\
\hline
\end{array}$$
Figure \[Fig:MomGrapgA3\] shows the moment graph of the degenerate flag variety ${\EuScript{F}}_4^a$ (We used Bernhard Keller’s quiver mutation applet to draw the picture [@quiver_mutation]). The 22 smooth torus fixed points are highlighted by a frame. These are the vertices adjacent to precisely $6={\rm dim }{\EuScript{F}}_4^a$ edges. An edge $p_{{\bf S}}$–$p_\mathbf{R}$ of the moment graph corresponds to a $T$–fixed curve between $p_{{\bf S}}$ and $p_\mathbf{R}$ in ${\EuScript{F}}_4^a$ whose direction around $p_{{\bf S}}$ and $p_\mathbf{R}$ is given by a standard basis vector of the tangent space at them. The edge is oriented $p_{{\bf S}}\rightarrow p_\mathbf{R}$ if and only if the direction around $p_{{\bf S}}$ has positive $T_0$–weight and it is labelled by the corresponding ${{\bf S}}$–effective pair (see theorem \[Thm:OneDimOrbits\] and remark \[Remark:S-effectiveTorus\]).
0;<1pt,0pt>:<0pt,-1pt>:: (190,0) \*+\[F\][1]{} =“0”, (1,141) \*+\[F\][2]{} =“1”, (220,89) \*+\[F\][3]{} =“2”, (80,104) \*+\[F\][4]{} =“3”, (122,95) \*+\[F\][5]{} =“4”, (269,90) \*+\[F\][6]{} =“5”, (341,121) \*+\[F\][7]{} =“6”, (0,234) \*+\[F\][8]{} =“7”, (29,187) \*+\[F\][9]{} =“8”, (106,216) \*+\[F\][10]{} =“9”, (170,170) \*+\[F\][11]{} =“10”, (216,213) \*+\[F\][12]{} =“11”, (257,215) \*+\[F\][13]{} =“12”, (299,232) \*+\[F\][14]{} =“13”, (332,230) \*+[15]{} =“14”, (372,236) \*+[16]{} =“15”, (8,320) \*+[17]{} =“16”, (43,280) \*+[18]{} =“17”, (87,277) \*+\[F\][19]{} =“18”, (148,215) \*+\[F\][20]{} =“19”, (56,334) \*+\[F\][21]{} =“20”, (248,310) \*+[22]{} =“21”, (213,302) \*+\[F\][23]{} =“22”, (364,302) \*+\[F\][24]{} =“23”, (2,398) \*+[25]{} =“24”, (24,438) \*+[26]{} =“25”, (109,428) \*+\[F\][27]{} =“26”, (161,300) \*+\[F\][28]{} =“27”, (155,351) \*+\[F\][29]{} =“28”, (302,374) \*+[30]{} =“29”, (356,429) \*+[31]{} =“30”, (360,383) \*+[32]{} =“31”, (237,429) \*+[33]{} =“32”, (37,497) \*+[34]{} =“33”, (175,483) \*+[35]{} =“34”, (184,390) \*+[36]{} =“35”, (348,487) \*+[37]{} =“36”, (174,564) \*+[38]{} =“37”, “0”, [“1”]{}, “0”, [“2”]{}, “0”, [“3”]{}, “0”, [“4”]{}, “0”, [“5”]{}, “0”, [“6”]{}, “1”, [“7”]{}, “1”, [“8”]{}, “1”, [“9”]{}, “1”, [“10”]{}, “1”, [“11”]{}, “2”, [“7”]{}, “2”, [“12”]{}, “2”, [“13”]{}, “2”, [“14”]{}, “2”, [“15”]{}, “3”, [“8”]{}, “3”, [“12”]{}, “3”, [“16”]{}, “3”, [“17”]{}, “3”, [“18”]{}, “4”, [“9”]{}, “4”, [“14”]{}, “4”, [“17”]{}, “4”, [“19”]{}, “4”, [“20”]{}, “5”, [“11”]{}, “5”, [“13”]{}, “5”, [“16”]{}, “5”, [“19”]{}, “5”, [“21”]{}, “6”, [“10”]{}, “6”, [“15”]{}, “6”, [“18”]{}, “6”, [“20”]{}, “6”, [“21”]{}, “7”, [“15”]{}, “7”, [“22”]{}, “7”, [“23”]{}, “7”, [“24”]{}, “8”, [“17”]{}, “8”, [“22”]{}, “8”, [“25”]{}, “8”, [“26”]{}, “9”, [“17”]{}, “9”, [“24”]{}, “9”, [“27”]{}, “9”, [“28”]{}, “10”, [“15”]{}, “10”, [“26”]{}, “10”, [“28”]{}, “10”, [“29”]{}, “11”, [“23”]{}, “11”, [“25”]{}, “11”, [“27”]{}, “11”, [“29”]{}, “12”, [“16”]{}, “12”, [“22”]{}, “12”, [“30”]{}, “12”, [“31”]{}, “13”, [“16”]{}, “13”, [“23”]{}, “13”, [“24”]{}, “13”, [“32”]{}, “23”, [“14”]{}, “24”, [“14”]{}, “27”, [“14”]{}, “14”, [“30”]{}, “14”, [“33”]{}, “15”, [“31”]{}, “15”, [“32”]{}, “15”, [“33”]{}, “16”, [“25”]{}, “16”, [“34”]{}, “16”, [“35”]{}, “17”, [“30”]{}, “17”, [“35”]{}, “17”, [“36”]{}, “18”, [“26”]{}, “18”, [“31”]{}, “18”, [“34”]{}, “18”, [“36”]{}, “19”, [“21”]{}, “19”, [“24”]{}, “19”, [“27”]{}, “19”, [“35”]{}, “20”, [“21”]{}, “20”, [“28”]{}, “20”, [“33”]{}, “20”, [“36”]{}, “21”, [“29”]{}, “21”, [“32”]{}, “21”, [“34”]{}, “22”, [“25”]{}, “22”, [“31”]{}, “22”, [“35”]{}, “23”, [“25”]{}, “23”, [“33”]{}, “24”, [“32”]{}, “24”, [“35”]{}, “25”, [“30”]{}, “25”, [“37”]{}, “26”, [“31”]{}, “26”, [“36”]{}, “26”, [“37”]{}, “27”, [“29”]{}, “27”, [“30”]{}, “28”, [“29”]{}, “28”, [“32”]{}, “28”, [“36”]{}, “29”, [“33”]{}, “29”, [“37”]{}, “35”, [“30”]{}, “30”, [“37”]{}, “31”, [“34”]{}, “31”, [“37”]{}, “32”, [“33”]{}, “32”, [“34”]{}, “33”, [“37”]{}, “35”, [“34”]{}, “36”, [“34”]{}, “34”, [“37”]{}, “36”, [“37”]{},
To illustrate, let us describe in detail the graph around vertex $(22)$. There are $7$ edges connected to this vertex as depicted in figure \[Fig:MomGraph22\]. In particular this T–fixed point is not smooth.
$$\begin{array}{ccccccc}
20=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt"20"{&&*[F]{\cdot}\\&*{\cdot}\ar@{-}[r]&*{\cdot}\\*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*{\cdot}&&}\\\hline
\end{array}
&
21=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}&\\*{\cdot}&&}\\\hline
\end{array}
&
6=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt{&&*[F]{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*{\cdot}&&}\\\hline
\end{array}
&
7=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}&\\*{\cdot}&&}\\\hline
\end{array}\\&&&&\\(1,2)\downarrow&(4,5)\downarrow&(1,4)\downarrow&(2,5)\downarrow
\end{array}$$ $$\begin{array}{c}
22=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*{\cdot}&&}\\\hline
\end{array}
\end{array}$$ $$\begin{array}{ccccccc}
&&&&&&\\&(3,4)\downarrow&&(3,5)\downarrow&&(3,6)\downarrow&\\
&
30=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*{\cdot}&&}\\\hline
\end{array}
&
&
33=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*{\cdot}&&}\\\hline
\end{array}
&&
35=\begin{array}{|c|}
\hline
\xymatrix@R=3pt@C=3pt{&&*{\cdot}\\&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*{\cdot}\ar@{-}[r]&*[F]{\cdot}\\*{\cdot}\ar@{-}[r]&*[F]{\cdot}&\\*[F]{\cdot}&&}\\\hline
\end{array}
\end{array}$$
The arrow $(20)\rightarrow (22)$ corresponds to the following curve (in the basis ) $$(\langle v_1\rangle,\langle v_1, v_3\rangle,\langle v_3+\lambda v_2, v_1, v_4\rangle)\;\;\; \lambda\in{\mathbb{P}}^1$$ For $\lambda=0$ one gets the starting point $(20)$ of $\alpha$, for $\lambda=\infty$ one gets the end point $(22)$ of $\alpha$. Its direction around $(22)$ has negative $T_0$ weight while around $(20)$ it has positive weight. The corresponding (20)–effective pair is $(1, 2)$. The remaining labelings of figure \[Fig:MomGraph22\] are obtained similarly.
Acknowledgements {#acknowledgements .unnumbered}
================
The work of Evgeny Feigin was partially supported by the Russian President Grant MK-3312.2012.1, by the Dynasty Foundation, by the AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023, and by the RFBR grants 12-01-00070 and 12-01-00944. This study comprises research fundings from the ‘Representation Theory in Geometry and in Mathematical Physics’ carried out within The National Research University Higher School of Economics’ Academic Fund Program in 2012, grant No 12-05-0014. This study was carried out within the National Research University Higher School of Economics Academic Fund Program in 2012-2013, research grant No. 11-01-0017.
GCI thanks Francesco Esposito for helpful discussions.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We derive Thouless-Anderson-Palmer (TAP) equations for quantum disordered systems. We apply them to the study of the paramagnetic and glassy phases in the quantum version of the spherical $p$ spin-glass model. We generalize several useful quantities (complexity, threshold level, etc.) and various ideas (configurational entropy crisis, etc), that have been developed within the classical TAP approach, to quantum systems. The analysis of the quantum TAP equations allows us to show that the phase diagram (temperature-quantum parameter) of the $p$ spin-glass model should be generic. In particular, we argue that a crossover from a second order thermodynamic transition close to the classical critical point to a first order thermodynamic transition close to the quantum critical point is to be expected in a large class of systems.'
author:
- |
Giulio Biroli$^{1,2}$ and Leticia F. Cugliandolo$^{2,3}$\
\
[$^1$Center for Material Theory, Department of Physics and Astronomy]{},\
[Rutgers University, Piscataway, NJ 08854 USA]{}\
[$^2$Laboratoire de Physique Th[é]{}orique de l’Ecole Normale Sup[é]{}rieure[^1]]{}\
[24 rue Lhomond, 75231 Paris cedex 05, France.]{}\
[$^3$Laboratoire de Physique Th[é]{}orique et Hautes Energies, Jussieu, ]{}\
[4, Place Jussieu, 75252 Paris Cedex 05, France ]{}
title: Quantum TAP equations
---
LPT-ENS/0033, LPTHE/0034.
Introduction {#intro}
============
Glassy systems of extremely diverse types exist in nature. They all share several common features like a very slow, non-equilibrium dynamics. The development of a full theoretical description of the glassy phase is one of the most important challenges in condensed matter physics. A variety of techniques, that range from scaling arguments to mean-field approaches have been, and are still used, with the aim of attempting a satisfactory description of the glassy properties.
One of these techniques is due to Thouless, Anderson and Palmer (TAP) [@TAP], who introduced an approach to classical disordered systems based on the study of a free-energy landscape. The key object is the Legendre transform of the free-energy $F(\beta) =-\ln Z / \beta $ with respect to a number of order parameters that are sufficient to describe the transition and the different phases in the system. This function behaves as an effective potential whose minima represent different possible phases. In a classical fully-connected Ising model only one order parameter is needed, the global magnetization $m=\sum_i \langle s_i\rangle/N$. The two possible minima of $F(\beta,m)$ correspond to the two possible states of positive and negative magnetization, $m = \pm m_o(T)$. Focusing on the Sherrington-Kirkpatrick (SK) mean-field model for spin-glasses, TAP showed that [*all*]{} the local magnetizations $m_i=\langle s_i\rangle$, $i=1,.\dots,N$, have to be included in order to derive the relevant free-energy landscape. The extremization condition of the TAP free-energy on the $m_i$s leads to the TAP equations. It was soon after realized by Bray and Moore [@deDo; @Brmo_TAP] that the number of solutions to the TAP equations for the SK model is exponential in the number of spins in the system, for temperatures below the spin-glass transition [@Ku]. A very useful alternative derivation of the TAP equations was given by Plefka [@Pl] who showed that these equations can also be obtained from a power expansion of the Gibbs potential up to second order in the exchange couplings. The advantage of this derivation is twofold: it allows to show convergence of the power expansion for all temperatures and it is easily applicable to other mean-field glassy models. Moreover, Georges and Yedidia [@Geye] showed that the high temperature series, at fixed order parameter, of the free-energy can be used to derive TAP-like equations, and its corrections, for models in finite dimensions or, equivalently, with finite range interactions. The connection between the TAP approach and the more standard analysis of the partition function of a disordered model has been exhibited by De Dominicis and Young [@deDoyo] who showed that, for the SK model, one recovers the equilibrium results of the replica or the cavity method [@Beyond] via weighted Boltzmann averages over solutions of the TAP equations. More recently, the TAP approach has been applied to other classical disordered models. In particular, two models that we shall discuss in the following, the spherical and Ising $p$ spin-glass models [@Kiwo-TAP; @Rieger; @Kupavi; @Crso2; @Cagipa1; @Cagagi] and the Ghatak-Sherrington (GS) model [@Gash; @Mosh] have been analyzed with this method [@Yo; @Reop; @deAr3].
Glassy systems, and in particular disordered ones, are characterized by having a very slow dynamics with non equilibrium effects at low temperatures [@Bocukume; @glassy_exps]. Mean-field models, like the the spherical $p$ spin-glass model [@Cuku] or the SK spin-glass [@Cuku2], capture this phenomenology. The dynamic solution for the evolution starting from random initial conditions, that represent a quench from high temperatures analytically, is intimately connected to the structure and organization of TAP solutions. One of the most striking results of the dynamic analysis of $p$ spin-glass-like models is that the energy density (and other one time-quantities) converges asymptotically to the energy density of high lying solutions of the TAP equations. This level has been called [*threshold*]{}. The energy-density in equilibrium is different. This and other related results suggest that an interpretation of the dynamics in terms of a motion in a TAP free-energy landscape can be given [@Cuku]. The generalization of the TAP approach to dynamics that has been developed in [@Bi] allows one to make this statement precise: the evolution is determined by a gradient-descent in the TAP free-energy landscape with the most important addition of non-Markovian terms.
Usually, glasses can be analyzed with a fully classical approach since their transition temperatures are rather high. Nevertheless, in many cases of great interest the critical temperature can be lowered by tuning another external parameter and quantum fluctuations become very important. This is the case for the insulating magnetic compound LiHo$_x$Y$_{1-x}$F$_4$, that is an experimental realization of a quantum spin-glass, and presently receives much attention [@Ro]. Other examples where glassy properties in the presence of quantum fluctuations have been observed are mixed hydrogen bonded ferro-antiferro electric crystals [@exp_q2], interacting electron systems [@Zvi], cuprates like La$_{2-x}$Sr$_x$CuO$_4$ [@cuprates], amorphous insulators [@Osheroff], etc.
The quantum fluctuations in LiHo$_x$Y$_{1-x}$F$_4$ can be controlled by tuning the strength of an external field that is transverse to the preferred direction of the randomly located magnetic impurities. After a series of experiments presented in [@Ro] the authors’ conclusions are: (1) The samples undergo a paramagnetic to spin-glass transition in the $(T,\Gamma)$ plane, where $\Gamma\propto H_t^2$ and $H_t$ is the strength of the transverse field. (2) The transition is of second order (in the thermodynamic sense) close to the classical critical point $(T=T_c,\Gamma=0)$ but crosses over to first order close to the quantum critical point $(T=0,\Gamma=\Gamma_c)$. (3) The system undergoes out of equilibrium dynamics in the glassy phase as demonstrated by the fact that the dynamics strongly depends on the preparation of the sample for all subsequent times explored experimentally.
The theoretical study of quantum spin-glasses started with Bray and Moore’s analysis of the equilibrium properties of the fully connected Heisenberg model [@Brmo]. In this article, Bray and Moore introduced a path-integral representation in imaginary time of the partition function that they analyzed with the replica trick. Many articles on the equilibrium of this, and related, mean-field models have been published since [@quantum_mf; @Isya; @rotors; @Cesare; @Cesare1; @Niri; @Culo; @Cugrsa; @Chamon]. The static properties of low dimensional models have been studied and it has been shown that, in finite dimensions, Griffiths-McCoy singularities are very important close to the quantum critical point [@quantum_finite]. In all these models, the transition from the paramagnetic to the spin-glass phase has been reported to be of second order throughout.
In most classical disordered models studied so far the transition from the disordered to the ordered phase is of second order in the thermodynamic sense. In the exact solution of the SK model, the spin-glass order parameter $q(x)$ is continuous at the transition which is of second order in the thermodynamic sense [@Beyond]. In other classical glassy models like the Potts glass [@Kiwo-TAP] or the spherical [@Crso1] and Ising [@De; @Kith] $p$ spin-glasses, the order parameter jumps at the transition which, however, is still of second-order in the thermodynamic sense since there is neither a jump in the susceptibility nor a latent heat. A classical model that exhibits a first-order transition is the anisotropic $p$ spin-glass, $p\geq 3$, in which the spins take integer values between $-S$ and $S$ and there is an extra term in the Hamiltonian $-D\sum_i s_i^2$, proportional to a coupling constant $D$, that controls the crystalline tendency. In this case, a crossover from a second-order to a first-order thermodynamic transition in the plane $(T/J,D/J)$ has been exhibited in the exact solution [@Mo]. The classical Ghatak-Sherrington (GS) model [@Gash] is another candidate to exhibit a second to first order crossover in the thermodynamic transition. It is the anisotropic extension of the SK model, or the $p=2$ limit of the previous model. In this case, a crossover from a second order to a first order transition in the plane $(T/J,D/J)$ has been exhibited in an [*approximate*]{} solution (one step replica symmetry breaking) [@Gash; @Mosh]. The exact solution has not been derived yet and it is then not well established if it has a true first-order thermodynamic transition.
In quantum problems, first order transitions have been reported in three models. The first one is the so-called “fermionic Ising spin-glass” analyzed by Oppermann and collaborators [@Opper]. This model, however, is thermodynamically equivalent to the classical GS model discussed above [@Opper1]. The other two models are very similar indeed and they are different ways of extending the classical spherical $p$ spin-glass model [@Crso1] to include quantum fluctuations. In one case, the continuous spins are generalized to $M$ component vectors and a global spherical constraint as well as commutation relations are imposed [@Niri]. The other one uses the fact that the spherical $p$ spin-glass model can be interpreted as a particle moving in an infinite dimensional hyper-sphere with a random potential. Quantization is then done by imposing commutation relations between coordinates and momenta [@Culo; @Cugrsa]. The latter can also be interpreted as an extension of the a quantum rotor model [@rotors] that includes $p$ interactions. The relation between the critical properties of the quantum versions of $p$ spin-glass models and the experiments in [@Ro] has been put forward in [@Cugrsa]. In addition, the connection between the static calculation supplemented by the marginality condition and the analysis of the out of equilibrium dynamics in contact with an environment developed in [@Culo] was also discussed in [@Cugrsa]. However, the reason why the transition changes from second to first order close to the quantum critical point was not clear from this analysis. It is one of the aims of this article to clarify this point, and study to what extent one can claim it to be general, with the use of the TAP approach.
Quantum TAP equations for the SK model in a transverse field have been presented by Ishii and Yamamoto [@Isya] and Cesare [*et al*]{} [@Cesare1]. The former use a perturbative expansion of the free-energy in the strength of the transverse field, and then follow closely TAP’s techniques; the latter implement a cavity method. The TAP equations derived by Rehker and Oppermann [@Reop] for the fermionic spin-glass model coincide with the ones presented by Yokota [@Yo] for the classical GS model since these two models are thermodynamically equivalent [@Opper1].
Hence, our aim is twofold. On the one hand we want to present a quantum extension of the TAP approach to the statistical properties of disordered systems. Thus, after a short revision of the classical TAP approach in Section \[introtap\], we discuss in Section \[sec1:formalism\] the derivation of the quantum TAP free-energy and TAP equations using a general approach that extends the ones developed by Plefka [@Pl] and Georges and Yedidia [@Geye]. The advantage with respect to previous derivations of quantum TAP equations is that this method can be applied to any quantum disordered model and it allows to obtain the TAP equations as well as the TAP free energy. In Section \[p-spin\] we present, as an example, the TAP free-energy and TAP-equations for the quantum extension of the $p$ spin spherical spin-glass model studied in [@Cugrsa; @Culo]. We show that the TAP equations can be easily related to the equations for the order parameter in the Matsubara replica approach and also to some of the equations appearing in the real-time dynamic approach. The TAP analysis of this model furnishes a benchmark to study the generalization to the quantum case of the methods and interpretations developed for classical systems. Section \[transition\] is devoted to the second aim of this article. Via the TAP approach we show that the same type of phase diagram naturally emerges for all systems having a discontinuous phase transition in their classical limit (these are models solved by a one-step replica symmetry breaking Ansatz within the replica analysis). In particular we relate the first-order transition close to the quantum critical point to the structure of metastable states. Finally, we present our conclusions in Section \[conclu\].
The classical TAP equations: a short revision {#introtap}
=============================================
In this section we present a short revision of the classical TAP approach to mean-field disordered spin models. The classical TAP free-energy [@TAP] is the Legendre transform of the free-energy with respect to local magnetic fields, $$\label{tapcl1}
-\beta F (\beta,m_{i})
=
{\mbox{Tr}}\exp \left(-\beta H -\sum_{i}h_{i}(s_{i}-m_{i}) \right)
\; ,$$ where $h_{i}$ are Lagrange parameters enforcing the condition $\langle s_{i}\rangle=m_{i}$. $-\beta F(\beta,m_i)$ is an effective potential that depends on the local magnetizations. The Lagrange conditions $-\partial \beta F /\partial m_{i}=h_{i}$, called TAP equations, fix the local magnetizations as functions of the local magnetic fields [^2]. The solutions $\{m_{i}^{\alpha } \}$ of the TAP equations are stationary points of $F (\beta,m_{i})$. If they are also stable (all the corresponding eigenvalues of the free-energy Hessian are positive), they are identified [@Beyond] with pure states, also called TAP states. This interpretation was put forward by De Dominicis and Young [@deDoyo] who showed that the partition function in the classical SK model can be written as a weighted sum over the stable solutions of the TAP equations: $$Z=\sum_{\alpha} \exp(-\beta F(\beta,{\bf m}^\alpha))
\; ,
\label{eq:weighted}$$ where the index $\alpha$ labels different TAP states, ${\bf m}^\alpha$ is an $N$-vector encoding the local magnetization in the solution $\alpha$, $F$ is the extensive TAP free-energy of such solution and the sum runs over all TAP solutions. Consequently, the static average of any observable can be computed from Eq. (\[eq:weighted\]). At low temperatures the TAP free-energy has a large number of minima. If one groups different TAP states with the same free-energy in sets ${\cal C}$ then the partition function can be written as $$Z=\sum_{{\cal C}}
{\cal N}(f,\beta ) \exp(-\beta N f)
\label{eq:weighted1}$$ where the factor ${\cal N}(\beta,f )$ is the number of solutions with TAP free-energy density $F(\beta,{\bf m}^\alpha)/N=f$. One can now replace the sum by an integral and exponentiate the factor ${\cal N}(\beta,f )$; this yields $$\lim_{N\rightarrow \infty }\frac{1}{N}\ln Z=\lim_{N\rightarrow \infty }\frac{1}{N}\ln\int df \exp(-N (\beta f-\sigma(\beta,f )))
\label{eq:weighted3}$$ where we have taken the continuous limit and introduced the [*complexity*]{} $$\label{complexity0}
\sigma(\beta,f) \equiv \lim_{N\to\infty} \frac{1}{N}
\ln\left( {\cal N}(\beta,f) \right)
\; .$$ The configurations that dominate the sum are those having a free-energy density such that it minimizes $\beta f - \sigma(\beta,f)$. The identity between the partition function and the weighted sum over TAP solutions has been demonstrated for many others models [@Crso2; @Kiwo-TAP; @Mon] and it is generally believed to hold for any mean-field disordered system.
In the following we focus on “discontinuous glassy systems” [@Bocukume] that are characterized by having a discontinuous transition (the Edwards-Anderson order parameter, $q_{\sc ea}$, jumps) that is still of second order thermodynamically. Within the replica analysis of the partition function these models are characterized by a one-step replica symmetry breaking solution below a static transition temperature $T_s$ and a replica symmetric (RS) solution, that corresponds to the paramagnetic phase, at $T> T_s$. However, for intermediate temperatures $T_s < T < T_d$ there are an exponential in $N$ number of non-trivial TAP solutions that combine themselves in such a way that the sum (\[eq:weighted3\]) is identical to the RS result.
The relationship between metastable states and replicas has been put forward in [@Kith; @Mon; @PaFr]. Indeed, consider $x$ different identical systems (“clones”) coupled by an attractive, infinitesimal (but extensive) interaction. When there exist many pure states all the clones fall into the same state and the free energy for the system of $x$ clones reads: $$\label{reltaprep0}
\lim_{N\rightarrow \infty }\frac{-1}{\beta N}\ln Z_{x}=\lim_{N\rightarrow \infty }\frac{-1}{\beta N}\ln \int df \exp(-N (\beta xf-\sigma(\beta,f)))
\; .$$ On the other hand the computation of the left hand-side of (\[reltaprep0\]) can be performed within the replica formalism: $$\label{reltaprep1}
\lim_{N\rightarrow \infty }\frac{-1}{\beta N}\ln Z_{x}=\lim_{N\rightarrow \infty }\frac{-1}{\beta N}\overline{\ln Z_{x}}=\lim_{N\rightarrow \infty,n\rightarrow 0 }\frac{-1}{\beta N n}\ln \overline{Z_{x}^{n}}$$ where the overline represents the average over disorder. Since the attractive coupling between the $x$ clones is infinitesimal, the computation of the right-hand side of (\[reltaprep1\]) reduces simply to the calculation of $\lim_{n'\rightarrow 0}(x/n')
\ln \overline{Z^{n'}}$, where the replica symmetry between the $n$ groups of $x$-replicas ($n'=nx$) is [*explicitly*]{} broken. When the system is in the replica symmetric phase ($T_{s}<T$), this reduces to study one-step solutions non-optimized with respect to $x$: $$\label{reltaprep}
-\lim_{N\rightarrow \infty}\frac{1}{\beta N }\ln \int df \exp(-N (\beta xf-\sigma(\beta,f)))
={x}{\mbox{Extr}}_{q_{\sc ea}}f_{\sc rep}(q_{\sc ea};\beta,x)$$ where $f_{\sc rep}$ is the free-energy computed by using replicas, $q_{\sc ea}$ is the Edwards-Anderson parameter and $x$ is the breakpoint (or the size of the blocks in the replica matrix). For simplicity we consider that the inter-state overlap $q_{0}$ equals zero. The definitions of these parameters are standard in the replica approach [@Beyond] and they will appear in the analysis of the quantum $p$ spin-glass model in Section 3. Since the integral on the left-hand-side of (\[reltaprep\]) is dominated by a saddle point contribution, one finds that, for a given temperature, fixing the value of $x$ is equivalent to summing over states with a given energy density $f$. The relationship between $f$ and $x$ reads $$\label{rela:xf}
\beta x=\frac{\partial\sigma(\beta,f) }{\partial f }
\; .$$ Note that within this framework one does not have to optimize with respect to $x$. Instead, $x$ is a free parameter and, by changing the value of $x$, one can consider different groups of metastable states.
The analysis of the TAP equations reveals three temperature regimes for discontinuous glassy systems:
- [*High temperatures*]{} $T_d< T$. The system is in the paramagnetic phase, the paramagnetic TAP solution $m_i=0$, for all $i$, dominates the sum and $f_{pm} = -\ln Z/(\beta N)$. $T_d$ is the dynamic critical temperature. Above $T_d$ the dynamics starting from a random initial condition converges asymptotically to the paramagnetic solution.
- [*Intermediate temperatures*]{} $T_s <T< T_d$. The replica analysis of the partition function indicates that the system is still in the paramagnetic phase. However, the study of the TAP equations and the dynamics show that at $T_d$ the paramagnetic solution is fractured in an exponentially large in $N$ number of minima of the TAP free-energy [@Kupavi; @Kiwo-TAP; @Alain; @PaFr]. Indeed one can recover these results also by the replica method. A careful replica analysis shows that there exist one-step solutions in these temperature regime other than the paramagnetic one. These solutions are in one to one correspondence with groups of states with a given free-energy density (through the relationship (\[rela:xf\])). For instance, one can follow the evolution of the threshold states (the states with highest free-energy) by tuning the parameter $x$. For these states, $x=1$ when $T=T_{d}$ and decreases at lower temperatures. Moreover, the dominant contribution to Eq. (\[eq:weighted3\]) is given by the states characterized by $x=1$, [*i.e.*]{} those with free-energy density such that $$\beta=\frac{\partial\sigma(\beta,f)}{\partial f}
\; .$$ These are the threshold states at $T_d$ and other groups when $T<T_d$. Hence, between $T_s<T<T_d$ saddle-point solutions (corresponding to $x=1$) that are not absolute minima of $F$ dominate the integral since their number scales exponentially with $N$. The final result for the free-energy density in this temperature range coincides with the one of the prolongation of the paramagnetic solution (that actually does not exist!). A naive replica computation fails to signal the difference between a true paramagnetic solution and the ensemble of non trivial TAP solutions with $m_i\neq 0$. The dynamic approach detects the change in free-energy landscape at $T_d$ since the system cannot reach equilibrium for any temperature below $T_d$ [@Cuku].
- [*Low temperatures*]{} $T<T_s$. At the static transition temperature the complexity of the TAP solutions, which dominates the sum (\[eq:weighted3\]), vanishes. The static transition appears as an [*entropy crisis*]{} since the part of the total entropy that is related to the large number of states disappears. For $T<T_{s}$ the TAP states which dominate the integral in Eq. (\[eq:weighted3\]) correspond to the equilibrium glassy phase. Dynamically, $T_s$ does not play any role. The out of equilibrium dynamics is dominated by the threshold states, which are the highest ones in free-energy and that are characterized by flat directions in the free-energy landscape.
Note that via the TAP approach one can obtain a reasonable justification of the marginality condition [@replicon] often used to obtain information about the out of equilibrium dynamics starting from a pure equilibrium computation [@Kith-replicon]. Indeed the value of $x$ fixed by the marginality condition corresponds to the TAP states which are marginally stable (the threshold states): the flatness of the free-energy landscape around these states is responsible for aging [@Cuku].
The quantum TAP equations {#sec1:formalism}
=========================
In this section we present a simple procedure to derive TAP equations for generic completely connected quantum systems. We also expose the physical meaning of quantum TAP equations by the cavity method [@Beyond].
We are aware of two publications where TAP equations for quantum systems have been already presented [@Isya; @Cesare1]. With respect to these works our derivation is more systematic, simple and it allows one to obtain the TAP equations as well as the TAP free-energy for any completely connected quantum disordered system.
Formalism, notations and models {#introtapeq}
-------------------------------
The formalism that we use to derive TAP equations for generic quantum problems is very similar to the one described in [@Pl; @Geye] and, it follows even more closely, the one used in [@Bi] to obtain the dynamical TAP equations for classical disordered models.\
We focus on systems characterized by the potential energy: $$\label{potential}
H_{p}=-\sum_{i_{1}<\dots <i_{p}}\sum _{\alpha }J_{i_{1},\dots ,i_{p}}s_{i_{1}}^{\alpha }
\cdots s_{i_{p}}^{\alpha } \qquad i=1,\dots ,N \quad \alpha =1,\dots ,m$$ where ${\mathbf{s}}_{i}$ may represent an SU(2) spin ($m=3$), a rotor ($m>1$), a spherical spin ($m=1$) or a space-coordinate ($m=1$) and $J_{i_{1},\dots ,i_{p}}$, the couplings between the different ${\mathbf{s}}_{i}$, are independent random variables with zero mean and variance $$\label{coupling}
\overline{\left(J_{i_1...i_p}\right)^2} = {{\tilde{J}}^2 p!\over 2N^{p-1}}.$$ As a consequence the following derivation applies to (completely connected) Heisenberg models, quantum rotor models and quantum continuous systems[^3]. Without loss of generality and to simplify the notation we shall suppress the index $\alpha $ in the rest of this section.
For classical spin-glasses TAP showed that all the local magnetizations $m_i$, $i=1,.\dots,N$, are needed to derive the relevant free-energy density to describe the metastable properties [@TAP]. If one is interested in the dynamics of classical disordered mean-field systems, one has to Legendre transform not only with respect to all time-dependent local magnetizations $m_i(t)$, but also with respect to the autocorrelation $C(t,t')=$ $1/N \sum_i \langle s_i(t) s_i(t')\rangle$ and the linear response $R(t,t')=1/N \sum_i \delta \langle s_i(t) \rangle/\delta h_i(t')|_{h=0}$ [@Bi].
In order to describe the metastable properties of a quantum disordered model we shall show that it is necessary to Legendre transform with respect to the local average coordinates, $m_i(\tau)$, and the autocorrelation function in imaginary time, $C(\tau,\tau')$. The quantum TAP free-energy reads $$\begin{aligned}
& & \left.-\beta F(\beta,m_i(\tau),C(\tau,\tau'),\alpha)
\right|_{\alpha =1}
=
\nonumber\\
& &
\;\;\;\;\;\;\;\;\;\;
\ln
\int {\cal D}{\bf s}(\tau) \,
\exp\left[ \;\; -\frac{1}{\hbar} \int_0^{\beta\hbar} d\tau \left(
H_{k} + \alpha H_{p}({\bf s}) \right)
\right.
\nonumber\\
& &
\;\;\;\;\;\;\;\;\;\;
\left.
+
\frac{1}{2 \hbar^2} \int_0^{\beta\hbar} d\tau \int_0^{\beta\hbar} d\tau' \; \sum_{i}\Lambda (\tau,\tau')
\left( C(\tau,\tau') - s_i(\tau) s_i(\tau') \right)
\right.
\nonumber\\
& &
\;\;\;\;\;\;\;\;\;\;
+
\left.
\left.\frac{1}{\hbar} \int_0^{\beta\hbar} d\tau
\sum_i h_i(\tau) ( m_i(\tau) - s_i(\tau))
\;\; \right]\right|_{\alpha =1}
\; .
\label{eq:Gamma}\end{aligned}$$ where $H_{k}$ is the kinetic energy, ${\cal D}{\bf s}(\tau)$ indicates the functional measure on the configuration space and $\alpha $ is a parameter whose role will be clarified in the following. For instance, if ${\mathbf{s}}_{i}$ are SU(2) spins $H_{k}$ is the Berry phase and the functional measure is restricted to periodic functions ${\mathbf{s}}_{i}(\tau )$ (with period $\beta $) satisfying the constraint ${\mathbf{s}}^{2}_{i}(\tau )=1$. The sources $h_i(\tau)$ and $\Lambda (\tau,\tau')$ have the role of Lagrange multipliers fixing the average value of the coordinates and the correlation: $$\begin{aligned}
m_i(\tau) &=& \langle s_i(\tau)\rangle
\; ,
\\
C(\tau,\tau') &=& \frac{1}{N}\sum_{i}\langle s_i(\tau) s_i(\tau')\rangle
\; .\end{aligned}$$ Once the TAP free-energy $F $ is known, one can derive the TAP equations as Legendre relations, $$\label{legendre}
-\frac{\delta\beta F}{\delta m_{i}(\tau)} = h_{i}(\tau )\; ,
\qquad \qquad
-\frac{2}{N} \frac{\delta\beta F}{\delta C(\tau,\tau')} = \Lambda(\tau,\tau')
\; .$$ Until now we have not not used the scaling (\[coupling\]) and all these definitions can be equally applied to finite dimensional systems. The great simplification due to the mean field character of the interactions in (\[coupling\]) is unveiled if one performs a perturbative expansion of Eq. (\[eq:Gamma\]) in $\alpha $ and writes $-\beta F(\beta,m_i(\tau),C(\tau,\tau'),\alpha)$ as a power series in $\alpha$: $$-\beta F(\beta,m_i(\tau),C(\tau,\tau'),\alpha)
=
\sum_{n=0}^\infty \frac{1}{n!} \left.
\frac{\partial^n(-\beta F(\beta,m_i(\tau),C(\tau,\tau'),\alpha))}{\partial \alpha^n} \right|_{\alpha=0}
\alpha^n
\; .
\label{series}$$ In fully-connected models, if one chooses the correct order parameters (which are $m_i(\tau)$, $C(\tau,\tau')$ in the quantum case), the perturbative expansion (\[series\]) around the pure kinetic theory is actually a simple sum over three terms. Higher order terms in the series vanish in the thermodynamic limit due to the scaling of $J_{i_{1},\dots ,i_{p}}$ with respect to $N$. In more general cases, in finite dimensions, this will not be the case and (\[series\]) becomes a $1/d$ expansion around mean field theory, where $d$ is the spatial dimension [@Geye].
Let us consider in more detail the terms arising from the expansion (\[series\]). The zeroth-order one is simply $-\beta F(\beta,m_i(\tau),C(\tau,\tau'),0)$, i.e. the free-energy of $N$ free spins constrained to have local magnetizations $m_{i}(\tau )$ and a global correlation function $C(\tau ,\tau ')$. This term depends only on the nature of the degrees of freedom, whether they are $SU(2)$ spins, rotors or space-coordinates. In particular it can be analytically computed only if $s_{i}$ are spherical spins or space-coordinates. In the other cases one has to resort to approximations or numerical computations.
The first-order term is the naive mean field free-energy: $$\label{naivemfe}
\left. \frac{\partial (-\beta F)}{\partial \alpha} \right|_{\alpha=0}
=\frac{1}{\hbar}\int_{0}^{\beta \hbar }\sum_{i_{1}<\dots <i_{p}}J_{i_{1},\dots ,i_{p}}\left<
s_{i_{1}}\cdots s_{i_{p}}\right>_{\alpha =0}=
\frac{1}{\hbar}\int_{0}^{\beta \hbar }\sum_{i_{1}<\dots <i_{p}}J_{i_{1},\dots ,i_{p}}
m_{i_{1}}\cdots m_{i_{p}}
\; .$$ Note that the decoupling of the spins for $\alpha =0$ is essential to obtain the last identity. The second-order term depends on the correlation function and the overlap function $Q(\tau ,\tau ')=\sum_{i}m_{i}(\tau )m_{i}(\tau ')/N$ only and equals $$\label{secondterm}
\frac{N\tilde{J}^{2}}{4\hbar ^{2}} \int_0^{\beta \hbar } d\tau \int_0^{\beta \hbar }
d\tau' \left(
C^p(\tau,\tau') - Q^p(\tau,\tau') - p (C(\tau,\tau') - Q (\tau,\tau') )
Q^{p-1}(\tau,\tau') \right)
\, ,$$ Using the scaling of the couplings with $N$ and by the same arguments developed for classical systems [@Pl; @Crso2], we have verified that all orders $n\geq 3$ in the series (\[series\]) are suppressed in the thermodynamic limit.
A cavity interpretation {#meaning}
-----------------------
First of all, let us write the TAP equations in a way which allows one to clarify the physical meaning of the different terms: $$\begin{aligned}
\label{tap1mean}
\left. \frac{\delta (-\beta F)}{\delta m_{i}(\tau )}
\right|_{\alpha =0}&=&h_{i}^{cav}(\tau )=
- \sum_{i_2 < \dots < i_p} J_{i, i_2,\dots,i_p} m_{i_2}(\tau) \dots
m_{i_p}(\tau)\\
&&-\frac{1}{\hbar }\int_0^{\beta \hbar } d\tau'
\left[\frac{p(p-1)}{2}
\left( Q(\tau,\tau') - C(\tau,\tau') \right) Q^{p-2}(\tau,\tau')
\right] \, m_i(\tau') \; ,
\nonumber \\
\label{tap2mean}
\left. \frac{2}{N}\frac{\delta (-\beta F)}{\delta C(\tau, \tau' )}
\right|_{\alpha =0}&=&G^{cav}(\tau,\tau ')=\frac{p}{2} \left[
Q^{p-1}(\tau,\tau') - C^{p-1}(\tau,\tau') \right]
\; .\end{aligned}$$ The solutions to these equations are expected to be time-translation invariant since we are developing a description of equilibrium and metastable properties. Therefore $h_{i}^{cav}$ is indeed independent of the imaginary time and $G^{cav}$ depends only on the difference between $\tau $ and $\tau '$.
An understanding of the meaning of the quantum TAP equations follows from the analysis of $F$ for $\alpha =0$. Indeed, by tracing out all the spins except $s_{i}$ in the partition function produces a single-site measure (for $s_{i}$) whose action reads: $$\label{singlesite}
-\frac{1}{\hbar}\int_{0}^{\beta \hbar }d\tau \left[
H_{k}(s_{i}(\tau ))+h^{cav}_{i}(\tau )s_{i}(\tau )\right]
-\frac{1}{2 \hbar^2} \int_0^{\beta\hbar} d\tau \int_0^{\beta\hbar} d\tau' \;
s_i(\tau)G^{cav} (\tau,\tau') s_i(\tau') \;,$$ where $H_{k}(s_{i}(\tau ))$ is the kinetic energy for the spin $s_{i}$. As a consequence the TAP solutions are the self-consistent relations that relate $G^{cav}(\tau ,\tau ')$ and $h_{i}^{cav}(\tau )$ (which are functions of $C(\tau ,\tau ')$ and $m_{i}(\tau )$) to $C(\tau ,\tau ')$ and $m_{i}(\tau )$ obtained from the single-site action (\[singlesite\]).
Equations (\[tap1mean\]) and (\[tap2mean\]) show that the action on the [*i*]{}th spin of the $N-1$ remaining ones reduces simply to $h_{i}^{cav}$ and $G^{cav}$. This implies that tracing out all the spins but the [*i*]{}th one produces a Gaussian measure for the instantaneous magnetic fields $h_{i}(\tau)=-\sum_{i_{2}<\dots <i_{p}}
J_{i_{1},\dots ,i_{p}}s_{i_{1}}\cdots s_{i_{p}}$, whose mean and connected two-point correlation function equal respectively $h_{i}^{cav}$ and $G^{cav}(\tau -\tau ')$.
The expression of $h_{i}^{cav}$ and $G^{cav}(\tau -\tau ')$ can be justified within the cavity method [@Beyond]. Let us focus for simplicity on the $p=2$ case for which $$\label{cavp2}
h_{i}^{cav}=-\sum_{k}J_{i, k }
\langle s_{k}\rangle _{N-1}$$ where $\langle\cdot \rangle _{N-1}$ represents the thermal average with respect to the system with the [*i*]{}th site removed. $\langle s_k \rangle _{N-1}$ is not simply equal to $m_{k}$, which is the mean magnetization for the system of N spins. A correction term, first discovered by Onsager, appears: $$\label{onsager}
\langle s_k \rangle _{N-1}=m_{k}-\frac{1}{\hbar}
\int_{0}^{\beta \hbar}d\tau \frac{\delta m_{k}(\tau )}{h_{k}(\tau ')}
J_{i,k}m_{i}(\tau ')=m_{k}-\frac{1}{\hbar}\int_{0}^{\beta \hbar}
d\tau \left[ C(\tau) -Q\right]J_{i,k} m_{i}$$ Plugging Eq. (\[onsager\]) into Eq. (\[cavp2\]) and using the scaling of the couplings with $N$ one recovers the expression for $h_{i}^{cav}$ given in (\[tap1mean\]) in the $p=2$ case. Whereas for $G^{cav}$ a similar computation [^4] gives back the expression given in (\[tap2mean\]).
Finally, we remark that the main difference between the classical and the quantum TAP approach is that in the latter the cavity interaction consists not only in a cavity field but also in the “Weiss function” $G^{cav}(\tau -\tau ')$, which is a function of (imaginary) time. This already happens in the mean-field theory of quantum non-disordered systems [@Kotliar] for which local quantum fluctuations are taken into account exactly, whereas the spatial ones are frozen. For disordered systems, even in the limit of infinite dimensions, one has to take into account not only the local quantum fluctuations but also some spatial fluctuations: all the instantaneous magnetic fields have the same variance but their averaged values fluctuate from site to site.
A continuous disordered quantum model {#p-spin}
=====================================
In this Section we apply the method of Section \[sec1:formalism\] to the study of the quantum spherical $p$ spin-glass model. We derive and analyze the TAP free-energy density and the TAP equations for the local magnetization and correlation function in imaginary time. We relate these equations to the equation for the order parameter in the Matsubara replicated approach to equilibrium and in the Schwinger-Keldysh approach to the non-equilibrium dynamics.
The model and its TAP equations {#model}
-------------------------------
A model of a quantum particle with position ${\bf s}$ and momentum ${\bf p}$ that moves on an $N$-dimensional random environment is defined as $$H[{\bf p},{\bf s},J]= \frac{{\bf p}^2}{2M} - \sum^{N}_{i_1<...<i_p} J_{i_1...i_p}
s_{i_1} ... s_{i_p}
\; .
\label{eq:action}$$ A Lagrange multiplier $z$ enforces the averaged spherical constraint $${1\over N}\sum^{N}_{i=1} \langle s_i^2 \rangle = 1
\; .
\label{eq:lm}$$ The random interaction strengths $J_{i_1...i_p}$ are taken with zero mean and variance defined in Eq. (\[coupling\]). This model is a possible quantum extension of the spherical $p$ spin-glass model introduced in [@Crso1] and it is a particular realization of the class defined in (\[potential\]) corresponding to space-coordinates $s_{i}$ constrained to move on a N-dimensional sphere.
The zero-th order term of the expansion (\[series\]) can be readily computed for this model. By setting $\alpha=1$, rescaling time according to $\tau \to \tau\hbar/\tilde{J}$, and defining the “quantum parameter” $\Gamma \equiv \hbar^2/(\tilde{J}M)$ we obtain the following expression for the quantum TAP free-energy (\[eq:Gamma\]) : $$\begin{aligned}
& & -\beta F =
\frac{N}{2} \mbox{Tr} \, \mbox{ln} (C-Q)
+
\frac{N}{2\Gamma} \mbox{Tr} \left( \frac{ \partial^2 C}{ \partial \tau^2 } \right)
+
\int_0^\beta d\tau \sum_{i_1< \dots< i_p}
J_{i_1,\dots,i_p} m_{i_1}(\tau) \dots m_{i_p}(\tau)
\nonumber\\
& & \;\;\;\;\;\;
+
\frac{N}{4} \int_0^\beta d\tau \int_0^\beta d\tau'
\left(
C^p(\tau,\tau') - Q^p(\tau,\tau') - p (C(\tau,\tau') - Q (\tau,\tau') ) Q^{p-1}(\tau,\tau') \right)
\nonumber\\
& &
\;\;\;\;\;\;
- \frac{N\beta}{2} \int_0^\beta d\tau
z(\tau) \left( C(\tau,\tau)-1 \right)\end{aligned}$$ The physical parameters $m_i(\tau)$ and $C(\tau,\tau')$ are fixed by the quantum TAP equations (\[legendre\]) $$\begin{aligned}
\label{eq2}
& & h_i(\tau) =
\sum_{i_2 < \dots < i_p} J_{i, i_2,\dots,i_p} m_{i_2}(\tau) \dots
m_{i_p}(\tau) \\
& &
+ \int_0^\beta d\tau'
\left[
-(C-Q)^{-1}(\tau,\tau') + \frac{p(p-1)}{2}
\left( Q(\tau,\tau') - C(\tau,\tau') \right) Q^{p-2}(\tau,\tau')
\right] \, m_i(\tau')
\; ,\nonumber
\\
& &
z(\tau)\delta (\tau -\tau') =
(C-Q)^{-1}(\tau,\tau') + \delta(\tau-\tau') \frac{1}{\Gamma} \frac{\partial^2}{\partial\tau^2} +
\frac{p}{2} \left[ C^{p-1}(\tau,\tau') - Q^{p-1}(\tau,\tau') \right]
\label{eq1}
\; .\nonumber\end{aligned}$$
Finally, setting $h_i(\tau)=0$ and using that at stationarity, $m_i(\tau) = m_i$, $Q(\tau,\tau')= q_{\sc ea}$, $z(\tau)=z$ and $C(\tau,\tau')=C(\tau-\tau')$, the previous equations are simplified to $$\begin{aligned}
\frac{1}{\Gamma} \frac{\partial^2 C(\tau) }{\partial \tau^2}
&=&
-\frac{p}{2} \int_0^\beta d\tau'
\left( C^{p-1}(\tau-\tau')-q_{\sc ea}^{p-1}\right) \left( C(\tau')-q_{\sc ea} \right)
\nonumber\\
& &
+z \left(C(\tau)-q_{\sc ea}\right) -\delta(\tau)
\; ,
\label{eqC-Qstat}
\\ \hphantom{a}\nonumber \\
z m_i
&=&
\sum_{i_2< \dots < i_p}
J_{i, i_2,\dots,i_p} m_{i_2}\dots m_{i_p}
+
\nonumber \\
& &
m_i \; \frac{p}{2} \int_0^\beta d\tau' \;
\left(
C^{p-1}(\tau') +(p-2) q_{\sc ea}^{p-1}
-
(p-1) C(\tau') q_{\sc ea}^{p-2}
\right)
\label{eqmstat}
\; .\end{aligned}$$
Analysis of the quantum TAP equations {#analysis}
-------------------------------------
In the classical case the TAP equations admit a large number of solutions at low temperatures. In the following we shall show that this remains the case in a certain regime of $T$ and $\Gamma $. Furthermore we shall classify them by their Edwards-Anderson parameters. Finally, we shall exhibit several properties of the TAP solutions valid at low temperatures.
### A simple equation on the Edwards-Anderson parameter {#eqqea}
Let us analyze in detail the equations for the local magnetizations, $m_{i}$. First of all we note that a simple equation that relates $q_{\sc ea}$ to the potential energy density derives from Eq. (\[eq2\]). In fact, by multiplying Eq. (\[eq2\]) by $m_i/N$ and summing over $i=1,\dots N$ one obtains (for $h_{i}=0$) $$0 = - \frac{q_{\sc ea}}{\tilde C(0)-\beta q_{\sc ea}} +
\frac{p}{N}\sum_{i_1< \dots< i_p} J_{i_1\dots i_p} m_{i_1}
\dots m_{i_p} - \frac{p(p-1)}{2} \left( \tilde C(0)-\beta q_{\sc ea}\right) q_{\sc ea}^{p-1}
\label{eqq1}$$ where we introduced the discrete Fourier transform of the correlation $$\tilde C(\omega) \equiv \int_0^\beta d\tau \; e^{i\omega\tau} C(\tau)
\; .$$ Following Kurchan [*et al*]{} [@Kupavi], we introduce the [*angular*]{} variables $\sigma_i=m_i/\sqrt{q_{\sc ea}}$ and define the [*angular potential energy density*]{} $${\cal E}({\bf \sigma}) \equiv - \frac{1}{N} \sum_{i_1<\dots <i_p} J_{i_1\dots i_p} \,
\sigma_{i_1} \dots \sigma_{i_p}
\; .$$ For any fixed energy level ${\cal E}$, Eq. (\[eqq1\]) becomes a second-order polynomial equation for $q_{\sc ea}$; the solution is determined by $$q_{\sc ea}^{p/2-1} (\tilde C(0)-\beta q_{\sc ea}) = z_\pm =\frac{1}{p-1}
\left(
-{\cal E}(\sigma) \pm \sqrt{{\cal E}^2(\sigma)- {\cal E}_{\sc th}^2}
\right)
\label{eqq1final}$$ and ${\cal E}_{\sc th}$, called the threshold value, is given by $${\cal E}_{\sc th} = -\sqrt{\frac{2(p-1)}{p}}
\; .$$ The right-hand-side of Eq. (\[eqq1final\]) has to be real. This imposes the condition ${\cal E} \leq {\cal E}_{\sc th}$ since ${\cal E}$ is a negative quantity.
For each sign in Eq. (\[eqq1final\]), its left-hand-side has a bell-shape as a function of $q_{\sc ea}$. It vanishes at $q_{\sc ea}=0$ and $q_{\sc ea}=\tilde C(0)/\beta$ and attains its maximum at $q_{\sc ea}=(1-2/p)\tilde C(0)/\beta$. Hence, at fixed values of ${\cal E}$ and $T$, Eq. (\[eqq1final\]) has none or [*two*]{} solutions, $q_{\sc ea}=q', q''$, with $$\begin{aligned}
0 \leq &q'& \leq \frac{(1-2/p) \tilde C(0)}{\beta}
\; ,
\\
\frac{(1-2/p) \tilde C(0)}{\beta} \leq &q''& \leq \frac{\tilde C(0)}{\beta} \end{aligned}$$ (we assume, as expected, that $C(\tau )$ is positive for all $\tau $). In the classical case, the minus sign in Eq. (\[eqq1final\]) leads to a value of $q_{\sc ea}$ that is a minimum of the TAP free-energy for all ${\cal E} < {\cal E}_{\sc th}$. The Edwards-Anderson parameter determined in this way has the expected physical behavior [@Crso2]. In Appendix A we show that in the quantum case one has to choose the minus sign in Eq. (\[eqq1final\]), too. Thus, $q_{\sc ea}$ is determined by $$\label{eqq1finalnew}
q_{\sc ea}^{p/2-1} (\tilde C(0)-\beta q_{\sc ea}) = \frac{1}{p-1}
\left( -{\cal E}(\sigma) - \sqrt{{\cal E}^2(\sigma)- {\cal E}_{\sc th}^2} \right)
\; .$$ This equation still has two solutions. It can be proven that the solution with the larger absolute value of $q_{\sc ea}$ has the correct physical properties. In particular, it is connected to the classical solution, and it is then the solution to be kept. Thus there is a one to one correspondence between $q_{\sc ea}$ and ${\cal E}$.
It is of particular interest, as we shall show below, the [*threshold solution*]{} ${\cal E}={\cal E}_{\sc th}$. In this case the equation for $q_{\sc ea}$ becomes $$\label{qea-eq}
1 = \frac{p(p-1)}{2} \; (\tilde C(0) - \beta q_{\sc ea})^2 q_{\sc ea}^{p-2}
\; .$$
Note that this equation coincides with the one found with the Matsubara formalism using the marginality condition to fix the block size $x$ in the replica matrix [@Cugrsa]. Furthermore, it coincides with the equation for the dynamic value of the Edwards-Anderson parameter $q_{\sc ea} \equiv \lim_{t\to\infty} \lim_{t_w\to\infty} C(t+t_w,t_w)$ obtained from the study of the real-time dynamics of the quantum model evolving in contact with an Ohmic quantum environment [@Culo], when one takes first the thermodynamic limit, next the long-time limit of the system’s dynamics in contact with the environment and, finally, the strength of the coupling to the environment to zero. The relationship between TAP, Matsubara and dynamical approach will be discussed in Section \[reltapmatdyn\].
### Multiplicity of TAP solutions {#subsec:complexity}
The equations (\[eqq1final\]) reveal an interesting structure of the TAP equations (\[eqC-Qstat\]) and (\[eqmstat\]). For a given value of the angular potential energy ${\cal E}$ the TAP equations decouple in two different sets: Eqs. (\[eqC-Qstat\]) and (\[eqq1final\]), with the spherical condition on $C$, determine the correlation function, the spherical parameter and the Edwards-Anderson parameter; whereas Eqs. (\[eqmstat\]) determine the angular variables only. They read $$\begin{aligned}
\label{angular}
\mu q_{\sc ea}^{1-p/2} \sigma_i &=& -p {\cal E}({\bf \sigma})\sigma_i=
p \sum_{i_2 < \dots < i_p}
J_{i, i_2,\dots,i_p} \sigma_{i_2}\dots \sigma_{i_p}
\; ,
\\
\mu & \equiv &z- \frac{p(p-2)\beta}{2} q_{\sc ea}^{p-1} +
\frac{p(p-1)}{2} \tilde C(0) q^{p-2} - \tilde \Sigma(0)
\; ,
\label{eq-mu}\end{aligned}$$ where we have defined $$\tilde \Sigma(0) \equiv \frac{p}{2} \int_0^\beta C^{p-1}(\tau)
\; .$$ For a given value of the angular potential energy ${\cal E}$ Eqs. (\[angular\]) allow one to determine the angular part of the TAP solutions.
In general for a given value of ${\cal E}$, Eqs. (\[eqC-Qstat\]) and (\[eqq1final\]) determine the correlation function, the spherical parameter and the Edwards-Anderson parameter [*in a unique way*]{}. (An exception to this rule are the paramagnetic solutions which however do not correspond to any ${\cal E}$.) As a consequence, the multiplicity of TAP solutions is entirely due to Eq. (\[angular\]) which, for certain values of ${\cal E}$, can admit an exponential (in $N$) number of solutions ${\cal N}({\cal E})$. The complexity as a function of ${\cal E}$ is then defined as $$\sigma({\cal E}) \equiv \lim_{N\to\infty} \frac{1}{N}
\overline{\ln( {\cal N}({\cal E}) )}
\; ,$$ Equation (\[angular\]) already appears at the classical level. The complexity has been computed by Crisanti and Sommers [@Crso2] and Cavagna [*et al.*]{} [@Cagipa1] with the following result. There are typically no solutions for ${\cal E}<{\cal E}_{\sc eq}$, whereas for ${\cal E}_{\sc eq}
<{\cal E}<{\cal E}_{\sc th}$ the complexity reads $$\begin{aligned}
\label{complexity}
\sigma({\cal
E})&=&\frac{1}{2}\left(1+\ln\left(\frac{p}{2}\right)\right)
-
{\cal E}^2+
\left(\frac{{\cal E}- \sqrt{{\cal E}^2-{\cal E}_{\sc th}
^2}}{\sqrt{2}{\cal E}_{\sc th}}\right)^2 +
\ln\left(-{\cal E}- \sqrt{{\cal E}^2-{\cal E}_{\sc th} ^2}\right)\nonumber
\\
&&\qquad \qquad\qquad \qquad\qquad \qquad {\mbox {for}}
\quad \quad
{\cal E}_{\sc eq}<{\cal E}<{\cal E}_{\sc th}\end{aligned}$$ where ${\cal E}_{\sc eq}$ is the value at which $\sigma({\cal E})$ vanishes. A plot of this function is traced in Fig. \[comp.fig\] for $p=3$.
### A low temperature and low $\Gamma $ approximation {#approximate}
In the classical case the TAP equations separate in two sets: $N$ equations for the angular variables and one for the Edwards-Anderson parameter. The former admit an exponential number of solution and are studied from a statistical point of view (one computes the number of solution, and the typical properties of solution corresponding to a given ${\cal {E}}$), whereas the latter can be easily solved. In the quantum case the analysis of the equations for the angular variables is identical to the one used for classical systems. The analog of the equation that determines $q_{\sc ea}$ becomes now a differential equation for $C(\tau)$ that has to be studied numerically. This differential equation can be mapped exactly onto the ones analysed with the replica method, as we shall show in Section (\[reltapmatdyn\]), and its numerical solution can be found in [@Cugrsa]. In this section we perform a low-temperature and low $\Gamma $ approximation, also discussed in [@Cugrsa], that allows one to obtain some qualitative results that remain valid for the exact solution.
At low temperature and low $\Gamma $, the extension of the imaginary time-interval diverges $[0,\beta\to\infty]$ and the periodic correlation $C(\tau)$ is expected to have a rapid decay, over a short time-interval, from $1$ to its “asymptotic” value, say, at $\tau=\beta/2$. Moreover the “regular” part of the correlation, that we define as [@Cugrsa] $$q_{\sc reg}(\tau) \equiv C(\tau) - q_{\sc ea}$$ can be assumed to be small. Therefore we can expand the TAP free-energy in powers of $q_{\sc reg}(\tau)$. Up to terms of the order of $q_{\sc reg}(\tau)^{3}$ we obtain $$\begin{aligned}
\label{tapapprox}
& & -\frac{\beta F}{N} =
\frac{1}{2} \mbox{Tr} \, \mbox{ln} (q_{\sc reg}(\tau))
+
\frac{1}{2\Gamma} \mbox{Tr}
\left( \frac{ \partial^2 q_{\sc reg}(\tau)}{ \partial \tau^2 } \right)
+
\frac{\beta }{N} \sum_{i_1< \dots< i_p}
J_{i_1,\dots,i_p} m_{i_1} \dots m_{i_p}
\nonumber\\
& & \;\;\;\;\;\;
+
\beta \frac{p(p-1)}{4}q_{\sc ea}^{p-2} \int_0^\beta d\tau q_{\sc reg}^{2}(\tau )
- \frac{\beta}{2}
z \left(q_{\sc reg}(0)+q_{\sc ea}-1 \right)
\; ,\end{aligned}$$ where we have focused on the space of time translation invariant (TTI) functions (since the TAP solutions are TTI this does not imply a loss of generality).
Within this approximation the TAP equations become quadratic in Fourier space, $$1- \left( \frac{w^2_k}{\Gamma} + z \right) \tilde q_{\sc reg}(\omega_k) +
\frac{p(p-1)}{2} \, q_{\sc ea}^{p-2} \, \tilde q_{\sc reg}^2(\omega_k) =0
\; ,$$ and yield $$\label{qapprox}
\tilde q_{\sc reg}(\omega_k) =
\frac{z + \omega_k^2/\Gamma \pm \sqrt{(z + \omega_k^2/\Gamma)^2
- 2 p (p-1) q_{\sc ea}^{p-2}}}{p (p-1) q_{\sc ea}^{p-2}}
\; .$$ By taking $\omega_k=0$ and comparing to Eq. (\[eqq1finalnew\]) one obtains $${\cal E} = - \frac{z q_{\sc ea}^{1-p/2}}{p}
\; .$$ The spherical constraint reads $$1-q_{\sc ea} = \frac{1}{\beta} \sum_k \tilde q_{\sc reg}(\omega_k) = \int_0^\infty \frac{d\omega}{\pi} \; \chi''(\omega) \; \coth\left(\frac{\beta\omega}{2}\right)
\label{eq:constaint}$$ where $$\chi''(\omega) \equiv \mbox{Im} \; \tilde q_{\sc reg}(\omega_k=-i\omega)
=
\frac{q_{\sc ea}^{1-p/2}}{p-1} \sqrt{{\cal E}^2_{\sc th} -
\left({\cal E}+\frac{\omega^2 q_{\sc ea}^{1-p/2}}{p\Gamma} \right)^2 }
\; .
\label{eq:Gamma-q1}$$ The integral in Eq. (\[eq:constaint\]) has to be taken on the interval $\omega \in [\omega_-,\omega_+]$ such that the square root is real.
In the low temperature limit, we approximate $\coth(\beta\omega/2)\sim 1$ and, by changing variables in the integral, we obtain $$\Gamma I^2({\cal E}, p) = \frac{\pi^2 (p-1)^2}{p} (1-q_{\sc ea})^2 q_{\sc ea}^{(p-2)/2}$$ with $$I({\cal E}, p)=
2\int_{\sqrt{-{\cal E}+{\cal E}_{\sc th}}/2}^{\sqrt{-{\cal E}-{\cal E}_{\sc th}}}
dx \, \sqrt{{\cal E}^2_{\sc th} - \left({\cal E}+ x^2\right)^2 }
\; .$$ Equation (\[eq:Gamma-q1\]) yields a relation between $\Gamma$, $q_{\sc ea}$ and ${\cal E}$ of the form $$\Gamma I^2({\cal E},p) = \mbox{ct} (1-q_{\sc ea})^2 q_{\sc ea}^{(p-2)/2}
\; ,$$ with $\mbox{ct}$ a numerical constant. For each ${\cal E}$, there is a solution with a physically meaningful value of $q_{\sc ea}$ that is close to $1$, until reaching a critical $\Gamma_{\sc max}({\cal E})$. This value tells us when the TAP solutions associated to ${\cal E}$ disappear. It can be easily proven that $I({\cal E},p)$ is a growing function of ${\cal E}$; hence, $\Gamma_{\sc max}({\cal E})$ is a decreasing function of ${\cal E}$. This implies that the TAP solutions that are at the threshold level disappear first than those that are at lower values of ${\cal E}$. This is again similar to the dependence of the classical TAP solutions with temperature [@Kupavi]: the solutions corresponding to the threshold level disappear at a lower temperature than the ones corresponding to the equilibrium level $T_{\sc max}({\cal E}_{\sc th})<T_{\sc max}({\cal E}_{\sc eq})$ and, more generally, $T_{\sc max}({\cal E}_1)<T_{\sc max}({\cal E}_2)$ if ${\cal E}_1 > {\cal E}_2$.
The low frequency behavior of the spectral density $\chi''(\omega) \equiv \mbox{Im} \; \tilde q_{\sc reg}(\omega_k=-i\omega)$ of the threshold states is gapless, $$\label{chinogap}
\chi''(\omega)\sim \omega \qquad {\mbox{for}} \quad \omega \rightarrow 0^{+}
\; ,$$ whereas all the other states (${\cal {E}}<{\cal {E}}_{\sc th}$) have a gap $\Delta $ in their excitation spectrum, $$\label{chinogap2}
\chi''(\omega)\sim \sqrt{\omega-\Delta } \qquad {\mbox{for}}
\quad \omega \rightarrow \Delta^{+}
\; .$$
Furthermore we have studied the dependence of the free-energy on $\cal {E} $ and $\Gamma $ in the low temperature limit. Plugging the solution (\[qapprox\]) into (\[tapapprox\]) we find, after a tedious computation, $$\label{freeenergyapprox}
-\frac{\beta F}{N} =-\int \frac{d\omega}{\pi \Gamma}
\ln \left(2\sinh \left(\frac{\beta \omega }{2}\right)\right)
\; \omega \chi''(\omega)+\frac{\beta q_{\sc ea}^{p/2}}{2}
\left(p-2-\frac{p}{q_{\sc ea}} \right) {\cal {E}}$$ where $q_{\sc ea}$ satisfies Eq. (\[eqq1finalnew\]). This expression allows one to study the evolution of the free-energy of the TAP states as a function of $\Gamma $. We have found that if one knows a TAP solution at zero temperature and zero $\Gamma $ one can follow it continuously in $\Gamma$. As in the classical case TAP solutions do not cross, merge nor divide in this model.
Figure \[fgamma.fig\] summarizes these results in a schematic way. Finally, note the special role of the threshold states, which are gapless (contrary to the others), the first ones to disappears and the ones with highest free-energy density.
### Stability of TAP states {#properties}
In the classical case one can check the stability of TAP states. In the quantum case this is a difficult task that has to be performed numerically. In the following we shall limit ourself to prove that the TAP solutions characterized by ${\cal {E}}={\cal {E}}_{\sc th}$ (threshold states) are characterized by zero modes and are hence marginally stable. We expect that a complete stability analysis will confirm that the TAP states characterized by ${\cal {E}}<{\cal {E}}_{\sc th}$ are stable.
Let us focus on the reduced free-energy Hessian $\partial^{2}F
/\partial m_{i}(\tau )\partial m_{j}(\tau' )$ evaluated in a TAP solution $\{m_{i}^{\alpha } \}$. This matrix depends on $\tau, \tau '$ only through their difference. Therefore it is diagonal in Fourier space. Focusing on zero frequency, the original problem reduces to the diagonalization of the following matrix: $$\label{hessian}
A_{i,j}=-\sum_{i_{3}<\dots <i_{p}}J_{i,j,i_{3},\dots ,i_{p}}m_{i_{3}}^{\alpha }
\cdots m_{i_{p}}^{\alpha }-p{\cal {E}}q_{\sc ea}^{p/2-1}\delta _{i,j}
\; ,$$ where $q_{\sc ea}=\sum_{i}(m_{i}^{\alpha })^{2}/N$. The density of eigenvalues of $\mathbf{A}$ has been computed in [@replicon] and, except for the isolated eigenvalue corresponding to the eigenvector $m_{i}^{\alpha }$, it is a semicircular law centered in $-p{\cal {E}}q_{\sc ea}^{p/2-1}$ with width $-p{\cal {E}}_{\sc th}q_{\sc ea}^{p/2-1}$. Consequently, threshold states are characterized by a vanishing fraction of zero modes.
### The classical limit {#classical}
The classical limit of Eqs. (\[eqC-Qstat\]) and (\[eqmstat\]) yields the classical TAP equations computed by Kurchan [*et al.*]{} [@Kupavi]. In fact, in the classical limit, $C(\tau)=1$ and the parameter $z$ is fixed by integrating Eq. (\[eqC-Qstat\]) between $0^+$ and $\beta^+$. This yields $$z = \frac{1}{\beta(1-q)} + \frac{p\beta}{2} \left( 1- q^{p-1}\right
)
\; .$$ By inserting this value of $z$ in Eq. (\[eqmstat\]) we obtain $$\left( \frac{1}{\beta(1-q)} + \frac{p(p-1)\beta}{2} (1-q) q^{p-2}
\right) m_i =
p \sum_{i_2 < \dots < i_p}
J_{i, i_2,\dots,i_p} m_{i_2} \dots m_{i_p}
\; ,
\label{eq-sigma_class}$$ that coincide with the classical TAP equations for the local magnetizations. The equation that fixes $q_{\sc ea}$ as a function of ${\cal E}$ in the classical limit is simply obtained from Eq. (\[eqq1final\]) by setting $\tilde C(0)=1$.
Relation between TAP, Matsubara and dynamic approaches {#reltapmatdyn}
------------------------------------------------------
In Section \[introtap\] we have recalled the relationship between TAP, replica and dynamical approaches in the classical case. In this Subsection we show how these connections are generalized to quantum systems.
### TAP-Matsubara
Via the replica analysis in the Matsubara imaginary-time framework and within a 1step RSB Ansatz, the order parameter is the $n\times n$ matrix $Q_{ab}$ which is fully described by: $n$ identical diagonal elements $q_d(\tau)$ that depend on the the imaginary time $\tau$, $n(x^2-1)$ constant elements $q_{\sc ea}$ that occupy the $x\times x$ blocks around the diagonal, the remaining $n^2-n(x^2-1)$ elements $q_0$ that in the absence of an external field are identically zero. In the $n\to 0$ limit, the three parameters $q_d(\tau)$, $q_{\sc ea}$ and $x$, together with the value of the Lagrange multiplier that enforces the averaged spherical constraint determine the full solution of the problem [@Cugrsa].
The connection between TAP and the Matsubara approaches is obtained by identifying the Edwards-Anderson parameters $q_{\sc ea}$ in the two approaches, $C(\tau)$ with the $\tau$-dependent diagonal parameter $q_d(\tau)$ in $Q_{ab}$, and the Lagrange multipliers. In particular we have shown that subtracting the equation obtained for $a\neq b$ from the one corresponding to $a=b$ one obtains Eq. (\[eqC-Qstat\]). In the Matsubara approach one has another equation for $q_{\sc ea}$ in which $x$ acts as an external parameter. Therefore by fixing the value of the breakpoint one fixes the value of $q_{\sc ea}$. As in the classical case two different recipes to fix the breaking point parameter $x$, namely optimization and the marginality condition, lead to the static and dynamic transitions, respectively. In the TAP approach the role of $x$ is played by ${\cal E}$ that enters the equation for $q_{\sc ea}$ as a parameter. We have found that the relationship between $x$ and $\cal E$ is encoded in $$\label{mE}
\beta x=\frac{\partial\sigma(\beta, f) }{\partial f}
\; ,$$ where $\sigma $ is the complexity defined in (\[complexity0\]), see Appendix B. This suggests that in the a quantum problem the relationship (\[reltaprep\]) is generalized to $$\begin{aligned}
\label{reltaprepquantum}
-\lim_{N\rightarrow \infty }\frac{1}{\beta N}\sum_{\alpha }e^{-\beta x Nf_{\alpha }}
&=&-\lim_{N\rightarrow \infty }\frac{1}{\beta N}\ln \int df e^{N(-\beta xf+\sigma (\beta,f))}
\\
&=&
x{\mbox {Extr}}_{q_{\sc ea},q_{d}(\tau )}f_{\sc rep}(q_{\sc ea},q_{d}(\tau );x,\beta,\Gamma )\nonumber\end{aligned}$$ Using Eq. (\[mE\]) we have found that the Matsubara equations for $q_{\sc ea}$ and $q_{d}(\tau )$ and the TAP equations for $q_{\sc ea}$ and $C(\tau )$ coincide. For instance, the TAP equations for the highest TAP states (threshold states) and the lowest TAP states coincide with the ones obtained in the Matsubara approach by using the marginality condition and the extremization with respect to $x$, respectively. Moreover, as another confirmation of Eq. (\[reltaprepquantum\]) we have verified that the free-energy obtained from the Matsubara computation [@Cugrsa] equals the one obtained in the TAP approach for all values of $\beta $ and $\Gamma $. In other words, we have checked that $$\label{check}
-\beta F=\ln \sum_{\alpha }e^{-\beta F_{\alpha }}
\; .$$ As a consequence the phase diagram that follows from the TAP approach coincides with the one obtained in [@Cugrsa].
### TAP-out of equilibrium dynamics
.
The study of the real-time dynamics of the p-spin quantum model evolving in contact with an Ohmic quantum environment has been performed in [@Culo]. The dynamical behavior is characterized by two regimes. At high temperature and high $\Gamma $ the system equilibrates in the paramagnetic state via an equilibrium dynamics. Whereas at low temperature and low $\Gamma $ the systems ages and remains out of equilibrium also at infinite times. As in the classical case we have found that the long-time out of equilibrium dynamics is dominated by the threshold states. This is proven by the fact that the equations for $q_{\sc ea}$ and $C(\tau)$ within the TAP approach coincide with the ones obtained from the study of the real-time dynamics, when the following limits are taken in its precise order: $\lim_{\gamma\to 0} \lim_{t\to\infty} \lim_{N\to \infty}$. In order words, when one takes first the thermodynamic limit, next the long-time limit of the system’s dynamics in contact with the environment and, finally, the strength of the coupling to the environment $\gamma$ to zero. Notice that this equivalence holds for the paramagnetic states also. Finally, we have shown that the relationship between the effective temperature [@Cukupe] arising in the asymptotic out of equilibrium regime [@Culo] and the complexity is, as classically, $$\label{Teff}
\frac{1}{T_{\sc eff}}=\left. \frac{\partial\sigma(\beta, f) }
{\partial f}\right|_{f=f_{\sc th}}
\; .$$
Note that the connection between TAP and real-time dynamics is done by identification of several equations. A more precise analysis, along the lines of [@Bi], should prove the full equivalence of the two methods.
The phase diagram of discontinuous glassy systems {#transition}
=================================================
In this section we present some general arguments that allow one to predict the phase diagram of discontinuous glassy systems. Since the free-energy landscape plays a key role, we expect these results to have a certain degree of universality and to apply to this entire class of disordered systems.
The static transition
---------------------
Let us focus on two limiting regions of the phase diagram: around the classical phase transition ($\Gamma =0$) and around the quantum phase transition ($T=0$).
In the former case the physics is well known and it is reviewed in Section \[introtap\]. The effect of switching on weak quantum fluctuations consists only in a weak variation of the complexity $\sigma $. For this reason the effect of quantum fluctuations reduces simply to a variation of the thermodynamic ($T_{s}$) and the dynamic ($T_{d}$) transition temperatures (respectively lines $(1)$ and $(3)$ in Fig. \[diag.fig\]).
At zero temperature and low $\Gamma $ the system is in the glassy phase (GP), whereas at very high $\Gamma $ quantum fluctuations destroy the glassy phase and the system is a quantum paramagnet (QPM). As a consequence one expects that a quantum phase transition should divide these two regimes at a certain value $\Gamma_{c}$.
At zero temperature the complexity is expected to remain a smooth function of the free-energy density[^5]. Consequently, equation (\[eq:weighted3\]) implies that the sum over the exponential number of glassy states is always dominated by the lowest ones in free-energy since the $\beta f$ term in the exponential largely dominates in this limit (this is different from the classical problem in which other states dominate between $T_s$ and $T_d$). At zero temperature these are the states with lower angular potential energy, [*i.e.*]{} with ${\cal E}={\cal E}_{\sc eq}$. Now, from Fig. \[comp.fig\] we conclude that $\sigma({\cal E}_{\sc eq})=0$ at zero temperature and for all $\Gamma$. For this reason the mechanism behind the transition must be totally different from the classical one. The transition cannot be related a configurational entropy that vanishes when approaching $\Gamma_c$ from above (“entropy crisis”) since this quantity is always zero at zero temperature.
Indeed, according to Eq. (\[mE\]), if we assume that $\partial
\sigma(\beta,f)/\partial f < +\infty$ when $T\to 0$, then $x\to 0$ for all $\Gamma$ in the glassy phase. In the paramagnetic phase instead $x=1$. Thus, $x$ must jump at the transition. If the Edwards-Anderson parameter also jumps at $\Gamma_c$, the susceptibility is discontinuous, and the transition is of first order thermodynamically. As in the previous case the effect of switching on thermal fluctuations reduces simply, for low $T$, to a variation of $\Gamma _{c}$ (line (2) in Fig. \[diag.fig\]).
Another hint on the difference between classical and quantum phase transition can be gained by a technical remark. It is well known that the paramagnetic solution of the classical problem remains stable in the low temperature phase. This is a spurious solution of the mean field equations which has to be discarded in the analysis of the low temperature regime. In the quantum case, one also expects to find a spurious paramagnetic solution, which is the continuation of the classical paramagnet to low temperatures. This solution exists to the left of line $(1)$ in Fig. \[diag.fig\], consequently one expects coexistence of two paramagnetic solutions: a physical one which is the continuation of the quantum paramagnet valid at low temperatures and high $\Gamma$ and a spurious one which is the continuation of the classical paramagnet.
In the classical case the transition is of second order even if the order parameter jumps discontinuously. This peculiar behavior is due to the fact that near the transition the paramagnetic state is fractured into an exponential number of states which continuously become the ones responsible for the glassy phase at low temperature. This is not possible at zero temperature (the quantum paramagnet is not formed by a collection of glassy states) and therefore it is reasonable to expect a quantum first order phase transition between the glass phase and the quantum paramagnet.[^6]
Finally, note that $F_{\sc qpm}=F_{\sc gp}=F_{\sc cpm}$ on the point B. We then expect that a first order transition line separating the QPM from the CPM starts at this point. This line should end on a point $C$ given that for very large value of $\Gamma $ and $T$ the quantum and thermal fluctuations are so strong that the system becomes non interacting and in this case only one paramagnetic phase exists. In the analysis of the $p=3$ spherical spin-glass model [@Cugrsa] the line $BC$ has not been found. We conjecture that in this case the line $BC$ is so short that it is very difficult to find numerically. Furthermore, within the accuracy of the algorithm, the dynamic and static critical lines collapse at the point B. In the quantum model studied in [@Niri] instead this line has been detected and it was demonstrated in this paper that its length increases with $p$.
The dynamic transition
----------------------
Now that the equilibrium phase diagram is completely predicted from a qualitatively point of view, we can focus on the non-equilibrium regime. As noted previously, low quantum fluctuations simply change the values of $T_{d}$ but do not change qualitatively the dynamic transition which remains second order in the sense that the asymptotic energy is continuous across the transition, but its derivative is not. This remains true until the line $(2)$ reaches the line $BC$. After this point the dynamic transition between the the quantum paramagnet and the threshold states becomes first-order, i.e. the asymptotic energy is not continuous across the transition. This, of course, is very difficult to see numerically since the discontinuity has a very small value.
Summary
-------
In summary, through some general arguments based on the TAP approach we have predicted a phase diagram that should have a certain degree of universality since its form is determined by the qualitative form of the free-energy landscape. Indeed, not only the quantum $p$ spin spherical model exhibits the phase diagram displayed in Fig. \[diag.fig\] but some other classical and quantum models share exactly the organization of phases and transitions [@Gash; @Mosh; @Niri; @Opper].
Conclusion {#conclu}
==========
In this paper we have derived TAP equations for a large class of mean field disordered quantum system. Moreover we have applied the TAP approach to the quantum version of the spherical $p$ spin model. The study of this system, whose real-time dynamics and statics have been analyzed in [@Culo; @Cugrsa], has furnished an ideal benchmark to generalize to the quantum case several concepts developed for classical disordered systems. Armed with this knowledge, founded on the study of the free-energy landscape, we have shown that the same phase diagram, presented in Fig. \[diag.fig\], naturally emerges in a large class of quantum disordered systems, those having a classical discontinuous transition.
Whether other models like the SK model in a transverse field, or its soft spin version studied in [@Chamon], also have such crossover in the transition from the disordered to the ordered phase is an issue that deserves revision. For the moment, no study of models with classical continuous transition has shown this feature. However, it might have been masked by the methods used in previous studies. The soft SK model might be the easiest example where to answer this equation via, [*e.g.*]{}, a careful application of the replicated Matsubara approach.
We would like to stress that the TAP approach furnishes an alternative and more transparent route to replicas which has also the advantage of showing explicitly the weakness of the mean field description. Let us cite one example. The marginality prescription in the replica approach becomes the more transparent statement that the non-equilibrium dynamics is dominated by the TAP states which are marginally stable, [*i.e.*]{} the flatness of the free-energy landscape around these states is responsible for aging. Concerning one of the weaknesses of the mean field description we would like to underline that the enormous number of pure states (with different free-energy densities) found for mean field models cannot persist in finite dimensions and the majority of them should become [*metastable*]{} states. How this changes the mean field scenario is an active domain of research for classical systems [@Biku].
We remark that interesting continuations of our work concern, on the one hand, the application of the TAP approach to different quantum mean-field models [@quantum_mf; @rotors; @Cesare] and, on the other hand, the generalization of the static quantum TAP approach to real-time dynamics (for classical systems this has been done in [@Bi]). This would allow one to show the relationship between long-time dynamics and free-energy landscape for quantum systems directly. Finally, the precise definition of a “quantum state” is a delicate matter and merits further analysis. In this paper we have simply called “state” a minimum of the TAP free-energy density. One possible way to verify the existence and stability of these states is by studying the dynamics of this system starting from particular initial conditions as done in [@Alain; @PaFr] for the classical model. This study is underway [@Cugrsa1].
[**Acknowledgments**]{} We wish to thank D. R. Grempel, J. Kurchan, G. Lozano and C. A. da Silva Santos for very useful discussions. G. B. is supported by the Center of Material Theory, Rutgers University, NJ, USA. L. F. C. thanks ECOS-Sud for a travel grant and financial support from the grant “Algorithmes d’optimisation et syst[è]{}mes quantiques d[é]{}sordonn[é]{}s”, ACI-Jeunes Chercheurs, 2000.
[**Appendix A**]{}
In this appendix we show that the equation for $q_{\sc ea}$ that leads to a solution with the correct physical properties is Eq. (\[eqq1final\]) with the minus sign. Indeed, we search a solution that corresponds to a minimum of the TAP free-energy. The full stability analysis, that involves the evaluation of the complete Hessian of the TAP free-energy, is rather hard and has to be done numerically (for instance, the form of $\tilde C(\omega)$ can only be obtained numerically).\
However, we can still perform a partial analysis that suffices to justify the choice of the minus sign. Let us concentrate on the following diagonal elements of the Hessian: $$\begin{aligned}
\label{q-stab}
\frac{\delta(-\beta F)}{\delta q_{\sc ea}^2}
&=&
-\frac{1}{2}\left[\left(1-\frac{p}{2}\right)\tilde{C}(0)+\frac{p}{2}\beta q_{\sc ea}
\right]
\left[\frac{1}{(\tilde{C}(0)-\beta q_{\sc ea})^{2}q_{\sc ea}}-\frac{p(p-1)}{2}q_{\sc ea}^{p-3}
\right]
\; ,
\\
\label{C(0)-stab}
\frac{\partial (-\beta F) }{\partial \tilde{C}(0)
\partial \tilde{C}(0)}
&=&
\frac{-1}{(\tilde{C}(0)-q\beta )^{2}}
+\frac{p(p-1)}{2\beta }\int_{0}^{\beta } d\tau C^{p-2}(\tau)
\; .\end{aligned}$$ From $z_\pm $’s definition we find $$\label{ineg}
z_{-}\le \frac{-{\cal E}_{\sc th}}{p-1}\; , \qquad\qquad
z_{+}\ge \frac{-{\cal E}_{\sc th}}{p-1} \; .$$ Since $q_{\sc ea}$ is fixed by Eq. (\[eqq1final\]), the second factor on the right-hand-side of Eq. (\[q-stab\]) is positive (negative) for $z_{-}$ ($z_{+}$). A stable solution corresponds to a negative value of (\[q-stab\]) and (\[C(0)-stab\]), therefore one has to take the solution $q''$ for $z_{-}$ and $q'$ for $z_{+}$. Moreover, since for ${\cal E}<{\cal E}_{\sc th}$ the right hand side of (\[eqq1final\]) is positive then we obtain that $\tilde{C}(0)-\beta q'$ and $\tilde{C}(0)-\beta q''$ are positive and using that $$\left(\frac{1}{\beta }\int_{0}^{\beta }C^{\alpha }(t)dt \right)\ge
\left(\frac{1}{\beta }\int_{0}^{\beta }C(t)dt \right)^{\alpha }
\qquad \alpha >1$$ and imposing that (\[C(0)-stab\]) has to be negative, we obtain: $$\frac{1}{(\tilde{C}(0)-\beta q_{\sc ea})
^{2}}-\frac{p(p-1)}{2}q_{\sc ea}^{p-2}\ge 0\qquad \mbox{for}\quad q=q',q''$$ But this is impossible for $z_{+}$, i.e. it is not possible to have a consistent stable $z^{+}$ solution.
[**Appendix B**]{}
In this Appendix we give an argument in favor of the relation $\beta x=\partial
\sigma(\beta,f)/\partial f$ in Eq. (\[mE\]). Let us start by assuming that it does hold and see that it leads to the equation linking $x$, $q_{\sc ea}$ and $C(\tau)$ in the Matsubara approach. First of all we write the derivative of $\sigma $ with respect to $f$ as a derivative with respect to $\cal E$. This can be easily done by noticing that differentiating $f$ in Eq. (\[eq:Gamma\]) with respect to $\cal E$ at $\beta$ and $\Gamma $ fixed is equivalent to differentiating $f$ with respect to $\cal E$ at $\beta ,\Gamma,q_{EA}$ and $C(\tau ) $ fixed because $f$ is stationary in $q_{EA}$ and $C(\tau ) $. As a consequence we find $$\beta x = \frac{\partial \sigma}{\partial f} =
\frac{\partial \sigma}{\partial {\cal E}}
\frac{\partial {\cal E}}{\partial f} =
\frac{\partial \sigma}{\partial {\cal E}} \; q_{\sc ea}^{-p/2}
\; .$$ The derivative $\partial \sigma/\partial {\cal E}$ can be easily computed from Eq. (\[complexity\]). One arrives at $$\frac{p}{2} = \left( \tilde C(0)^2 -\beta^2 q_{\sc ea} (x-1) + \beta
q_{\sc ea} \tilde C(0) (x-2) \right)^{-1}
\; .$$ In summary, starting from the TAP approach at fixed ${\cal E}$ we obtain $q_{\sc ea}$ and $C(\tau)$. Assuming then that Eq. (\[mE\]) holds we obtain the equation linking $q_{\sc ea}$ and $x$ in the Matsubara approach. The equations for $C(\tau)$ in the TAP and Matsubara approaches, once $q_{\sc ea}$ is fixed, are identical. Hence we have proven that Eq. (\[mE\]) leads to the Matsubara results in [@Cugrsa].
The proof will be complete if we showed the other sense of the implication.
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[^1]: Unit[é]{} Mixte de Recherche du Centre National de la Recherche Scientifique et de l’Ecole Normale Sup[é]{}rieure.
[^2]: Note that within the TAP approach one does not average over disorder from the beginning as in the replica method. Consequently the TAP free energy depends on the particular realization of disorder.
[^3]: The results obtained in this section can be straightforwardly generalized to more complicate potentials containing different monomials or characterized by a more complicate tensorial coupling between $s_{i_{1}}^{\alpha _{1}}, \cdots , s_{i_{p}}^{\alpha _{p}}$.
[^4]: Indeed easier since there is no reaction term for $G^{cav}$ on a completely connected lattice.
[^5]: For instance the complexity does not blow up for $T\rightarrow 0$ since it is bounded by the logarithm of the number of energy minima divided by $N$, which is a finite quantity independent of temperature.
[^6]: Note however that other scenarios are possible. For example the number of glassy states could diminish when $\Gamma$ increases and vanish exactly at $\Gamma_{c}$. In this case the glassy states could be grown up continuously from the quantum paramagnet.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The Minority Game (MG) is a basic multi-agent model representing a simplified and binary form of the bar attendance model of Arthur. The model has an informationally efficient phase in which the agents lack the capability of exploiting any information in the winning action time series. We illustrate how a theory can be constructed based on the ranking patterns of the strategies and the number of agents using a particular rank of strategies as the game proceeds. The theory is applied to calculate the distribution or probability density function in the number of agents making a particular decision. From the distribution, the standard deviation in the number of agents making a particular choice (e.g., the bar attendance) can be calculated in the efficient phase as a function of the parameter $m$ specifying the agent’s memory size. Since situations with tied cumulative performance of the strategies often occur in the efficient phase and they are critical in the decision making dynamics, the theory is constructed to take into account the effects of tied strategies. The analytic results are found to be in better agreement with numerical results, when compared with the simplest forms of the crowd-anticrowd theory in which cases of tied strategies are ignored. The theory is also applied to a version of minority game with a networked population in which connected agents may share information.
[**Paper to be presented in the 10th Annual Workshop on Economic Heterogeneous Interacting Agents (WEHIA 2005), 13-15 June 2005, University of Essex, UK.**]{}
author:
- 'K.P. Chan$^*$, Pak Ming Hui$^*$, and Neil F. Johnson$^+$'
title: 'Decision Making, Strategy dynamics, and Crowd Formation in Agent-based models of Competing Populations'
---
Introduction
============
Agent-based models represent an efficient way in exploring how individual (microscopic) behaviour may affect the global (macroscopic) behaviour in a competing population. This theme of relating macroscopic to microscopic behaviour has been the focus of many studies in physical systems, e.g., macroscopic magnetic properties of a material stem from the local microscopic interactions of magnetic moments between atoms making up of the material. In recent years, physicists have constructed interesting models for non-traditional systems and established new branches in physics such as econophysics and sociophysics. The Minority Game (MG) proposed by Challet and Zhang [@Challet; @Challetbook] and the Binary-Agent-Resource (B-A-R) model proposed by Johnson and Hui [@BAR1; @book; @BAR3; @preprint; @BAR2], for example, represent a typical physicists’ binary abstraction of the bar attendance problem proposed by Arthur [@Arthur; @volatility]. In MG, agents repeatedly compete to be in a minority group. The agents have similar capabilities, but are heterogeneous in that they use different strategies in making decisions. Decisions are made based on the cumulative performance of the strategies that an agent holds. The performance is a record of the correctness of the predictions of a strategy on the winning action which, in turn, is related to the collective behaviour of the agents. Thus, the agents interact through their decision-making process, creation of the record of winning actions, and strategy selection process. Interesting quantities for investigations include the statistics of the fraction of agents making a particular choice $A(t)$ every time step and the variance or standard deviation (SD) $\sigma$ of this number [@Challet; @book]. These quantities are related in that knowing the distribution of $A$, one may obtain $\sigma$. The MG, suitably modified, can be used to model financial markets and reproduce stylized facts. The variance, for example, is a quantity related to the volatility in markets [@book].
Recently, we proposed a theory of agent-based models based on the consideration of decision-making and strategy dynamics [@NETMG1]. The importance of the strategy selection dynamics has been pointed out by D’Hulst and Rodgers [@rodgers]. This approach [@rodgers; @NETMG1], which we refer to as the strategy-ranking theory (SRT), emphasizes on how the strategies performance ranking pattern changes as the game proceeds and the number of agents using a strategy in a certain rank for making decisions. It is recognized that the SRT has the advantages of including tied strategies into consideration and avoiding the troublesome in considering each strategy’s performance separately. The theory, thus, represents a generalization of the crowd-anticrowd theory [@book; @preprint; @crowd; @crowd1] to cases with tied strategies and strategy ranking evolutions – two factors that are particularly important in the so-called informationally efficient phase of the MG. The theory has been applied successfully to explain non-trivial features in the mean success rate of the agents in (i) MG with a population of non-networked [@rodgers] or networked agents [@NETMG1; @NETMG2], (ii) MG with some randomly participating agents [@RPA], and (iii) B-A-R model with a tunable resource level [@BAR2]. In this conference paper, we aim to illustrate the basic ideas of SRT. In particular, we present results based on SRT in evaluating the distribution of $A(t)$ and $\sigma$, in the efficient phase of MG in non-networked and networked populations. Validity of the results of our theory is tested against results obtained by numerical simulations. While the SRT was developed within the context of MG, many of the ideas are should also be appliable to a wide range of agent-based models.
Model: The Minority Game
========================
The basic MG [@Challet; @Challetbook] comprises of $N$ agents competing to be in a minority group at each time step. The only information available to the agents is the history. The history is a bit-string of length $m$ recording the minority (i.e., winning) option for the most recent $m$ time steps. There are a total of $2^{m}$ possible history bit-strings. For example, $m=2$ has $2^2=4$ possible histories of the winning outcomes: $00$, $01$, $10$ and $11$. At the beginning of the game, each agent picks $s$ strategies, with repetition allowed. They make their decisions based on their strategies. A strategy is a look up table with $2^{m}$ entries giving the predictions for all possible history bit-strings. Since each entry can either be ‘0’ or ‘1’, the full strategy pool contains $2^{2^{m}}$ strategies. Adaptation is built in by allowing the agents to accumulate a merit (virtual) point for each of her $s$ strategies as the game proceeds, with the initial merit points set to zero for all strategies. Strategies that predicted the winning (losing) action at a given time step, are assigned (deducted) one virtual point. At each turn, the agent follows the prediction of her best-scoring strategy. In case of tied best-scoring strategies, a random choice will be made to break the tie.
In the present work, we will focus on the regime where $2\cdot
2^{m} \ll N\cdot s$, i.e., the efficient phase. In MG literature, a parameter $\alpha = 2^{m}/N$ is defined with $\alpha <
\alpha_{c} \approx 0.34$ characterizing the efficient phase [@Challet1]. Features in this regime is known to be dominated by the crowd effect [@crowd; @crowd1]. A quantitative theory in this regime would have to include the consideration of frequently occurred tied strategies into account, as the dynamics in this regime is highly sensitive to the agents’ strategy selection. In what follows, we introduce the basic physical picture of the strategy ranking theory and apply it to evaluate the distribution in the fraction of agents making a particular choice $P(A)$ and the variance $\sigma^{2}$ from an analytic expression for non-networked and networked populations.
Numerical and analytical results: Non-networked Agents
======================================================
To put our discussions into proper context, we will first present the numerical results of the quantities that we are focusing on. Let $A(t)$ be the fraction of agents taking the action “1" (or “0") at time step $t$. As the game proceeds, there will be a time series $A(t)$. We may then analyze these values of $A(t)$ by considering the distribution or probability density function $P(A)$, where $P(A)dA$ is the probability of having a value within the interval $A$ to $A+dA$. In using the MG for market modelling, $A(t)$ can be taken to be the fraction of agents deciding to buy (or sell) an asset at time $t$. In the context of the El Farol bar attendance problem [@Arthur; @volatility], $A(t)$ may be taken to be the fraction of agents attending the bar. Note that every realization of the MG may have a different distribution of strategies among the agents and a different initial bit-string to start the game. These details do not affect the main results reported here, especially when we consider cases deep into the efficient phase, i.e., when $2\cdot 2^{m} \ll N$. To illustrate the point, we have carried out detailed numerical simulations for the simplest case of $m=1$ and $s=2$. Figure 1 shows the numerical results (squares) of $P(A)$ for systems with two different sizes ($N=129$ and $N=4097$), with the aim of emphasizing the size effect on $P(A)$. Notice that the distribution consists of a few peaks (five peaks for the case of $m=1$ and $s=2$), indicating that as the game proceeds the number $A(t)$ jumps among values characterized by these peak values. For larger population, the peaks are sharper. Also shown in Fig.1 are the results of the strategy ranking theory (lines). The theoretical results are in reasonably agreement with numerical results. We defer the discussion on obtaining the theoretical results to the next section.
Besides the typical results shown in Fig.1, we have studied the variance $\sigma^{2}$ in the following way. We carried out numerical simulations in many realizations using different values of $m$ and $N$, with $N$ up to $8193$ and $m$ up to $8$. For each run, a value of $\sigma^{2}$ is obtained. To facilitate comparison with theory, we select those data that are deep in the efficient phase, i.e., with $2\cdot 2^{m}/N < 0.125$ and plotted them (black dots) in Fig.2 to show the dependence of $\sigma^{2}/N^{2}$ on $m$. The data points do not show significant scatter, and essentially fall on a line. Also included in the figure are two (dashed) lines corresponding to two approximations within the crowd-anticrowd theory [@preprint; @crowd; @crowd1]. These approximations assume that all the strategies can be ranked at every time step without tied virtual points. One of them assumes that the popularity rankings, i.e., ranking based on the number of agents using a strategy, of a strategy and its anti-correlated partner are uncorrelated and gives an expression for $\sigma^{2}/N^{2}$ for cases with $s=2$ as [@crowd] $$\frac{\sigma_{flat}^2}{N^{2}} = \frac{1}{24 \times 2^m} \left[1 -
(\frac{1}{2 ^ {m+1}})^2 \right].$$ Another approximation is that the ranking of strategies are highly correlated. For example, the anti-correlated partner of the momentarily most-popular strategy is the least-popular one, and so on. This leads to another expression within the crowd-anticrowd theory [@crowd]: $$\frac{\sigma_{delta}^2}{N^{2}} = \frac{1}{12 \times 2^m} \left[1
- (\frac{1}{2 ^ {m+1}})^2 \right].$$ We note that for small values of $m$, the numerical data fall within the two crowd-anticrowd approximations, with neither of the approximations capturing the $m$-dependence of $\sigma^{2}/N^{2}$. As will be discussed later, the strategy ranking theory gives an [*analytic*]{} expression for $\sigma^{2}/N^{2}$ that captures the $m$-dependence very well in the small $m$ regime where the criteria $2\cdot 2^{m}/N \ll 1$ is satisfied to a fuller extent.
Strategy ranking theory: Key ideas
==================================
We proceed to discuss how we could obtain the analytic results shown in Figs.1 and 2, within the strategy ranking theory. Details of the theory can be found in [@rodgers; @NETMG1; @RPA]. Here we briefly summarize the key ideas, with the aim to make the theory physically transparent. We note that in MG and other agent-based models of competing populations, it is the interplay between decision-making, strategy selections, and collective response that leads to the non-trivial and often interesting global behaviour of a system. With this in mind, the strategy performance ranking pattern is of crucial importance. At any time step, the strategies can be classified into $\kappa+1$ ranks, according to the virtual points of the strategies. The momentarily best-performing strategy (or strategies) belongs (belong) to rank-1, and so on. At the beginning of the game, all strategies are tied that thus they all belong to the same rank. This is also the case when the strategies are all tied during the game. Thus, the lower bound of $\kappa$ is zero. It is also noted that there are two different kinds of behaviour in the ranking pattern [*after*]{} a time step: (i) the number of different ranks [*increases*]{} and such a time step is called an “even" time step, and (ii) the number of different ranks [*decreases*]{} and such a time step is called an “odd" time step. Take, for example, a time step at which the strategies are all tied before decision. Regardless of the history based on which the agents decide [*and*]{} the wining outcome after the agents decided, the strategies split into two ranks, i.e., $\kappa$ increases from 0 to 1 after the time step. Half of the strategies belong to the better rank and half to the worse rank, as half of the strategies would have predicted the correct winning outcome for the history concerned. Generally speaking, the underlying mechanism for this splitting is that [*there is no registered virtual point or stored information in the strategies for the history concerned*]{}. We call this kind of time steps “even" time steps because this is what would happen when the population encounters a history for decision that had occurred an even number of times since the beginning of the game, not counting the one that is currently in use for decisions.
The parameter $\kappa$ has another physical meaning. It is the number of history bit-strings that have occurred an odd number of times since the beginning of the game, regardless the current history in use for decisions. Since there are at most $2^{m}$ history bit-strings for a given $m$, the upper bound of $\kappa$ is $2^{m}$. Thus we have $0 \leq \kappa \leq 2^{m}$. Therefore, every time step as the game proceeds can be classified as “even" or “odd", together with a parameter $\kappa$. For $\kappa=0$ when all the strategies are tied, the time step is necessarily an even time step. For $\kappa = 2^{m}$ where there are $2^{m}+1$ ranks, the time step is necessarily an odd time step since all the histories have occurred an odd number of times, including the current history in use for decisions. Noting that the total number of strategies is $2^{2^{m}}$, there are in general several strategies in a certain ranking. In this way, the theory takes explicit account of cases of tied strategies.
For even time steps (regardless of the value of $\kappa$), there is no registered virtual points in the strategies for the current history. Therefore, [*even time steps are characterized by agents making random decisions*]{} [@NETMG1; @rodgers; @NETMG2; @RPA]. Using a random walk argument, the distribution $P_{even,\kappa}(A)
= P_{even}(A)$ is a normal distribution independent of $\kappa$, with a mean $\mu_{even} = 0.5$ and a variance $\sigma_{even}^{2} =
1/(4N)$, i.e., $$P_{even}(A) = \frac{1}{\sqrt{2\pi} \sigma_{even}} \exp\left( -
\frac{(A - \mu_{even})^{2}}{2\sigma_{even}^{2}} \right).$$ It turns out that the part of the distribution around $A=0.5$ shown in Fig.1 originates from the even time steps.
For odd time steps, there are registered virtual points or stored information in the strategies for the current history. This is the origin of the crowd effect [@preprint; @crowd; @crowd1], which is fundamental to the understanding of collective response in the class of agent-based models based on MG. In this case, the momentarily better performing strategies have predicted the correct action in the last occurrence of the current history in use for decision. There will then be more agents using these better-performing strategies for decisions. However, the number is too large, hence forming a crowd, that the winning action in the last occurrence becomes the losing action in this turn. This is the anti-persistent nature or double periodicity of MG [@Challet1; @Challet2; @Marsili; @Savit; @Jefferies; @Zheng]. Using the strategy ranking theory, we know that there are $(\kappa +1)$ ranks among the strategies for time steps labelled $\kappa$. The ratio of the fractions of strategies in different ranks is given by [@NETMG1] $C_{0}^{\kappa} : C_{1}^{\kappa} : \cdots :
C_{\ell}^{\kappa} : \cdots : C_{\kappa}^{\kappa}$, which are simply the numbers in the Pascal triangles. Given that the agents use their best-performing strategy for decision, we can readily count the number of agents using a strategy in a particular rank. As mentioned, the better-performing strategies are more likely to lead to wrong predictions at odd time steps. This can be modelled by a winning probability at odd time steps of the form of $(\ell
-1)/\kappa$ for a strategy belonging to rank-$\ell$, for a given value of $\kappa$ [@NETMG1; @rodgers]. Putting the information together, we arrive at the probability density function $P_{odd,\kappa}(A)$ for $1 \leq \kappa \leq 2^{m}$. The distribution $P_{odd,\kappa}(A)$ is given by normal distributions centered at the mean values of $$\mu_{odd,\kappa}^{\pm} = 0.5 \pm \frac{C_{\kappa-1}^{2 \kappa
-1}}{ 2 ^ {2 \kappa}}$$ with a variance $$\sigma_{odd,\kappa}^2 = \frac{C_{\kappa-2}^{2 \kappa -2}}{ 2^{2
\kappa - 1}} \frac{1}{4N}.$$ Applying Eq. (4) to the results for $m=1$ in Fig.1, we immediately identify that the peaks in $P(A)$ at $A=1/4$ and $A=3/4$ are originated from odd time steps corresponding to $\kappa =1$ and the peaks at $A=0.6875$ and $0.3125$ are originated from odd time steps corresponding to $\kappa =2$. These peaks are more noticeable in Fig. 1(b) when the population size is large. In Eq. (5), the binomial coefficients should formally be expressed in terms of Gamma functions, so that when the lower index in the coefficient becomes negative, $\sigma_{odd,\kappa}$ vanishes. This is the case for $\kappa=1$, and the corresponding distribution will then be very sharp. This is, for example, the case for the sharp peaks at $A=1/4$ and $A=3/4$ in Fig. (1).
To obtain an expression for the overall $P(A)$, including both even and odd time steps and all possible values of $\kappa$, we need to take a weighted average over the occurrence of odd and even time steps [@NETMG1]. The resulting expression is $$P(A) = \sum_{\kappa=0}^{2^m}
\frac{{C_{\kappa}^{2^{m}}}}{{2^{2^{m}}}}
[(\frac{\kappa}{2^m})P_{odd,\kappa}(A)+(1-\frac{\kappa}{2^m})P_{even}(A)],$$ where the factor $C_{\kappa}^{2^{m}}/2^{2^{m}}$ is the probability of having $\kappa$ history bit-strings occurred an odd number of times. The factor $\kappa/2^{m}$ is the probability that given $\kappa$, the time step is odd. Applying Eq. (6) to the case of $m=1$, we obtain the results (lines) shown in Fig. 1. We note that the expression in Eq. (6) is also applicable to $m>1$, as long as the efficient phase criteria is satisfied.
The calculation of the variance follows from the definition $$\sigma^{2} = N^{2} \langle (A - \overline{A})^{2} \rangle_{t},$$ where $\overline{A}=0.5$ is the mean value of $A$ and the average $\langle \cdots \rangle_{t}$ represents a time average. Replacing the time average by invoking the probability density function $P(A)$, we have $$\begin{aligned}
\frac{\sigma^2}{N^{2}} & = & \int_{0}^{1} (A - 0.5)^{2} P(A) dA
\nonumber \\
&=& \sum_{\kappa=0}^{2^m} \frac{{C_{\kappa}^{2^{m}}}}{{2^{2^{m}}}}
\left\{(\frac{\kappa}{2^m})
[\frac{1}{2}(0.5-\mu_{odd,\kappa}^{+})^2 +
\frac{1}{2}(0.5-\mu_{odd,\kappa}^{-})^{2}
+\sigma_{odd,\kappa}^2] + (1-\frac{\kappa}{2^m}) \sigma_{even}^2 \right\}\\
&\approx& \sum_{\kappa=0}^{2^m} \frac{C_{\kappa}^{2^m}}{2^{2^m}}
(\frac{\kappa}{2^m}) (\frac{C_{\kappa-1}^{2 \kappa -
1}}{2^{2\kappa}})^2 \nonumber \\
& = & \sum_{\kappa=0}^{2^m} \frac{C_{\kappa}^{2^m}}{2^{2^m}}
(\frac{\kappa}{2^m}) \left(\frac{1}{2} \prod_{q=1}^{\kappa} (1 -
\frac{1}{2q}) \right)^{2},\end{aligned}$$ where the approximation is valid for $2.2^m / N << 1$. Eq. (9) is an [*analytic*]{} expression for $\sigma^{2}$. The last two expressions are equivalent and one may use whichever convenient in obtaining numerical values from Eq. (9).
Several remarks are worth mentioning. Firstly, we note that the expression of $\sigma^{2}$ is closely related to the analytic expression for the winning probability reported in [@NETMG1], from which an alternative approach arriving at the same result is possible [@thesis]. Secondly, the results from Eq. (9) are plotted (open squares) in Fig.2. We note that the strategy ranking theory does capture the $m$-dependence of $\sigma^{2}/N^{2}$, with good agreement with numerical simulation results in the range where the criteria $2\cdot 2^{m}/N \ll 1$ is better fulfilled [@rodgers]. The success of the theory stems from the inclusion of tied strategies, as each rank typically consists of a number of strategies. In the simplest case of $m=1$, for example, there are tied strategies in [*every*]{} time step of the game. The better agreement with numerical results when compared with the crowd-anticrowd approximations is thus an indication of the importance of (i) the tied strategies and (ii) the time evolution of the ranking pattern from time step to time step. In MG, both the number of tied strategies, i.e., number of strategies belonging to the same rank, and the time evolution of strategy ranking pattern can be readily found. Thirdly, the result Eq. (9) is interesting in that there have been much effort in trying to re-scale numerical results of $\sigma^{2}$ as a function of the parameter $\alpha = 2^{m}/N$ so that results from systems of different values of $N$ and $m$ can be collapsed onto a single curve. Eq. (9) suggests that $\sigma^{2}/N^{2}$ is a complicated function of $m$, deep in the efficient phase. In particular, as one increases the population size at fixed and small $m$, one should approach the result given by Eq.(9) assuming a uniform initial distribution of strategies to the agents. It is, in fact, possible to include the effects of a finite population size $N$ into the strategy-ranking theory starting from Eq. (8) by incorporating the so-called market impact effects [@rodgers; @NETMG2; @thesis].
Networked agents
================
Systems in the real world are characterized by connected agents [@networks]. The connections are often used for collecting information from the neighbours. Recently, several interesting attempts [@NETMG1; @NETMG2; @anghel; @NETMG3; @NETMG4] have been made to incorporate information sharing mechanisms among the agents into MG and B-A-R models. As an illustration of the application of SRT to networked MG, we focus on the model proposed by Anghel [*et al.*]{} [@NETMG2; @anghel]. As in the MG, Anghel [*et al.*]{}’s model [@anghel] features $N$ agents who repeatedly compete to be in a minority group. Communications between agents are introduced by assuming that the agents are connected by an undirected random network, i.e., classical random graph, with a connectivity $p$ being the probability that a link between two randomly chosen agents exists. The links are used as follows. Each agent compares the cumulated performance of his predictor, which is the suggested action from his own best-performing strategy at each time step, with that of his neighbours, and then follows the suggested action of the best performing predictor among his neighbours and himself. The $p=0$ limit of the model reduces to the MG. Note that the identity of the best-performing strategy changes over time. For $p>0$ the predictor’s performance is generally [*different*]{} from the agent’s performance. It has been reported that the efficiency of the population as a whole, characterized by either $\sigma^{2}$ [@anghel] or by the average winning probability per agent per turn [@NETMG2], shows a [*non-monotonic*]{} dependence on the connectivity $p$ with the most efficient performance occurring at a small but finite value of $p$. In other words, a small fraction of links is beneficial but too many of them are bad. We have explained the feature successfully within the framework of SRT [@NETMG2]. The most important point is that, from our understanding of the non-networked MG (e.g., see Fig. 1), the performance of an agent actually depends on how similar the $s=2$ strategies that he is holding, with the best performing ones holding two identical strategies. The links then act in two ways depending on the connectivity. For low connectivity, the links bring the agents with two anti-correlated strategies to have the chance to use other strategies so that these agents will not always join the crowd at odd time steps and hence with their winning probability enhanced. For high connectivity, there are so many links that many agents are linked to the momentarily best-performing predictor or predictors. As discussed in previous section, the higher ranking strategies have a smaller chance of predicting the correct minority outcome. When the connectivity is high, there are many links so that agents have access to strategies that are more likely to lose. This leads to a drop in the average winning probability of the agents [@NETMG2].
Figure 3 shows how the distribution $P(A)$ changes with the connectivity $p$ at two small values of $p$. The range of small $p$ is particularly of interest since for a large population ($N=1001$) the non-monotonic feature occurs for $p < 0.01$. The symbols (open circles) give the results from numerical simulations. The peaks of the distribution $P(A)$ shifts as $p$ is varied. Applying SRT and incorporating the effects of the presence of links, we found that $P(A)$ can again be represented by a weighted sum of distributions characterized by different kinds of time steps. In particular, for $p=0.01$, the parameters of the distributions in Eq. (4) can be found [@NETMG2; @thesis] to be $\mu_{odd,\kappa=1}^{-} = 0.207$, $\mu_{odd,\kappa=2}^{-} =
0.278$, and $\mu_{odd,\kappa=1,2}^{+} = 1 -
\mu_{odd,\kappa=1,2}^{-}$. The variances are given by Eq. (5) as $\sigma^{2}_{odd,\kappa=1}=0$ and $\sigma^{2}_{odd,\kappa=2} =
1/32N$. Similarly for $p=0.02$, we have $\mu_{odd,\kappa=1}^{-} =
0.075$ and $\mu_{odd,\kappa=2}^{-} = 0.160$, with the same variances. The values of these parameters are obtained by considering the different winning probabilities of the strategies in different ranks and the change in the number of agents using a strategy of a certain rank due to the presence of the links. The solid lines in Fig. 3 show the distributions obtained by SRT. The theory captures the shifts in $P(A)$ with the connectivity $p$.
summary
=======
In the present work, we illustrated the basic ideas in constructing a strategy ranking theory for a class of multi-agent models incorporating the effects of tied strategies and strategy selections. We showed how the theory can be applied to MG in the efficient phase to evaluate the distribution $P(A)$ in the fraction of agents making a particular decision and the associated variance $\sigma^{2}$. In particular, an analytic expression is given for $\sigma^{2}$ in a non-networked population. The theory is also applied to a version of networked MG in which there exists non-trivial dependence on the performance of the agents as a function of the connectivity. Besides $P(A)$ and $\sigma^{2}$, the theory can also be applied to evaluate other quantities such as the average winning probability of the agents. In closing, while SRT is developed with models based on the MG in mind, the general approach, namely that of focusing on the ranking pattern of the strategies and how the pattern evolves in time, should be a key ingredient in the construction of theories for a large class of agent-based models.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Yisheng Huang$^{a},$[^1]Zeng Liu$^{b},$[^2]Yuanze Wu$^{c}$[^3]\
$^{a}$[*Department of Mathematics, Soochow University,*]{}\
\
$^{b}$[*Department of Mathematics, Suzhou University of Science and Technology,*]{}\
\
$^{c}$[*College of Sciences, China University of Mining and Technology,*]{}\
title: '**Positive solutions to an elliptic equation in ${\mathbb{R}}^N$ of the Kirchhoff type**'
---
[**Abstract:**]{} In this paper, we consider the following Kirchhoff type problem $$\left\{\aligned&-\bigg(a+b\int_{{\mathbb{R}}^N}|\nabla u|^2dx\bigg)\Delta u+V(x) u=|u|^{p-2}u&\text{ in }{\mathbb{R}}^N,\\&u\in{H^1({\mathbb{R}}^N)},\endaligned\right.\eqno{(\mathcal{P}_{a,b})}$$ where $N\geq3$, $2<p<2^*=\frac{2N}{N-2}$, $a,b>0$ are parameters and $V(x)$ is a potential function. Under some mild conditions on $V(x)$, we prove that $(\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the variational method, a non-existence result is also established in the cases $N\geq4$. Our results in the case $N=3$ partial improve the results in [@G15; @LY14] and our results in the cases $N\geq4$ are totally new to the best of our knowledge. By combining the scaling technique, we also give a global description on the structure of the positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$, that is $V(x)\equiv\lambda>0$. This result can be seen as a partial complement of the studies in [@A12; @A13].
[**Keywords:**]{} Positive solution; Kirchhoff problem; Variational method; Scaling technique. [**AMS**]{} Subject Classification 2010: 35B09; 35B38; 35J20; 35J61.
0.26in
Introduction
============
In this paper, we consider the following Kirchhoff type problem $$\left\{\aligned&-\bigg(a+b\int_{{\mathbb{R}}^N}|\nabla u|^2dx\bigg)\Delta u+V(x)u=|u|^{p-2}u&\text{ in }{\mathbb{R}}^N,\\&u\in{H^1({\mathbb{R}}^N)},\endaligned\right.\eqno{(\mathcal{P}_{a,b})}$$ where $N\geq3$, $2<p<2^*=\frac{2N}{N-2}$, $a,b>0$ are parameters and $V(x)$ is a potential function satisfying the following condition:
1. There exist two positive constants $v_0$ and $v_\infty$ such that $$\begin{aligned}
v_0\leq V(x)\leq v_\infty=\lim_{|y|\to+\infty}V(y)\quad \text{for all }x\in{\mathbb{R}}^N\end{aligned}$$ and there exists a subset of ${\mathbb{R}}^N$ with positive Lebesgue measure such that $V(x)<v_\infty$ on this set.
The operator $-\bigg(a+b\int_{\Omega}|\nabla u|^2dx\bigg)\Delta u$ first appears in the following model: $$\label{eq001}
\left\{\aligned &u_{tt}-\bigg(a+b\int_{\Omega}|\nabla u|^2dx\bigg)\Delta u=h(x,u)\quad\text{in }\Omega\times(0, T),\\
&u=0\quad\text{on }\partial\Omega\times(0, T),\\
&u(x,0)=u_0(x),\quad u_t(x,0)=u^*(x),\endaligned\right.$$ where $\Omega\subset{\mathbb{R}}^N$ is a bounded domain, $T>0$ is a constant, $u_0, u^*$ are continuous functions. Such model was first proposed by Kirchhoff in 1883 as an extension of the classical D’Alembert’s wave equations for free vibration of elastic strings, Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Due to this reason, the operators as $-\bigg(a+b\int_{\Omega}|\nabla u|^2dx\bigg)\Delta u$ are always called as the Kirchhoff type operators, and the equations including the Kirchhoff type operators are always called as the Kirchhoff type problems. In , $u$ denotes the displacement, the nonlinearity $h(x,u)$ denotes the external force and the parameter $a$ denotes the initial tension while the parameter $b$ is related to the intrinsic properties of the string (such as Young¡¯s modulus). For more details on the physical background of the Kirchhoff type problems, we refer the readers to [@A12; @K83].
Under some suitable assumptions on the nonlinearities, the elliptic type Kirchhoff problems have variational structures in some proper Hilbert spaces. Due to this reason, such problems have been studied extensively in the literatures by the variational method, see for example [@A12; @A13; @ACM05; @AF12; @CWL12; @CKW11; @G15; @HZ12; @HLP14; @HLW151; @LLS12; @LLS14; @LY14; @LLT152; @N141; @PZ06; @R15; @SW14; @WHL151; @W15] and the references therein. In particular, in [@LY14], Li and Ye studied the Kirchhoff type problem $(\mathcal{P}_{a,b})$ in ${\mathbb{R}}^3$. Under some further assumptions on $V(x)$, the authors proved that $(\mathcal{P}_{a,b})$ has a positive ground state solution in ${\mathbb{R}}^3$ for $3<p<6$ by the variational method. Their proof is dependent heavily on the following Pohozaev type condition:
1. $V(x)\in C({\mathbb{R}}^3, {\mathbb{R}})$ is weakly differentiable and $(D V(x),x)\in L^\infty({\mathbb{R}}^3)\cup L^{\frac32}({\mathbb{R}}^3)$ and $$\begin{aligned}
V(x)-(D V(x), x)\geq0\quad\text{a.e. in }{\mathbb{R}}^3,\end{aligned}$$ where $(\cdot,\cdot)$ is the usual inner product in ${\mathbb{R}}^3$.
Li and Ye’s result was partially improved by Guo in [@G15], where the following non-autonomous Kirchhoff type problem was considered $$\begin{aligned}
\label{eqnew0003}
\left\{\aligned&-\bigg(a+b\int_{{\mathbb{R}}^3}|\nabla u|^2dx\bigg)\Delta u+V(x)=f(u)&\text{ in }{\mathbb{R}}^3,\\
&u\in H^1({\mathbb{R}}^3),\endaligned\right.\end{aligned}$$ where $a,b>0$ are parameters, $V(x)$ is a potential function and $f(u)$ is a nonlinearity involving the power-type $|u|^{p-2}u$ for $2<p<6$. Under some further assumptions on $V(x)$ and $f(u)$, Guo proved that has a positive ground state solution, in particular, when $f(u)=|u|^{p-2}u$, then has a positive ground state solution for all $2<p<6$. Guo’s proof is also dependent heavily on the following Pohozaev type condition:
1. $V(x)\in C^1({\mathbb{R}}^3, {\mathbb{R}})$ and there exists a positive constant $A<a$ such that $$\begin{aligned}
|(\nabla V(x), x)|\leq\frac{A}{|x|^2}\quad\text{for all $x$ in }{\mathbb{R}}^3\backslash\{0\}.\end{aligned}$$
Since involves $(\mathcal{P}_{a,b})$, a natural question inspired by the above facts is that
1. Are the Pohozaev type conditions as $(V_0)$ or $(V_1)$ necessary in finding the positive solution of $(\mathcal{P}_{a,b})$?
It is worth to point out that, as pointed out by Li and Ye in [@LY14], by using a similar method in [@HZ12], we can obtain the following.
\[thmnew001\] Let $N=3$ and $4<p<6$. If $V(x)$ satisfies the condition $(V)$, then $(\mathcal{P}_{a,b})$ has a positive solution.
It follows from Theorem \[thmnew001\] that the question $(Q_1)$ has a positive answer for $N=3$ and $4<p<6$. However, to the best of our knowledge, the question $(Q_1)$ for the cases $N\geq3$ and $p\in(2,4]\cap(2,2^*)$ is still open. Thus, the first purpose of this paper is to explore the question $(Q_1)$ in the cases $N\geq3$ and $p\in(2,4]\cap(2,2^*)$.
Before we state our first result, we need to introduce some notations. Let $$\begin{aligned}
\label{eq0104}
\mathcal{J}_V(u)=\frac{b}{4}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^4+\frac{a}{2}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2+\frac12\int_{{\mathbb{R}}^N}V(x)u^2dx-\frac1p\|u\|_{L^p({\mathbb{R}}^N)}^p,\end{aligned}$$ where $\|\cdot\|_{L^q({\mathbb{R}}^N)}$ is the usual norm in $L^q({\mathbb{R}}^N)(q\geq1)$. Then it is easy to check that $\mathcal{J}_V(u)$ is of $C^2$ in ${H^1({\mathbb{R}}^N)}$ and the critical points of $\mathcal{E}(u)$ are equivalent to the weak solutions to $(\mathcal{P}_{a,b})$ under the condition $(V)$. Let the Nehari type manifold of $\mathcal{J}_V(u)$ be $$\begin{aligned}
\label{eq0102}
\mathcal{N}_V=\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid \mathcal{J}_V'(u)u=0\}.\end{aligned}$$ Then it is easy to see that all nontrivial critical points are contained in $\mathcal{N}_V$. Let $$\begin{aligned}
G_u(t)&=&\mathcal{J}_V(tu)\\
&=&\frac{bt^4}{4}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^4+\frac{at^2}{2}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2\\
&&+\frac{t^2}2\int_{{\mathbb{R}}^N}V(x)u^2dx-\frac{t^p}{p}\|u\|_{L^p({\mathbb{R}}^N)}^p.\end{aligned}$$ Then by a direct calculation, we can see that $G_u(t)$ is of $C^2$ in ${\mathbb{R}}^+$ for every $u\in{H^1({\mathbb{R}}^N)}$ and $G_u'(t)=0$ if and only if $tu\in\mathcal{N}_V$. Thus, it is natural to divide the Nehari type manifold $\mathcal{N}_V$ into the following three parts: $$\begin{aligned}
\mathcal{N}_V^-&=&\{u\in\mathcal{N}_V\mid G_u''(1)<0\};\label{eq1005}\\
\mathcal{N}_V^0&=&\{u\in\mathcal{N}_V\mid G_u''(1)=0\};\\
\mathcal{N}_V^+&=&\{u\in\mathcal{N}_V\mid G_u''(1)>0\}.\label{eq1007}\end{aligned}$$ Now, our first result can be stated as follows.
\[thm1001\] Let $N\geq3$, $a>0$, $p\in(2, 4]\cap(2, 2^*)$ and $V(x)$ satisfy the condition $(V)$. Then there exist $0<b_*(a)\leq b_{**}(a)<+\infty$ such that $(\mathcal{P}_{a,b})$ has a positive solution for $0<b<b_*(a)$, which minimizes the functional $\mathcal{J}_V(u)$ on $\mathcal{N}_V^-$. Moreover, $(\mathcal{P}_{a,b})$ only has trivial solutions for $b>b_{**}(a)$ in the cases $N\geq4$.
1. For the case $N=3$, Theorem \[thm1001\] can be seen as an improvement of the results in [@G15; @LY14] to $(\mathcal{P}_{a,b})$ in the sense that we totally remove the conditions $(V_0)$ or $(V_1)$ for $b$ small enough to obtain positive solutions of $(\mathcal{P}_{a,b})$. To the best of our knowledge, Theorem \[thm1001\] is totally new for the cases $N\geq4$.
2. By Theorem \[thm1001\], we can see that the Pohozaev type conditions are not needed in finding positive solutions of $(\mathcal{P}_{a,b})$ for the parameter $b$ small enough. Thus, Theorem \[thm1001\] gives a partial answer to the question $(Q_1)$.
3. It is also worth to point that in [@LLS12], Li et al. studied the existence of a positive solution for the following autonomous Kirchhoff problem $$\begin{aligned}
\left\{\aligned&\bigg(a+b(\int_{{\mathbb{R}}^N}(|\nabla u|^2+\lambda u^2)dx)\bigg)(-\Delta u+\lambda u)=f(u)&\text{ in }{\mathbb{R}}^N,\\&u\in{H^1({\mathbb{R}}^N)},\endaligned\right.\end{aligned}$$ where $N\geq3$, $a,b>0$ are parameters, $\lambda>0$ is a constant and $f(u)$ is a nonlinearity involving the power-type $|u|^{p-2}u$ for $2<p<2^*$. By using a truncation argument combined with a monotonicity trick introduced by Jeanjean [@J99] (see also Struwe [@S85]), the authors proved that such equation has a positive radial solution for $b$ small enough. Their method is heavily dependent on two facts. One fact is that $H^1_r({\mathbb{R}}^N)$ is embedded compactly into $L^q({\mathbb{R}}^N)$ for $2\leq q<2^*$, where $H^1_r({\mathbb{R}}^N)=\{u\in{H^1({\mathbb{R}}^N)}\mid u\text{ is radial symmetric}\}$. The other fact is that the order $4$ term $(\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2+\|u\|_{L^2({\mathbb{R}}^N)}^2)^2$ can totally control the nonlinearity $f(u)$ by the Sobolev embedding theorem. Since $D^{1,2}({\mathbb{R}}^N)$ can not be embedded into $L^p({\mathbb{R}}^N)$, the order $4$ term $\|\nabla u\|_{L^2({\mathbb{R}}^N)}^4$ can not control the order $p$ term $\|u\|_{L^p({\mathbb{R}}^N)}^p$ totally for $(\mathcal{P}_{a,b})$. On the other hand, since $(\mathcal{P}_{a,b})$ is non-autonomous, $H^1_r({\mathbb{R}}^N)$ is not a good choice for the variational setting of $(\mathcal{P}_{a,b})$. Due to these reasons, their method can not be used for $(\mathcal{P}_{a,b})$.
4. From the view point of the fibering maps, it seems that there exists another positive solution to $(\mathcal{P}_{a,b})$ for $b$ small enough in the cases $2<p<\min\{4, 2^*\}$, which minimizes the functional $\mathcal{J}_V(u)$ on $\mathcal{N}_V^+$. However, we actually observe in Theorem \[thm0004\] below that for the autonomous form of $(\mathcal{P}_{a,b})$, there exists a unique positive solution in the cases $N=3,4$. Thus, it seems that $\inf_{\mathcal{N}_V^+}\mathcal{J}_V(u)$ can not be attained in the cases $N=3,4$. For the cases $N\geq5$, we believe that there exists another positive solution to $(\mathcal{P}_{a,b})$ for $b$ small enough, which minimizes the functional $\mathcal{J}_V(u)$ on $\mathcal{N}_V^+$. However, since the energy values of bubbles to the $(PS)$ sequence of $\mathcal{J}_V(u)$ may be negative at the energy level $\inf_{\mathcal{N}_V^+}\mathcal{J}_V(u)$, it is hard to exclude the dichotomy case in the concentration-compactness principle (see Lions [@L84]). Due to this reason, we do not obtain the second solution to $(\mathcal{P}_{a,b})$ for $b$ small enough in the cases $N\geq5$.
5. The condition $(V)$ can be weaken to some other ones which ensure that the Schrödinger operator $-\Delta+V(x)$ is definite. However, we do not want to go further in that direction.
6. Even though the positive solution obtained by Theorem \[thm1001\] minimizes the functional $\mathcal{J}_V(u)$ on $\mathcal{N}_V^-$, this solution also may not be a ground state solution for $(\mathcal{P}_{a,b})$, since $\mathcal{N}_V^0$ and $\mathcal{N}_V^+$ may be nonempty sets. Thus, how to find the ground state solution of $(\mathcal{P}_{a,b})$ without the Pohozaev type conditions as $(V_0)$ or $(V_1)$ is still an interesting question and also open to us.
On the other hand, Azzollini introduced the scaling technique $$\begin{aligned}
u(x)\to u_t(x):=u(tx)\end{aligned}$$ to deal with the autonomous Kirchhoff type problem in ${\mathbb{R}}^N$ in [@A12; @A13]. Such method is much more simple than the variational method and can be used to establish the relation between the solutions of the autonomous Kirchhoff type problem and that of the related local problem. Applying Azzollini’s scaling technique to the autonomous form of $(\mathcal{P}_{a,b})$, we can easily to obtain the following.
\[thm1002\] Let $N\geq3$, $a,b>0$, $2<p<2^*$ and $V(x)\equiv\lambda>0$. Then we have the following.
1. In the case $N=3$, the autonomous form of $(\mathcal{P}_{a,b})$ has a unique positive radial solution $u_{a,b,\lambda}$ with the expression $$\begin{aligned}
\label{eq2008}
u_{a,b,\lambda}(x)=U_\lambda(tx),\end{aligned}$$ where $t$ is a positive constant satisfying $$\begin{aligned}
at^2+b\bigg(\int_{{\mathbb{R}}^N}|U_{\lambda}|^2dx\bigg)t^{4-N}=1\end{aligned}$$ and $U_\lambda$ is the unique positive radial solution of the equation $$\begin{aligned}
\label{eqnew1006}
\left\{\aligned&-\Delta u+\lambda u=|u|^{p-2}u&\text{ in }{\mathbb{R}}^3,\\
&u\in H^1({\mathbb{R}}^3).\endaligned\right.\end{aligned}$$
2. In the case $N=4$, if $b\int_{{\mathbb{R}}^4}|\nabla U_\lambda|^2dx<1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has a unique positive radial solution $u_{a,b,\lambda}$ with the same expression of , where $U_\lambda$ is the unique positive radial solution of the equation in ${\mathbb{R}}^4$. If $b\int_{{\mathbb{R}}^4}|\nabla U_\lambda|^2dx\geq1$, then the autonomous form of $(\mathcal{P}_{a,b,\lambda})$ has no solution.
3. In the cases $N\geq5$, if $\mathcal{F}_{a,b}(U_\lambda)<1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has exact two positive radial solutions $u_{a,b,\lambda}^\pm$ with the same expressions of , where $$\begin{aligned}
\mathcal{F}_{a,b}(U_\lambda)=\bigg(\frac{(N-4)b\int_{{\mathbb{R}}^N}|\nabla U_{\lambda}|^2dx}{2}\bigg)^{\frac{2}{N-2}}\frac{(N-2)a^{\frac{N-4}{N-2}}}{N-4}
\end{aligned}$$ and $U_\lambda$ is the unique positive radial solution of the equation in ${\mathbb{R}}^N(N\geq5)$. If $\mathcal{F}_{a,b}(U_\lambda)=1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has a unique positive radial solution $u_{a,b,\lambda}^0$ with same expression of , where $U_\lambda$ is the unique positive radial solution of the equation in ${\mathbb{R}}^N(N\geq5)$. If $\mathcal{F}_{a,b}(U_\lambda)>1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has no solution.
Since the autonomous form of $(\mathcal{P}_{a,b})$ can also be studied by the variational method, a natural question for the autonomous form of $(\mathcal{P}_{a,b})$ due to Theorem \[thm1002\] is that
1. Can the solutions to the autonomous form of $(\mathcal{P}_{a,b})$ founded in Theorem \[thm1002\] also be founded by the variational method, that is, does the solutions to the autonomous form of $(\mathcal{P}_{a,b})$ founded by the scaling technique coincide with that founded by the variational method?
Due to the uniqueness of the positive solution to $(\mathcal{P}_{a,b})$ given by Theorem \[thm1002\], this question has a positive answer for the cases $N=3,4$ by the results in [@A12]. However, since the autonomous form of $(\mathcal{P}_{a,b})$ has two positive solutions for $\mathcal{F}_{a,b}(U_\lambda)<1$ due to Theorem \[thm1002\], this question is still open for the cases $N\geq5$ to the best of our knowledge. Thus, the second purpose of this paper is to study the question $(Q_2)$ to the autonomous form of $(\mathcal{P}_{a,b})$ for the cases $N\geq5$.
Before we state our results on the question $(Q_2)$, we also need to introduce some notations. Let $$\begin{aligned}
\label{eq0004}
\mathcal{E}(u)=\frac{b}{4}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^4+\frac{a}{2}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2+\frac{\lambda}{2}\|u\|_{L^2({\mathbb{R}}^N)}^2-\frac1p\|u\|_{L^p({\mathbb{R}}^N)}^p.\end{aligned}$$ Then it is easy to check that $\mathcal{E}(u)$ is of $C^2$ in ${H^1({\mathbb{R}}^N)}$ and the critical points of $\mathcal{E}(u)$ are equivalent to the weak solutions to the autonomous form of $(\mathcal{P}_{a,b})$. Let the Pohozaev type manifold of $\mathcal{E}(u)$ be $$\begin{aligned}
\label{eq0002}
\mathcal{M}=\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid \Psi(u)=0\},\end{aligned}$$ where $$\begin{aligned}
\Psi(u)&=&\frac{N-2}{2N}(a\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2+b\|\nabla u\|_{L^2({\mathbb{R}}^N)}^4)\notag\\
&&+\frac{\lambda}{2}\|u\|_{L^2({\mathbb{R}}^N)}^2-\frac{1}{p}\|u\|_{L^p({\mathbb{R}}^N)}^p.\label{eq0003}\end{aligned}$$ Then by the Pohozaev identity of the autonomous form of $(\mathcal{P}_{a,b})$ (see [@A12; @LY14]), every critical point of $\mathcal{E}(u)$ is contained in $\mathcal{M}$. Let $$\begin{aligned}
F_u(t)&=&\mathcal{E}(u_t)\\
&=&\frac{b}{4}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^4t^{4-2N}+\frac{a}{2}\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2t^{2-N}\\
&&+(\frac{\lambda}{2}\|u\|_{L^2({\mathbb{R}}^N)}^2-\frac{1}{p}\|u\|_{L^p({\mathbb{R}}^N)}^p)t^{-N},\end{aligned}$$ where $u_t(x)=u(tx)$. Then by a direct calculation, we can see that $F_u(t)$ is of $C^2$ in ${\mathbb{R}}^+$ for every $u\in{H^1({\mathbb{R}}^N)}$ and $F_u'(t)=0$ if and only if $u_t\in\mathcal{M}$. Thus, it is natural to divide the Pohozaev type manifold $\mathcal{M}$ given by into the following three parts: $$\begin{aligned}
\mathcal{M}^-&=&\{u\in\mathcal{M}\mid F_u''(1)<0\};\label{eq0005}\\
\mathcal{M}^0&=&\{u\in\mathcal{M}\mid F_u''(1)=0\};\label{eq0006}\\
\mathcal{M}^+&=&\{u\in\mathcal{M}\mid F_u''(1)>0\}.\label{eq0007}\end{aligned}$$ Now, our second result can be stated as follows.
\[thm0001\] Let $N\geq5$, $a,b>0$, $2<p<2^*$ and $V(x)\equiv\lambda>0$. Let $U_\lambda$ be the unique positive radial solution of in ${\mathbb{R}}^N(N\geq5)$. Then we have the following.
1. If $\mathcal{F}_{a,b}(U_\lambda)<1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has two positive radial solutions $u_{a,b,\lambda}^\pm$. Moreover, we also have $$\begin{aligned}
\mathcal{E}(u_{a,b,\lambda}^-)=\inf_{\mathcal{M}^-}\mathcal{E}(u)\quad\text{and}\quad
\mathcal{E}(u_{a,b,\lambda}^+)=\inf_{\mathcal{M}^+}\mathcal{E}(u)=\inf_{\mathcal{M}}\mathcal{E}(u).
\end{aligned}$$
2. If $\mathcal{F}_{a,b}(U_\lambda)=1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has a positive radial solution $u_{a,b,\lambda}^0$. Moreover, $\mathcal{M}=\mathcal{M}^0$ and $\mathcal{E}(u_{a,b,\lambda}^0)=\inf_{\mathcal{M}}\mathcal{E}(u)$.
3. If $\mathcal{F}_{a,b}(U_\lambda)>1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has no solution.
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1. The proof of Theorem \[thm0001\] is pure variational. Thus, Theorem \[thm0001\] gives a positive answer of the question $(Q_2)$ in the cases $N\geq5$.
2. In [@A12], it has been proved that the ground state solution to the autonomous form of $(\mathcal{P}_{a,b})$ in the cases $N=3,4$ also minimizes the energy functional $\mathcal{E}(u)$ on the Pohozaev type manifold $\mathcal{M}$. However, to the best of our knowledge, such property to the autonomous form of $(\mathcal{P}_{a,b})$ in the cases $N\geq5$ has not been obtained in the literatures. Now, since every critical point of $\mathcal{E}(u)$ is contained in $\mathcal{M}$ by the Pohozaev identity of the autonomous form of $(\mathcal{P}_{a,b})$, $u_{a,b,\lambda}^+$ and $u_{a,b,\lambda}^0$ obtained by Theorem \[thm0001\] must be the the ground state solution to the autonomous form of $(\mathcal{P}_{a,b})$ respectively in the cases $\mathcal{F}_{a,b}(U_\lambda)<1$ and $\mathcal{F}_{a,b}(U_\lambda)=1$. Thus, by Theorem \[thm0001\], we can see that the ground state solution to the autonomous form of $(\mathcal{P}_{a,b})$ in the cases $N\geq5$ also minimizes the energy functional $\mathcal{E}(u)$ on the Pohozaev type manifold $\mathcal{M}$.
3. Some other $C^1$-manifolds were used to find the positive ground state solution to the autonomous form of $(\mathcal{P}_{a,b})$ in [@G15; @LY14] for the case $N=3$. Such manifolds can be seen as some kinds of the unifications of the Nehari type manifold and the Pohozaev type manifold. Thus, the energy level of the ground state solution to the autonomous form of $(\mathcal{P}_{a,b})$ in the case $N=3$ has some other expressions in the view point of the calculus of variation. However, such manifolds are not good choices for the high dimensions $(N\geq4)$ due to the fact that $D^{1,2}({\mathbb{R}}^N)$ can not be embedded into $L^p({\mathbb{R}}^N)$. Indeed, if we consider such manifolds, then we will trap in the trouble that we can not describe the manifold very clear for all $a,b>0$ in the high dimensions $(N\geq4)$. Thus, we can not describe the positive solution to the autonomous form of $(\mathcal{P}_{a,b})$ totally as Theorem \[thm0001\].
In [@CR92; @CL97], Chipot et al. introduced another scaling technique $u(x)\to tu(x)$ to deal with the elliptic equations of the Kirchhoff type with power-type nonlinearity (see also [@A13; @ACM05; @HLW15; @LLT152]). This method was further developed in our previous paper [@WHL15]. Note that the nonlinearity to the autonomous form of $(\mathcal{P}_{a,b})$ is also power-type. Thus, the autonomous form of $(\mathcal{P}_{a,b})$ also can be studied by the scaling technique $u(x)\to tu(x)$. Due to this fact, the following question is also natural.
1. What is the relation between the two differential scaling technique for the autonomous form of $(\mathcal{P}_{a,b})$?
In order to study the question $(Q_3)$, we introduce a more general scaling technique $u\to su(tx)$, $s,t>0$, which can be seen as a unification of the two differential scaling technique used in the literatures. By such scaling technique, we observe the following.
\[thm0002\] Let $N\geq3$, $a,b>0$, $2<p<2^*$ and $V(x)\equiv\lambda>0$. Then the solution of the autonomous form of $(\mathcal{P}_{a,b})$ must be of the form $sU_{\frac{\lambda}{s^{p-2}}}(tx)$, where $U_{\frac{\lambda}{s^{p-2}}}$ is the unique positive radial solution of for $\lambda=\frac{\lambda}{s^{p-2}}$ in ${\mathbb{R}}^N(N\geq3)$, $s$ and $t$ satisfy $\frac{t}{s^{\frac{p-2}{2}}}=\gamma>0$ and $\gamma$ is the solution of the following equation $$\begin{aligned}
\label{eqnew0005}
a\gamma^2+b\bigg(\int_{{\mathbb{R}}^N}|U_{\lambda}|^2dx\bigg)\gamma^{4-N}=1.\end{aligned}$$ Moreover, the number of positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$ equals to the number of solutions to the equation .
Theorem \[thm0002\] gives all expressions of the solutions to the autonomous form of $(\mathcal{P}_{a,b})$ obtained by the scaling technique. Furthermore, note that it is well known that $U_{\lambda}(x)=\lambda^{\frac{1}{p-2}}U_1(\sqrt{\lambda}x)$. Thus, we must have $$\begin{aligned}
\label{eqnew0010}
sU_{\frac{\lambda}{s^{p-2}}}(tx)=\lambda^{\frac{1}{p-2}}U_1((\frac{\lambda}{s^{p-2}})^{\frac12}tx)=U_\lambda(\frac{1}{s^{\frac{p-2}{2}}}tx)
=U_\lambda(\gamma x).\end{aligned}$$ It follows that all scaling technique coincide with the special one $u(x)\to u(tx)$ for the autonomous form of $(\mathcal{P}_{a,b})$ due to Theorem \[thm1002\], which also gives the answer to the question $(Q_3)$.
Combing Theorems \[thm0001\]–\[thm0002\] and the results in [@A12; @A13], we can obtain the following.
\[thm0004\] Let $N\geq3$£¬ $a,b>0$, $2<p<2^*$ and $V(x)\equiv\lambda>0$. Then we have the following.
1. In the case $N=3$, the autonomous form of $(\mathcal{P}_{a,b})$ has a unique positive radial solution $u_{a,b,\lambda}$ with the expression $$\begin{aligned}
\label{eq0008}
u_{a,b,\lambda}(x)=\bigg(\frac{\lambda}{\alpha}\bigg)^{\frac{1}{p-2}}U_\alpha(tx),\end{aligned}$$ where $\alpha$ and $t$ are two positive constants satisfying $\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac12}t=\gamma>0$ and $\gamma$ is the solution of , moreover, $u_{a,b,\lambda}$ is the ground state solution and $\mathcal{E}(u_{a,b,\lambda})=\inf_{\mathcal{M}}\mathcal{E}(u)$.
2. In the case $N=4$, if $b\int_{{\mathbb{R}}^N}|\nabla U_\lambda|^2dx<1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has a unique positive radial solution $u_{a,b,\lambda}$ with the same expression of , moreover, $u_{a,b,\lambda}$ is the ground state solution and $\mathcal{E}(u_{a,b,\lambda})=\inf_{\mathcal{M}}\mathcal{E}(u)$; if $b\int_{{\mathbb{R}}^N}|\nabla U_\lambda|^2dx\geq1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has no solution.
3. In the cases $N\geq5$, if $\mathcal{F}_{a,b}(U_\lambda)<1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has exact two positive radial solutions $u_{a,b,\lambda}^\pm$ with the same expressions of , moreover, $u_{a,b,\lambda}^+$ is the ground state solution and $\mathcal{E}(u_{a,b,\lambda}^-)=\inf_{\mathcal{M}^-}\mathcal{E}(u)$ and $\mathcal{E}(u_{a,b,\lambda}^+)=\inf_{\mathcal{M}^+}\mathcal{E}(u)=\inf_{\mathcal{M}}\mathcal{E}(u)$; if $\mathcal{F}_{a,b}(U_\lambda)=1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has a unique positive radial solution $u_{a,b,\lambda}^0$ with same expression of , moreover, $u_{a,b,\lambda}^0$ is the ground state solution and $\mathcal{M}=\mathcal{M}^0=\{u_{a,b,\lambda}^0\}$; if $\mathcal{F}_{a,b}(U_\lambda)>1$, then the autonomous form of $(\mathcal{P}_{a,b})$ has no solution.
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1. To the best of our knowledge, Theorem \[thm0004\] is the first result which describe the positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$ totally. Thus, Theorems \[thm0001\]–\[thm0002\] can be seen as a complement of the results in [@A12; @A13] for the autonomous form of $(\mathcal{P}_{a,b})$.
2. By making some further observations on the function , we can obtain some concentration behaviors of the positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$ for the parameters $a,b$ due to the precise expressions given by Theorem \[thm0004\]. However, we do not want to go further in that direction in the current paper.
Through this paper, $o_n(1)$ will always denote the quantities tending to zero as $n\to\infty$ and $C_i$ will denote the positive constants which may be different and independent of the parameter $b$. For the sake of simplicity, we respectively denote $\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2$, $\|u\|_{L^p({\mathbb{R}}^N)}^p$, $\|u\|_{L^2({\mathbb{R}}^N)}^2$ and $\int_{{\mathbb{R}}^N}V(x)u^2dx$ by $\mathfrak{A}_u$, $\mathfrak{B}_u$, $\mathfrak{C}_u$ and $\mathfrak{C}_{u,V}$ in the remaining of this paper.
The autonomous case
===================
The Pohozaev manifold $\mathcal{M}$ in $N\geq5$
-----------------------------------------------
As we stated in the introduction, the fibering map $F_u(t)=\mathcal{E}(u_t)$ can be used to observe the the Pohozaev manifold $\mathcal{M}$ and the divisions $\mathcal{M}^\pm, \mathcal{M}^0$, where $u_t(x)=u(tx)$. Let $$\begin{aligned}
\mathcal{C}&=&\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u>0\}\end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^*&=&\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid\frac{N-2}{2N}\mathfrak{A}_u+\frac\lambda2\mathfrak{C}_u-\frac1p\mathfrak{B}_u=0\}.\end{aligned}$$ Then we have the following.
\[lem0002\] For every $u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}$, there exists a unique $t>0$ such that $u_t=u(tx)\in\mathcal{M}^*$ in the case $u\in\mathcal{C}$ while $u_t=u(tx)\not\in\mathcal{M}$ for all $t>0$ in the case $u\not\in\mathcal{C}$. Moreover, if $u\in\mathcal{C}$, then $I(u_t)=\max_{s>0}I(u_s)$ and $I(u_s)$ is strictly increasing on $(0, t)$ and strictly decreasing on $(t, +\infty)$, , where $I(u)$ is the corresponding functional of and given by $I(u)=\frac12\|\nabla u\|_{L^2({\mathbb{R}}^N)}^2+\frac{\lambda}{2}\|u\|_{L^2({\mathbb{R}}^N)}^2-\frac{1}{p}\|u\|_{L^p({\mathbb{R}}^N)}^p$.
Let $u\in {H^1({\mathbb{R}}^N)}\backslash\{0\}$ and consider the fibering map $L_u(t)=I(u_t)$, where $u_t=u(tx)$. By a direct calculation, we can see that $$\begin{aligned}
\label{eq0013}
L_u'(t)=Nt^{-N-1}(\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u-\frac{N-2}{2N}\mathfrak{A}_ut^2).\end{aligned}$$ Clearly, there exists a unique $t>0$ such that $L_u'(t)=0$ in the case $u\in\mathcal{C}$; while $L_u'(t)<0$ for all $t>0$ in the case $u\not\in\mathcal{C}$. It follows from $t^{-N}\mathfrak{B}_u=\mathfrak{B}_{u_t}$, $t^{-N}\mathfrak{C}_u=\mathfrak{C}_{u_t}$ and $t^{2-N}\mathfrak{A}_u=\mathfrak{A}_{u_t}$ that $L_u'(t)=t^{-1}L_{u_t}'(1)$, which implies that there exists a unique $t>0$ such that $u_t=u(tx)\in\mathcal{M}^*$ in the case $u\in\mathcal{C}$; while $u_t=u(tx)\not\in\mathcal{M}$ for all $t>0$ in the case $u\not\in\mathcal{C}$. Now, by , we can see that if $u\in\mathcal{C}$, then $I(u_t)=\max_{s>0}I(u_s)$ and $I(u_s)$ is strictly increasing on $(0, t)$ and strictly decreasing on $(t, +\infty)$.
Let $$\begin{aligned}
&\mathcal{B}_{-}=\{u\in\mathcal{C}\mid \mathcal{B}(u)<1\};\label{eq0022}\\
&\mathcal{B}_{0}=\{u\in\mathcal{C}\mid \mathcal{B}(u)=1\};\label{eqnew0011}\\
&\mathcal{B}_{+}=\{u\in\mathcal{C}\mid \mathcal{B}(u)>1\},\label{eqnew0012}\end{aligned}$$ where $$\begin{aligned}
\mathcal{B}(u)=\frac{a^{\frac{N-4}{N-2}}b^{\frac{2}{N-2}}\mathfrak{A}_u^{\frac{N}{N-2}}}
{2^{\frac{N}{N-2}}N(N-4)^{\frac{N-4}{N-2}}(\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u)}.\label{eq0023}\end{aligned}$$ Then our first observation on $\mathcal{M}$ can be stated as follows.
\[lem0001\] Let $N\geq5$. Then we have the following.
1. For every $u\in\mathcal{B}_{-}$, there exist unique $0<t_+<t_-$ such that $u_{t,-}=u(t_-x)\in\mathcal{M}^-$ and $u_{t,+}=u(t_+x)\in\mathcal{M}^+$, where $\mathcal{M}^\pm$ are respectively given by and .
2. For every $u\in\mathcal{B}_{0}$, there exists a unique $t>0$ such that $u_t=u(tx)\in\mathcal{M}^0$, where $\mathcal{M}^0$ is given by .
3. For every $u\in\mathcal{B}_{+}$, $u_t=u(tx)\not\in\mathcal{M}$ for all $t>0$.
Let $u\in {H^1({\mathbb{R}}^N)}\backslash\{0\}$ and consider the fibering map $F_u(t)$. By a direct calculation, we have that $F_u'(t)=Nt^{-N-1}F_{1,u}(t)$, where $$\begin{aligned}
F_{1,u}(t)=\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u-\frac{N-2}{2N}\mathfrak{A}_u(at^2+b\mathfrak{A}_ut^{4-N}).\end{aligned}$$ It is easy to see that $F_u'(t)<0$ for all $t>0$ when $u\not\in\mathcal{C}$. For every $u\in\mathcal{C}$, by Lemma \[lem0002\], there exists $t_0>0$ such that $u_{t_0}\in\mathcal{M}^*$. Thus, $$\begin{aligned}
F_{1,u}(t)=t_0^NF_{1,u_{t_0}}(s)=t_0^N\frac{N-2}{2N}\mathfrak{A}_{u_{t_0}}(1-b\mathfrak{A}_{u_{t_0}}s^{4-N}-as^2),\end{aligned}$$ where $s=\frac{t}{t_0}$. Set $F_{2, u_{t_0}}(s)=1-b\mathfrak{A}_{u_{t_0}}s^{4-N}-as^2$. Then by a direct calculation, we can see that there exist unique $0<s_1<s_2$ such that $F_{2, u_{t_0}}(s_i)=0$ for $i=1,2$, $F_{2, u_{t_0}}(s)<0$ for $0<s<s_1$, $F_{2, u_{t_0}}(s)>0$ for $s_1<s<s_2$ and $F_{2, u_{t_0}}(s)<0$ for $s>s_2$ when $\mathcal{F}_{a,b}(u_{t_0})<1$, there exists a unique $s_0=\bigg(\frac{(N-4)b\mathfrak{A}_{u_{t_0}}}{2a}\bigg)^{\frac{1}{N-2}}$ such that $F_{2, u_{t_0}}(s_0)=0$ when $\mathcal{F}_{a,b}(u_{t_0})=1$ and $F_{2, u_{t_0}}(s)<0$ for all $s>0$ when $\mathcal{F}_{a,b}(u_{t_0})>1$. By , we can see that $t_0=\bigg(\frac{\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u}{\frac{N-2}{2N}\mathfrak{A}_u}\bigg)^{\frac12}$. Now, by a direct calculation, we have $$\begin{aligned}
\mathcal{F}_{a,b}(u_{t_0})=\frac{a^{\frac{N-4}{N-2}}b^{\frac{2}{N-2}}\mathfrak{A}_u^{\frac{N}{N-2}}}
{2^{\frac{N}{N-2}}N(N-4)^{\frac{N-4}{N-2}}(\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u)}=\mathcal{B}(u).\label{eq0053}\end{aligned}$$ Thus, the conclusions follows immediately from the relation between the Pohozaev manifold $\mathcal{M}$ and the fibering map $F_u(t)$.
\[rmk0001\] By checking the proof of Lemma \[lem0001\], we can also see that for every $u\in\mathcal{B_{-}}$, $F_u(t_+)=\min_{0<s\leq t_-}F_u(s)$ and $F_u(t_-)=\max_{t_+\leq s}F_u(s)$ and $F_u(s)$ is strictly decreasing for $0<s<t_+$, strictly increasing for $t_+<s<t_-$ and strictly decreasing for $s>t_-$, where $t_\pm$ are given in Lemma \[lem0001\].
Our second observation on the Pohozaev manifold $\mathcal{M}$ is the following.
\[lem0003\] Let $u\in\mathcal{M}$ and $N\geq5$, then we have $$\begin{aligned}
\label{eq0015}
\bigg(\frac{(N-2)ap}{2N}\bigg(\frac{\lambda p}{2}\bigg)^{\frac{2^*-p}{p-2}}\mathcal{S}^{\frac{2^*}{2}}\bigg)^{\frac{2}{2^*-2}}\leq\mathfrak{A}_u\leq\bigg(\frac{2N}{(N-2)bp}\bigg(\frac{2}{\lambda p}\bigg)^{\frac{2^*-p}{p-2}}\mathcal{S}^{-\frac{2^*}{2}}\bigg)^{\frac{2}{4-2^*}}.\end{aligned}$$ Moreover, we also have $\mathfrak{A}_u<\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^-$, $\mathfrak{A}_u=\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^0$ and $\mathfrak{A}_u>\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^+$.
By the Hölder and Sobolev inequalities, for every $u\in{H^1({\mathbb{R}}^N)}$, we have $$\begin{aligned}
\label{eqnew9100}
\mathfrak{B}_u\leq\mathcal{S}^{-\frac{2^*(p-2)}{2(2^*-2)}}\mathfrak{C}_u^{\frac{2^*-p}{2^*-2}}\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}.\end{aligned}$$ Therefore, for $u\in\mathcal{M}$, we can see that $$\begin{aligned}
\frac\lambda2\mathfrak{C}_u&\leq&a\frac{N-2}{2N}\mathfrak{A}_u+b\frac{N-2}{2N}\mathfrak{A}_u^2+\frac\lambda2\mathfrak{C}_u\\
&=&\frac1p\mathfrak{B}_u\\
&\leq&\frac1p\mathcal{S}^{-\frac{2^*(p-2)}{2(2^*-2)}}\mathfrak{C}_u^{\frac{2^*-p}{2^*-2}}\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}},\end{aligned}$$ which deduces that $$\begin{aligned}
\label{eq0014}
\frac{\lambda p}{2}\mathcal{S}^{\frac{2^*(p-2)}{2(2^*-2)}}\mathfrak{C}_u^{\frac{p-2}{2^*-2}}\leq\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}.\end{aligned}$$ It follows from $u\in\mathcal{M}$ once more that $$\begin{aligned}
a\frac{N-2}{2N}\mathfrak{A}_u&\leq& a\frac{N-2}{2N}\mathfrak{A}_u+b\frac{N-2}{2N}\mathfrak{A}_u^2\notag\\
&=&\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u\notag\\
&\leq&\frac1p\mathcal{S}^{-\frac{2^*(p-2)}{2(2^*-2)}}\mathfrak{C}_u^{\frac{2^*-p}{2^*-2}}\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}-\frac\lambda2\mathfrak{C}_u\notag\\
&\leq&\frac1p\bigg(\frac{2}{\lambda p}\bigg)^{\frac{2^*-p}{p-2}}\mathcal{S}^{-\frac{2^*}{2}}\mathfrak{A}_u^{\frac{2^*}{2}}\label{eqnew0006}.\end{aligned}$$ On the other hand, by and the fact that $u\in\mathcal{M}$ once more, we can see that $$\begin{aligned}
b\frac{N-2}{2N}\mathfrak{A}_u^2&\leq& a\frac{N-2}{2N}\mathfrak{A}_u+b\frac{N-2}{2N}\mathfrak{A}_u^2\notag\\
&=&\frac1p\mathfrak{B}_u-\frac\lambda2\mathfrak{C}_u\notag\\
&\leq&\frac1p\mathcal{S}^{-\frac{2^*(p-2)}{2(2^*-2)}}\mathfrak{C}_u^{\frac{2^*-p}{2^*-2}}\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}-\frac\lambda2\mathfrak{C}_u\notag\\
&\leq&\frac1p\bigg(\frac{2}{\lambda p}\bigg)^{\frac{2^*-p}{p-2}}\mathcal{S}^{-\frac{2^*}{2}}\mathfrak{A}_u^{\frac{2^*}{2}}.\label{eq0017}\end{aligned}$$ Since $2<2^*<4$ for $N\geq5$, follows immediately from and . It remains to prove that $\mathfrak{A}_u<\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^-$, $\mathfrak{A}_u=\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^0$ and $\mathfrak{A}_u>\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^+$. We only give the proof of the conclusion that $\mathfrak{A}_u<\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^-$, since the other two conclusions can be proved in a similar way. Indeed, for every $u\in\mathcal{M}^-$, by the definition of $\mathcal{M}^-$ given by , we can see that $$\begin{aligned}
\frac{(N-2)(2N-3)b}{2N}\mathfrak{A}_u^2+\frac{(N-2)(N-1)a}{2N}\mathfrak{A}_u-(N+1)(\frac1p\mathfrak{B}_u-\frac{\lambda}{2}\mathfrak{C}_u)<0.\end{aligned}$$ Since $\mathcal{M}^-\subset\mathcal{M}$, we must have that $$\begin{aligned}
\frac1p\mathfrak{B}_u-\frac{\lambda}{2}\mathfrak{C}_u=a\frac{N-2}{2N}\mathfrak{A}_u+b\frac{N-2}{2N}\mathfrak{A}_u^2.\end{aligned}$$ It follows that $$\begin{aligned}
0&>&\frac{(N-2)(2N-3)b}{2N}\mathfrak{A}_u^2+\frac{(N-2)(N-1)a}{2N}\mathfrak{A}_u\\
&&-a\frac{(N-2)(N+1)}{2N}\mathfrak{A}_u-b\frac{(N-2)(N+1)}{2N}\mathfrak{A}_u^2\\
&=&\frac{N-2}{2N}\bigg((N-4)b\mathfrak{A}_u^2-2a\mathfrak{A}_u\bigg).\end{aligned}$$ Note that $N\geq5$. Thus, we must have that $\mathfrak{A}_u<\frac{2a}{(N-4)b}$ for $u\in\mathcal{M}^-$, which completes the proof.
Our third observation on the Pohozaev manifold $\mathcal{M}$ is the following.
\[lem0004\] Let $u_0\in\mathcal{M}$ be a local minimum point of $\mathcal{E}(u)$ on $\mathcal{M}$ and $N\geq5$. If $u_0\not\in\mathcal{M}^0$, then $\mathcal{E}'(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$, where $H^{-1}({\mathbb{R}}^N)$ is the dual space of ${H^1({\mathbb{R}}^N)}$ and $\mathcal{M}^0$ is given by .
The main idea of this proof comes from [@R06], which was also used in [@A12; @LY14]. However, as we will see, since we need to deal with the high dimensions, we also need to borrow some ideas from [@HWW15]. Suppose $u_0\in\mathcal{M}$ be a local minimum point of $\mathcal{E}(u)$ on $\mathcal{M}$. Since $\Psi(u)$ is of $C^1$ in ${H^1({\mathbb{R}}^N)}$, by the method of Lagrange multipliers, there exists $\sigma\in{\mathbb{R}}$ such that $\mathcal{E}'(u_0)-\sigma\Psi'(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$. For the sake of clarity, we divide the following proof into two claims.
[**Claim 1**]{}We have $\Psi'(u_0)\not=0$ in $H^{-1}({\mathbb{R}}^N)$.
Indeed, suppose the contrary that $\Psi'(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$. Then recalling the definition of $\Psi(u)$ given by , we can see that $u_0$ satisfies the following equation in the weak sense $$\begin{aligned}
\label{eq0018}
-\bigg(\frac{a(N-2)}{N}+\frac{2b(N-2)}{N}\mathfrak{A}_{u_0}\bigg)\Delta u_0+\lambda u_0=|u_0|^{p-2}u_0.\end{aligned}$$ Thus, by the Pohozaev identity of , we have $$\begin{aligned}
\bigg(\frac{a(N-2)}{N}+\frac{2b(N-2)}{N}\mathfrak{A}_{u_0}\bigg)\frac{N-2}{2N}\mathfrak{A}_{u_0}+\frac\lambda 2\mathfrak{C}_{u_0}-\frac1p\mathfrak{B}_{u_0}=0,\end{aligned}$$ which together with the fact that $u_0\in\mathcal{M}$, implies $$\begin{aligned}
\bigg(\frac{a(N-2)}{N}+\frac{2b(N-2)}{N}\mathfrak{A}_{u_0}\bigg)\frac{N-2}{2N}\mathfrak{A}_{u_0}=\frac{a(N-2)}{2N}\mathfrak{A}_{u_0}+\frac{b(N-2)}{2N}\mathfrak{A}_{u_0}^2.\end{aligned}$$ It follows that $$\begin{aligned}
\label{eq0019}
\frac{N-2}{2N}\mathfrak{A}_{u_0}\bigg(\frac{b(N-4)}{N}\mathfrak{A}_{u_0}-\frac{2a}{N}\bigg)=0\end{aligned}$$ Since $N\geq5$, by , we must have that $\mathfrak{A}_{u_0}=\frac{2a}{b(N-4)}$. By Lemma \[lem0003\], we must have that $u_0\in\mathcal{M}^0$, which is a contradiction.
[**Claim 2**]{}We have $\sigma=0$.
Indeed, suppose the contrary that $\sigma\not=0$. Then recalling the definition of $\mathcal{E}(u)$ given by and the fact that $\mathcal{E}'(u_0)-\sigma\Psi'(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$, we can see that $u_0$ satisfies the following equation in the weak sense $$\begin{aligned}
\label{eq0020}
&&-\bigg((1-\frac{(N-2)\sigma}{N})a+(1-\frac{2(N-2)\sigma}{N})b\mathfrak{A}_{u_0}\bigg)\Delta u_0+(1-\sigma)\lambda u_0\notag\\
&&=(1-\sigma)|u_0|^{p-2}u_0.\end{aligned}$$ By the Pohozaev identity of , we have $$\begin{aligned}
&&\bigg((1-\frac{(N-2)\sigma}{N})a+(1-\frac{2(N-2)\sigma}{N})b\mathfrak{A}_{u_0}\bigg)\frac{N-2}{2N}\mathfrak{A}_{u_0}\\
&&+\frac{\lambda(1-\sigma)}{2}\mathfrak{C}_{u_0}-\frac{1-\sigma}{p}\mathfrak{B}_{u_0}=0,\end{aligned}$$ which together with the fact that $u_0\in\mathcal{M}$, implies $$\begin{aligned}
&&\bigg((1-\frac{(N-2)\sigma}{N})a+(1-\frac{2(N-2)\sigma}{N})b\mathfrak{A}_{u_0}\bigg)\frac{N-2}{2N}\mathfrak{A}_{u_0}\\
&&=\frac{a(N-2)(1-\sigma)}{2N}\mathfrak{A}_{u_0}+\frac{b(N-2)(1-\sigma)}{2N}\mathfrak{A}_{u_0}^2.\end{aligned}$$ It follows that $$\begin{aligned}
\label{eq0021}
\frac{(N-2)}{2N}\mathfrak{A}_{u_0}(\frac{b(N-4)}{N}\mathfrak{A}_{u_0}-\frac{2a}{N})\sigma=0.\end{aligned}$$ Since $N\geq5$, by , we must have $\mathfrak{A}_{u_0}=\frac{2a}{b(N-4)}$, which together with Lemma \[lem0003\], implies $u_0\in\mathcal{M}^0$. It is also a contradiction.
Now, combining the above two claims and the fact that $\mathcal{E}'(u_0)-\sigma\Psi'(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$, we must have $\mathcal{E}'(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$.
Proof of Theorem \[thm0001\]
----------------------------
We respectively denote $\inf_{\mathcal{M}}\mathcal{E}(u)$, $\inf_{\mathcal{M}^-}\mathcal{E}(u)$ and $\inf_{\mathcal{M}^+}\mathcal{E}(u)$ by $m$, $m^-$ and $m^+$. Then we have the following.
\[lem0005\] Let $N\geq5$. If $\mathcal{B}_-\not=\emptyset$, then $m^\pm$ can be attained by some $u^\pm_{a,b,\lambda}$, which are both radial and nonnegative in ${\mathbb{R}}^N$.
Since $\mathcal{B}_-\not=\emptyset$, by Lemma \[lem0001\], $\mathcal{M}^\pm\not=\emptyset$. Let $\{u_n^\pm\}\subset\mathcal{M}^\pm$ respectively be a minimizing sequence of $\mathcal{E}(u)$ for $m^\pm$. Then by the Schwartz symmetrization, there exists $\{u_n^{*,\pm}\}\subset H^1_r({\mathbb{R}}^N)$ such that $$\begin{aligned}
\label{eq0026}
\mathfrak{A}_{u_n^{*,\pm}}\leq\mathfrak{A}_{u_n^\pm},\quad \mathfrak{B}_{u_n^{*,\pm}}=\mathfrak{B}_{u_n^\pm}\quad\text{and}\quad\mathfrak{C}_{u_n^{*,\pm}}=\mathfrak{C}_{u_n^\pm}.\end{aligned}$$ Thus, by the definitions of $\mathcal{B}_-$ and $\mathcal{B}(u)$ respectively given by and , we must have that $\{u_n^{*,\pm}\}\subset\mathcal{B}_-$. It follows from Lemma \[lem0001\] that there exist unique $0<t_{n,+}<t_{n,-}$ such that $v_n^\pm=u_n^{*,\pm}(t_{n,\pm}x)\in\mathcal{M}^\pm$. Since holds, we must have from $\{u_n^\pm\}\subset\mathcal{M}^\pm$ that $F_{u_n^{*,\pm}}'(1)\geq0$. It follows from Remark \[rmk0001\] that $t_{n,+}\leq1\leq t_{n,-}$, which together with Remark \[rmk0001\] once more, implies $$\begin{aligned}
m^-+o_n(1)&=&F_{u_n^-}(1)\notag\\
&\geq&F_{u_n^-}(t_{n,-})\notag\\
&=&\frac{b}{4}\mathfrak{A}_{u_n^-}^2t_{n,-}^{4-2N}+\frac{a}{2}\mathfrak{A}_{u_n^-}t_{n,-}^{2-N}+(\frac\lambda 2\mathfrak{B}_{u_n^-}-\frac1p\mathfrak{C}_{u_n^-})t_{n,-}^{-N}\notag\\
&\geq&\frac{b}{4}\mathfrak{A}_{u_n^{*,-}}^2t_{n,-}^{4-2N}+\frac{a}{2}\mathfrak{A}_{u_n^{*,-}}t_{n,-}^{2-N}+(\frac\lambda 2\mathfrak{B}_{u_n^{*,-}}-\frac1p\mathfrak{C}_{u_n^{*,-}})t_{n,-}^{-N}\notag\\
&=&\frac{b}{4}\mathfrak{A}_{v_n^-}^2+\frac{a}{2}\mathfrak{A}_{v_n^-}+\frac\lambda 2\mathfrak{B}_{v_n^-}-\frac1p\mathfrak{C}_{v_n^-}\notag\\
&\geq&m^-\label{eq0027}\end{aligned}$$ and $$\begin{aligned}
m^++o_n(1)&=&F_{u_n^+}(1)\notag\\
&=&\frac{b}{4}\mathfrak{A}_{u_n^+}^2+\frac{a}{2}\mathfrak{A}_{u_n^+}+\frac\lambda 2\mathfrak{B}_{u_n^+}-\frac1p\mathfrak{C}_{u_n^+}\notag\\
&\geq&\frac{b}{4}\mathfrak{A}_{u_n^{*,+}}^2+\frac{a}{2}\mathfrak{A}_{u_n^{*,+}}+\frac\lambda 2\mathfrak{B}_{u_n^{*,+}}-\frac1p\mathfrak{C}_{u_n^{*,+}}\notag\\
&=&F_{u_{n}^{*,+}}(1)\notag\\
&\geq&F_{u_{n}^{*,+}}(t_{n,+})\notag\\
&\geq&m^+.\label{eq0028}\end{aligned}$$ Therefore, $\{v_n^\pm\}$ are also minimizing sequences of $\mathcal{E}(u)$ for $m^\pm$, respectively. By Lemma \[lem0003\] and , $\{v_n^\pm\}$ are bounded in $H^1_r({\mathbb{R}}^N)$. Therefore, without loss of generality, we may assume that $v_n^\pm=v_0^\pm+o_n(1)$ weakly in $H^1_r({\mathbb{R}}^N)$. Thanks to the Sobolev embedding theorem, we also have that $v_n^\pm=v_0^\pm+o_n(1)$ strongly in $L^q({\mathbb{R}}^N)(2\leq q<2^*)$. Clearly, $v_0^\pm\in\mathcal{B}_-$, which together with Lemma \[lem0001\], implies that there exist unique $0<t_{0,+}<t_{0,-}$ such that $v_0^{*,\pm}=v_0^\pm(t_{0,\pm}x)\in\mathcal{M}^\pm$. Since $v_n^\pm=v_0^\pm+o_n(1)$ weakly in $H^1_r({\mathbb{R}}^N)$, we must have that $F_{v_0^{\pm}}'(1)\geq0$. It follows from Remark \[rmk0001\] that $t_{0,+}\leq1\leq t_{0,-}$. Now, by similar arguments as used for and , we can see that $m^\pm$ can be attained by $v_0^{*,\pm}$. Note that $|v_0^{*,\pm}|$ also attain $m^\pm$ by the definitions of $\mathcal{M}^\pm$, respectively. Thus, $m^\pm$ can be attained by some $u^\pm_{a,b,\lambda}$, which are both radial and nonnegative in ${\mathbb{R}}^N$.
Now, we can give the proof of Theorem \[thm0001\].
**Proof of Theorem \[thm0001\].**$(1)$Since $N\geq5$, if $\mathcal{F}_{a,b}(U_\lambda)<1$, then by the fact that $U_\lambda\in\mathcal{M}^*$, we can see that $U_\lambda\in\mathcal{B}_-$. It follows from Lemma \[lem0005\] that $m^\pm$ can be attained by some $u^\pm_{a,b,\lambda}$, which are both radial and nonnegative in ${\mathbb{R}}^N$. By a direct calculation, we can see that the function $H(s)=\frac{a}{N}s+\frac{(4-N)b}{4N}s^2$ is strictly increasing on $(0, \frac{2a}{(N-4)b})$ and strictly decreasing on $(\frac{2a}{(N-4)b}, +\infty)$. Note that $\mathcal{E}(u)=\frac{a}{N}\mathfrak{A}_{u}+\frac{(4-N)b}{4N}\mathfrak{A}_{u}^2$ for all $u\in\mathcal{M}$. Thus, by Lemma \[lem0003\], $u_{a,b,\lambda}^\pm$ are both local minimum points of $\mathcal{E}(u)$ on $\mathcal{M}$. Moreover, by Remark \[rmk0001\], we also have that $m^+=m$. Thanks to Lemma \[lem0004\] and the maximum principle, $u_{a,b,\lambda}^\pm$ are two radial positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$.
$(2)$If $\mathcal{F}_{a,b}(U_\lambda)=1$, then by the fact that $U_\lambda\in\mathcal{M}^*$, we can see that $U_\lambda\in\mathcal{B}_0$, where $\mathcal{B}_0$ is given by . Consider the function $v_t(x)=U_\lambda(tx)$, where $t=\bigg(\frac{(N-4)b\mathfrak{A}_{U_\lambda}}{2a}\bigg)^{\frac{1}{N-2}}$. Then by a direct calculation, we can see that the following equation holds in the weak sense $$\begin{aligned}
-\bigg(a+b\mathfrak{A}_{v_t}\bigg)\Delta v_t&=&-\bigg(a+b\mathfrak{A}_{U_\lambda}t^{2-N}\bigg)t^{2-N}\Delta U_\lambda\\
&=&\bigg(a+b\mathfrak{A}_{U_\lambda}t^{2-N}\bigg)t^{2-N}(U_\lambda^{p-1}-\lambda U_\lambda)\\
&=&\bigg(a+b\mathfrak{A}_{U_\lambda}t^{2-N}\bigg)t^{2}(v_t^{p-1}-\lambda v_t)\\
&=&\frac{(N-2)a}{N-4}t^2(v_t^{p-1}-\lambda v_t)\\
&=&v_t^{p-1}-\lambda v_t.\end{aligned}$$ Thus, $v_t(x)$ is a radial positive solution to the autonomous form of $(\mathcal{P}_{a,b})$. Suppose $\mathcal{M}^+\cup\mathcal{M}^-\not=\emptyset$. Then by Lemma \[lem0001\], there exists $u\in\mathcal{B}_-\cap\mathcal{C}$. It follows from Lemma \[lem0002\] that there exists $t=\bigg(\frac{\frac1p\mathfrak{B}_{u}-\frac\lambda2\mathfrak{C}_{u}}{\frac{N-2}{2N}\mathfrak{A}_{u}}\bigg)^{\frac12}$ such that $u_t\in\mathcal{M}^*$. By a similar argument as used in the proof of Lemma \[lem0005\], we can see that $I(U_\lambda)=\inf_{\mathcal{M}^*}I(u)$. Note that $I(u)=\frac{1}{N}\mathfrak{A}_{u}$ for $u\in\mathcal{M}^*$. Therefore, we must have $$\begin{aligned}
\label{eq0031}
\mathfrak{A}_{U_\lambda}=\min\{\mathfrak{A}_{u}\mid u\in \mathcal{M}^*\},\end{aligned}$$ which together with a similar argument as used in and the fact that $u\in\mathcal{B}_-$, implies $U_\lambda\in\mathcal{B}_-$. It is impossible since we already have $U_\lambda\in\mathcal{B}_0$. Thus, thanks to Lemma \[lem0001\], if $\mathcal{F}_{a,b}(U_\lambda)=1$, then $\mathcal{M}=\mathcal{M}^0$.
$(3)$Suppose the autonomous form of $(\mathcal{P}_{a,b})$ has a solution $u$ if $\mathcal{F}_{a,b}(U_\lambda)>1$. Then by the fact that $U_\lambda\in\mathcal{M}^*$, we can see that $U_\lambda\in\mathcal{B}_+$, where $\mathcal{B}_+$ is given by . It follows that $(\mathcal{P}_{a,b,\lambda})$ has a solution $u$ if $U_\lambda\in\mathcal{B}_+$. Note that we must have that $u\in\mathcal{M}\cap\mathcal{C}$. Thus, by Lemmas \[lem0002\] and \[lem0001\], we can see that $\mathcal{B}(u)\leq1$ and there exists $t=\bigg(\frac{\frac1p\mathfrak{B}_{u}-\frac\lambda2\mathfrak{C}_{u}}{\frac{N-2}{2N}\mathfrak{A}_{u}}\bigg)^{\frac12}$ such that $u_t\in\mathcal{M}^*$. By a direct calculation, we can see that $$\begin{aligned}
1\geq\mathcal{B}(u)=\mathcal{F}_{a,b}(u_t)\geq\mathcal{F}_{a,b}(U_\lambda),\end{aligned}$$ which together with , implies that $U_\lambda\in\mathcal{B}_-\cup\mathcal{B}_0$. It is impossible since we have $U_\lambda\in\mathcal{B}_+$.
The scaling technique
---------------------
In this section, we will study $(\mathcal{P}_{a,b,\lambda})$ by the scaling technique. Our main observation is the following two lemmas.
\[lem0006\] Let $a,b,\lambda>0$, $N\geq3$ and $2<p<2^*$. Suppose $u$ is a solution of $(\mathcal{P}_{a,b,\lambda})$, then there exist $s,t$ and $\alpha>0$ such that $U_{\alpha}(x)=su(tx)$ up to a translation.
Let $v(x)=su(tx)$, where $s,t>0$ are constants. Then for every $\varphi\in {H^1({\mathbb{R}}^N)}$, we have $$\begin{aligned}
&&\int_{{\mathbb{R}}^N}\nabla v(x)\nabla \varphi(x)dx\\
&=&\int_{{\mathbb{R}}^N}\nabla su(tx)\nabla \varphi(x)dx\\
&=&t^2s\int_{{\mathbb{R}}^N}\nabla u(x)\nabla \varphi(\frac{x}{t})d\frac{x}{t}\\
&=&\frac{t^{2-N}s}{(a+b\mathfrak{A}_{u})}\int_{{\mathbb{R}}^N}(u^{p-1}(x)-\lambda u(x))\varphi(\frac{x}{t})dx\\
&=&\frac{t^{2}s}{(a+b\mathfrak{A}_{u})}\int_{{\mathbb{R}}^N}(u^{p-1}(x)-\lambda u(x))\varphi(\frac{x}{t})d\frac{x}{t}\\
&=&\frac{t^{2}}{(a+b\mathfrak{A}_{u})}\int_{{\mathbb{R}}^N}(s^{2-p}v^{p-1}(x)-\lambda v(x))\varphi(x)dx.\end{aligned}$$ Since $a,b>0$, for fixed $s>0$, the equation $s^{2-p}t^2=a+b\mathfrak{A}_{u}$ must have a unique solution $t>0$. Let $\alpha=\frac{\lambda t^2}{a+b\mathfrak{A}_{u}}$. Then by the uniqueness of $U_\alpha$, we can see that $U_{\alpha}(x)=su(tx)$ up to a translation.
\[lemnew0001\] Let $a,b,\lambda>0$, $N\geq3$ and $2<p<2^*$. Then $\bigg(\frac{\lambda}{\alpha}\bigg)^{\frac{1}{p-2}}U_\alpha(tx)$ is a positive solution to the autonomous form of $(\mathcal{P}_{a,b})$ if and only if there exist two positive constants $t$ and $\alpha$ satisfying $\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac12}t=\gamma>0$ and $\gamma$ is the solution of .
Let $u(x)=sU_{\alpha}(tx)$, then for every $\varphi\in{H^1({\mathbb{R}}^N)}$, we have $$\begin{aligned}
&&(a+b\mathfrak{A}_{u})\int_{{\mathbb{R}}^N}\nabla u(x)\nabla \varphi(x)dx\\
&=&(a+bt^{2-N}s^2\mathfrak{A}_{U_{\alpha}})\int_{{\mathbb{R}}^N}\nabla u(x)\nabla \varphi(x)dx\\
&=&(a+bt^{2-N}s^2\mathfrak{A}_{U_{\alpha}})t^{2-N}\int_{{\mathbb{R}}^N}\nabla sU_{\alpha}(x)\nabla \varphi(\frac{x}{t})dx\\
&=&(a+bt^{2-N}s^2\mathfrak{A}_{U_{\alpha}})t^{2-N}s\int_{{\mathbb{R}}^N}(U_{\alpha}^{p-1}(x)-\alpha U_{\alpha}(x))\varphi(\frac{x}{t})dx\\
&=&(a+bt^{2-N}s^2\mathfrak{A}_{U_{\alpha}})t^{2}\int_{{\mathbb{R}}^N}(s^{2-p}u^{p-1}(x)-\alpha u(x))\varphi(x)dx.\end{aligned}$$ Thus, $u(x)$ is a solution to the autonomous form of $(\mathcal{P}_{a,b})$ if and only if $$\left\{\aligned &1=(a+bt^{2-N}s^2\mathfrak{A}_{U_{\alpha}})t^{2}s^{2-p},\\
&\lambda=\alpha(a+bt^{2-N}s^2\mathfrak{A}_{U_{\alpha}})t^{2},
\endaligned
\right.$$ which is equivalent to $s=\bigg(\frac{\lambda}{\alpha}\bigg)^{\frac{1}{p-2}}$ and $t$ is a solution of the following equation $$\begin{aligned}
\label{eq9956}
\frac{\lambda}{\alpha}=a t^2+b(\frac{\lambda}{\alpha})^{\frac{2}{p-2}}\mathfrak{A}_{U_{\alpha}}t^{4-N}.\end{aligned}$$ Thanks to , we can see that $\mathfrak{A}_{U_{\alpha}}=\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac{2}{p-2}-\frac{N-2}{2}}\mathfrak{A}_{U_{\lambda}}$. It follows from that $$\begin{aligned}
1&=&a\bigg(\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac12}t\bigg)^2+b\mathfrak{A}_{U_{\lambda}}\bigg(\frac{\alpha}{\lambda}\bigg)^{1-\frac{N-2}{2}}t^{4-N}\\
&=&a\bigg(\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac12}t\bigg)^2+b\bigg(\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac12}t\bigg)^{4-N}\mathfrak{A}_{U_{\lambda}}.\end{aligned}$$ Thus, let $\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac12}t=\gamma$, then $\gamma$ is the solution of if and only if $\bigg(\frac{\lambda}{\alpha}\bigg)^{\frac{1}{p-2}}U_\alpha(tx)$ is a positive solution to the autonomous form of $(\mathcal{P}_{a,b})$.
Due to Lemmas \[lem0006\] and \[lemnew0001\], we can give a proof of Theorem \[thm0002\].
**Proof of Theorem \[thm0002\].**By Lemmas \[lem0006\] and \[lemnew0001\], we can see that the solution to the autonomous form of $(\mathcal{P}_{a,b})$ must be of the form $\bigg(\frac{\lambda}{\alpha}\bigg)^{\frac{1}{p-2}}U_\alpha(tx)$, where $U_\alpha$ is the unique positive radial solution of for $\lambda=\alpha$, $\alpha$ and $t$ are two positive constants satisfying $\bigg(\frac{\alpha}{\lambda}\bigg)^{\frac12}t=\gamma>0$ and $\gamma$ is the solution of . Moreover, due to Lemma \[lemnew0001\], the number of positive solutions to $(\mathcal{P}_{a,b,\lambda})$ equals to the number of solutions to the equation .
We close this section by
**Proof of Theorem \[thm0004\].**By making a direct observation on the equation , we can see that has a unique solution for all $a,b,\lambda>0$ in the case $N=3$; has a unique solution for $b\mathfrak{A}_{U_{\lambda}}<1$ and has no solution for $b\mathfrak{A}_{U_{\lambda}}\geq1$ in the case $N=4$; has exact two solutions for $\mathcal{F}_{a,b}(U_\lambda)<1$, has a unique solution for $\mathcal{F}_{a,b}(U_\lambda)=1$ and has no solution for $\mathcal{F}_{a,b}(U_\lambda)>1$ in the cases $N\geq5$. Thus, the conclusion follows immediately from Theorems \[thm0001\] and \[thm0002\] and the results in [@A12; @A13].
The non-autonomous case
=======================
The Nehari manifold $\mathcal{N}_V$ for $p\in(2, 4]\cap(2, 2^*)$
----------------------------------------------------------------
We first consider the cases $p\in(2, 4)\cap(2,2^*)$. Let $$\begin{aligned}
\mathcal{D}&=&\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid \mathcal{G}(u)<0\}.\label{eqnew9001}\end{aligned}$$ where $$\begin{aligned}
\mathcal{G}(u)=a\mathfrak{A}_u+\mathfrak{C}_{u,V}-\frac{4-p}{2}\bigg(\frac{(p-2)\mathfrak{B}_u}{2b\mathfrak{A}_u^2}\bigg)^{\frac{p-2}{4-p}}\mathfrak{B}_u.\end{aligned}$$ Then our first observation on the Nehari manifold by the fibering map $G_u(t)$ is the following.
\[lemnew0002\] Let $p\in(2, 4)\cap(2,2^*)$ and the condition $(V)$ hold. Then there exist unique $0<t^-<t^+<+\infty$ such that $t^\pm u\in\mathcal{N}_V^\pm$ for $u\in\mathcal{D}$, where $\mathcal{N}_V^\pm$ are given by and . Moreover, $G_u(s)$ is strictly increasing on $(0, t^-)$, strictly decreasing on $(t^-, t^+)$ and strictly increasing on $(t^+, +\infty)$.
By a direct calculation, we can see that $G'_u(t)=t(b\mathfrak{A}_u^2t^2-\mathfrak{B}_ut^{p-2}+a\mathfrak{A}_u+\mathfrak{C}_{u,V})$. Set $$\begin{aligned}
\label{eqnew9002}
g_u(t)=b\mathfrak{A}_u^2t^2-\mathfrak{B}_ut^{p-2}+a\mathfrak{A}_u+\mathfrak{C}_{u,V}.\end{aligned}$$ Then by the condition $(V)$, we can see that $g_u(t)$ is strictly decreasing on $(0, t^*)$ and strictly increasing on $(t^*, +\infty)$, where $$\begin{aligned}
\label{eqnew9007}
t^*=\bigg(\frac{(p-2)\mathfrak{B}_u}{2b\mathfrak{A}_u^2}\bigg)^{\frac{1}{4-p}}.\end{aligned}$$ It follows that $g_u(t^*)=\min_{t\geq0}g_u(t)=\mathcal{G}(u)$. Now, the conclusion follows immediately from the definition of $\mathcal{D}$.
Next, we consider the case $p=4<2^*$, which implies $N=3$. We define $$\begin{aligned}
\label{eqnew9003}
\mathcal{Q}=\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid b\mathfrak{A}_u^2-\mathfrak{B}_u<0\}.\end{aligned}$$ Then we have the following.
\[lemnew0003\] Let $N=3$, $p=4$ and the condition $(V)$ hold. Then there exists unique $0<t^0<+\infty$ such that $t^0 u\in\mathcal{N}_V$ for $u\in\mathcal{Q}$, where $\mathcal{N}_V$ is given by . Moreover, $G_u(s)$ is strictly increasing on $(0, t^0)$ and strictly decreasing on $(t^0, +\infty)$.
The proof is similar to that of Lemma \[lemnew0002\].
\[rmknew0001\] By checking the proof of Lemmas \[lemnew0002\] and \[lemnew0003\], we can see that $tu\not\in\mathcal{N}_V^\pm$ for all $t\geq0$ if $u\not\in\mathcal{D}$ in the cases $p\in(2, 4)\cap(2,2^*)$ and $tu\not\in\mathcal{N}_V$ for all $t\geq0$ if $u\not\in\mathcal{Q}$ in the case $N=3$ and $p=4$.
Let $$\begin{aligned}
\label{eqnew9006}
\mathcal{S}_{p,a,V}=\inf_{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}}\frac{a\mathfrak{A}_u+\mathfrak{C}_{u,V}}{\mathfrak{B}_u^{\frac2p}}.\end{aligned}$$ Then by the condition $(V)$, we can see that $\mathcal{S}_{p,a,V}>0$ is well defined.
\[lemnew0004\] Let $a>0$ and $p\in(2, 4]\cap(2,2^*)$. If the condition $(V)$ holds, then there exists $b_*(a)>0$ such that $\mathcal{D}$ and $\mathcal{Q}$ are both nonempty sets for $0<b<b_*(a)$, where $\mathcal{D}$ and $\mathcal{Q}$ are respectively given by and .
Let ${u_n}$ be a minimizing sequence of $\mathcal{S}_{p,a,V}$. Then for $p\in(2, 4)\cap(2,2^*)$, we have from the condition $(V)$ that $$\begin{aligned}
\mathcal{G}(u_n)&\leq&a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V}
-\frac{(4-p)a^{\frac{2(p-2)}{4-p}}}{2}\bigg(\frac{(p-2)}{2b}\bigg)^{\frac{p-2}{4-p}}\frac{\mathfrak{B}_{u_n}^{\frac{2}{4-p}}}
{(a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V})^{\frac{2(p-2)}{4-p}}}\\
&=&\frac{\mathfrak{B}_{u_n}^{\frac{2}{4-p}}}{(a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V})^{\frac{2(p-2)}{4-p}}}
\bigg(\mathcal{S}_{p,a,V}^{\frac{p}{4-p}}
-\frac{(4-p)a^{\frac{2(p-2)}{4-p}}}{2}\bigg(\frac{(p-2)}{2b}\bigg)^{\frac{p-2}{4-p}}+o_n(1)\bigg).\\\end{aligned}$$ For $N=3$ and $p=4$, we also have from the condition $(V)$ that $$\begin{aligned}
b\mathfrak{A}_{u_n}^2-\mathfrak{B}_{u_n}&\leq&\frac{b}{a^2}(a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V})^2-\mathfrak{B}_{u_n}\\
&=&\mathfrak{B}_{u_n}(\frac{b}{a^2}\mathcal{S}_{4,a,V}^2+o_n(1)-1).\end{aligned}$$ Thus, there exists $b_*(a)>0$ such that $\mathcal{D}$ and $\mathcal{Q}$ are both nonempty sets for $0<b<b_*(a)$.
By Lemmas \[lemnew0002\]–\[lemnew0004\], we can see that $\mathcal{N}_V^-\not=\emptyset$ for $0<b<b_*(a)$ in all the cases $p\in(2, 4]\cap(2, 2^*)$ and $\mathcal{N}_V^+\not=\emptyset$ for $0<b<b_*(a)$ in the cases $p\in(2, 4)\cap(2, 2^*)$. Thus, $m^\pm=\inf_{\mathcal{N}_V^\pm}\mathcal{J}_V(u)$ are both well defined respectively in these cases, where $\mathcal{J}_V(u)$ is given by .
\[lemnew0005\] Let $a>0$, $p\in(2, 4]\cap(2, 2^*)$ and the condition $(V)$ hold. Then $m^->0$ for $0<b<b_*(a)$. Moreover, $m^+<0$ for $0<b<b_*(a)$ in the cases $p\in(2, 4)\cap(2, 2^*)$.
Let $u\in\mathcal{N}_V^-$. Then we have $$\begin{aligned}
b\mathfrak{A}_u^2-\mathfrak{B}_u+a\mathfrak{A}_u+\mathfrak{C}_{u,V}=0\label{eqnew9004}\\
3b\mathfrak{A}_u^2-(p-1)\mathfrak{B}_u+a\mathfrak{A}_u+\mathfrak{C}_{u,V}<0.\label{eqnew9005}\end{aligned}$$ By , and the condition $(V)$, we can see that $$\begin{aligned}
a\mathfrak{A}_u+\mathfrak{C}_{u,V}&\leq&\mathfrak{B}_u\\
&\leq&C_1\mathfrak{C}_u^{\frac{2^*-p}{2^*-2}}\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}\\
&\leq&C_2\mathfrak{C}_{u,V}^{\frac{2^*-p}{2^*-2}}\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}\end{aligned}$$ It follows from the Young inequality that $$\begin{aligned}
\label{eqnew9017}
a\mathfrak{A}_u\leq C_2\mathfrak{C}_{u,V}^{\frac{2^*-p}{2^*-2}}\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}-\mathfrak{C}_{u,V}
\leq C_3\mathfrak{A}_u^{\frac{2^*}{2}},\end{aligned}$$ which implies $\mathfrak{A}_u\geq C_4$. On the other hand, by and , we must have that $(4-p)\mathfrak{A}_u^2<(p-2)(a\mathfrak{A}_u+\mathfrak{C}_{u,V})$. Thus, for $p\in(2, 4)\cap(2, 2^*)$, we can see that $$\begin{aligned}
\mathcal{J}_V(u)&=&\frac{p-2}{2p}(a\mathfrak{A}_u+\mathfrak{C}_{u,V})-\frac{4-p}{4p}b\mathfrak{A}_u^2\notag\\
&>&\frac{4-p}{4p}b\mathfrak{A}_u^2\label{eqnew9025}\\
&\geq&C_5.\notag\end{aligned}$$ For the case $p=4$, by a similar argument as used for , we can see that $\mathcal{J}_V(u)\geq\frac{(p-2)a}{2p}\mathfrak{A}_u\geq C_6$. Since $u\in\mathcal{N}_V^-$ is arbitrary, by Lemmas \[lemnew0002\]–\[lemnew0004\], we must have that $m^->0$ for $0<b<b_*(a)$ in all the cases $p\in(2, 4]\cap(2, 2^*)$. Next, we prove that $m^+<0$ for $0<b<b_*(a)$ in the cases $p\in(2, 4)\cap(2, 2^*)$. Indeed, by choosing $b_*(a)$ small enough if necessary, we can see that $$\begin{aligned}
\mathcal{S}_{p,a,V}<a^{\frac{2(p-2)}{p}}\bigg(\frac{(4-p)(p+2)}{4p}\bigg)^{\frac{4-p}{p}}\bigg(\frac{(p-2)}{2b}\bigg)^{\frac{p-2}{p}}\end{aligned}$$ for $0<b<b_*(a), $where $\mathcal{S}_{p,a,V}$ is given by . It follows that there exists $u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}$ such that $$\begin{aligned}
\label{eqnew9011}
a\mathfrak{A}_u+\mathfrak{C}_{u,V}<a^{\frac{2(p-2)}{p}}\bigg(\frac{(4-p)(p+2)}{4p}\bigg)^{\frac{4-p}{p}}\bigg(\frac{(p-2)}{2b}\bigg)^{\frac{p-2}{p}}\mathfrak{B}_u^{\frac2p}.\end{aligned}$$ Since $2<p<4$, $\frac{p+2}{2p}<1$. Thus, we must have that $u\in\mathcal{D}$. By Lemma \[lemnew0002\], there exists $t^+>0$ such that $t^+u\in\mathcal{N}_V^+$. Thanks to Lemma \[lemnew0002\] once more, we also have that $\mathcal{J}_V(t^+u)\leq\mathcal{J}_V(t^*u)$, where $t^*$ is given by . It follows from that $$\begin{aligned}
\mathcal{J}_V(t^+u)&\leq&\mathcal{J}_V(t^*u)\\
&=&\frac{(t^*)^2}{2}\bigg(a\mathfrak{A}_u+\mathfrak{C}_{u,V}-\frac{(4-p)(p+2)(t^*)^{p-2}}{4p}\mathfrak{B}_u\bigg)\\
&\leq&\frac{(t^*)^2}{2}\bigg(a\mathfrak{A}_u+\mathfrak{C}_{u,V}\\
&&-\frac{(4-p)(p+2)a^{\frac{2(p-2)}{4-p}}}{4p}\bigg(\frac{(p-2)\mathfrak{B}_u}{2b(a\mathfrak{A}_u+\mathfrak{C}_{u,V})^2}\bigg)^{\frac{p-2}{4-p}}\mathfrak{B}_u\bigg)\\
&=&\frac{(t^*)^2}{2(a\mathfrak{A}_u+\mathfrak{C}_{u,V})^{\frac{2(p-2)}{4-p}}}\bigg((a\mathfrak{A}_u+\mathfrak{C}_{u,V})^{\frac{p}{4-p}}\\
&&-\frac{(4-p)(p+2)a^{\frac{2(p-2)}{4-p}}}{4p}\bigg(\frac{(p-2)}{2b}\bigg)^{\frac{p-2}{4-p}}\mathfrak{B}_u^{\frac{2}{4-p}}\bigg)\\
&<&0,\end{aligned}$$ which implies $m^+<0$ for $0<b<b_*(a)$ in the cases $p\in(2, 4)\cap(2, 2^*)$.
We close this section by the following observation on $\mathcal{N}_V$ for $0<b<b_*(a)$ in all the cases $p\in(2, 4]\cap(2, 2^*)$.
\[lemnew0006\] Let $a>0$ and the condition $(V)$ holds. If $u_0\in\mathcal{N}_V^-$ minimizes $\mathcal{J}_V(u)$ on $\mathcal{N}_V^-$ for $0<b<b_*(a)$ in the cases $p\in(2, 4]\cap(2, 2^*)$, then $u_0$ is also a critical point of $\mathcal{J}_V(u)$ in ${H^1({\mathbb{R}}^N)}$.
Let $u_0\in\mathcal{N}_V^-$ be a minimum point of $\mathcal{J}_V(u)$ on $\mathcal{N}_V^-$ for $0<b<b_*(a)$ in the cases $p\in(2, 4]\cap(2, 2^*)$. by the definition of $\mathcal{N}_V^-$, $u_0$ is also a minimum point of $\mathcal{J}_V(u)$ on $\mathcal{N}_V$ for $0<b<b_*(a)$. Thanks to the method of Lagrange multipliers, there exists $\tau\in{\mathbb{R}}$ such that $\mathcal{J}'_V(u_0)-\tau\Phi'_V(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$, where $\Phi_V(u)=\mathcal{J}'_V(u)u$. It follows from $u_0\in\mathcal{N}_V$ and a direct calculation that $0=\mathcal{J}'_V(u_0)u_0=\tau\Phi'_V(u_0)u_0=\tau\mathcal{G}_{u_0}''(1)$. Note that $u_0\in\mathcal{N}_V^-$, thus, $\mathcal{G}_{u_0}''(1)<0$, which implies $\tau=0$. It follows that $\mathcal{J}'_V(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$.
A local compactness result
--------------------------
Let $$\begin{aligned}
\label{eqnew9018}
\mathcal{J}_\infty(u)=\frac{b}{4}\mathfrak{A}_u^2+\frac{a}{2}\mathfrak{A}_u+\frac{v_\infty}{2}\mathfrak{C}_u-\frac1p\mathfrak{B}_u\end{aligned}$$ be the corresponding functional of the following equation $$\begin{aligned}
\label{eqnew9019}
\left\{\aligned&-\bigg(a+b\int_{{\mathbb{R}}^N}|\nabla u|^2dx\bigg)\Delta u+v_\infty u=|u|^{p-2}u&\text{ in }{\mathbb{R}}^N,\\&u\in{H^1({\mathbb{R}}^N)},\endaligned\right.\end{aligned}$$ Then $\mathcal{J}_\infty(u)$ and can respectively be seen as the “limit” functional and equation of $\mathcal{J}_V(u)$ and $(\mathcal{P}_{a,b})$. Let $\mathcal{D}_\infty=\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid\mathcal{G}_\infty(u)<0\}$, where $$\begin{aligned}
\mathcal{G}_\infty(u)=a\mathfrak{A}_u+v_\infty\mathfrak{C}_{u}-\frac{4-p}{2}\bigg(\frac{(p-2)\mathfrak{B}_u}{2b\mathfrak{A}_u^2}\bigg)^{\frac{p-2}{4-p}}\mathfrak{B}_u.\end{aligned}$$ Let $$\begin{aligned}
\label{eqnew9020}
\mathcal{N}_\infty=\{u\in{H^1({\mathbb{R}}^N)}\backslash\{0\}\mid \mathcal{J}_\infty'(u)u=0\}.\end{aligned}$$ Then it is easy to see that all nontrivial critical points of $\mathcal{J}_\infty(u)$ are contained in $\mathcal{N}_\infty$. Let $$\begin{aligned}
G_{u,\infty}(t)&=&\mathcal{J}_\infty(tu)\\
&=&\frac{bt^4}{4}\mathfrak{A}_u^2+\frac{at^2}{2}\mathfrak{A}_u+\frac{t^2v_\infty}2\mathfrak{C}_u^2-\frac{t^p}{p}\mathfrak{B}_u.\end{aligned}$$ Then by a direct calculation, we can see that $G_{u,\infty}(t)$ is of $C^2$ in ${\mathbb{R}}^+$ for every $u\in{H^1({\mathbb{R}}^N)}$ and $G_{u,\infty}'(t)=0$ if and only if $tu\in\mathcal{N}_\infty$. Thus, it is natural to divide the Nehari type manifold $\mathcal{N}_{\infty}$ into the following three parts: $$\begin{aligned}
\mathcal{N}_\infty^-&=&\{u\in\mathcal{N}\mid G_{u,\infty}''(1)<0\};\label{eqnew9021}\\
\mathcal{N}_\infty^0&=&\{u\in\mathcal{N}\mid G_{u,\infty}''(1)=0\};\label{eqnew9022}\\
\mathcal{N}_\infty^+&=&\{u\in\mathcal{N}\mid G_{u,\infty}''(1)>0\}.\label{eqnew9023}.\end{aligned}$$ Then choosing $b_*(a)$ small enough if necessary and by similar arguments as used in Lemmas \[lemnew0002\]–\[lemnew0006\], we can obtain the following.
\[lemnew0007\] Let $a>0$, $0<b<b_*(a)$ and $p\in(2, 4]\cap(2, 2^*)$. Then we have the following.
1. $\mathcal{D}_\infty$ and $\mathcal{Q}$ are both nonempty sets.
2. If $p\in(2, 4)\cap(2,2^*)$. Then there exist unique $0<t_\infty^-<t_\infty^+<+\infty$ such that $t_\infty^\pm u\in\mathcal{N}_\infty^\pm$ for $u\in\mathcal{D}_\infty$, where $\mathcal{N}_\infty^\pm$ are given by and . Moreover, $G_{u,\infty}(s)$ is strictly increasing on $(0, t_\infty^-)$, strictly decreasing on $(t_\infty^-, t_\infty^+)$ and strictly increasing on $(t_\infty^+, +\infty)$.
3. If $N=3$ and $p=4$. Then there exists unique $0<t_\infty^0<+\infty$ such that $t_\infty^0 u\in\mathcal{N}_\infty$ for $u\in\mathcal{Q}$, where $\mathcal{N}_\infty$ is given by . Moreover, $G_{u,\infty}(s)$ is strictly increasing on $(0, t_\infty^-)$ and strictly increasing on $(t_\infty^-, +\infty)$.
4. $m_\infty^->0$ in the cases $p\in(2, 4]\cap(2, 2^*)$ and $m_\infty^+<0$ in the cases $p\in(2, 4)\cap(2, 2^*)$ for $0<b<b_*(a)$, where $m_\infty^\pm=\inf_{\mathcal{N}_\infty^\pm}\mathcal{J}_\infty(u)$.
5. If $u_0\in\mathcal{N}_\infty^-$ minimizes $\mathcal{J}_\infty(u)$ on $\mathcal{N}_\infty^-$ in the cases $p\in(2, 4]\cap(2, 2^*)$ for $0<b<b_*(a)$, then $u_0$ is also a critical point of $\mathcal{J}_\infty(u)$ in ${H^1({\mathbb{R}}^N)}$.
Now, by Lemma \[lemnew0007\], we can obtain the following.
\[propnew0001\] Let $a>0$, $0<b<b_*(a)$ and $p\in(2, 4]\cap(2, 2^*)$. Then there exists $u_0\in\mathcal{N}_\infty^-$ such that $u_0>0$, $\mathcal{J}_\infty(u_0)=m_\infty^-$ and $\mathcal{J}'_\infty(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$.
By $(4)$ of Lemma \[lemnew0007\], $m_\infty^-$ is well defined. Let $\{u_n\}\subset\mathcal{N}_\infty^-$ be a minimizing sequence of $\mathcal{J}_\infty(u)$ for $m^-$. Then by the Schwartz symmetrization, there exists $\{u_n^{*}\}\subset H^1_r({\mathbb{R}}^N)$ such that $$\begin{aligned}
\label{eqnew0026}
\mathfrak{A}_{u_n^{*}}\leq\mathfrak{A}_{u_n},\quad \mathfrak{B}_{u_n^{*}}=\mathfrak{B}_{u_n}\quad\text{and}\quad\mathfrak{C}_{u_n^{*}}=\mathfrak{C}_{u_n}.\end{aligned}$$ It follows from $\{u_n\}\subset\mathcal{N}_\infty^-$ that $G_{u_n^*,\infty}'(1)\leq0$, which together with $(1)$ and $(2)$ of Lemma \[lemnew0007\], implies there exists $t_{n,\infty}^-\leq1$ such that $\{t_{n,\infty}^-u_n^{*}\}\subset\mathcal{N}_\infty^-$. Thus, by $(1)$ and $(2)$ of Lemma \[lemnew0007\] and once more, we can see that $$\begin{aligned}
m_\infty^-+o_n(1)=\mathcal{J}_\infty(u_n)\geq\mathcal{J}_\infty(t_{n,\infty}^-u_n)\geq\mathcal{J}_\infty(t_{n,\infty}^-u_n^*)\geq m_\infty^-.\end{aligned}$$ Therefore, $\{t_{n,\infty}^-u_n^{*}\}\subset H^1_r({\mathbb{R}}^N)\cap\mathcal{N}_\infty^-$ is also a minimizing sequence of $\mathcal{J}_\infty(u)$ for $m^-$. Without loss of generality and for the sake of simplicity, we assume $\{u_n\}\subset H^1_r({\mathbb{R}}^N)\cap\mathcal{N}_\infty^-$ is a minimizing sequence of $\mathcal{J}_\infty(u)$ for $m^-$. For the sake of clarity, we divide the following proof into several steps.
[**Step. 1**]{}We prove that $\{u_n\}$ is bounded in ${H^1({\mathbb{R}}^N)}$.
Indeed, if $p=4$, then by calculating $\mathcal{J}_\infty(u_n)-\frac{1}{4}\mathcal{J}_\infty'(u_n)u_n$, we can easily to show that $\{u_n\}$ is bounded in ${H^1({\mathbb{R}}^N)}$. Next, we consider the cases $2<p<\min\{2^*, 4\}$. Since $\{u_n\}\subset\mathcal{N}_\infty^-$, by a similar argument as used for , we can see that $\mathcal{J}_\infty(u_n)\geq\frac{4-p}{4p}b\mathfrak{A}_{u_n}^2$. Thus, $\{u_n\}$ is bounded in $D^{1,2}({\mathbb{R}}^N)$. On the other hand, since $\{u_n\}\subset\mathcal{N}_\infty^-$, by a similar argument as used for , we can see that $v_\infty\mathcal{S}^{\frac{2^*(p-2)}{2(2^*-2)}}\mathfrak{C}_u^{\frac{p-2}{2^*-2}}\leq\mathfrak{A}_u^{\frac{2^*(p-2)}{2(2^*-2)}}$. It follows that $\{u_n\}$ is bounded in ${H^1({\mathbb{R}}^N)}$.
[**Step. 2**]{}We prove that there exists $u_0\in\mathcal{N}_\infty^-$ such that $\mathcal{J}_\infty(u_0)=m_\infty^-$.
Indeed, by Step. 1, there exists $u_0\in H^1_r({\mathbb{R}}^N)$ such that $u_n=u_0+o_n(1)$ weakly in $H^1_r({\mathbb{R}}^N)$. Thanks to the Sobolev embedding theorem, $u_n=u_0+o_n(1)$ strongly in $L^q({\mathbb{R}}^N)(2\leq q<2^*)$. Note that $\{u_n\}\subset\mathcal{N}_\infty^-$, thus, we must have that $G_{u_0,\infty}'(1)\leq0$. It follows from $(1)$ and $(2)$ of Lemma \[lemnew0007\] that there exists $t_{\infty}^-\leq1$ such that $\{t_{\infty}^-u_0\}\subset\mathcal{N}_\infty^-$. Thus, by $(1)$ and $(2)$ of Lemma \[lemnew0007\] once more, we can see that $$\begin{aligned}
m_\infty^-+o_n(1)=\mathcal{J}_\infty(u_n)\geq\mathcal{J}_\infty(t_{\infty}^-u_n)\geq\mathcal{J}_\infty(t_{\infty}^-u_0)+o_n(1)\geq m_\infty^-+o_n(1),\end{aligned}$$ which implies $\mathcal{J}_\infty(t_{\infty}^-u_0)=m_\infty^-$.
Note that $|t_{\infty}^-u_0|\in\mathcal{N}_\infty^-$ and $\mathcal{J}_\infty(|t_{\infty}^-u_0|)=\mathcal{J}_\infty(t_{\infty}^-u_0)$, by $(5)$ of Lemma \[lemnew0007\] and the maximum principle, $|t_{\infty}^-u_0|$ is a positive solution of .
Based upon Proposition \[propnew0001\], we can obtain the following local compactness result.
\[lemnew0009\] Let $a>0$, $0<b<b_*(a)$ and $p\in(2, 4]\cap(2, 2^*)$. If the condition $(V)$ holds, then for every $(PS)_{m^-}$ sequence of $\mathcal{J}_V(u)$ contained in $\mathcal{N}_V^-$, there exists a subsequence which is compact in ${H^1({\mathbb{R}}^N)}$.
Let $\{u_n\}\subset\mathcal{N}_V^-$ is a $(PS)_{m^-}$ sequence of $\mathcal{J}_V(u)$. Then $\mathcal{J}_V(u_n)=m^-+o_n(1)$ and $\mathcal{J}_V'(u_n)=o_n(1)$ strongly in $H^{-1}({\mathbb{R}}^N)$. For the sake of clarity, we divide the following proof into two steps.
[**Step. 1**]{}We prove that $m^-<m_\infty^-$.
Indeed, by Proposition \[propnew0001\], there exists $u_0\in\mathcal{N}_\infty^-$ such that $u_0>0$, $\mathcal{J}_\infty(u_0)=m_\infty^-$ and $\mathcal{J}'_\infty(u_0)=0$ in $H^{-1}({\mathbb{R}}^N)$. By the condition $(V)$, we can see that $\mathcal{J}_V'(u_0)u_0<0$. It follows from Remark \[rmknew0001\] and Lemmas \[lemnew0002\] and \[lemnew0003\] that there exists $t^-<1$ such that $t^-u_0\in\mathcal{N}_V^-$. Now, thanks to $(1)$ and $(2)$ of Lemma \[lemnew0007\], we have $$\begin{aligned}
m^-\leq\mathcal{J}_V(t^-u_0)<\mathcal{J}_\infty(t^-u_0)<\mathcal{J}_\infty(u_0)=m_\infty^-.\end{aligned}$$
[**Step. 2**]{}We prove that there exists a subsequence of $\{u_n\}$ and $u^*\in{H^1({\mathbb{R}}^N)}$, still denoted by $\{u_n\}$, such that $u_n=u^*+o_n(1)$ strongly in ${H^1({\mathbb{R}}^N)}$.
Indeed, by a similar argument as used in Step. 1 of the proof to Proposition \[propnew0001\], we can show that $\{u_n\}$ is bounded in ${H^1({\mathbb{R}}^N)}$. Thus, without loss of generality, we assume that $u_n=u^*+o_n(1)$ weakly in ${H^1({\mathbb{R}}^N)}$ for some $u^*\in{H^1({\mathbb{R}}^N)}$. Clearly, one of the following two cases must happen:
1. $u^*=0$;
2. $u^*\not=0$.
We first consider the case $(a)$. Since $\{u_n\}\subset\mathcal{N}_V^-$, by , we have that $\mathfrak{B}_{u_n}\geq C_1+o_n(1)$. Thanks to the Lions lemma, there exist $R>0$ and $\{x_n\}\subset{\mathbb{R}}^N$ such that $$\begin{aligned}
\label{eqnew9026}
\int_{B_R(x_n)}|u_n|^2dx\geq C_2+o_n(1),\end{aligned}$$ where $B_R(x_n)=\{x\in{\mathbb{R}}^N\mid|x-x_n|<R\}$. Let $w_n=u_n(\cdot-x_n)$. Then $\mathfrak{A}_{u_n}=\mathfrak{A}_{w_n}$ and $\mathfrak{C}_{u_n}=\mathfrak{C}_{w_n}$. By and the Sobolev embedding theorem, we can see that $|x_n|\to+\infty$ as $n\to\infty$, $w_n=w_0+o_n(1)$ weakly in ${H^1({\mathbb{R}}^N)}$ and $w_0\not=0$. Let $w_n^1=w_n-w_0$, then by the Brezís–Lieb lemma and the condition $(V)$, we have $$\begin{aligned}
\label{eqnew9027}
\mathcal{J}_V(u_n)=\mathcal{J}_\infty(w_n)+o_n(1)=\mathcal{J}_\infty(w_n^1)+\mathcal{J}_\infty(w_0)+\frac b2\mathfrak{A}_{w_n^1}\mathfrak{A}_{w_0}+o_n(1).\end{aligned}$$ Moreover, since $\mathcal{J}_V'(u_n)=o_n(1)$ strongly in $H^{-1}({\mathbb{R}}^N)$, by a standard argument, we can see that $\mathcal{J}_\infty'(w_n)=o_n(1)$ strongly in $H^{-1}({\mathbb{R}}^N)$ due to the condition $(V)$. It follows that $\mathcal{J}_\infty'(w_0)w_0\leq0$. Clearly, there are also two cases:
1. $\mathcal{J}_\infty'(w_0)w_0=0$;
2. $\mathcal{J}_\infty'(w_0)w_0<0$.
In the case $(a1)$, since $\mathcal{J}_\infty'(w_n)w_0=o_n(1)$ and $w_n=w_0+o_n(1)$ weakly in ${H^1({\mathbb{R}}^N)}$, we have that $\mathfrak{A}_{w_0}=\mathfrak{A}_{w_n}+o_n(1)$. It follows from $w_n=w_0+o_n(1)$ weakly in ${H^1({\mathbb{R}}^N)}$ once more that $w_n=w_0+o_n(1)$ strongly in $D^{1,2}({\mathbb{R}}^N)$. Now, by applying the Sobolev embedding theorem and the fact that $(\mathcal{J}_\infty'(w_n)-\mathcal{J}_\infty'(w_0))(w_n-w_0)=o_n(1)$ in a standard way, we can see that $w_n=w_0+o_n(1)$ strongly in ${H^1({\mathbb{R}}^N)}$, which implies that $\mathcal{J}_\infty'(w_0)=0$ in $H^{-1}({\mathbb{R}}^N)$. Since $u_n\in\mathcal{N}_V^-$ and $w_n=u_n(\cdot-x_n)$, we can see from the condition $(V)$ that $G_{w_0,\infty}''(1)=G_{w_n,\infty}''(1)+o_n(1)=G_{u_n}''(1)+o_n(1)$. We claim that $G_{u_n}''(1)\leq-C_1+o_n(1)$. Indeed, let $w^*$ be the positive solution of $(\mathcal{P}_{1, 0})$. Since $w^*$ is independent of $b$, by choosing $b_*(a)$ small enough if necessary and the condition $(V)$, we have $$\begin{aligned}
\mathcal{G}(w^*)&\leq&a\mathfrak{A}_{w^*}+\mathfrak{C}_{w^*,V}\\
&&-\frac{(4-p)a^{\frac{2(p-2)}{4-p}}}{2}\bigg(\frac{(p-2)}{2b}\bigg)^{\frac{p-2}{4-p}}\frac{\mathfrak{B}_{w^*}^{\frac{2}{4-p}}}
{(a\mathfrak{A}_{w^*}+\mathfrak{C}_{w^*,V})^{\frac{2(p-2)}{4-p}}}<0\end{aligned}$$ for $0<b<b_*(a)$. By Lemmas \[lemnew0002\] and \[lemnew0003\], there exists $t^-_b>0$ such that $t^-_bw^*\in\mathcal{N}_V^-$. Suppose $t^-_b\to+\infty$ as $b\to0^+$, then by the fact that $t^-_bw^*\in\mathcal{N}_V^-$, we can see that $$\begin{aligned}
0&=&(t^-_b)^4b\mathfrak{A}_{w^*}^2+(t^-_b)^2a\mathfrak{A}_{w^*}+(t^-_b)^2\mathfrak{C}_{w^*,V}-(t^-_b)^p\mathfrak{B}_{w^*}\\
&=&(t^-_b)^p\bigg((t^-_b)^{4-p}b\mathfrak{A}_{w^*}^2+(t^-_b)^{2-p}(a\mathfrak{A}_{w^*}+\mathfrak{C}_{w^*,V})-\mathfrak{B}_{w^*}\bigg)\\
&<&(t^-_b)^p\bigg(\frac{p-4}{2}\mathfrak{B}_{w^*}+(t^-_b)^{2-p}(a\mathfrak{A}_{w^*}+\mathfrak{C}_{w^*,V})\bigg)\\
&<&0\end{aligned}$$ for $b$ small enough. In this inequality, we use the fact that $t^-_b<t^*$, where $t^*=\bigg(\frac{(p-2)\mathfrak{B}_{w^*}}{2b\mathfrak{A}_{w^*}^2}\bigg)^\frac{1}{4-p}$ is given by . Thus, by choosing $b_*(a)$ small enough if necessary, we must have that $t^-_b\leq C_2$ for $0<b<b_*(a)$. Now, by a standard argument, we can show that $\mathcal{J}_V(u_n)\leq C_3+o_n(1)$. On the other hand, suppose that dist$(u_n, \mathcal{N}_V^+\cup\mathcal{N}_V^0)=o_n(1)$, where dist$(u, \mathcal{N}_V^+\cup\mathcal{N}_V^0)=\inf\{w\in \mathcal{N}_V^+\cup\mathcal{N}_V^0\mid (\mathfrak{A}_{u-w}+\mathfrak{C}_{u-w})^\frac12\}$. Then by the Sobolev embedding theorem and the condition $(V)$, we must have that $$\begin{aligned}
b\mathfrak{A}_{u_n}^2-\mathfrak{B}_{u_n}+a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V}=0\label{eqnew9012}\\
3b\mathfrak{A}_{u_n}^2-(p-1)\mathfrak{B}_{u_n}+a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V}=o_n(1).\notag\end{aligned}$$ It follows that $$\begin{aligned}
\label{eqnew9101}
(4-p)b\mathfrak{A}_{u_n}^2=(p-2)(a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V})+o_n(1).\end{aligned}$$ Now, by and , we can see that $$\begin{aligned}
o_n(1)+C_3&\geq&\mathcal{J}_V(u_n)\\
&=&\frac{p-2}{2p}(a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V})-\frac{4-p}{4p}b\mathfrak{A}_{u_n}^2\notag\\
&=&\frac{4-p}{4p}b\mathfrak{A}_{u_n}^2+o_n(1),\end{aligned}$$ which implies $\mathfrak{A}_{u_n}\leq C_4b^{-\frac{1}{2}}+o_n(1)$. Note that by , we also have that $\mathfrak{A}_{u_n}\geq C_5b^{-1}+o_n(1)$. Thus, it is a contradiction for $0<b<b_*(a)$ by choosing $b_*(a)$ small enough if necessary. Hence, we must have that $G_{u_n}''(1)\leq-C_1+o_n(1)$. Now, by the definition of $\mathcal{N}_\infty^-$, we have that $w_0\in\mathcal{N}_\infty^-$, which together with , implies that $m^-=m_\infty^-$. It contradicts to Step. 1. In the case $(a2)$, by Lemma \[lemnew0007\], there exists $t^-_\infty<1$ such that $t^-_\infty w_0\in\mathcal{N}_\infty^-$. By Lemma \[lemnew0002\] and a similar argument as used for , we can see that $$\begin{aligned}
\mathcal{J}_V(u_n)&\geq&\mathcal{J}_V(t_\infty^-u_n)\notag\\
&=&\mathcal{J}_\infty(t_\infty^-w_n)+o_n(1)\notag\\
&=&\mathcal{J}_\infty(t_\infty^-w_n^1)+\mathcal{J}_\infty(t_\infty^-w_0)+o_n(1).\label{eqnew9028}\end{aligned}$$ Clearly, there also two cases:
1. $\mathcal{J}_\infty(t_\infty^-w_n^1)\geq0$;
2. $\mathcal{J}_\infty(t_\infty^-w_n^1)<0$.
In the case $(a2_1)$, we obtain that $m^-\geq m_\infty^-$ by , which is also a contradiction to Step. 1. In the case $(a2_2)$, by Lemma \[lemnew0007\] once more, there exist $t_{\infty,n}^-<t_\infty^-<1$ such that $t_{\infty,n}^-w_n^1\in\mathcal{N}_\infty^-$. Since $t_{\infty,n}^-<t_\infty^-$, we must have from Lemma \[lemnew0007\] once more that $\mathcal{J}_\infty(t_{\infty,n}^-w_0)\geq0$. Now, by a similar argument as used for , we also have that $$\begin{aligned}
\mathcal{J}_V(u_n)&\geq&\mathcal{J}_V(t_{\infty,n}^-u_n)\notag\\
&=&\mathcal{J}_\infty(t_{\infty,n}^-w_n)+o_n(1)\notag\\
&=&\mathcal{J}_\infty(t_{\infty,n}^-w_n^1)+\mathcal{J}_\infty(t_{\infty,n}^-w_0)+o_n(1),\notag\end{aligned}$$ which implies $m^-\geq m_\infty^-$. It also contradicts to Step. 1. Hence, we must have the case $(b)$. Let $w_n=u_n-u^*$. Then $w_n=o_n(1)$ weakly in ${H^1({\mathbb{R}}^N)}$. Moreover, by the Brezís–Lieb lemma and the condition $(V)$, we have $$\begin{aligned}
\mathcal{J}_V(u_n)=\mathcal{J}_V(u^*)+\mathcal{J}_\infty(w_n)+\frac12 \mathfrak{A}_{u^*}\mathfrak{A}_{w_n}+o_n(1).\end{aligned}$$ Since $\mathcal{J}_V'(u_n)=o_n(1)$ strongly in $H^{-1}({\mathbb{R}}^N)$, we also have that $\mathcal{J}_V'(u^*)u^*\leq0$. Now, thanks to Lemmas \[lemnew0002\] and \[lemnew0003\], we can apply similar arguments as used for the case $(a2)$ to obtain that $m^-\geq m_\infty^-$ if $\mathcal{J}_V'(u^*)u^*<0$. Thus, we must have $\mathcal{J}_V'(u^*)u^*=0$. By a similar argument as used for the case $(a1)$, we can see that $u_n=u^*+o_n(1)$ strongly in ${H^1({\mathbb{R}}^N)}$, which completes the proof.
Proof of Theorem \[thm1001\]
----------------------------
By the Ekeland variational principle, there exists $\{u_n\}\subset\mathcal{N}_V^-$ such that
1. $\mathcal{J}_V(u_n)=m^-+o_n(1)$;
2. $\mathcal{J}_V(v)-\mathcal{J}_V(u_n)\geq-\frac1n\bigg(\mathfrak{A}_{u_n-v}+\mathfrak{C}_{u_n-v}\bigg)^{\frac12}$ for all $v\in\mathcal{N}_V^-$.
\[lemnew0010\] Let $a>0$, $0<b<b_*(a)$ and $p\in(2, 4]\cap(2, 2^*)$. If the condition $(V)$ holds, then there exist ${\varepsilon}_n>0$ and $t_n(l):[-{\varepsilon}_n, {\varepsilon}_n]\to[\frac12, \frac32]$ such that $t_n(l)u_n+lw\in\mathcal{N}_V^-$ for all $w\in\mathbb{B}_1:=\{u\in{H^1({\mathbb{R}}^N)}\mid\mathfrak{A}_{u}+\mathfrak{C}_{u}=1\}$. Moreover, $t_n(l)$ are of $C^1$ and $$\begin{aligned}
\label{eqnew9030}
t_n'(0)=\frac{(4b\mathfrak{A}_{u_n}+2a)\langle\nabla u_n, \nabla w\rangle+2\langle u_n, w\rangle-p\int_{{\mathbb{R}}^N}|u_n|^{p-2}u_nwdx}{3b\mathfrak{A}_{u_n}+a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V}-\mathfrak{B}_{u_n}},\end{aligned}$$ where $\langle\cdot, \cdot\rangle$ is the usual inner product in $L^2({\mathbb{R}}^N)$.
Let $$\begin{aligned}
\mathcal{T}_n(t,l)=b\mathfrak{A}_{tu_n+lw}^2+a\mathfrak{A}_{tu_n+lw}+\mathfrak{C}_{tu_n+lw,V}-\mathfrak{B}_{tu_n+lw},\end{aligned}$$ where $w\in\mathbb{B}_1$. Clearly, $\mathcal{T}_n(1,0)=0$. By applying the implicit function theorem to $\mathcal{T}_n(t,l)$, we can show that there exist ${\varepsilon}_n>0$ and $t_n(l):[-{\varepsilon}_n, {\varepsilon}_n]\to[\frac12, \frac32]$ such that $t_n(l)u_n+lw\in\mathcal{N}_V$ for all $w\in\mathbb{B}_1$. Moreover, $t_n(l)$ are of $C^1$ and $t_n'(0)$ satisfy . It remains to show that $t_n(l)u_n+lw\in\mathcal{N}_V^-$. Indeed, by a similar argument as used in the proof of Lemma \[lemnew0009\], we can see that $3b\mathfrak{A}_{u_n}^2-(p-1)\mathfrak{B}_{u_n}+a\mathfrak{A}_{u_n}+\mathfrak{C}_{u_n,V}\leq-C_1+o_n(1)<0$. Now, by choosing ${\varepsilon}_n$ small enough if necessary, we actually have that $t_n(l)u_n+lw\in\mathcal{N}_V^-$.
Now, we can give the proof of Theorem \[thm1001\].
**Proof of Theorem \[thm1001\].**Let $v$ in $(2)$ be $t_n(l)u_n+lw$, then we have
1. $\mathcal{J}_V(u_n)=m^-+o_n(1)$;
2. $\mathcal{J}_V(t_n(l)u_n+lw)-\mathcal{J}_V(u_n)\geq-\frac1n\bigg(\mathfrak{A}_{(t_n(l)-1)u_n+lw}+\mathfrak{C}_{(t_n(l)-1)u_n+lw}\bigg)^{\frac12}$.
By a similar argument as used in the proof of Step. 1 to Proposition \[propnew0001\], we can show that $\{u_n\}$ is bounded in ${H^1({\mathbb{R}}^N)}$. Thanks to , we actually have that $|t_n(0)|\leq C_2(\mathfrak{A}_w+\mathfrak{C}_w)^{\frac12}$. It follows from $(2)$, Lemma \[lemnew0010\] and a standard argument that $\mathcal{J}_V'(u_n)=o_n(1)$ strongly in $H^{-1}({\mathbb{R}}^N)$. By Lemma \[lemnew0009\], $u_n=u_*+o_n(1)$ strongly in ${H^1({\mathbb{R}}^N)}$ for some $u_*$ up to a subsequence. It follows from a similar argument as used in the proof of Lemma \[lemnew0009\] that $u_*\in\mathcal{N}_V^-$ and $\mathcal{J}_V(u_*)=m^-$. Note that $|u_*|\in\mathcal{N}_V^-$ and $\mathcal{J}_V(|u_*|)=m^-$, thus, by Lemmas \[lemnew0005\] and \[lemnew0006\], $|u_*|$ is a nonnegative solution to $(\mathcal{P}_{a, b})$. Thanks to the maximum principle, $(\mathcal{P}_{a, b})$ has a positive solution for $a>0$, $p\in(2, 2^*)\cap(2, 4]$ and $0<b<b_*(a)$. It remains to show that $(\mathcal{P}_{a, b})$ only has trivial solution for $a>0$, $p\in(2, 2^*)\cap(2, 4)$ and $b$ large enough in the case $N\geq4$. Indeed, let $u$ be a nontrivial solution of $(\mathcal{P}_{a, b})$ in the cases $N\geq4$, then by a similar argument as used for , we can see that $$\begin{aligned}
0&=&b\mathfrak{A}_{u}^2+a\mathfrak{A}_{u}+\mathfrak{C}_{u,V}-\mathfrak{B}_{u}\notag\\
&\geq&b\mathfrak{A}_{u}^2-C_1\mathfrak{A}_{u}^{\frac{2^*}{2}}.\label{eqnew9031}\end{aligned}$$ Since $2^*=4$ for $N=4$, is impossible for $b$ large enough. In the cases $N\geq5$, we have that $2^*<4$. Thus, by and the Young inequality, we can see that $$\begin{aligned}
\label{eqnew9032}
0\geq\frac b2\mathfrak{A}_{u}^2-C_2 b^{-\frac{2^*}{4-2^*}}.\end{aligned}$$ Note that by a similar argument as used for , we also have that $\mathfrak{A}_{u}\geq C_3$. Thus, is impossible for $b$ large enough.
Acknowledgement
===============
Y. Huang was Natural Science Foundation of China (11471235 and 11171247). Z. Liu was supported by Suzhou University of Science and Technology foundation grant (331412104). Y. Wu was supported by the Fundamental Research Funds for the Central Universities (2014QNA67).
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[^1]: E-mail address: [email protected](Yisheng Huang)
[^2]: E-mail address: [email protected](Zeng Liu)
[^3]: Corresponding author. E-mail address: [email protected] (Yuanze Wu).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Online Social Networks (OSNs) allow personalities and companies to communicate directly with the public, bypassing filters of traditional medias. As people rely on OSNs to stay up-to-date, the political debate has moved online too. We witness the sudden explosion of harsh political debates and the dissemination of rumours in OSNs. Identifying such behaviour requires a deep understanding on how people interact via OSNs during political debates. We present a preliminary study of interactions in a popular OSN, namely Instagram. We take Italy as a case study in the period before the 2019 European Elections. We observe the activity of top Italian Instagram profiles in different categories: politics, music, sport and show. We record their posts for more than two months, tracking “likes” and comments from users. Results suggest that profiles of politicians attract markedly different interactions than other categories. People tend to comment more, with longer comments, debating for longer time, with a large number of replies, most of which are not explicitly solicited. Moreover, comments tend to come from a small group of very active users. Finally, we witness substantial differences when comparing profiles of different parties.'
author:
- 'Martino Trevisan, Luca Vassio, Idilio Drago, Marco Mellia'
- 'Fabricio Murai, Flavio Figueiredo, Ana Paula Couto da Silva, Jussara M. Almeida'
bibliography:
- 'main.bib'
title: Towards Understanding Political Interactions on Instagram
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We show how to modify any Bratteli diagram $E$ for an AF-algebra $A$ to obtain a Bratteli diagram $KE$ for $A$ whose graph algebra $C^*(KE)$ contains both $A$ and $C^*(E)$ as full corners.'
address: |
School of Mathematical and Physical Sciences\
University of Newcastle\
NSW 2308\
Australia
author:
- Jason Tyler
date: 3 June 2003
title: |
Every AF-algebra is Morita equivalent\
to a graph algebra.
---
[^1]
An elegant theorem of Drinen says that every AF-algebra $A$ is isomorphic to a corner in a graph algebra [@dri Theorem 1], and hence is Morita equivalent to the graph algebra. The graph in question is a Bratteli diagram for $A$, but it needs to be a carefully chosen one; two constructions of such a diagram were described in [@dri], one attributed to Kumjian. Here we show that applying Kumjian’s construction to an arbitrary Bratteli diagram $E$ for $A$ gives a graph $KE$ whose $C^*$-algebra contains both $A$ and $C^*(E)$ as full corners, so that $A$ is Morita equivalent to the $C^*$-algebra $C^*(E)$ of the original Bratteli diagram $E$.
A *directed graph* $E$ consists of countable sets $E^0$ of vertices and $E^1$ of edges, along with functions $r,s:E^1\to E^0$ which map edges to their range and source vertices. The graph is *row-finite* if each vertex emits at most finitely many edges. Given a row-finite graph $E$, a *Cuntz-Krieger $E$-family* in a $C^*$-algebra consists of a set of mutually orthogonal projections $\{p_v:v\in E^0\}$ and a set of partial isometries $\{s_e:e\in E^1\}$ satisfying the *Cuntz-Krieger relations*: $$s_e^*s_e = p_{r(e)} \text{ for }e\in E^1\text{ and }
p_v=\sum_{e\in s^{-1}(v)}s_es_e^*\text{ whenever }
s^{-1}(v)\not=\emptyset.$$ The graph algebra $C^*(E)$ is the universal $C^*$-algebra generated by a Cuntz-Krieger $E$-family $\{s_e,p_v\}$ [@kpr Theorem 1.2]. We denote by $E^*$ the set of all finite paths in $E$; that is, sequences of edges $\mu_1\mu_2\ldots\mu_n$ such that $r(\mu_i)=s(\mu_{i+1})$ for $1\le i<n$. We include the vertices as paths of length zero. Given $\mu=\mu_1\mu_2\ldots\mu_n\in E^*$, define $s_\mu:=
s_{\mu_1}s_{\mu_2}\ldots s_{\mu_n}$. It follows from [@kpr Lemma 1.1] that $$C^*(E) =\overline{\newspan}\{s_\mu s_\nu^*: \mu,\nu\in E^*,
r(\mu)=r(\nu)\}.$$
A *Bratteli diagram* is a directed graph $E$ such that:
- $E^0$ is the disjoint union of finite sets $\{V_n\}$,
- every edge with source in $V_n$ has range in $V_{n+1}$, and
- each $v\in E^0$ is labelled with a positive integer $d_v$ satisfying $d_v\ge \sum_{e\in r^{-1}(v)} d_{s(e)}$.
We say that $E$ is a Bratteli diagram for a sequence of $C^*$-algebras $A_1\subset A_2\subset \ldots$ if each $A_n$ is isomorphic to $\bigoplus_{v\in V_n} M_{d_v}(\C)$ and the embedding of each $M_{d_v}(\C)\subset A_n$ in each $M_{d_w}(\C)
\subset A_{n+1}$ scales the trace by $\#(s^{-1}(v)\cap r^{-1}(w))$. We say that $E$ is a Bratteli diagram for an AF-algebra $A$ if there exists a sequence of $C^*$-subalgebras $\{A_n\}$ of $A$ such that $A=\overline{\bigcup A_n}$ and $E$ is a Bratteli diagram for $\{A_n\}$.
Let $E$ be a Bratteli diagram for an AF-algebra $A$. Then there exists a Bratteli diagram $KE$ for $A$ such that $C^*(KE)$ contains $A$ and $C^*(E)$ as complementary full corners.
The projection $p$ defining the corner is the sum $p=\sum_{v\in S} p_v $ where $S\subset KE^0$; this sum converges strictly to a projection in $M(C^*(KE))$ by [@bprs Lemma 1.1]. Crucial for us is the observation that for $\mu,\nu\in KE^*$, $$ps_\mu s_\nu^*=\begin{cases}
s_\mu s_\nu^*&\text{if $s(\mu)\in S$}\\
0&\text{otherwise}
\end{cases}$$ so that $$pC^*(KE)p=\overline{\newspan}\{s_\mu s_\nu^*: s(\mu),s(\nu)\in S,
r(\mu)=r(\nu)\}.$$
For $n>0$, denote by $V_n$ the set of vertices on the $n$th level of $E$, and let $V_0=\emptyset$. For each $v\in E^0$, let $d_v$ be the rank of the matrix algebra corresponding to $v$. For every vertex $v\in E^0$, calculate $\sigma_v:=d_v -
\sum_{e\in r^{-1}(v)}d_{s(e)}$. We define $KE^0=\bigcup_{n=0}^\infty KV_n$, where $$KV_n := \begin{cases}
V_n & \text{if $\sigma_v=0$ for all $v\in V_{n+1}$}\\
V_n \cup \{w_n\} & \text{if $\sigma_v>0$ for some $v\in V_{n+1}$,}
\end{cases}$$ and define $KE^1$ to be $E^1$ together with, for every $w_n$ and $v\in V_{n+1}$, $\sigma_v$ edges from $w_n$ to $v$. Denote by $S$ the set $KE^0\backslash E^0=\bigcup \{w_n\}$, and set $d_w=1$ for all $w\in S$. Constructing $KE$ in this fashion ensures that for all $v\in {KE}^0$, the number of paths beginning in $S$ and ending at $v$ is $d_v$. Note that if $A$ is unital, so $\sigma_v=0$ for all $v\in E^0\backslash V_1$, then we add only one vertex to $E$ and $KE^0 = E^0 \bigcup \{w_0\}$.
Since $E$ is a Bratteli diagram for $A$, there is an increasing sequence of $C^*$-subalgebras $F_n$ of $A$ such that $A=\overline{\bigcup F_n}$ and $E$ is a Bratteli diagram for the sequence $\{F_n\}$. For those $n$ where $KV_n \not= V_n$, we define a subalgebra $F'_n$ of $A$ by $F'_0:= \C1$ and $$F'_n:=F_n\oplus\C(1_{F_{n+1}}-1_{F_n})\cong\bigoplus_{v\in
V_n} M_{d_v}(\C)\oplus \C\text{\hskip1cm for $n>0$.}$$ For all other $n$, define $F'_n=F_n$. The graph $KE$ is then a Bratteli diagram for the sequence $\{F'_n\}$. Since $F_n\subseteq
F'_n\subseteq F_{n+1}$ for all $n$, we have $\overline{\bigcup F'_n} =
\overline {\bigcup F_n} = A$; thus $KE$ is a Bratteli diagram for $A$.
Let $\{s_e,p_v\}$ be the universal Cuntz-Krieger $KE$-family generating $C^*(KE)$. Define a projection $p\in M(C^*(KE))$ by $p:=\sum_{v\in S}p_v$. We aim to show that the corner $pC^*(KE)p$ is isomorphic to $A$. Since two algebras with the same Bratteli diagram are isomorphic [@dav Proposition III.2.7], we can achieve this by identifying a sequence of subalgebras of $pC^*(KE)p$ for which $E$ is a Bratteli diagram and whose union is dense in $pC^*(KE)p$. For each $n>0$ define $D_n:= \newspan\{D^v:v\in V_n\}$, where $$D^v:= \newspan\{s_\mu s_\nu^*: \mu,\nu\in KE^*,s(\mu),s(\nu)\in S,
r(\mu)=r(\nu)=v\}$$ for each $v\in KE^0$. Note that $$pC^*(KE)p =\overline{\newspan}\{s_\mu s_\nu^*:
\mu,\nu\in KE^*,s(\mu),
s(\nu)\in S,r(\mu)=r(\nu)\} = \overline{\bigcup D_n}.$$ Given $v\in E^0$ and paths $\mu,\nu,\alpha,\beta$ with source in $S$ and range $v$, observe that none of $\mu,\nu,\alpha,\beta$ can extend any other since $KE$ contains no loops; [@kpr Lemma 1.1] then gives $$s_\mu s_\nu^*s_\alpha s_\beta^* = \begin{cases}
s_\mu s_\beta^*&\text{if $\nu=\alpha$}\\
0 &\text{otherwise.}
\end{cases}$$ Also, $(s_\mu s_\nu^*)^* = s_\nu s_\mu^*$, so $$\{s_\mu s_\nu^*: \mu,\nu\in KE^*,s(\mu),s(\nu)\in S, r(\mu)=r(\nu)=v\}$$ is a family of matrix units. Since there are $d_v$ paths $\mu$ with $s(\mu)\in S$ and $r(\mu)=v$, $D^v$ is isomorphic to $M_{d_v}(\C)$. Further, note that for distinct $v,w\in V_n$, no path ending at $v$ may extend one ending at $w$, so $D^vD^w=0$ and $D_n = \bigoplus_{v\in
V_n}D^v\cong \bigoplus_{v\in V_n} M_{d_v}(\C)$. It remains only to check that the embedding of each $D_n$ in $D_{n+1}$ matches that described by $E$; specifically, for $v\in V_n$ and $w\in
V_{n+1}$ we need that $D^v$ is embedded in $D^w$ with multiplicity $\#(s^{-1}(v)\cap r^{-1}(w))$. This follows from the Cuntz-Krieger relations at $v$: take paths $\mu,\nu$ with source in $S$ and range $v$, decompose the matrix unit $s_\mu s_\nu^*\in D^v$ as $$s_\mu s_\nu^*=s_\mu p_v s_\nu^*
=s_\mu \Bigl(\sum_{e\in s^{-1}(v)}s_e s_e^*\Bigr) s_\nu^*
=\sum_{e\in s^{-1}(v)}s_{\mu e}s_{\nu e}^*$$ and note that $s_{\mu e}s_{\nu e}^* $ is a matrix unit in $D^w$ precisely when $e\in r^{-1}(w)$.
Consider now the complementary corner $$(1-p)C^*(KE)(1-p)=
\overline{\newspan}\{s_\mu s_\nu^*:\mu,\nu\in KE^*,
s(\mu),s(\nu)\in E^0,r(\mu)=r(\nu)\}.$$ Since $KE^1 \backslash E^1$ contains only edges from $S$ to $E^0$, paths beginning in $E^0$ never leave $E^0$. Thus $(1-p)C^*(KE)(1-p)$ is generated by the Cuntz-Krieger $E$-family $$\{s_e,p_v:e\in E^1,v\in E^0\}.$$ Further, $E$ contains no loops, so the Cuntz-Krieger uniqueness theorem [@bprs Theorem 3.1] implies that $(1-p)C^*(KE)(1-p)$ is isomorphic to $C^*(E)$.
Finally, we must show that $p$ and $1-p$ are full. Note that for every $v\in KE^0$ there is a path beginning in $S$ and ending at $v$. Suppose that $I$ is an ideal in $C^*(KE)$ containing $pC^*(KE)p$; then $I$ certainly contains the projections $\{p_w:w\in S\}$. Given a vertex $v$ in $E^0$, choose a path $\alpha$ beginning at some $w\in S$ and ending at $v$. Then $s_\alpha= p_{w}s_\alpha\in I$, so $p_v=s_\alpha^*s_\alpha\in
I$, every generator $\{s_e,p_v\}$ of $C^*(KE)$ is in $I$, and $I=C^*(KE)$. Now suppose that $J$ is an ideal in $C^*(KE)$ containing $(1-p)C^*(KE)(1-p)$, so for every $v\in E^0$ we have $p_v\in J$. Given a vertex $v\in S$, note that every edge $e$ with $s(e)=v$ satisfies $r(e)\in E^0$; so for all $e\in s^{-1}(v)$, we know that $p_{r(e)}=s_e^*s_e\in J$, implying $s_e= s_es_e^*s_e\in
J$ and $s_es_e^*\in J$. Thus $p_v=\sum_{e\in s^{-1}(v)}s_es_e^*\in J$, the universal $KE$-family $\{s_e,p_v\}$ is contained in $J$, and $J=C^*(KE)$.
[1]{}
T. Bates, D. Pask, I. Raeburn and W. Szymański, [*The $C^*$-algebras of row-finite graphs*]{}, New York J. Math. [**6**]{} (2000), 307–324
K. Davidson, [*$C^*$-algebras by example*]{}, Amer. Math. Soc., Providence, 1996.
D. Drinen, [*Viewing AF-algebras as graph algebras*]{}, Proc. Amer. Math. Soc. [**128**]{} (1999), 1991–2000.
A. Kumjian, D. Pask and I. Raeburn, [*Cuntz-Krieger algebras of directed graphs*]{}, Pacific J. Math. [**184**]{} (1998), 161–174.
[^1]: My thanks go to my indefatigable supervisor Iain Raeburn for his guidance throughout this work.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We study the spin stiffness of stacked triangular antiferromagnets using both heat bath and broad histogram Monte Carlo methods. Our results are consistent with a continuous transition belonging to the chiral universality class first proposed by Kawamura.'
author:
- 'A. Peles and B.W. Southern'
bibliography:
- 'p1.bib'
title: Spin Stiffness of Stacked Triangular Antiferromagnets
---
Introduction
============
The magnetic ordering of geometrically frustrated antiferromagnets differs substantially from the conventional magnetic ordering in nonfrustrated magnets[@n1; @n2]. Indeed, the nature of the phase transition in the case of stacked triangular antiferromagnets remains controversial[@n3; @n4; @n5; @n6; @n9; @n10; @n11; @n17; @n18; @n19]. The geometry of the stacked triangular lattice has triangles as the elementary units and this arrangement inhibits an anti-parallel alignment of the spins in each triangular layer. Consequently, the system is said to be geometrically frustrated. This frustration leads to a comprise where the spins on each triangle adopt a non-collinear spin ordering at low temperatures. The spins form a planar configuration in which nearest neighbours are oriented at an angle of 120 degrees with respect to one another. The ground state can be described by a matrix like order parameter giving the orientation of each spin on the elementary triangles and forms an SO(3) parameter space[@n7]. This unusual symmetry of the order parameter and the appearance of ’chiral’ degrees of freedom which correspond to the ground state having two possible realizations, left and right handed, lead Kawamura[@n8] to conjecture the existence of a new chiral universality class . The chiral degrees of freedom are believed to be responsible for the novel critical behaviour but they are not decoupled from the spin degrees of freedom and the two quantities order simultaneously. While recent field-theoretic renormalization group studies of this system using an expansion up to six loops in fixed dimension $d=3$ indicate the existence of a stable fixed point that corresponds to the proposed chiral universality class[@n17; @n18], similar analyses using a three loop perturbation technique as well as an epsilon expansion approach to the same order, did not find a stable fixed point and hence exclude the possibility of a continuous phase transition for this frustrated system[@n9; @n10]. Non perturbative RG approaches find that the phase transition is possibly a very weak first order transition with effective critical exponents[@n11].
In the present work we use both a standard heat bath Monte Carlo method as well as a recently developed broad histogram method[@n14] to study the classical isotropic antiferromagnet on this geometry. In particular, we study the spin stiffness which provides a direct measure of the correlation length exponent $\nu$. The spin stiffness is a convenient quantity to study since it does not require knowledge of the order parameter but does measure the rigidity of the ordered phase against fluctuations. Our results confirm the picture of a continuous transition belonging to a new chiral universality class.
Model and Methods
=================
The model is described by the following Hamiltonian $$H = -\sum_{i<j} J \vec{S}_{i} \cdot \vec{S}_{j} -
\sum_{k<l} J' \vec{S}_{k} \cdot \vec{S}_{l}.$$ where $\vec S_{i}$ is a classical three component vector of unit length located at the sites $i$ of a hexagonal lattice. The first sum is over nearest neighbours in the triangular planes which interact with an antiferromagnetic coupling $J <0$ and the second sum is over inter-plane nearest neighbours which are taken to have a ferromagnetic coupling $J' >0$ with $|J'| =|J|=1$. Hence all energies and temperatures are measured in units of $|J|$.
We study the response of the system to a virtual twist of the spin system. The spin stiffness , or helicity modulus[@n13] , measures the increase in free energy associated with twisting the order parameter in spin space by imposing a gradient of the twist angle about some axis $\hat{n}$ in spin space along some direction $\hat{u}$ in the lattice. The spin stiffness can be written as a second derivative of the free energy with respect to the strength of the gradient and can be calculated as an equilibrium response function[@n12] . Finite size scaling theory predicts that the spin stiffness should vanish at the critical point with an exponent related to the correlation length exponent.
We calculate the diagonal elements of the spin stiffness tensor corresponding to twists about three principal directions in spin space. If we divide the lattice sites into three equivalent sublattices $A, B$ and $C$ corresponding to the vertices of the elementary triangles, then a chirality vector can be defined to characterize the non-collinear ordering of the spins. The chirality is defined locally for each upward (downward) triangle by the following expression $$\vec{K}_{\triangle}=\vec{S}_{A}\times\vec{S}_{B}+\vec{S}_{B}\times\vec{S}_{C}+
\vec{S}_{C}\times\vec{S}_{A}$$
In the ground state, the chirality is uniform and perpendicular to the spin planes. This symmetry of the order parameter suggests that the average chirality direction ($\hat{K}$) be chosen for one of the principal axes and the other two directions ($\hat{\perp}_{1},\hat{\perp}_{2}$) are chosen to be in the spin plane perpendicular to the average chirality vector such that the three vectors form an orthonormal triad. The spin stiffness component $\rho_{\alpha}$ at temperature $T$ can be written as [@n12]
$$\rho_{\alpha}=
\frac{1}{N}\sum_{i<j} J_{ij} (\hat{e}_{ij} \cdot \hat{u})^{2}
\left < S_{i}^{\beta}S_{j}^{\beta} +
S_{i}^{\gamma}S_{j}^{\gamma} \right>
- \frac{1}{NT}
\left < \left ( \sum_{i<j} J _{ij} (\hat{e}_{ij} \cdot \hat{u})
\left[
S_{i}^{\beta}S_{j}^{\gamma} - S_{i}^{\gamma}S_{j}^{\beta}\right]
\right)^{2}\right>$$
where $\alpha,\beta,\gamma= \hat{K},\hat{\perp}_{1},\hat{\perp}_{2}$ and the indices are taken in cyclic order. The twist is taken to be along the $\hat{u}$ direction in the lattice and $\hat{e}_{ij}$ is a unit vector directed along the nearest neighbour bond from site $i$ to $j$ . The angular brackets indicate a thermal average in the canonical ensemble. Since the ground state is a planar spin arrangement, the stiffnesses satisfy a perpendicular axis theorem $\rho_{\hat{K}}= \rho_{\hat{\perp}_{1}} + \rho_{\hat{\perp}_{2}}$ at zero temperature. Deviations from this relationship are a measure of fluctuations of spins from the planar order.
We perform numerical simulations using both a conventional Monte Carlo (MC) heat bath method and the more recent broad histogram method (BHM) introduced by Oliveira et. al. The latter method is based on the microcanonical ensemble approach to statistical sampling at fixed energy and allows an accurate estimate of the energy density of states[@n14; @n15; @n16] $g(E)$ . By knowing the density of states $g(E)$ and the microcanonical averages of various quantities $<Q>_E$, their temperature dependence can be determined by using the following expression for the canonical averages $$<Q>_{T}=\frac{ \sum_{E}<Q>_{E}g(E)e^{-E/T}}{\sum_{E}g(E)e^{-E/T}}$$
In the conventional heat bath method temperature is tuned as an external parameter and number of temperature points is limited by number of computer runs. The BHM method allows us to probe the system in a continuous range of $T$ but requires a large number of energy bins for large system sizes. We simulate spin systems of size $N=L^3$ with $L=24, 30, 42,60$ and $66$ for the heat bath method and only up to $L=60$ for the BHM method. Periodic boundary conditions are employed for both methods. We find excellent agreement between these two numerical methods.
Results
=======
{width="3.5in"}
{height="2.5in"} {height="2.5in"}
The broad histogram method(BHM) is based on the relation $$\begin{aligned}
g(E) < N_{up}(E) > = g(E+ \Delta E) < N_{dn}(E+ \Delta E) >\end{aligned}$$ where $<N_{up}(E)>, <N_{dn}(E)>$ are microcanonical averages which measure the number of moves which increase (decrease) the energy by the amount $\Delta E$. Once these microcanonical averages are known, the microcanonical temperature $T_m(E)$ can be determined from $$\begin{aligned}
1/T_m(E) & \equiv & \frac{d \ln g(E)}{dE} \nonumber \\
&\simeq & \frac{1}{\Delta E} \ln \frac{<N_{up}(E)>}{<N_{dn}(E+ \Delta E)>}\end{aligned}$$ and we can then integrate this expression in some range of energy to obtain the energy density of states $\ln g(E)$ as a function of $E$. In our case the energy is a continuous variable and we divide the energy axis into bins of a fixed size $\Delta E = 1.8 $ such that $\Delta E << E$, where $E$ is [*total*]{} energy of interest. We employ a simple microcanonical dynamics to sample phase space and the energy density of states $g(E)$ (up to a multiplicative constant) is determined using the BHM relation above. One microcanonical sweep consists of a random sweep through the lattice sites and generating a new configuration of the spins by restricting the choice of a new random orientation of the spin at site $i$ with respect to the local field of the nearest neighbours such that the total energy of the system remains within the energy interval $E, E + \Delta E$. At any given value of $E$ , 75 microcanonical sweeps were performed and 25 sample measurements were taken of various thermodynamic quantities such as the energy, specific heat and spin stiffness. Before sampling the next energy interval, 40 initial microcanonical sweeps were performed to avoid correlations. This procedure was repeated using different seeds for random numbers and errors were determined using the standard deviations for these separate measurements.
Figure 1 shows our results for $\ln g(e)$ as a function of $e=E/N$ in the case of a $42 \times 42 \times 42$ lattice. The units are arbitrary since we integrate equation (6) starting from $e = -2.1$ and not the ground state value $e_0 =-2.5$ .The number of energy bins used for this energy range was 61740. For general values of $L$, the number of energy bins required to study this same range with the same fixed size of energy bin is $5 L^3 /6$ and is thus of the same order as the number of sites. When the energy density of states is combined with the microcanonical averages $<Q>_E$ for various thermodynamic quantities, we can then plot them as continuous functions of $T$ using equation (4). Figure 2 shows the energy per site and the specific heat obtained using the BHM method for various linear sizes $L$. The energy displays strong finite size effects near the temperature where the specific heat has a maximum. The figures clearly indicate that a phase transition occurs near $T \sim 0.96$ in agreement with previous MC studies.[@n1]
{height="2.5in"} {height="2.5in"}
{width="3.5in"}
We have used both the BHM method and a Monte Carlo heat bath method at fixed values of $T$ to calculate the spin stiffness. In the heat bath method, we discard the first 1000 sweeps and perform 45000 MC steps in each run. Figure 3(a) shows both our heat bath results, indicated by points, and the BHM results, indicated by lines, for the three stiffnesses for various lattice sizes $L$ as a function of the temperature $T$. The relation $\rho_{\hat{K}}= \rho_{\hat{\perp}_{1}} + \rho_{\hat{\perp}_{2}}$ is well satisfied for all values of $T < 0.95$ indicating that there is a relatively small deviation from the planar spin configuration. All three stiffnesses are nonzero at low $T$ and vanish near $T \sim .96$ which corresponds to the specific heat divergence in figure 2. Figure 3(b) shows the heat bath data for $\rho_{\hat{K}}$ on an enlarged temperature scale. The stiffnesses clearly show large finite size effects and approach zero near $T \sim .96$. The points labelled infinity are obtained by plotting $\rho_{\hat{K}}$ versus $1/L$ at various values of $T$ and extrapolating to the large $L$ limit as shown in figure 4.
{width="3.5in"}
These finite size effects can be used to determine the correlation length exponent $\nu$ directly. Finite size scaling considerations for $\rho(T,L)$ predict $$\begin{aligned}
\rho(T,L) = \frac{1}{L} f(L/ \xi) = \frac{1}{L} f(L^{1/ \nu} |t|)\end{aligned}$$ where $t$ is the reduced temperature. This form suggests that we can plot $L \rho(T,L)$ versus $T$ to identify $T_c$ as the temperature where the curves for different values of $L$ intersect. Figure 5 shows our heat bath results for $L \rho_K$ as a function of $T$ for lattice sizes $L=24,30,42,60,66$. Linear interpolations of neigbouring temperature points indicate that the curves intersect at a value of $T_c = 0.958 \pm 0.002$. We have also used our BHM results in the same temperature range and we obtain the same estimate for $T_c$.
In the limit as $L \rightarrow \infty$, the scaling form predicts $\rho \sim |t|^{\nu}$. Using the values of the stiffness obtained by extrapolating to large values of $L$ as in figure 4 and then plotting these versus $ |t|$ on a ln-ln scale, we can obtain an estimate of $\nu$. Figure 6 shows our results for $\rho_K$ which yields the value $\nu = .589 \pm .007$. This value agrees very well with previous Monte Carlo estimates[@n1] but is slightly larger than the value found by the recent six loop renormalization group calculations in three dimensions.[@n17; @n18]
{width="3.5in"}
{width="3.5in"}
Figure 7 shows a finite size scaling plot of our stiffness results using the values of $T_c$ and $\nu$ quoted above. The data obtained from both the heat bath MC method for sizes $L=24,30,42,60,66$ and the BHM method for $L=24,30,42$ collapse very well to a universal function for temperatures below $T_c$. The value $\nu = .589 \pm .007$ is certainly very different from the value $\nu= 0.7113$ which describes the three dimensional Heisenberg universality class.[@n20]
Summary
=======
We have calculated the spin stiffness of the isotropic Heisenberg antiferromagnet on the stacked triangular geometry using both a MC heat bath and BHM method. The spin stiffness has the advantage that it measures the rigidity of the ordered phase in response to a virtual twist without having to specify the order parameter. The results obtained from both numerical approaches agree and predict a continuous phase transition which belongs to the new chiral universality class proposed by Kawamura.
This work was supported by the Natural Sciences and Research Council of Canada and the HPC facility at the University of Manitoba.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Many experiments that interrogate fundamental theories require detectors whose sensitivities are limited by the laws of quantum mechanics. In cavity-based searches for axionic dark matter, vacuum fluctuations in the two quadratures of the cavity electromagnetic field limit the sensitivity to an axion-induced field. In an apparatus designed to partially mimic existing axion detectors, we demonstrate experimentally that such quantum limits can be overcome through the use of squeezed states. By preparing a microwave cavity in a squeezed state and measuring just the squeezed quadrature, we enhance the spectral scan rate by a factor of $2.12 \pm 0.08$. This enhancement is in excellent quantitative agreement with a theoretical model accounting for both imperfect squeezing and measurement.'
author:
- 'M. Malnou'
- 'D. A. Palken'
- 'B. M. Brubaker'
- 'Leila R. Vale'
- 'Gene C. Hilton'
- 'K. W. Lehnert'
date:
-
-
title: Squeezed vacuum used to accelerate the search for a weak classical signal
---
[^1]
[^2]
INTRODUCTION
============
In searches for dark matter axions [@peccei1977cp; @peccei1977constraints; @preskill1983cosmology; @abbott1983a; @dine1983the], the quadratures $\hat X$ and $\hat Y$ of a resonant cavity’s electromagnetic field carry the imprint of the dark matter signal as a slight excess in their power spectra. Problematically, the quantum noise [@caves1982quantum] intrinsic to a measurement of these observables overpowers the signal by roughly three orders of magnitude, and the signal frequency is *a priori* unknown [@bradley2003microwave]. Even at the quantum limit, scanning the 1–10 GHz frequency band at the pessimistic benchmark DFSZ [@zhitnitsky1980possible; @dine1981simple] coupling with present detector technologies [@du2018search; @zhong2018results] will take upwards of 20,000 years of experimental live time. Realizing the several thousandfold speed-up required to put these detectors on expedient schedules will require many parallel innovations in detector design and sensitivity. Among the most promising advances are those which allow access to a fundamentally distinct parameter regime in which axion searches are no longer limited by quantum noise.
In detectors measuring the quadratures of a resonant mode, quantum noise can be circumvented by preparing the mode in a squeezed state [@caves1981quantum]. Squeezing unbalances the uncertainties in the two quadrature observables, thereby permitting precise knowledge of one at the expense of the other. Leveraging quantum squeezing to enhance measurement sensitivity has been a long-standing goal in the optical domain for the sensing of gravitational waves [@caves1981quantum; @abadie2011gravitational; @aasi2013enhanced; @kimble2001conversion]. At microwave frequencies, meanwhile, squeezing has been demonstrated in principle [@murch2013reduction; @clark2016observation; @bienfait2017magnetic], but it has yet to aid in a search for new physical phenomena.
But is microwave squeezing beneficial in an axion dark matter search given current experimental constraints? This question involves both matters of principle and practice. Specifically, in axion haloscope [@sikivie1983experimental] searches, the dark matter signal is a persistent tone of unknown frequency and phase, but more coherent than the resonant cavity itself. As such, a haloscope uses a tunable cavity to scan a resonance through frequency, with the critical parameter being the rate at which the cavity can be tuned with some specified sensitivity to an axion. The suitability of squeezing to this type of search has not received specific theoretical attention. Furthermore, if squeezing is in principle beneficial in this case, can microwave losses be made sufficiently low to yield a practical improvement?
In this article, we show both theoretically and experimentally that squeezing increases the scan rate in a search for a weak, axion-like signal of unknown frequency. This improvement exists in spite of the fact that squeezing does not improve the sensitivity of a haloscope to a tone of known frequency. We demonstrate this speed-up in an apparatus designed to mimic the behavior of existing haloscopes. By delivering a maximum of $4.5\pm 0.1$ dB of squeezing from one Josephson parametric amplifier (JPA) [@yamamoto2008flux; @castellanos2008amplification; @zhou2014high] to another in a squeezed state receiver (SSR) configuration, we perform a realistic acquisition and processing protocol both with and without squeezing enabled. We demonstrate that the SSR accelerates the scan rate at constant sensitivity to axion-like signals by a factor of $2.12\pm0.08$. Equipping the SSR to a dark matter detector such as HAYSTAC [@brubaker2017first; @zhong2018results] or ADMX [@du2018search] will mark the transition of searches for physics beyond the standard model to the sub-quantum limited noise regime.
THEORY OF SQUEEZING-ENHANCED SCAN RATE {#quantsq}
======================================
Figure \[fig:schematic\] shows a representative experimental apparatus in which a resonant cavity is coupled to an SSR comprising a pair of JPAs. We start by considering the cavity, whose internal mode (frequency $\omega_c$) has Hamiltonian $\hat{H} = \hbar\omega_c(\hat{X}^2 + \hat{Y}^2)/2$, where $\hat{X}$ and $\hat{Y}$ are the noncommuting quadratures of the cavity field, obeying $[\hat X, \hat Y] = i$. This cavity is modeled as exchanging energy with three ports. First, a measurement port couples the cavity mode to the propagating modes of a transmission line with power decay rate $\kappa_m$. Along this line, a microwave circulator spatially separates incoming and outgoing propagating modes. Second, a loss port, connected to a fictitious transmission line, models the cavity’s internal energy dissipation at rate $\kappa_l$. Third, the cavity’s coupling to the signal of interest at rate $\kappa_a$ is modeled as occurring through another fictitious transmission line; the signal itself is modeled as a microwave generator characterized by its frequency $\omega_a$, spectral width $\Delta_a$, and amplitude $\mathcal{E}_a$. We assume $\mathcal{E}_a \gg 1$, implying that the displacement of the cavity mode by the signal is classical (i.e., the contribution of the signal to the cavity’s quantum fluctuations can be neglected in comparison to those coming from the loss and measurement ports). We also assume a narrow band signal ($\Delta_a \ll \{\kappa_l,\kappa_m\}$) so weakly coupled ($\mathcal{E}_a^2 \kappa_a \ll \kappa_l$) that the time required to resolve the displacement is much longer than the signal’s phase coherence time. Therefore, on average, this displacement yields a small excess power above the vacuum fluctuations, isotropic in quadrature space (see bottom panel in Fig.\[fig:schematic\]). These inequalities are very well satisfied in the case of the axion field (see Appendix \[sup:axionmodel\]).
![Schematic of the SSR and cavity. Two JPAs, SQ and AMP, respectively squeeze and read out a microwave field interacting with a cavity at rate $\kappa_m$. An axion-like field $\mathcal{E}_a$, coupled to the cavity at rate $\kappa_a$, displaces the cavity state. Energy leaves the cavity through internal absorption at a rate $\kappa_l$. Bottom panel: quadrature representation of a vacuum state detuned from the cavity resonant frequency, traveling in the SSR. At the SQ’s input (a) it is Gaussian, azimuthally equiprobable in the $(\hat{X},\hat{Y})$ plane (red disk). The state is squeezed along $\hat{X}$ by the SQ (b), displaced along a random phase within the cavity (c), and amplified along $\hat{X}$ by the AMP (d). Comparing to what happens without squeezing (e)–(g), the size of the signal-plus-noise (green) relative to the noise (red) in this quadrature is larger with (d) than without (g) squeezing.[]{data-label="fig:schematic"}](schematic_exp_v41.pdf)
The SSR itself comprises the two JPAs shown in Fig.\[fig:schematic\], which couple respectively to incoming and outgoing modes at the cavity’s measurement port. This configuration exploits the fact that a portion of the vacuum noise exiting the measurement port arises from vacuum noise incident on that same port. The first JPA (called SQ) squeezes these input fluctuations along the $\hat{X}$ quadrature, reducing the observable’s variance below vacuum levels: $\sigma_X^2 < 1/2$ [@mallet2011quantum; @bienfait2017magnetic; @menzel2012path; @boutin2017effect; @malnou2018optimal]. To satisfy the uncertainty principle, the opposing quadrature’s fluctuations exceed vacuum, $\sigma_Y^2 > 1/2$, but are not measured. The squeezed input field subsequently enters the cavity, where a small displacement by the signal would yield a small excess power in both quadratures. At the measurement port output, the second JPA (called AMP) noiselessly amplifies only the $\hat{X}$ quadrature with sufficient gain to overwhelm the noise added by following amplifiers and mixers [@yamamoto2008flux; @zhou2014high; @pogorzalek2017hysteretic]. Note that in the absence of squeezing there is neither a benefit nor a penalty associated with measuring only one quadrature compared to the usual two-quadrature case (see Appendix \[sup:singledouble\]).
The benefit of squeezing can be understood by analyzing the microwave network formed by the combination of SSR and cavity using input-output theory (see Appendix \[sup:SSR\]). The signal spectral density at the measurement port output is equal to the signal spectral density at the measurement port input, weighted by the susceptibility of the measurement port output to the signal port input. Similarly, the noise density at the measurement port output is a susceptibility-weighted sum of squeezed and unsqueezed noise from the measurement and loss ports, respectively. In the absence of transmission losses between the JPAs and the cavity, the ratio of output signal spectral density to total output noise spectral density (hereafter called the signal visibility) is thus $$\label{SNR}
\alpha(\omega) \approx
\frac{n_A\kappa_a\kappa_m}{\left(n_T + \frac{1}{2}\right)\left(\kappa_l\kappa_m + \frac{\beta(\omega)}{G_s}\right)}.$$ Here, $\omega$ is the frequency relative to cavity resonance, $n_T = 1/[\exp{(\hbar\omega_c/k_BT)} - 1]$ is the mean thermal photon number incident on the cavity from the measurement and loss ports, $n_A = \mathcal{E}_a^2$ is the mean photon number sourced by the fictitious generator, $\beta(\omega) = (\kappa_m - \kappa_l)^2/4 + \omega^2$, and $G_s$ is the power gain of the SQ (ideally equal to the reduction of the squeezed state variance below the vacuum value).
When optimizing $\alpha(\omega)$ in Eq., we assume that $\kappa_m$ and $G_s$ can be freely varied, whereas $n_T$, $n_A$, $\kappa_l$, and $\kappa_a$ are fixed by the physics of the signal source or technical constraints of the detector. On cavity resonance ($\omega=0$), $\alpha$ is maximized at critical coupling ($\kappa_m = \kappa_l$). Because $\beta(0) = 0$ at critical coupling, $\alpha(0)$ is independent of $G_s$ and there is no benefit from squeezing; physically, the squeezed state injected into the cavity is completely absorbed in it, while all the unsqueezed noise from the loss port reaches the AMP. For $\omega \neq 0$, squeezing increases $\alpha(\omega)$ for any value of $\kappa_m$, as $G_s$ reduces the amount of measurement port noise reaching the output. In the limit where $G_s\rightarrow\infty$, the $\beta(\omega)$ term in the denominator can be neglected, and $\alpha(\omega)$ approaches the critically coupled resonant value $\alpha(\omega=0,\kappa_m=\kappa_l)$ for *any* $\kappa_m$ and *all* $\omega$. This illustrates an important point of principle: squeezing cannot improve the peak sensitivity of a haloscope, but there is no *fundamental* limit to how much it can enhance the detector bandwidth over which this peak sensitivity is achieved. When $G_s$ is finite, overcoupling ($\kappa_m > \kappa_l$) increases the cavity bandwidth at the cost of reducing $\alpha(0)$. This can be a favorable trade-off (even in the absence of squeezing) because the signal’s frequency $\omega_a$ is *a priori* unknown [@irastorza2018new; @chaudhuri2018fundamental], and broader bandwidth enables larger cavity tuning steps. Moreover, squeezing mitigates the reduction of $\alpha(0)$ from overcoupling, thus enabling faster tuning without significant degradation of sensitivity.
To quantify this speed-up, we calculate the rate $R$ (in Hz/s) at which we can tune the cavity resonance through frequency space in search for a signal. This scan rate is inversely proportional to the measurement time at each tuning step, which in turn scales with $\alpha^{-2}$ as a consequence of Gaussian noise statistics (see Appendix \[sup:SSRnoloss\]). Hence $R\propto\Delta_a\int_{-\infty}^\infty\alpha^2(\omega) d\omega$; carrying out the integral we obtain $$\label{R}
R \propto \frac{\Delta_a\sqrt{G_s}n_A^2\kappa_a^2\kappa_m^2}{\left(n_T+\frac{1}{2}\right)^2\left[\kappa_l\kappa_m + \frac{1}{G_s}\left(\frac{\kappa_l-\kappa_m}{2}\right)^2\right]^{3/2}}.$$ Without squeezing ($G_s=1$), $R$ is maximized when the cavity is twice overcoupled ($\kappa_m=2\kappa_l$), and at this optimal coupling the scan rate scales as $R_u^\mathrm{max}\propto\kappa_a^2/\kappa_l$, a known result [@alkenany2017design]. When $G_s\gg1$, the optimal coupling is $\kappa_m=2G_s\kappa_l$ and the scan rate scales as $R_s^\mathrm{max}\propto G_s\kappa_a^2/\kappa_l$. Comparing the two situations, the scan rate is improved by $G_s$, which shows that an ideal SSR greatly accelerates the search for a weak classical signal when squeezing and overcoupling.
In practice, however, losses in microwave components reduce the SQ-to-AMP transmission efficiency $\eta$ and hence the benefit of squeezing, because part of the squeezed state is replaced with unsqueezed vacuum (see Appendix \[sup:SSRwloss\]). Figure \[fig:Rtheo\] compares the theoretical scan rate enhancement $E_t=R_s/R_u^\mathrm{max}$ (a) when $\eta=1$ (perfect transmission), and (b) when $\eta=0.69$ (efficiency we observe in practice), as a function of $G_s$ and $\kappa_m/\kappa_l$. In the first case, $E_t$ improves arbitrarily as squeezing and coupling are together increased. In the second case, it plateaus at $E_t^\mathrm{max}\approx2.2$ for $G_s>20$ when optimally coupled. In Secs. \[scanrate\] and \[sec:faxion detection\], we present experimental results for the scan rate enhancement consistent with this theory.
![The scan rate enhancement $E_t$, calculated as a function of SQ single-quadrature power gain $G_s$ and coupling ratio $\kappa_m/\kappa_l$. With perfect efficiency, i.e. $\eta=1$, (a) $E_t$ grows steadily with $G_s$ and $\kappa_m$. When $\eta=0.69$ (b), it plateaus at $2.2$ for $\kappa_m/\kappa_l=5$ and $G_s>20$. The color scales for (a) and (b) differ by close to a factor of $10$.[]{data-label="fig:Rtheo"}](R_map_theo_eta0p69_v6.png)
EXPERIMENTALLY IMPROVED SIGNAL RECOVERY WITH SQUEEZING {#scanrate}
======================================================
The scan rate can be degraded not only by transmission losses, but also by distortions in the squeezed state [@malnou2018optimal; @boutin2017effect] and added noise in the amplifier chain. In order to investigate how accurately Eq. predicts the performance of a squeezing-enhanced haloscope, we construct an apparatus that mimics its microwave network, but without some of the cumbersome features. Specifically, in an axion haloscope the mechanically tunable cavity must reside in a large static magnetic field to enhance $\kappa_a$ and $n_A$. In our apparatus, the cavity has a fixed frequency $\omega_c = 2\pi\times7.146$GHz and there is no applied field. Consequently our cavity, constructed from superconducting aluminum, has much lower intrinsic loss than a copper haloscope cavity. We create additional loss through an explicit port that extracts energy from the cavity at a rate $\kappa_l =2\pi \times\ 100$kHz, a typical value for a haloscope cavity. We introduce an axion-like signal into our cavity through a microwave generator connected to a second port weakly coupled at rate $\kappa_a = 2\pi \times 100$Hz. Finally, a third port couples the cavity mode to an SSR with a rate chosen to be close to the optimum value for the case with ($\kappa_m =10 \kappa_l$) or without ($\kappa_m =1.5 \kappa_l$) squeezing, creating a physical realization of the model in Fig.\[fig:schematic\].
For the largest increase in scan rate, the SSR should be attached to the cavity measurement port with as little transmission loss as possible. To investigate the transmission loss independent of the cavity loss, we use the fact that the JPAs are narrow band ($\sim 5$ MHz), tunable amplifiers and detune both the SQ and AMP far off cavity resonance ($\sim 10$ MHz) so that the squeezed state is promptly reflected from the cavity. Figure \[fig:squeezing\] illustrates our ability to efficiently generate, transport, and amplify a squeezed state in this off-resonance configuration. It shows histograms of the measured voltage in the AMP’s amplified quadrature $\hat{X}_{\mathrm{out},m}$ as a function of $\theta$, the phase between the amplified quadrature of the SQ and the amplified quadrature of the AMP. When $\theta=\pi/2$ or $3\pi/2$, one quadrature is first squeezed and then amplified. At these points, comparing the output noise variances $\sigma^2_\mathrm{on}$ and $\sigma^2_\mathrm{off}$ measured with SQ on and off (not pumped), respectively [@malnou2018optimal], we obtain a squeezing $S=\sigma^2_\mathrm{on}/\sigma^2_\mathrm{off}=-4.5\pm0.1$dB. We emphasize that this is not an inferred squeezing; rather, we directly measure an overall $4.5$dB reduction in the noise floor of $\hat{X}_{\mathrm{out},m}$ over the whole bandwidth of the quadrature measurement. This amount of squeezing (consistent with our estimate of $\eta=0.69\pm0.01$, see Appendix \[sup:losslines\]) required particular care in reducing the transmission losses between the two JPAs, which was facilitated by using flux-pumped JPAs [@yamamoto2008flux; @zhou2014high; @pogorzalek2017hysteretic].
![Histograms of voltage fluctuations $V_X$ (a) and corresponding vacuum squeezing (b) $S=\sigma_\mathrm{on}^2/\sigma_\mathrm{off}^2$ measured along $\hat{X}_{\mathrm{out},m}$, as a function of the SQ-AMP relative phase $\theta$.[]{data-label="fig:squeezing"}](squeezing_jet_v7.pdf)
This large amount of delivered squeezing implies that the SSR should improve our ability to resolve a weakly coupled signal detuned from cavity resonance. To demonstrate this improvement, we center the SQ and AMP bands on cavity resonance, overcouple the cavity’s measurement port ($\kappa_m=10\kappa_l$), and set the phase $\theta$ to $\pi/2$ (see Fig.\[fig:squeezing\]). We complete the single-quadrature measurement by mixing down $\hat{X}_{\mathrm{out},m}$ with a local oscillator (LO) at the cavity resonant frequency and computing its power spectral density. As for any such measurement, the frequency component at $\omega$ in the down-mixed output is a linear combination of two frequency components at the mixer’s high-frequency input, $\pm\omega$ detuned from the LO frequency. Figure \[fig:SNRimpr\]a shows the spectral density when the tone is $1$MHz detuned from the cavity’s resonance. Comparing two situations, SQ on and SQ off, the signal visibility improves by roughly $4$dB in the presence of squeezing, with $0.5$-dB degradation from cavity loss.
![Improvement in microwave tone’s signal visibility due to squeezing. Power spectra normalized to the unsqueezed vacuum power, with (red) and without (purple) squeezing are shown in (a) for a tone $1$MHz detuned from the resonant frequency of the overcoupled cavity ($\kappa_m=10\kappa_l$). The $x$-axis has been shifted between the two situations for visual clarity. In (b), the visibility $\alpha(\omega)$ is measured as a function of the tone’s detuning from cavity resonance ($\omega=0$) for two cases: no squeezing while near critical coupling ($\kappa_m=1.5\kappa_l$, blue circles), and squeezing while strongly overcoupled ($\kappa_m=10\kappa_l$, red circles). In both cases $\alpha(\omega)$ is normalized to the expected maximum value $\alpha_\mathrm{max}=\alpha(0)$ evaluated at $\kappa_m=\kappa_l$. The theoretical expectation for each case is superimposed as a black curve, calculated using $\kappa_l=2\pi\times100$kHz, $\eta=0.69$ and $G_s=13$dB.[]{data-label="fig:SNRimpr"}](SNR_improvement_unshapped_v6.pdf)
To estimate the associated increase in scan rate from the SSR, we step the tone across the cavity’s resonance and measure the visibility $\alpha(\omega)$ at each detuning $\omega$. In order to compare the optimal squeezed and unsqueezed cases, we measure $\alpha(\omega)$ for two cases, displayed in Fig.\[fig:SNRimpr\]b: for $\kappa_m=10\kappa_l$ with squeezing, and for $\kappa_m=1.5\kappa_l$ without squeezing (see Appendix \[sup:dual\] for two other complementary cases). Without squeezing, $\alpha(0)$ is greater but the bandwidth is poor. With squeezing, $\alpha$ remains relatively high as $\omega$ increases. We extract $R\propto\int_{-\infty}^\infty\alpha(\omega)^2d\omega$ for the two cases, and obtain the estimated scan rate enhancement $E_e=2.05\pm0.04$. By independently measuring $\eta$, $\kappa_m$, and $G_s$, we calculate expected values for $\alpha(\omega)$, shown as solid lines, in excellent agreement with the measured values in Fig.\[fig:SNRimpr\]b. Finally, from the expected $\alpha(\omega)$ we calculate $E_t=2.11\pm0.07$, also in quantitative agreement with the data-based estimate $E_e$.
SQUEEZING-ENHANCED SEARCH FOR AN AXION-LIKE SIGNAL {#sec:faxion detection}
==================================================
In a real axion search, the aim is to detect a signal many orders of magnitude smaller than that in Fig.\[fig:SNRimpr\]. Inferring the presence or absence of an axion at each frequency requires combining the measured powers from many adjacent cavity tunings [@brubaker2017first; @du2018search]. Furthermore, over a long integration time the benefit inferred from Fig.\[fig:SNRimpr\] is vulnerable to practical non-idealities. Drifts of either JPA’s gain, drifts of the SQ-AMP relative phase $\theta$, non-Gaussian noise processes, and interfering rf or IF tones are of particular concern. In this section, we demonstrate that our SSR indeed matches the performance presented in the previous section when searching for a feeble tone over a wide frequency range.
We attempt to detect a fake axion, or “faxion," sent through the cavity weakly coupled port. It is synthesized from a randomly modulated microwave tone, whose power is adjusted such that the faxion spectral density emerging from the cavity is roughly $1\%$ of vacuum, and whose width is broadened to roughly $\Delta_a\approx9$kHz, comparable to expectations for a realistic axion at frequency $\omega_a \approx 2\pi\times7$GHz [@brubaker2017haystac]. Stepping the faxion tone frequency “backwards" past a stationary cavity simulates a realistic axion search without the hardware demands imposed by a tunable cavity. The faxion’s initial frequency is chosen randomly within a $2$-MHz window around the cavity resonance, and is then tuned in discrete $-10$-kHz steps over a $4$-MHz window. At each tuning step, we record an output power spectral density as described in Sec.\[scanrate\]. These spectra are mixed down and referred to the mixer’s input by symmetrizing about $\omega=0$. Spectra are artificially shifted in steps of $+10$kHz to simulate cavity tuning. We then rescale the spectra by $\alpha(\omega)$ such that frequency bins with higher sensitivity to the faxion are weighted more. These rescaled, shifted spectra are then added into a grand spectrum (see Appendix \[sup:processing\] and Ref.[@brubaker2017haystac]). With this procedure, the faxion’s contributions in each individual spectrum add at its initial frequency as if we had tuned the cavity, creating a clear excess of power in the grand spectrum. Figure \[fig:faxionsearch\]a presents some symmetrized spectra, obtained while squeezing with an overcoupled cavity ($\kappa_m = 10\kappa_l$), in which the faxion excess power is too small to rise above vacuum fluctuations. The spectra were obtained at different fictitious cavity tunings, normalized to their measured standard deviations, and vertically offset from one another for visual clarity. In the resulting grand spectrum (Fig.\[fig:faxionsearch\]b), a prominent faxion peak emerges with $6\sigma$ visibility.
![Response of the SSR in the presence of a faxion. The acquired power spectra are symmetrized (a), rescaled and then shifted in frequency to align all the frequency bins containing the faxion with each other (red dots), thereby effectively tuning the cavity. The power excess of the processed spectra is plotted in units of their standard deviation $\sigma_p$. Combining all the spectra into a grand spectrum (b), whose excess power is plotted in units of its standard deviation $\sigma_g$, a large power excess is observed at the initial frequency of the faxion. Repeating this measurement allows us to compute distributions of faxion powers (c). When squeezing with $\kappa_m=10\kappa_l$, the powers (red points) are drawn from a Gaussian distribution $\mathcal{N}(\mu_s,\sigma_g^2)$ (solid line), with $\mu_s=6.05\pm0.07$. When not squeezing with $\kappa_m=1.5\kappa_l$, the powers (blue points) are drawn from $\mathcal{N}(\mu_u,\sigma_g^2)$ (solid line), with $\mu_u=4.15\pm0.07$. The powers (black points) drawn from the faxion-less noise distribution $\mathcal{N}(0,\sigma_g^2)$ (solid line) are also represented for comparison.[]{data-label="fig:faxionsearch"}](faxion_v14.pdf)
In order to extract a scan rate enhancement from such a realistic signal search, we acquire two distributions of faxion powers, one when squeezing and one when not squeezing, with near-optimal $\kappa_m$ for each case. We obtain each distribution by repeating the faxion injection and detection protocol $200$ times. Over the course of the measurement, which takes roughly nine hours per configuration, we keep the relative phase $\theta$ between SQ and AMP quadratures at $\pi/2$ via a feedback loop (see Appendix \[sup:PLL\]). Figure \[fig:faxionsearch\]c displays the two faxion power distributions, as well as the noise power distribution. The no-faxion distribution of mean zero and variance $\sigma_g^2$ is guaranteed by the central limit theorem to be Gaussian distributed, $\mathcal{N}(0,\sigma_g^2)$. The faxion adds a small mean excess of power $\mu$ insufficient to enlarge the variance. The faxion distribution is therefore $\mathcal{N}(\mu,\sigma_g^2)$. The signal and noise distributions separate as the total measurement time squared, and thus the speed-up due to squeezing is equal to $E_m = (\mu_s/\mu_u)^2$, where $\mu_s$ $(\mu_u)$ is the mean of the faxion power distribution obtained when squeezing (not squeezing). The two distributions of faxion powers are characterized by $\mu_s=6.05\pm0.07$ and $\mu_u=4.15\pm0.07$, leading to a measured scan rate enhancement of $E_m=2.12\pm0.08$, in quantitative agreement with the estimate $E_e$ obtained from visibility measurements in Sec.\[scanrate\].
CONCLUSION
==========
The low signal-to-noise ratios achievable in state-of-the-art axion haloscopes [@du2018search; @zhong2018results; @chung2016cultask; @mcallister2017organ] and similar resonant searches for faint electromagnetic signals sourced by dark matter axions [@ouellet2018first; @silvafeaver2017design; @majorovits2017madmax] make the timescales over which they can sweep out appreciable fractions of parameter space unreasonably long. Partially replacing the vacuum noise in an axion haloscope with squeezed vacuum circumvents the standard quantum limit on the noise of the measurement apparatus, enabling high sensitivity over a broader bandwidth and a more rapid dark matter search. Squeezing moreover moves haloscopes into a qualitatively distinct design parameter space: whereas a quantum limited haloscope’s scan rate plateaus with improving microwave transmission efficiencies $\eta$, a sub-quantum limited haloscope will benefit almost arbitrarily as microwave losses are further reduced. Thus, haloscopes can reap larger benefits from increased efficiency (for example, a tenfold scan rate enhancement at $\eta=0.91$) as low-loss quantum technologies such as on-chip circulators and directional amplifiers mature [@sliwa2015reconfigurable; @macklin2015a; @chapman2017widely].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Kyle Thatcher for his help in the design and fabrication of the SSR mechanical parts and Felix Vietmeyer for his help in the design and fabrication of room temperature electronics. This work was supported by the National Science Foundation, under Grants No.PHY-1607223 and No.PHY-1734006, and by the Heising-Simons Foundation under Grant No.2014-183.
Theory of the SSR operation {#sup:SSR}
===========================
In this appendix, we track the propagating electromagnetic fields through the SSR and cavity, so as to derive the susceptibility matrix of the entire system. We then calculate the signal visibility $\alpha(\omega)$ and the scan rate $R$. We first consider the case where the propagating fields experience no loss (the cavity mode still decays partially out its loss port), which nonetheless faithfully illustrates the utility of the SSR, then we treat the full system in the presence of transmission losses.
Lossless case {#sup:SSRnoloss}
-------------
We model the energy exchange between the cavity’s ports in its own rotating frame. The time evolution of the cavity is governed by the Heisenberg-Langevin equation $$\label{EOM}
\frac{d\hat{A}}{dt} = -\frac{\kappa_T}{2} \hat{A}(t) + \sum_{j} \sqrt{\kappa_j} \hat{a}_{\mathrm{in},j}(t),$$ where $\hat{A}$ is the cavity ladder operator, $\kappa_T=\kappa_m+\kappa_l+\kappa_a$, and $\hat{a}_{\mathrm{in},j}$ ($j=m,l,a$) are the annihilation field operators of the input modes incident on ports indexed by $m,$ $l$, and $a$. At the measurement port, we physically separate input ($\hat{a}_{\mathrm{in},m}$) and output ($\hat{a}_{\mathrm{out},m}$) fields with a circulator. Given the input-output relations $\hat{a}_{\mathrm{out},j}(t) = \hat{a}_{\mathrm{in},j}(t) - \sqrt{\kappa_j}\hat{A}(t)$, the input and output field operators are related in the Fourier domain by $$\label{IOcav}
\hat{a}_{\mathrm{out},j}(\omega)=\sum_k\chi_{jk}(\omega)\hat{a}_{\mathrm{in},k}(\omega),$$ where $$\label{suceptibility}
\chi_{jk}(\omega)=\frac{-\sqrt{\kappa_j\kappa_k} + (\kappa_T/2+i\omega)\delta_{jk}}{\kappa_T/2+i\omega}$$ are the elements of a $3\times3$ susceptibility matrix fully describing the behavior of the cavity [@clerk2010introduction].
By cascading the input-output relations for each element presented in Fig.\[fig:schematic\], we can calculate the benefit from squeezing. We work in the quadrature basis and consider the vector of input quadratures $\vec{x}_\mathrm{in}=[\hat{X}_{\mathrm{in},m},\hat{X}_{\mathrm{in},l},\hat{X}_{\mathrm{in},a}]^\mathrm{T}$ aligned with our squeezing. We will calculate the SSR/cavity susceptibility matrix $\boldsymbol{\Xi_{X}}$, in terms of which the vector of output quadratures is $\vec{x}_\mathrm{out}=\boldsymbol{\Xi_{X}}\vec{x}_\mathrm{in}$.
The first element in the system is the SQ, which performs a one-mode squeezing (OMS) operation on the measurement port’s input quadrature: $$\label{IOSQ}
\vec{x}_s = \boldsymbol{S_X} \vec{x}_\mathrm{in} = \begin{bmatrix} \frac{1}{\sqrt{G_s}} && 0 && 0\\
0 && 1 && 0 \\
0 && 0 && 1 \\\end{bmatrix} \vec{x}_\mathrm{in},$$ where the subscript $s$ refers to the SQ output port and $G_s$ is the SQ single-quadrature power gain. The OMS operation also amplifies the other quadrature of the measurement port mode, $\hat{Y}_{\mathrm{in},m}$ by $\sqrt{G_s}$ in order to preserve the Heisenberg uncertainty relation. We do not track the evolution of the $\hat{Y}$ quadrature here, as it is irrelevant for SSR performance.
Next, the cavity transforms the quadrature operators. In the cavity rotating frame, the vectors of quadrature operators are obtained from the vectors of ladder operators by $$\label{quadladder}
\begin{bmatrix} \vec{x}\\\vec{y} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} \boldsymbol{I_3} && \boldsymbol{I_3} \\ -i\boldsymbol{I_3} && i\boldsymbol{I_3}\end{bmatrix} \begin{bmatrix} \vec{a}(\omega)\\\vec{a}^\dagger(-\omega) \end{bmatrix},$$ or $[\vec{x},\vec{y}]^\mathrm{T}=\boldsymbol{P}[\vec{a}(\omega),\vec{a}^\dagger(-\omega)]^\mathrm{T}$. Here $\boldsymbol{I_3}$ is the $3\times3$ identity matrix, and the mode ordering in each vector is $m,l,a$: for example, $\vec{a}(\omega) = [\hat{a}_m(\omega),\hat{a}_l(\omega),\hat{a}_a(\omega)]^\mathrm{T}$. Thus, the cavity susceptibility matrix in the quadrature basis $\boldsymbol{\Tilde{\chi}}$ is: $$\label{quadsusceptibility}
\boldsymbol{\Tilde{\chi}} = \boldsymbol{P} \begin{bmatrix} \boldsymbol{\chi}(\omega) && \boldsymbol{0_3} \\ \boldsymbol{0_3} && \boldsymbol{\chi}^*(-\omega)\end{bmatrix} \boldsymbol{P}^{-1} = \begin{bmatrix} \boldsymbol{\chi}(\omega) && \boldsymbol{0_3} \\ \boldsymbol{0_3} && \boldsymbol{\chi}(\omega)\end{bmatrix},$$ where $\boldsymbol{0_3}$ is the null matrix. We see that the cavity’s effect on field and quadrature operators is identical, which leads to $$\label{IOcavSSR}
\vec{x}_o = \boldsymbol{\chi}(\omega) \vec{x}_s,$$ where $o$ refers to the cavity output port.
Finally the AMP performs a second OMS operation so as to amplify the quadrature that the SQ originally squeezed: $$\label{IOAMP}
\vec{x}_\mathrm{out} = \boldsymbol{A_X} \vec{x}_o = \begin{bmatrix} \sqrt{G_a} && 0 && 0\\
0 && 1 && 0 \\
0 && 0 && 1 \\\end{bmatrix} \vec{x}_o,$$ where $G_a$ is the AMP power gain.
The SSR and cavity thus transform input to output quadratures according to $$\label{IOSSR}
\vec{x}_\mathrm{out} = \boldsymbol{A_X} \boldsymbol{\chi}(\omega) \boldsymbol{S_X} \vec{x}_\mathrm{in} = \boldsymbol{\Xi_{X}} \vec{x}_\mathrm{in}.$$
We can then calculate the single-quadrature output spectral density matrix $\boldsymbol{\Sigma_{\mathrm{out,}X}} =\langle [\vec{x}_\mathrm{out}]^\dagger [\vec{x}_\mathrm{out}]^\mathrm{T} \rangle / 2\pi$, where the Hermitian conjugation here does not transpose the vector, in order to determine the total output signal and noise powers. Substituting for $\vec{x}_\mathrm{out}$ yields $$\label{Sout}
\boldsymbol{\Sigma_{\mathrm{out,}X}} = \boldsymbol{\Xi_{X}}^* \boldsymbol{\Sigma_{\mathrm{in,}X}} \boldsymbol{\Xi_{X}}^\mathrm{T},$$ where $\boldsymbol{\Sigma_{\mathrm{in,}X}} =\langle [\vec{x}_\mathrm{in}]^\dagger [\vec{x}_\mathrm{in}]^\mathrm{T}\rangle / 2\pi$ is the input noise spectral density matrix. Since the SSR is connected to three incoherent and uncorrelated modes (see Fig.\[fig:schematic\]), $\langle \hat{X}_{\mathrm{in},j} \hat{X}_{\mathrm{in},k} \rangle/2\pi = \delta_{jk}(n_{\mathrm{in,}j} + 1/2)$, where $n_{\mathrm{in,}j}$ is the input mean photon number per unit time per unit bandwidth (henceforth simply mean photon number) of mode $j$. Thus, $$\label{Sin}
\boldsymbol{\Sigma_{\mathrm{in,}X}} = \begin{bmatrix} n_T+\frac{1}{2} && 0 && 0\\
0 && n_T+\frac{1}{2} && 0 \\
0 && 0 && n_A+\frac{1}{2} \\\end{bmatrix}.$$ Taking the first entry in the matrix $\boldsymbol{\Sigma_{\mathrm{out,}X}}$ we obtain the single-quadrature output spectral density at the measurement port: $$\label{SoutXm}
\begin{aligned}
\Sigma_{\mathrm{out,}X,m} \approx \frac{G_a}{B(\omega)} \bigg[ & n_A\kappa_a\kappa_m \\
+ & \left(n_T + \frac{1}{2}\right) \left(\kappa_l\kappa_m + \frac{\beta(\omega)}{G_s}\right) \bigg],
\end{aligned}$$ where $B(\omega) = (\kappa_m + \kappa_l)^2/4 + \omega^2$, $\beta(\omega) = (\kappa_m - \kappa_l)^2/4 + \omega^2$, and we used $\kappa_a \ll \{\kappa_l,\kappa_m\}$ and $n_A \gg 1/2$.
The visibility $\alpha(\omega)$, defined as the ratio of signal and noise spectral densities at the measurement port output, can be directly extracted from $\Sigma_{\mathrm{out,}X,m}$, since the signal is the term proportional to $n_A$ and the noise is $\Sigma_{\mathrm{out,}X,m}(n_A=0)$: $$\label{eq:SNRapp}
\alpha(\omega) \approx
\frac{n_A\kappa_a\kappa_m}{\left(n_T + \frac{1}{2}\right)\left(\kappa_l\kappa_m + \frac{\beta(\omega)}{G_s}\right)},$$ which is identical to Eq.. We now consider a search protocol comprising many measurements of $\Sigma_{\mathrm{out,}X,m}$, as the cavity frequency is tuned by discrete steps $\delta_c$. The spectral scan rate $R$ is obtained by taking the limit: $$\label{Rdef}
R = \lim_{\substack{\delta_c\to 0 \\ \tau\to 0}} \frac{\delta_c}{\tau},$$ where $\tau$ is the duration of each measurement. In the rest of this section we derive Eq. for the scan rate. We begin by defining the signal-to-noise ratio (SNR) at a single tuning step as $$\label{SNRdef}
\overline{\alpha}(\omega) = \sqrt{\tau\Delta_a}\frac{\alpha(\omega)}{2},$$ which scales as the square root of the number of independent measurements of the power contained in the bandwidth $\Delta_a$, centered on a frequency detuned from cavity resonance by $\omega/2\pi$ [@dicke1946the; @brubaker2017first]; the factor of 2 is a consequence of our single-quadrature measurement scheme (see Appendix \[sup:singledouble\]).
For $\delta_c\lesssim \kappa_T/2\pi$, multiple tuning steps will contribute to the SNR at each frequency. Without loss of generality, we evaluate the net SNR for some putative signal frequency $\omega_a$ which coincides exactly with the cavity resonance at a particular step. We then consider the contributions to the SNR of the surrounding $2n+1$ tuning steps, where $2\pi\delta_c=2\kappa_T/(2n+1)$. Contributions to the SNR add in quadrature, so the net squared SNR is $$\label{alpha_In}
\overline{\alpha}_{2n+1}^2 = \frac{\tau\Delta_a}{4}\cfrac{1}{\delta_c}\sum_{k=-n}^{k=n}\alpha^2(k2\pi\delta_c)\delta_c.$$ In the limit $n\to\infty$, $\tau\to0$, we obtain the integrated squared SNR $$\label{alpha_I}
\overline{\alpha}_{I}^2 = \frac{\Delta_a}{4}\frac{1}{R} \int_{-\infty}^\infty\alpha^2(\omega) \frac{d\omega}{2\pi},$$ where we have extended the limits of integration to $\pm\infty$: the contributions from tuning steps where the cavity is detuned by more than $\kappa_T$ are negligible.
Note that our arbitrary choice of $\omega_a$ appears nowhere explicitly on the RHS of Eq.; thus the integrated SNR will be frequency independent, insofar as the quantities contributing to $\alpha(\omega)$ remain constant as we tune the cavity. In practice, we set a certain target value of $\overline{\alpha}_I$ for which a real signal would appear as a sufficiently prominent peak in the grand spectrum (see e.g., Fig.\[fig:faxionsearch\]b); this choice determines the required spectral scan rate $R$. Solving for $R$, we obtain $$\label{Rfull}
\begin{aligned}
R &= \frac{\Delta_a}{4\overline{\alpha}_I^2} \int_{-\infty}^\infty\alpha^2(\omega) \frac{d\omega}{2\pi}\\
&= \frac{\Delta_a\sqrt{G_s}n_A^2\kappa_a^2\kappa_m^2}{16\overline{\alpha}_I^2\left(n_T+\frac{1}{2}\right)^2\left[\kappa_l\kappa_m + \frac{1}{G_s}\left(\frac{\kappa_l-\kappa_m}{2}\right)^2\right]^{3/2}},
\end{aligned}$$ in agreement with Eq.. In a comparison of squeezed and unsqueezed scan rates, the $\Delta_a/4\overline{\alpha}_I^2$ factor drops out; thus, we need only compare the integral in Eq. for the two cases.
SSR operation in the presence of transmission losses {#sup:SSRwloss}
----------------------------------------------------
Here we generalize the calculations of Appendix \[sup:SSRnoloss\] to account for imperfect power transmission efficiency $\eta$ between SQ and AMP. Integrating the squared SNR over all frequencies, we obtain the scan rate as a function of $\eta$, used in Fig.\[fig:Rtheo\]. For simplicity, we treat the transmission efficiency $\lambda$ between the SQ and the cavity as identical to that between the cavity and the AMP, hence $\eta = \lambda^2$. Note that, in principle, loss between SQ and cavity is slightly less harmful than loss between cavity and AMP, as loss that occurs after the cavity degrades the signal along with the squeezing.
As in Appendix \[sup:SSRnoloss\], we track the vector of input quadratures $\vec{x}_\mathrm{in}$ throughout the SSR and cavity system, neglecting the orthogonal quadratures. First, the SQ performs the OMS operation defined by Eq.: $\vec{x}_s=\boldsymbol{S_X} \vec{x}_\mathrm{in}$. Next we must account for transmission losses between the SQ and the cavity, modeled as a beamsplitter interaction between the quadrature operators of the measurement line at the SQ output $\{\hat{X}_{s,m},\hat{Y}_{s,m}\}$, and those of another, uncontrolled mode $\{\hat{X}_{s,\lambda},\hat{Y}_{s,\lambda}\}$. We could define generalized quadrature vectors $\vec{x}_s$ and $\vec{y}_s$ to include $\hat{X}_{s,\lambda}$ and $\hat{Y}_{s,\lambda}$ respectively, but we do not need to keep track of the electromagnetic field’s evolution in loss modes. Among the modes we do keep track of, only the measurement port experiences the loss: $$\label{IOloss}
\vec{x}_{i} = \begin{bmatrix} \sqrt{\lambda} && 0 && 0\\
0 && 1 && 0 \\
0 && 0 && 1 \\\end{bmatrix} \vec{x}_s,$$ or $\vec{x}_{i} = \boldsymbol{L_X} \vec{x}_s$. Here $i$ refers to the cavity’s input, and $0\leq\lambda\leq1$ is the single-sided transmission efficiency.
Since we are not tracking the SQ-to-cavity propagating loss mode, the unsqueezed vacuum that it introduces enters as an added noise term, $$\label{noise}
\boldsymbol{N_X} = \begin{bmatrix} (n_T+\frac{1}{2})(1-\lambda) && 0 && 0\\
0 && 0 && 0 \\
0 && 0 && 0 \\\end{bmatrix}.$$ We thus have the single-quadrature noise spectral density at the cavity input: $\boldsymbol{\Sigma_{i,X}} = [\boldsymbol{L_X} \boldsymbol{S_X}]^*\boldsymbol{\Sigma_{\mathrm{in,}X}}[\boldsymbol{L_X} \boldsymbol{S_X}]^\mathrm{T} + \boldsymbol{N_X}$, where $\boldsymbol{\Sigma_{\mathrm{in,}X}}$ is given by Eq.. This expression yields $$\label{Sbeta1}
\boldsymbol{\Sigma_{i,X}} = \begin{bmatrix} \left(n_T+\frac{1}{2}\right)\left(\frac{\lambda}{G_s}+1-\lambda\right) && 0 && 0 \\
0 && n_T+\frac{1}{2} && 0 \\
0 && 0 && n_A+\frac{1}{2} \\\end{bmatrix},$$ where both the partially attenuated squeezed vacuum term ($\propto\lambda/G_s$), and the unsqueezed contribution from the loss mode ($\propto 1-\lambda$) appear clearly. Note that in the absence of loss, $\boldsymbol{\Sigma_{i,X}} = \boldsymbol{S_X}^*\boldsymbol{\Sigma_{\mathrm{in,}X}}\boldsymbol{S_X}^\mathrm{T}$, in agreement with Eq..
We can now calculate $\boldsymbol{\Sigma_{\mathrm{out,}X}}$, the output spectral density matrix along $\vec{x}$. Given Eq. for the cavity’s susceptibility in the quadrature basis, we may write $\boldsymbol{\Sigma_{o,X}} = \boldsymbol{\chi}^*(\omega)\boldsymbol{\Sigma_{i,X}}\boldsymbol{\chi}(\omega)^\mathrm{T}$. Then losses between cavity and AMP are accounted for in the same manner as before, and finally the AMP performs another OMS operation, amplifying the SQ squeezed quadrature. We thus obtain $$\label{Soutloss}
\begin{aligned}
\boldsymbol{\Sigma_{\mathrm{out,}X}} & = \boldsymbol{A_X}^*[\boldsymbol{L_X}^*\boldsymbol{\Sigma_{o,X}}\boldsymbol{L_X}^\mathrm{T}+\boldsymbol{N_X}]\boldsymbol{A_X}^\mathrm{T}\\
& = \boldsymbol{A_X}^*[\boldsymbol{L_X}^*\boldsymbol{\chi}^*(\omega)\boldsymbol{\Sigma_{i,X}}\boldsymbol{\chi}(\omega)^\mathrm{T}\boldsymbol{L_X}^\mathrm{T}+\boldsymbol{N_X}]\boldsymbol{A_X}^\mathrm{T}
\end{aligned}$$ as the analog of Eq. in the presence of loss.
The first entry in the matrix $\boldsymbol{\Sigma_{\mathrm{out,}X}}$ is the output spectral density at the measurement port along $\hat{X}$: $$\label{SoutXmloss}
\begin{aligned}
\Sigma_{\mathrm{out,}X,m} = & \left(n_T+\frac{1}{2}\right)G_a(1-\lambda)\\
+ \frac{G_a\lambda}{B(\omega)}\bigg[ & \left(n_A + \frac{1}{2}\right)\kappa_a\kappa_m\\
+ & \left(n_T + \frac{1}{2}\right) \left(\kappa_l\kappa_m + \left(1-\lambda+\frac{\lambda}{G_s}\right)\beta(\omega)\right)\bigg],
\end{aligned}$$ where $B(\omega)$ and $\beta(\omega)$ are defined as in Eq., and the same approximations have been made.
From $\Sigma_{\mathrm{out,}X,m}$ we can extract the signal visibility $\alpha(\omega)$ in the presence of loss: $$\label{SNRloss}
\begin{aligned}
\alpha(\omega) = & \frac{\lambda n_A}{\left(n_T + \frac{1}{2}\right)} \times \\
& \frac{\kappa_a\kappa_m}
{B(\omega)\left(1-\lambda\right) + \lambda\left[\kappa_l\kappa_m + \left(1-\lambda+\frac{\lambda}{G_s}\right)\beta(\omega)\right]}.
\end{aligned}$$ When $\lambda=1$, Eq. reduces to Eq..
Finally, the scan rate enhancement $E_t$ in the presence of loss, presented in Fig.\[fig:Rtheo\]b, is $\int_{-\infty}^\infty\alpha^2(\omega) d\omega$, normalized by the same integral with $G_s=1$ and $\kappa_m=2\kappa_l$.
Model for the axion field {#sup:axionmodel}
=========================
Figure \[fig:schematic\] models the axion field as a fictitious generator that drives the cavity through a weakly coupled port. In this appendix, we relate the fictitious port coupling $\kappa_a$ and the power spectral density $n_A$ at the generator output to physical parameters normally found in the haloscope literature, and show that for representative values, the axion field acts as a classical force.
The measurable axion-sourced power is obtained from the first term in the output spectral density, Eq.. Referring to the cavity output and multiplying by the axion linewidth $\Delta_a$, the on-resonance ($\omega_a=\omega_c$) signal power is $$\label{gen_model_pow}
P_\mathrm{sig} = 4 \hbar \omega_a n_A \Delta_a \frac{\kappa_a \kappa_m}{(\kappa_m + \kappa_l)^2}.$$ The two model parameters $\kappa_a$ and $n_A$ can readily be related to physical parameters by comparing Eq., to e.g. Eq.(1) of Ref.[@alkenany2017design] evaluated on resonance. In our notation this expression takes the form $$\label{eq:haloscope2}
P_\text{sig} = \left(g_{a\gamma\gamma}^2\frac{\hbar c^3\rho_a}{\mu_0\omega_a^2}\right)\times\left(B_0^2VC_{mn\ell}\frac{\omega_c^2\kappa_m}{(\kappa_m + \kappa_l)^2}\right),$$ where, in the first set of parentheses, $g_{a\gamma\gamma}$ parametrizes the axion field’s coupling to electromagnetism, $\rho_a$ is the local dark matter energy density, $c$ is the speed of light, $\mu_0$ is the vacuum permeability, and we have related the axion rest mass $m_a$ to the frequency of axion-induced photons $\omega_a$ as $m_ac^2 = \hbar \omega_a$. The second parenthetical expression contains properties of the haloscope: $B_0$ is the static magnetic field, $C_{mnl}$ is the cavity mode-dependent form factor, and $V$ is the cavity volume. Equating Eqs. and , we obtain $$\label{n_A times kappa_a}
n_A \kappa_a = \frac{g_{a\gamma\gamma}^2 \rho_a c^3}{4 \omega_a \mu_0 \Delta_a}B_0^2 C_{mnl} V.$$
To derive a second expression relating $\kappa_a$ to $n_A$, we observe that the fictitious generator in our haloscope model may equivalently be represented as a second harmonic oscillator mode with very high occupancy but very weak coupling to the haloscope cavity. The axion field in any laboratory-scale volume constitutes such an oscillator mode, with resonant frequency $\omega_a$. Specifically, we model the oscillations of the axion field as a fictitious cavity occupying the same volume as the real haloscope cavity. The quanta of this fictitious cavity are axion particles, so its total occupancy is $N_A = V\rho_a/\hbar\omega_a$; it is coupled to the haloscope cavity with interaction Hamiltonian $\hat{H}_\mathrm{int} = \hbar g (\hat{A} + \hat{A}^\dagger)(\hat{B} + \hat{B}^\dagger)$, where $g$ is the interaction strength and $\hat A$ ($\hat B$) is the annihilation operator of the haloscope (fictitious) cavity. The Hamiltonian of the closed system is $\hat{H} = \hat{H}_0 + \hat{H}_\mathrm{int}$, where $\hat{H}_0 = \hbar \omega_c(\hat{A}^\dagger \hat{A} + 1/2) + \hbar \omega_a(\hat{B}^\dagger \hat{B} + 1/2)$. Coupling the haloscope cavity to measurement and loss ports at rates $\kappa_m$ and $\kappa_l$, respectively, we write down the Heisenberg-Langevin equations of motion for the open system: $$\begin{aligned}
\frac{d\hat{A}}{dt} &= -i\omega_c \hat{A}(t) -i g [\hat{B}(t) + \hat{B}^\dagger(t)] - \frac{\kappa_m + \kappa_l}{2} \hat{A} \label{dAdt} \\
& + \sqrt{\kappa_m} \hat{a}_{\mathrm{in},m}(t) + \sqrt{\kappa_l} \hat{a}_{\mathrm{in},l}(t) \nonumber \\
\frac{d\hat{B}}{dt} &= -i\omega_a \hat{B}(t) -i g [\hat{A}(t) + \hat{A}^\dagger(t)], \label{dBdt}
\end{aligned}$$ where we describe the input bath associated with the measurement (loss) port with the annihilation operator $\hat{a}_{\mathrm{in},m}$ ($\hat{a}_{\mathrm{in},l}$).
We restrict ourselves to the classical limit of the system, with operators demoted to complex amplitudes, and the case where the resonances coincide, $\omega_a = \omega_c$, with no power entering via the measurement and loss ports, $a_{\mathrm{in},m} = a_{\mathrm{in},l} = 0$. Transforming into the rotating frame of the haloscope cavity, $\{A(t), B(t)\}\rightarrow \{A(t)e^{-i\omega_c t}, B(t)e^{-i \omega_c t}\}$ and making a rotating wave approximation, Eqs. and reduce to $$\begin{aligned}
\frac{dA}{dt} &= -i g B(t) -\frac{\kappa_m + \kappa_l}{2}A(t) \label{dAdt simplified}\\
\frac{dB}{dt} &= -i g A(t) \label{dBdt simplified},
\end{aligned}$$ where $g$ plays the role of the field exchange rate corresponding to the power decay rate out of the axion cavity, $g = \kappa_a / 2$. These equations of motion describe an exchange of energy between the two cavities and a decay of that energy out from the haloscope cavity via the measurement and loss ports. We are interested in the steady-state ($dA/dt = 0$) output field $A_{\mathrm{out},m} = -\sqrt{\kappa_m} A$ when the occupancy of the axion cavity is $|B|^2 = N_A$. We find a steady-state occupancy of the haloscope cavity $|A|^2 = [\kappa_a / (\kappa_m + \kappa_l)]^2 N_A$, implying an output signal power of $$\label{cav model pow}
P_\mathrm{sig} =\hbar\omega_a |A_{\mathrm{out},m}|^2 = \frac{\hbar \omega_a N_A\kappa_a^2 \kappa_m}{(\kappa_m + \kappa_l)^2}.$$ Equations and for the output power must agree, implying $$\label{n_A over kappa_a}
\frac{n_A}{\kappa_a} = \frac{N_A}{4 \Delta_a} = \frac{V\rho_a}{4\hbar\omega_a\Delta_a}.$$ In terms of physical parameters, Eqs. and yield $$\begin{aligned}
n_A &= \frac{|g_{a\gamma\gamma}|\rho_a B_0 V}{4 \omega_a \Delta_a}\sqrt{\frac{C_{mnl} c^3}{\hbar \mu_0}}\\
\kappa_a &= |g_{a\gamma\gamma}| B_0 \sqrt{\frac{C_{mnl} \hbar c^3}{\mu_0}}.
\end{aligned}$$
**quantity** **value**
--------------------- -------------------------------------------
$\rho_a$ $0.45 \ \mathrm{GeV}/\mathrm{cm}^3$
$B_0$ $9 \ \mathrm{T}$
$g_{a\gamma\gamma}$ $-7.7 \times 10^{-24} \ \mathrm{eV}^{-1}$
$\Delta_a$ $5 \ \mathrm{kHz}$
$\omega_a / 2\pi$ $5 \ \mathrm{GHz}$
$V$ $1.5 \ \mathrm{L}$
$C_{mnl}$ $0.5$
: Representative physical values used to estimate our model parameters.[]{data-label="tab:phys vals"}
Using representative values for the HAYSTAC experiment [@alkenany2017design] in the presence of a 5-GHz KSVZ [@kim1979weak; @shifman1980can] axion shown in Table \[tab:phys vals\], we obtain the values for our model parameters shown in Table \[tab:param vals\]. We see that the fictitious generator is well into the classical regime, $n_A\gg 1/2$, while its extremely feeble coupling $\kappa_a$ nonetheless makes its presence a challenge to detect.
**parameter** **value**
------------------- ------------------------
$N_A$ $3.3 \times 10^{16}$
$\kappa_a / 2\pi$ $2.3\ \mathrm{\mu Hz}$
$n_A$ $2.4 \times 10^7$
: Model parameter values calculated using the physical values from Table \[tab:phys vals\].[]{data-label="tab:param vals"}
Single vs. double quadrature measurement {#sup:singledouble}
========================================
Single-mode squeezing can only enhance sensitivity to displacements along a single quadrature of the cavity field. In a situation such as an axion search, the signal distributes its excess power equally between the two quadratures. Thus switching to single-quadrature measurement from double-quadrature measurement, currently the operational mode of choice of axion haloscopes [@brubaker2017first; @du2018search], seems detrimental. In this appendix, we show that in the absence of squeezing, neither measurement scheme has an advantage over the other.
If we neglect amplifier added noise, the signal visibility $\alpha$ is independent of whether we measure one quadrature or both. Specifically, an axion signal characterized by its spectral density $S_a$ at the amplifier input divides itself equally as $S_a/2$ between the two quadratures. Similarly, the vacuum noise spectral density $\hbar\omega_a / 2$ divides its power equally as $\hbar\omega_a / 4$ between the quadratures.
However, we must account for the quantum limits on two-quadrature measurements. Any linear amplifier that measures both quadratures adds at least a second half-quantum of input-referred noise, evenly distributed between the two quadratures [@caves1982quantum]. An ideal double-quadrature measurement thus yields $\alpha_\mathrm{2Q} = S_a / \hbar \omega_a$ in each quadrature. By comparison, there is no quantum limit on single-quadrature amplification, so an ideal single-quadrature measurement yields $\alpha_\mathrm{1Q} = 2 S_a / \hbar \omega_a = 2\alpha_\mathrm{2Q}$ in the amplified quadrature.
To make a fair comparison between the single-quadrature and double-quadrature cases, we must consider the improvement in the SNR (defined in Appendix \[sup:SSRnoloss\]). Because all pertinent differences between the two cases enter when considering a single tuning step, we neglect tuning in the following discussion. The SNR $\overline{\alpha}$ is given in terms of the spectral density ratio $\alpha$ by $$\overline{\alpha}=\sqrt{\frac{N}{2}}\frac{\Delta_a}{\Delta}\alpha,$$ where $\Delta_a/\Delta$ is the ratio of signal to noise bandwidths, and $N$ is the number of independent measurements of the voltage contained in the noise bandwidth $\Delta$. In considering the appropriate values of $N$ and $\Delta$ for the two cases of interest, we find two independent effects, each reducing $\overline{\alpha}_{\mathrm{1Q}}$ by a factor of $\sqrt{2}$ relative to $\overline{\alpha}_{\mathrm{2Q}}$. Together, these effects cancel the apparent benefit stemming from the absence of quantum noise limits in the single-quadrature case.
First, the Nyquist theorem guarantees that there are $N_\mathrm{2Q}=2\tau\Delta_\mathrm{2Q}$ independent measurements of the noise voltage in a double-quadrature measurement of duration $\tau$ and bandwidth $\Delta_\mathrm{2Q}$, where the factor of 2 counts the two independent quadrature amplitudes for each resolved Fourier component. Thus there are $N_\mathrm{1Q}=\tau\Delta_\mathrm{1Q}$ measurements of the noise voltage in a single-quadrature measurement of bandwidth $\Delta_\mathrm{1Q}$.
Second, noiseless single-quadrature measurement with a parametric amplifier creates an irreversible ambiguity between the output signal and idler frequencies, equally spaced about the amplifier band center. This ambiguity necessitates mapping amplifier outputs at a given detuning from band center to the input signal and idler frequencies (see Appendix \[sup:processing\]). The consequent addition of spectral densities, half of which are guaranteed not to have an axion-induced excess power, effectively increases the noise bandwidth in a single-quadrature measurement by a factor of $2$ relative to the signal bandwidth: $\Delta_\mathrm{1Q}=2\Delta_a$. In the standard double-quadrature measurement scheme, the signal and noise bandwidths are equal: $\Delta_\mathrm{2Q}=\Delta_a$.
Putting this all together, the two measurement schemes are seen to be equivalent: $$\begin{aligned}
\overline{\alpha}_{\mathrm{1Q}} &= \sqrt{\frac{\tau\Delta_\mathrm{1Q}}{2}}\frac{\Delta_a}{\Delta_
\mathrm{1Q}}\alpha_\mathrm{1Q} \\
&= \sqrt{\frac{2\tau\Delta_\mathrm{2Q}}{2}}\frac{\Delta_a}{2\Delta_\mathrm{2Q}}2\alpha_\mathrm{2Q} \\
&= \overline{\alpha}_{\mathrm{2Q}}.
\end{aligned}$$ The first line agrees with Eq. for the single-quadrature SNR.
Characterization of the experimental apparatus
==============================================
In this appendix we discuss and characterize key features of the SSR, including its control electronics. As the performance of the SSR is primarily limited by its transmission losses, many of our design choices were driven by loss mitigation. Additionally, we designed the experiment to have a minimum number of adjustable controls, so that the faxion search could be easily automated.
Flux-pumped JPAs {#sup:fluxJPAs}
----------------
The requirement of running the SSR automatically for several days makes monochromatically current-pumped JPAs a poor choice for SQ and AMP, as the strong current pump tone remains present at the center of the amplification band output [@castellanos2008amplification; @mallet2011quantum; @malnou2018optimal]. If not canceled by a $\pi$-phase shifted tone of equal amplitude, the SQ pump would reach and saturate the AMP. Furthermore, with insufficient microwave isolation the AMP pump could also be reflected from the cavity and perturb the AMP operating point. Keeping both cancellation tones stable in amplitude and phase, while controlling both SQ and AMP pumps to obtain good gain and good squeezing is impractical over a long period of time.
Flux-pumped JPAs do not require cancellation tones, because they are pumped at twice their operating frequency, far outside the bandwidth of the receiver chain. In addition, their topology is such that pump and signal propagate along spatially distinct paths, ensuring good isolation between the two (see Fig.\[fig:FPJPA\]). This also negates the need for a directional coupler in front of the chip, and therefore reduces the insertion loss along the squeezed state path. When flux pumping, a JPA can be thought of as a linear, frequency-tunable $LC$ resonator, capacitively coupled to a transmission line. Josephson junctions arranged to form a series array of superconducting quantum interference devices (SQUIDs) provide the tunable inductance. The bare resonance $\omega_0$ of the $LC$ resonator is tunable via a dc magnetic flux. The pump generates an ac magnetic flux which modulates the SQUID inductance at $\omega_p = 2\omega_0$, generating parametric amplification.
We optimized several constraints to ensure efficient flux pumping. First, we took care in the design to ensure identical coupling between the pump line and each of the SQUIDs that collectively constitute the tunable inductor. Uniform coupling guarantees that the JPA’s dynamics are spatially homogeneous, resulting in improved power handling. Second, we placed the pump line close to the SQUID array, in order to minimize the pump power required to drive the $LC$ resonator. At the same time, the coupling of the $LC$ circuit to the pump line is kept much lower than the coupling to the line carrying the signal, in order to avoid losing part of the signal through the pump port: the transmission between the two ports is kept below $-20\,$dB. Finally, we shaped the flux line as a U around the SQUID array, as shown in Figs.\[fig:FPJPA\]a and b. With this configuration, the pump couples to the differential mode current circulating inside each SQUID loop and is isolated from the common-mode current, unidirectional across the SQUID array.
![A flux-pumped JPA, similar to those used in the SSR. It is fabricated in the standard niobium-aluminum-niobium process in a coplanar waveguide geometry. A scanning electron microscope image (a) shows, in false color, the main elements of the JPA: a $550$-fF interdigital capacitor (purple), an inductance comprising a SQUID array of $6$-[$\upmu$]{}A Josephson junctions (red), and a flux line, shorted to ground (blue). A closer view (b) shows several SQUID loops (red), surrounded by the flux line (blue). The pump current $I(\omega_p)$ is circulating back and forth along the flux line, thereby favoring the differential mode current and attenuating the common-mode current in the SQUID loops. The JPA equivalent circuit (c) represents the two-port device: an incoming signal on port 1 is reflected and amplified. On port 2, the pump at $\omega_p=2\omega_0$ modulates the SQUID inductance.[]{data-label="fig:FPJPA"}](FPJPA_v10_lowresforarxiv.pdf)
Experimental setup {#sup:setup}
------------------
The full experimental setup is represented in Fig.\[fig:fullsetup\]. The SQ-cavity-AMP ensemble is attached to the bottom plate of a dilution refrigerator. For each JPA, the pump tone is routed though a $10$-dB directional coupler connected to the pump port. The placement and configuration of these directional couplers ensures that the large pump tone required for flux pumping primarily heats up a $50$-$\ohm$ termination whose Johnson noise propagates back up the pump line, away from the JPA. A coil around each chip, connected to a dc current source at room temperature, generates a dc magnetic field, and the chip-coil ensemble is magnetically shielded with aluminum and cryoperm. Each JPA is connected to a circulator through superconducting NbTi cables in order to minimize transmission losses.
An ensemble of four circulators routes the squeezed state and provides microwave isolation between the SSR elements. Two circulators between the SQ and the cavity protect the SQ from power reflected back from the cavity. These circulators provide sufficient isolation when the SQ is operated with $13$dB of signal gain. Similarly, three circulators separate the AMP and the cavity, as the AMP is operated with higher signal gain, around $25$dB. We experimentally observed that with only two circulators, undesirable feedback between the cavity and AMP perturbs the AMP’s gain by effectively changing its pump’s power.
At the chain’s input, either vacuum noise or a probe tone can be injected via a $20$-dB directional coupler. This probe tone is useful when characterizing the JPA gain profiles or when characterizing the cavity. Finally, at the output, a double-junction isolator protects the AMP from signals reflected from the next amplifier, a HEMT at $4$K.
At room temperature, a single microwave generator (Keysight E8257D) drives both JPAs and also serves as LO for the in-phase/quadrature (IQ) mixer in the readout line. On the path leading to the SQ pump’s input, a voltage-controlled variable attenuator and phase shifter provide control of both the SQ pump amplitude and phase. On the path leading to the IQ mixer, a frequency divider converts $\omega_p$ to $\omega_0$, the bare resonant frequency of both JPAs and the cavity. This divider (Pasternack PE88D2000) can input a wide range of powers (from $-20$ to $5$dBm) while always outputting the same power (roughly $-4$dBm). Thus, the AMP gain can be tuned freely with the microwave generator’s output power. A second generator injects a tone through the weakly coupled port of the axion cavity. This tone can be shaped into a $9$-kHz-wide Lorentzian with an arbitrary waveform generator (AWG).
At the output, a $6$dB directional coupler routes a fraction of the power to a vector network analyzer (VNA), which monitors signals either from the SSR chain’s input or from the cavity’s weakly coupled port. The VNA is used to measure *in situ* the couplings $\kappa_m$ and $\kappa_l+\kappa_a$. The other portion of the output power reaches the IQ mixer’s rf port. After being mixed down, the in-phase and quadrature signals are amplified and directed onto $1.9$MHz anti-aliasing low-pass filters, then finally digitized by an analog-to-digital converter (ADC).
Phase locking of SQ and AMP pumps {#sup:PLL}
---------------------------------
Enhancing the scan rate with squeezing is only beneficial if the AMP amplifies the SQ squeezed quadrature. To this end, as suggested by Fig.\[fig:squeezing\], the phase $\theta$ between SQ and AMP pumps must be maintained at $\pi/2$ (or $3\pi/2$) to a high precision, given the sharp dependence of the degree of squeezing $S$ on $\theta$. When $S>0$dB, the output is noisier with the SQ on than off, and the SSR is therefore detrimental to the axion search. Furthermore $S$ also depends on the SQ gain $G_s$, i.e. on the SQ pump power, and there is an optimal $G_s$ for which $S$ reaches its minimum value $S_\mathrm{min}$. In fact, as the gain increases, $S$ first improves as the squeezed state elongates in phase space, but then saturates due to distortion effects [@malnou2018optimal; @boutin2017effect]. For this type of JPA, we experimentally find $S = S_\mathrm{min}$ at $G_s\approx13$dB.
In order to achieve $S = S_\mathrm{min}$ throughout a data run, we implemented a feedback loop that uses the output variance $\sigma_\mathrm{on}^2$ as its control parameter on the voltage-controlled variable attenuator and phase shifter. When initializing a $9$hr acquisition, a 2-dimensional sweep of the variable attenuation $A_s$ and phase shift $\theta$ is used to estimate the global minimum of $\sigma_\mathrm{on}^2$. Then, periodically throughout the data run, a fast gradient descent-type algorithm corrects for small drifts of $\sigma_\mathrm{on}^2$. Note that this approach is robust to possible phase shifts due to changes in the variable attenuation and vice versa. Empirically, we are able to automatically maintain $S$ to $S_\mathrm{min}\pm0.1$dB, over the course of the entire experiment.
We did not need to adjust the AMP pump power, as it remained stable around $25$dB. However, when implementing the SSR in HAYSTAC, it will have to be adjusted, in particular because the frequency of the JPAs will also be stepped in time. Note that in a practical haloscope run, sizable fluctuations of the net receiver gain in a SSR-integrated setup on timescales shorter than the raw spectrum acquisition time can be detrimental to axion detection. This is true even if all sources of added noise are overwhelmed. For our raw spectra acquisition times of $0.32$s, the receiver gain fluctuations are negligibly small.
Transmission loss between SQ and AMP {#sup:losslines}
------------------------------------
The presence of four circulators and several microwave connectors, including adapters from SMP (used on the JPAs chips) to SMA standards, inevitably reduces the transmission efficiency $\eta$ between SQ and AMP. We minimize $\eta$ through the use of a triple junction circulator, and superconducting SMA cables between the two JPAs and the circulators. However, $\eta$ still provides the primary limitation on the efficacy of the SSR in accelerating axionic dark matter searches.
We estimate $\eta$ by measuring the output power spectral density $P_\mathrm{out}$ of vacuum fluctuations in a single quadrature amplified by the AMP with the SQ off, then repeating this measurement with the role of the JPAs interchanged [@malnou2018optimal]. Assuming that the gain of the amplification chain from the HEMT to the ADC stays constant, we can then deduce the microwave loss between the two JPAs as the difference in the net gain between the two cases.
More precisely, having only vacuum noise at the chain’s input, we have, in one quadrature: $$P_\mathrm{out} = \frac{1}{4}\hbar\omega B G_\mathrm{JPA}^c G_\mathrm{JPA},$$ where $\omega$ is the rf angular frequency at the center of integration bandwidth $B$ for the power spectral density, $G_\mathrm{JPA}^c$ is the chain’s gain after the JPA, and $G_\mathrm{JPA}$ is the SQ or AMP gain. Thus, there is a linear relation between $P_\mathrm{out}$ and $G_\mathrm{JPA}$, whose slope gives $G_\mathrm{JPA}^c$. Figure \[fig:outputgain\] shows $4P_\mathrm{out}/(\hbar\omega B)$, when varying either the SQ or the AMP gain. We extract $\eta = G_\mathrm{SQ}^c/G_\mathrm{AMP}^c = 0.69\pm0.01$.
![Measurement of the single-quadrature output power spectral density $P_\mathrm{out}$, normalized, as a function of the JPA gain $G_\mathrm{JPA}$. Two configurations are represented: when operating only the SQ (red circles) and when operating only the AMP (blue circles). A linear fit (solid lines) to these linear responses allows us to extract $\eta$.[]{data-label="fig:outputgain"}](output_gain_chain_v4.pdf)
The efficiency $\eta$ observed in our mock-haloscope setup should not be significantly degraded by the large magnetic fields required in a real haloscope experiment such as HAYSTAC [@zhong2018results] or ADMX [@du2018search]. This is because the primary constraint imposed by the large field is an increased spatial separation between the axion cavity and the circulators and amplifiers that process the signal. This $\sim 1$-m distance is bridged with superconducting coaxial cables, whose attenuation [@kurpiers2017characterizing] is far subdominant to other sources of transmission loss, such as the microwave circulators in our setup, and is robust to large magnetic fields [@brubaker2017first].
From $\eta$ we can estimate the squeezing $S$ that we should obtain far off cavity resonance ($\sim 10$ MHz detuned). Considering the ideal case where the reduction of the squeezed state variance is equal to the SQ single-quadrature power gain $G_s$, we obtain: $$S = \frac{\eta}{G_s} + 1-\eta,$$ which, for $G_s=13$ dB leads to $S=-4.6$ dB, in quantitative agreement with what we measure in practice.
Complementary measurements of the microwave tone’s improvement in resolution {#sup:dual}
============================================================================
We presented in Fig.\[fig:SNRimpr\]b the visibility $\alpha(\omega)$ of a microwave tone as a function of $\omega$, the detuning between the tone’s frequency and the cavity bare resonance. We compared $\alpha(\omega)$ between the two relevant cases: squeezing with a strongly overcoupled cavity versus not squeezing with a near-critically coupled cavity. Figure \[fig:SNRimpr\_sup\] presents these, along with measurements of $\alpha(\omega)$ for the two complementary cases: strongly overcoupling without squeezing, and near-critically coupling with squeezing. In all four cases, there is excellent agreement with predictions from the theory, developed in Appendix \[sup:SSRwloss\] and shown in solid lines. The theory curves are not fits; we used parameter values for $\kappa_l$, $\eta$, and $G_s$ measured independently. Furthermore, the values are the same across all four cases.
Squeezing is beneficial, even when not overcoupling the cavity’s measurement port, as it enhances $\alpha$ off cavity resonance while having no effect on resonance. Experimentally, we obtain the complementary estimate $E^{c1}_e=1.72\pm0.03$ when squeezing and near-critically coupling, compared to the same situation without squeezing, in agreement with the theoretical prediction, $E^{c1}_t=1.77\pm0.03$. Conversely, the scan rate is worse when strongly overcoupling without squeezing, compared to near-critically coupling with squeezing: from the data, we obtain $E^{c2}_e=0.57\pm0.01$, not far from the theoretical value of $E^{c2}_t=0.52$.
![Signal visibility $\alpha(\omega)$, as a function of the signal’s detuning from cavity resonance. It is normalized by $\alpha_\mathrm{max}$, obtained at zero detuning for $\kappa_m=\kappa_l$. Four cases have been measured, with the corresponding theoretical predictions represented as solid lines: overcoupling ($\kappa_m=10\kappa_l$) with squeezing (red circles), near-critically coupling ($\kappa_m=1.5\kappa_l$) without squeezing (blue circles), overcoupling without squeezing (purple triangles), and near-critically coupling with squeezing (green triangles). The theory lines have been calculated for $\kappa_l=100$kHz, $\eta=0.69$, and $G_s=13$dB.[]{data-label="fig:SNRimpr_sup"}](SNR_improvement_sup_v6.pdf)
Processing of the SSR spectra {#sup:processing}
=============================
In order to detect the small cavity displacement of Sec.\[sec:faxion detection\], we acquire 401 “raw" spectra, which we process and combine into one “grand" spectrum. The faxion tone that we inject is sufficiently small relative to the size of the vacuum fluctuations that in any one raw spectrum it typically does not stand out. However, after processing the spectra, the faxion emerges often well above the level of vacuum noise, as in Fig.\[fig:faxionsearch\]b. This appendix provides an overview of the steps in the processing of the spectra. Our processing procedure is based closely on the work of Ref.[@brubaker2017haystac]. As such, we borrow the terminology used for the intermediate processing stages set forth in that work, and we also omit many details and rationales which are there covered extensively.
Using the setup of Fig.\[fig:schematic\], fluctuations emerging from the cavity, possibly in the presence of squeezed vacuum noise, are directed into a chain of amplifiers led by the AMP. The fluctuations are mixed down to dc, further amplified, low-pass filtered up to $1.9\ \mathrm{MHz}$, and sampled with an Alazar ATS9462 digitizer at $6\ \mathrm{MS/s}$. Each raw spectrum, distinguished as the least processed data saved to a hard disk, itself actually comprises the frequency-averaged power spectral densities of $32$ “subspectra," each acquired over $10\ \mathrm{ms}$ and fast Fourier transformed to provide a spectral resolution of $\Delta_b = 100\ \mathrm{Hz}$. Since $401$ raw spectra are acquired for each of the $200$ squeezed and $200$ unsqueezed data runs, the live acquisition time of all the spectra totals just over $14$ hours, not counting the dead time. In practice it took roughly $18$ hours.
Once the raw spectra are in hand, they are symmetrized. Since the output of a flux-pumped JPA at a given detuning from the center of its amplification band is, in the high-gain limit, identical to the output at the opposite detuning from band center, the best one can do is to infer the spectral density measured in the homodyne configuration as coming equally from both sides of the pump.
The symmetrized spectra are then averaged according to their real frequencies (i.e. not accounting for the fictitious stepping of the cavity used to simulate a haloscope search) in order to detect excess power in either the pertinent rf or IF band [^3]. This mean spectrum is then high-pass filtered by the profile of a Savitsky-Golay (SG) filter [@savitzky1964smoothing] applied to the average spectrum. A SG filter is simply a computationally quick means of applying the $d^\mathrm{th}$-degree polynomial generalization of a moving average within a window of $2W+1$ bins. If the width $W$ is much larger than the size of interesting spectral features, those features will be minimally attenuated when the spectrum is divided by the filter response, whereas features extending over wide bandwidths will be effectively removed. Following Ref.[@brubaker2017haystac], we use $d=10$, $W=500$ for this first filtering.
The resulting real-frequency-averaged spectrum is used to detect and remove excesses of power which do not act like our faxion. Bins that exhibit power fluctuations more than $4$ standard deviations above the mean at their real frequencies are discarded along with their close neighbors as contaminated for all spectra. In practice, we keep over $99\%$ of bins, only eliminating extreme outliers. Note that since in the real frequency space, the faxion tone is being stepped, its $400$ tunings do not combine over any one narrow block of bins, and this bin rejection procedure will therefore not eliminate a faxion signal. Once contaminated bins are removed, the SG filter is recalculated, as it will be slightly different without the presence of outlier bins, and reapplied to each spectrum individually.
This baseline removal leaves a set of nearly flat “normalized" spectra, each centered near $1$. Residual structure left from fluctuations in the overall receiver gain or its profile is then removed by dividing out the SG profile ($d = 4$, $W=500$) of each individual spectrum. The mean value of $1$ is then subtracted from each spectrum to form the “processed" spectra, several of which are shown in Fig.\[fig:faxionsearch\]a.
Next we must rescale the spectra to account for the varying sensitivity to a faxion tone as a function of its detuning from cavity resonance in any given spectrum. To form these “rescaled" spectra, the processed spectra are divided by the relative visibility profile of the squeezed state receiver: Eq. with an additional term accounting for the small contribution of the HEMT’s added noise referred to the input of the AMP, as the weak frequency dependence of AMP gain over the bandwidth of the cavity in the presence of spectrally flat HEMT added noise contributes a correspondingly weak frequency dependence to the noise. The rescaled spectra have the important property that a given power excess from the weakly coupled port of the cavity produces a constant expectation value of rescaled spectrum power excess, regardless of detuning from the center of the cavity. As a consequence, the variance of the power distribution for each bin grows with detuning from cavity resonance.
Aligning bins along the fictitious frequencies for which the stepped faxion tone would appear stationary, it is then possible to construct the maximum likelihood estimate of the power excess in each bin. Doing so yields the “combined" spectrum, a single spectrum whose bins contain as few as one (at the extreme edges) and as many as hundreds of contributions from individual rescaled spectra. The combined spectra bins each have a resolution of $\Delta_b$, far below the linewidth $\Delta_a$ of the faxion. The power within non-overlapping sets of $K^r = 10 \ll \Delta_a / \Delta_b$ bins are thus averaged, to form the “rebinned" spectrum, which has $K^r \Delta_b = 1\ \mathrm{kHz}$ spectral resolution.
Finally, overlapping sets of $K^g = 41 \gg \Delta_a / K^r \Delta_b$ rebinned spectrum bins are combined, accounting for the independently experimentally determined lineshape of the faxion, in order to produce the maximum likelihood estimate of the faxion-shaped power centered on each bin. The resulting spectrum of estimated powers is the grand spectrum, shown renormalized to have mean power excess 0 and standard deviation 1 in Fig.\[fig:faxionsearch\]b. In the grand spectrum, the faxion typically stands out well above the surrounding noise.
The acquisition and processing of sets of 401 raw spectra are repeated 200 times apiece for the optimal squeezed and unsqueezed cases. The two data sets require slightly different processing, chiefly because the visibility profile of the unsqueezed case does not include the frequency-dependent contribution from the squeezing. This results not from finite bandwidth effects of the SQ, but from the fact that squeezed noise is preferentially absorbed near cavity resonance by the cavity’s loss port, wherefrom it is replaced with unsqueezed vacuum.
Each of the 400 total acquisitions provides one measured faxion power. The powers are histogrammed for the optimal squeezed and unsqueezed cases, along with the far larger number of grand spectrum powers of bins not containing a faxion, in Fig.\[fig:faxionsearch\]c. While the absolute mean values of the squeezed and unsqueezed faxion distributions carry little meaning, as they scale with our somewhat arbitrary choice of faxion power, their squared ratio, which would be preserved for a faxion of any power, gives the scan rate enhancement obtained from squeezing, as discussed in Sec.\[sec:faxion detection\].
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[^1]: These two authors contributed equally
[^2]: These two authors contributed equally
[^3]: In a real haloscope search, since the local oscillator being used for homodyne measurement would be stepped along with the cavity, non-axion-induced power excesses in the rf are trickier to reject than their IF counterparts [@brubaker2017haystac].
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Recently there has been considerable interest in the fate of delocalized electronic states in a weak magnetic field in two dimensions (2D)[@glozman; @kravechenko; @furneaux; @shahbazyan; @liu]. In the limit of strong magnetic field, or equivalently weak randomness, it is believed that there exists a single critical energy within each Landau band where the localization length of electronic states diverges[@hk; @bodoreview]. In contrast, one electron localization theory[@gangof4] predicts that in the absence of magnetic field all states are localized in 2D. Consequently, it was argued[@lk; @laughlin] that in the limit of weak magnetic field or strong randomness, where Landau bands merge together, these extended states do not disappear discontinuously but “float up", tending to infinite energy in the $B\rightarrow 0$ limit[@kramer]. Thus, for a given electron density (and hence finite Fermi energy $E_F$), for sufficiently low $B$ all extended states are above $E_F$ and the system becomes insulating. This scenario is crucial to the global phase diagram for the quantum Hall effect proposed by Kivelson [*et al.*]{}[@klz] and has received strong experimental support[@glozman; @kravechenko; @furneaux]. Recently, however, based on numerical calculations of localization length on a tight binding model (TBM), Liu [*et al.*]{}[@liu] concluded that extended states do not float and simply become localized as randomness increases. This issue is more clearly posed, and its resolution well described, by studying certain topological properties of the electronic eigenstates, as we shall see below.
A second issue of interest is the divergence of the localization length when approaching the insulator-quantum Hall phase transition. A previous numerical study[@gammel] performed on a random site TBM with a magnetic field suggested that the localization length exponent $\nu_i\simeq 0.8$ in 2D at the localization transition point. Besides the fact that this value is much smaller than that at the transition between quantum Hall phases in the strong magnetic field limit[@bodoreview] $\nu_H\approx 2.4$, it violates the inequality $\nu\geq 2/d$[@chayes] which is widely believed to be satisfied in known random systems[@bockstedte]. To address both these issues, a more clear-cut numerical method appears warranted.
In the presence of a magnetic field, electronic states exhibit interesting topological properties[@tknn; @niu; @arovas; @yan]. In particular, each state can be labeled by an integer called the Chern number, which is its boundary condition averaged Hall conductance, in units of $e^2/h$[@niu; @arovas]. A state with nonzero Chern number carries Hall current and is necessarily extended. Thus by calculating the Chern numbers one is able to identify extended states unambiguously on [*finite*]{} size systems. This approach has proved very successful in addressing the localization problem in the lowest Landau band[@huo]. In this paper, we apply this approach to the TBM studied by Liu [*et al.*]{} and also by Ando[@ando]. Our results clearly support the “floating up" picture and are consistent with Thouless number calculations by Ando[@ando]. In fact, results of Liu [*et al.*]{}[@liu] are also consistent with ours, but our interpretation of their results is somewhat different, as we discuss later.
We have also studied the dependence of the number and energies of extended states on system size. We find just like the case of individual Landau bands, the localization length diverges only at individual energies. In the high field (weak randomness) limit, the localization exponent is found to be the same as that of an isolated lowest Landau band, $\nu_H\approx 2.4$ [@bodoreview]. For strong enough randomness we find that the localization length remains finite throughout the band and the number of states with nonzero Chern number goes to zero as the system size goes to infinity[@note1]. Using finite size scaling, we find the largest localization length of the system diverges as the critical randomness is reached with an exponent $\nu_i$ which is the same as $\nu_H$, contrary to previous suggestion[@gammel] that the strong randomness exponent may be different from that in the lowest Landau levels. Thus our data show that $\nu$ is a universal exponent for all spin polarized integer quantum Hall transitions, including the ultimate one to the insulating state.
We study the TBM on a square lattice with nearest neighbor hopping, a uniform magnetic field and random potential, described by the Hamiltonian: $$\begin{aligned}
H&=&\sum_{mn}\{-t(c^\dagger_{m+1,n}c_{m,n}+c^\dagger_{m,n+1}e^{i2\pi \alpha m}
c_{m,n}+h.c.)\nonumber\\
&+&\epsilon_{m,n}c^\dagger_{m,n}c_{m,n}\},
\label{hamilt}\end{aligned}$$ where the integers $m$ and $n$ are the $x$ and $y$ coordinates of the lattice site in terms of lattice constant, $c_{m,n}$ is the fermion operator on that site, $t$ is the hopping matrix element which we set as the unit of energy from now on, and $\epsilon$ is the random potential ranging [*uniformly*]{} from $-W$ to $W$ (as in the Anderson model[@anderson]). $\alpha$ is the amount of magnetic flux per plaquette in units of the flux quantum $hc/e$. The Landau gauge ${\bf A}=(0, Bx, 0)$ is used in Eq. (\[hamilt\]). Here we concentrate on the case $\alpha=1/N_f$, where $N_f$ is an integer. In this case, we have $N_f$ Landau subbands in the absence of random potential, and the lowest energy subbands map onto the lowest Landau levels in the limit $N_f\rightarrow\infty$, which is the continuum limit.
The Hall conductance of an individual eigenstate $|m\rangle$ can be obtained easily using the Kubo formula:[@yan] $$\begin{aligned}
\nonumber
\sigma_{xy}^{m}={ie^2\hbar\over A}\sum_{n\ne m}{\langle m|v_y|n\rangle
\langle n|v_x|m\rangle-\langle m|v_x|n\rangle\langle n|v_y|m\rangle\over
(E_n-E_m)^2},\nonumber\end{aligned}$$ where $A$ is the area of the system, $v_x$ and $v_y$ are the velocity operators in the $x$ and $y$ directions respectively. For a finite system with the geometry of a parallelogram with periodic boundary conditions (torus geometry), $\sigma_{xy}^m$ depends on the two boundary condition phases $\phi_1$ and $\phi_2$. As shown by Niu [*et al.*]{}, the boundary condition averaged Hall conductance takes the form[@niu] $$\langle\sigma_{xy}^m\rangle={1\over 4\pi^2}\int{d\phi_1d\phi_2\sigma_{xy}^{m}
(\phi_1,\phi_2)}=C(m)e^2/h,$$ where $C(m)$ is an integer called the Chern number of the state $|m\rangle$. States with nonzero Chern numbers carry Hall current and are necessarily extended states[@arovas; @huo]. Thus by numerically diagonalizing the Hamiltonian on a grid of $\phi_1$ and $\phi_2$, and calculating the Chern numbers of states of finite size systems by converting the integral in (2) to a sum over grid points, we are able to identify extended states unambiguously.
We have studied systems of square geometry with various size (from $3\times 3$ to $15\times 15$), strength of randomness ($W$) and magnetic field (equivalently, $N_f$). The number of samples explored for a given $W$ range from $2000$ to $30$ depending on system size. Most of our data were taken for $N_f=3$. We do not, however, see any qualitative difference in behavior of the extended states, for systems with $N_f$ as large as 13. Hence we believe our results with $N_f=3$ are generic and apply to the continuum limit $N_f\rightarrow
\infty$.
Fig. \[fig1\] shows the density of states \[$\rho(E)$\] and density of extended states with nonzero Chern numbers \[$\rho_c(E)$\], for two different strength of randomness for $1/3$ flux quantum per plaquette ($N_f=3$) on a square of lattice size $9\times 9$. For weak enough randomness ($W=1.0$), the three Landau subbands are broadened by randomness, but are still well separated. We see there are extended states in all subbands, with their densities peaked essentially at the center of each subband. This is consistent with the previous study on individual Landau bands[@huo]. As randomness increases, the subbands further broaden and start to merge, as is seen for $W=2.5$. In this case there are still three prominent peaks in $\rho(E)$ (we call them $E_1$, $E_2$ and $E_3$ respectively), which are (loosely) identified as centers of Landau subbands. $\rho_c(E)$, however, now looks very different: most of the extended states are near the center of the entire band ($E_2$) and there is no peak in $\rho_c(E)$ at $E_1$ or $E_3$, which are the centers of Landau subbands. There are nontrivial features in $\rho_c(E)$ which we discuss below, but it is clear from Fig. \[fig1\] that as the three subbands start to merge, the extended states in the lower and upper subbands move away from the centers of the subbands ($E_1$ and $E_3$) toward center of the band ($E_2$). This behavior is also seen in systems of $N_f$ as large as 13. We hence believe in the limit $N_f\rightarrow\infty$ (which can be mapped onto the continuum model), the extended states in the lowest subbands (which becomes Landau levels) float up toward the center of the band (which is at infinitely high energy relative to them in the continuum model). This provides unambiguous support for the floating up picture predicted theoretically[@lk] and seen experimentally[@glozman].
The fact that the extended states in the lower and upper subbands float toward the center of the band as randomness increases may be understood in the following manner. In finite size systems, the Chern number of a state can change only when it becomes degenerate with a another state under certain boundary conditions. This can be shown to occur only by tuning three parameters, including the two boundary condition angles plus the parameter characterizing the random potential. If such a degeneracy were to occur, the Chern numbers of the two states involved may change but their sum is conserved. Randomness tends to localize all states and annihilate the nonzero Chern numbers carried by the extended states. Thus states with nonzero Chern numbers of opposite signs “attract" each other and tend to move close in energy as randomness increases. It is believed that in the thermodynamic limit the localization length diverges and true current carrying (extended) states exist only at individual critical energies. (We will provide numerical evidence for this belief below.) Each such critical energy is characterized by its total Chern number which is [*invariant*]{} as randomness varies, unless merging between critical energies occur. For exactly the same reason, critical energies with total Chern numbers of opposite sign also “attract" each other as randomness increases. In the case of $N_f=3$ systems, the total Chern numbers for the three subbands are 1, -2 and 1 respectively. Due to the “attraction", we expect that as randomness is turned on, the extended states in the central subband with total Chern number $-2$ splits into two critical energies with total Chern number $-1$ each (by symmetry) and move toward the two band edges as randomness is increased further. Concurrently, the two critical energies of the upper and lower subbands with total Chern number +1 move away from the center of the subbands toward the center of the band. This is precisely what is seen in the $\rho_c(E)$ at $W=2.5$: There is a small dip at the center of the band indicating the splitting of the central critical energy; further, there are two less pronounced peaks from the two edge subbands, whose positions have clearly moved away from the corresponding peaks of $\rho(E)$.
Fig. \[fig2\] depicts the number of states with nonzero Chern number $N_c\equiv\int_{-\infty}^{\infty}{\rho_c(E)dE}$ versus the system size $N_s$ (number of sites), for different values of disorder $W$, for $N_f=3$, on a double logarithmic plot. We find the plot is essentially linear for small $W$ up to $W\approx3.0$, with slope $y=0.79\pm 0.01$ which is relatively independent of $W$[@linearnote], indicating that $N_c\sim (N_s)^y$ in this region. This power law behavior is exactly what is expected[@yan; @huo] where there are individual critical energies $E_c^i$ in the vicinity of which the localization length diverges with a power law of the form $\xi(E)\sim |E-E_c|^{-\nu}$. In a finite system with linear size $L_s=\sqrt{N_s}$, states with $\xi(E)>L_s$ look extended. The number of such states goes like $N_c\sim N_s\rho(E_c)L_s^{-1/\nu}\sim N_s^{1-1/2\nu}$, thus $y=1-1/2\nu$. This gives $\nu=2.4\pm 0.1$, in agreement with the $\nu_H$ for lowest Landau band[@bodoreview; @huo]. This suggests that $\nu$ is a universal exponent in all spin-polarized integer quantum Hall transitions.
For larger $W$, the dependence of $N_c$ on $N_s$ deviates from a power law and bends down as $N_s$ increases. This indicates that in this regime the two critical energies with total Chern number -1 have merged with the other two with Chern number +1; all extended states have disappeared and $\xi$ is finite throughout the band. For strong enough randomness and large $N_s$, $N_c$ [*decreases*]{} as $N_s$ increases; thus in the localized regime the average number of extended states per sample goes to zero in the thermodynamic limit. From the shape of the density of extended states and scaling of data we determine the critical randomness to be $W_c\approx 2.9\pm 0.1$. For $W$ greater than but close to $W_c$, and large sizes $N_s$, $N_c$ is expected to take the scaling form: $N_c\sim N_s^y \tilde{F}(L_s/\xi_m)\sim N_s^y F(N_s^{1/(2\nu_i)}(W-W_c))$, where $\xi_m$ is the largest localization length in the system that diverges as $W_c$ is approached with exponent $\nu_i$. The best scaling is achieved with $\nu_i\approx 2.3$, assuming $W_c=2.9$, and Fig. \[fig3\] shows the scaling function $F$. Taking into uncertainty in $W_c$ we estimate $\nu_i\approx 2.3\pm 0.3$. This suggests that the localization length exponents are the same in both the localized and extended regimes, in contrast to a previous suggestion that they may be different[@gammel]. The increasing negative slope of the scaling curve suggests that $N_c$ goes to zero faster than any power law as $N_s$ increases at large $N_s$.
We emphasize that the existence of this localized regime in the TBM is due to the facts that there exist critical energies with negative Chern numbers and the total Chern number of the system is zero. In the continuum however, the total Chern number of the critical energy of each Landau band is one, and there is [*no*]{} critical energy with negative Chern number at finite energy. Hence the extended states at these critical energies cannot annihilate their Chern numbers and become all localized as the randomness increases. They only float up and “disappear" at infinite energy. This becomes clear as one views the continuum system as the $N_f\rightarrow\infty$ limit of the TBM. In the TBM, the natural energy scale is hopping $t$ (set to be 1 previously), and the zero point of energy is the center of the band. In the continuum however, the energy scale is Landau level spacing $\hbar\omega_c$, and the zero point of energy is determined by identifying the center of the lowest energy band with energy $\hbar\omega_c/2$. In terms of TBM parameters we have $\hbar\omega_c=4t/N_f$. Based on our data up to $N_f=13$ we conclude that the critical randomness is almost $N_f$ independent and is about $W_c\approx 3t$, in agreement with Ando[@ando]. The energy at which the final merging and disappearance of critical energies measured from the [*bottom*]{} of the band is found to be of order $O(W_c)$, which is the only energy scale of the TBM at criticality. Therefore the number of Landau subbands below the lowest critical energy before it finally disappears is of order $O(W_c/\hbar\omega_c)\propto N_f$. We hence conclude in the continuum limit ($\hbar\omega_c$ finite, $N_f\rightarrow\infty$) the critical randomness strength ($W_c\propto N_f\hbar\omega_c$) is infinite, and extended states all float up to infinite energy in the strong randomness (or weak magnetic field) limit.
Liu [*at al.*]{}[@liu] interpret the localization transition in the TBM as indication of disappearance of extended states in the continuum. As discussed above, the continuum limit of the TBM is subtle. They also find the energies of the extended states do not shift much relative to the [*center*]{} of the band ($E=0$) and interpret it as evidence against floating. Our results for $N_f=3$ is consistent with very little shift of $E_1$ and $E_3$ relative to $E_2$. However, critical energies clearly float away from peaks of $\rho(E)$ (which are roughly at the centers of Landau subbands), and more so relative to the [*bottom*]{} of the band. This is because as randomness increases, both the bottom of the band and peaks in $\rho(E)$ move [*downward*]{}. We believe this [*relative*]{} movement is clear indication of floating of extended states which survives the continuum limit.
In summary, we have found unambiguous numerical evidence, using a tight binding model (TBM) and considering the passage to the continuum limit, which indicates that extended states float up in energy toward infinity in the weak magnetic field limit in 2D in the continuum. For the TBM, this can be heuristically understood in terms of an “attraction" between states with opposite Chern numbers. There is a critical randomness strength in the TBM at which all states become localized. The localization length diverges with the same exponent as that of the isolated lowest Landau band (high field limit) when approaching this critical point.
We thank M. Guo, F.D.M. Haldane, B. Huckestein and D. Shahar for helpful conversations. This work was supported by NSF grants DMR-9224077, DMR-9400362 and CDA-9121709. One of us (RNB) was also supported in part by a Guggenheim fellowship.
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This localized regime is special to this one band model. Continuum systems in a magnetic field always have some extended states. See below.
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Actually for $W\leq 2.5$ the curves seem to deviate slightly upwards from linear behavior. This may be due to the existence of close lying critical energies between which the localization length is larger than the system sizes we have studied, so the system appears to have a narrow band of extended states instead of isolated critical energies.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We have modelled the spectral energy distributions of the 13 HDF galaxies reliably detected by ISO. For 2 galaxies the emission detected by ISO is consistent with being starlight or the infrared ’cirrus’ in the galaxies. For the remaining 11 galaxies there is a clear mid-infrared excess, which we interpret as emission from dust associated with a strong starburst. 10 of these galaxies are spirals or interacting pairs, while the remaining one is an elliptical with a prominent nucleus and broad emission lines.
We give a new discussion of how the star formation rate can be deduced from the far infrared luminosity and derive star formation rates for these galaxies of 8-1000 $\phi M_{\sun}$ per yr, where $\phi$ takes account of the uncertainty in the initial mass function. The HDF galaxies detected by ISO are clearly forming stars at a prodigious rate compared with nearby normal galaxies. We discuss the implications of our detections for the history of star and heavy element formation in the universe. Although uncertainties in the calibration, reliability of source detection, associations, and starburst models remain, it is clear that dust plays an important role in star formation out to redshift 1 at least.
author:
- |
M. Rowan-Robinson$^1$, R.G. Mann$^1$, S.J. Oliver$^1$, A. Efstathiou$^1$, N. Eaton$^1$,\
[ P. Goldschmidt$^1$, B. Mobasher$^1$, S. Serjeant$^1$, T.J. Sumner$^1$, L. Danese$^2$, D. Elbaz$^3$, ]{}\
[ A. Franceschini$^4$, E. Egami$^5$, M. Kontizas$^6$, A. Lawrence$^7$, R. McMahon$^8$, ]{}\
[ H.U. Norgaard-Nielsen$^9$, I. Perez-Fournon$^{10}$, J.I. Gonzalez-Serrano$^{11}$ ]{}\
$^1$Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2BZ;\
$^2$SISSA, Via Beirut 2-4, Trieste, Italy;\
$^3$Service d’Astrophysique, Saclay, 91191, Gif-sur-Yvette, Cedex, France;\
$^4$Osservatorio Astronomico de Padova, Vicolo dell’Osservatorio 5, I-35 122, Padova, Italy;\
$^5$Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, D-8046, Garching bei Munchen, Germany;\
$^6$Astronomical Institute, National Observatory of Athens, P.O.Box 200048, GR-118 10, Athens, Greece;\
$^7$Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ;\
$^8$Institute of Astronomy, The Observatories, Madingley Road, Cambridge, CB3 0HA;\
$^9$Danish Space Research Institute, Gl. Lundtoftevej 7, DK-2800 Lyngby, Copenhagen, Denmark;\
$^{10}$Instituto Astronomico de Canarias, Via Lactea, E-38200 La Laguna, Tenerife, Canary Islands, Spain;\
$^{11}$Instituto de Fisica de Cantabria, Santander, Spain\
title: 'Observations of the Hubble Deep Field with the Infrared Space Observatory V: Spectral Energy Distributions, Starburst Models and Star Formation History'
---
0.0in 0.0in 9.0in 6.25in
infrared: galaxies - galaxies: evolution - star:formation - galaxies: starburst - cosmology: observations
Introduction
============
Because of its great depth, high resolution, and the intensive follow-up which has been carried out in it, the Hubble Deep Field (HDF) is an exceptional resource for cosmological studies. The central area of the HDF consists of 5 square arcmin of sky. It was imaged by the Hubble Space Telescope on 150 orbits in December 1995 and reaches to at least 29th magnitude in I (800nm), V (600nm) and B (450nm), and to 27th magnitude in U (300nm) (Williams et al 1996). We were successful in bidding for Director’s Time on the Infrared Space Observatory (ISO) and were awarded a total of 12.5 hours to map the HDF with ISO-CAM in the LW2 (6.7 $\mu$m) and LW3 (15 $\mu$m) filters. The observations were carried out in July 1996 and have been described by Serjeant et al (1997). The images have been searched for point sources by Goldschmidt et al (1997): a total of 15 sources were found in the central HDF area at 6.7 $\mu$m, and 5 at 15 $\mu$m, of which 6 and 4, respectively, are from complete and reliable sub-samples. A further 27 sources were found in the flanking fields around the HDF. The resulting source-counts have been discussed by Oliver et al (1997) and shown to be consistent with the strongly evolving starburst models previously used to model the 60 $\mu$m and 1.4 GHz counts (Franceschini et al 1994, Rowan-Robinson et al 1993, Pearson and Rowan-Robinson 1996). Associations for the 17 ISO sources in the central HDF area (2 were detected at both 6.7 and 15 $\mu$m) were sought with HDF galaxies using a likelihood method (Mann et al 1997) and 13 credible associations were found. In this paper we take the view that these associations tentatively confirm the reality of those sources which are not in the reliable and complete sub-samples. There is ambiguity about some of the associations (Mann et al 1997) and in some case the ISO flux may be due to more than one galaxy (this is particlularly so for the 15 $\mu$m detections). In this paper we have assumed that all the flux is assigned to the galaxy with the highest likelihood. This assumption does not have a great effect on our overall conclusions. For the two sources where the likelihoods did not completely resolve ambiguities (12 36 43.0 +62 11 52 and 12 36 48.4 +62 12 15), we have conservatively chosen the lower redshift galaxy as the association.
In this paper we discuss the spectral energy distribution of the 13 galaxies detected by ISO in the central HDF area and consider the implications for star formation rates and the overall history of star formation in the universe. Details of the 13 galaxies are given in Table 1.
A striking feature of the fainter HDF galaxies is their blue colours, indicative of high redshift galaxies undergoing bursts of star formation. This is confirmed both by systematic analyses of the colours of the HDF galaxies (Mobasher et al 1996) and by studies of the morphologies of the galaxies (Abraham et al 1996), which show a high proportion of interacting and merging systems. In both respects the HDF galaxies look like a higher redshift version of the starburst galaxies found by IRAS. This was one of the strong motivations for seeking observing time with ISO. The fact that HDF galaxies have been detected by ISO is sufficient to demonstrate that this analogy with IRAS galaxies is highly relevant.
An Einstein de Sitter model ( $\Omega_{0}$ = 1), with a Hubble constant $H_{0}$ = 50 km/s/Mpc, has been used throughout the paper.
Spectral Energy Distributions
=============================
For the 13 galaxies reliably detected by ISO in the HDF (Goldschmidt et al 1997, Mann et al 1997) we have modelled their spectral energy distributions (seds) from 0.3 to 15 $\mu$m. Spectroscopic redshifts are available for 9 of the galaxies (Cowie 1996, Cohen et al 1996, Phillips et al 1997). For the remaining 4 we have used photometric redshifts determined by the method of Mobasher et al (1996). The latter analysis has been repeated using the total magnitudes and colours given in the STScI HDF catalogue (Williams et al 1996). The resulting redshifts agree well with those determined in a small, fixed aperture by Mobasher et al (1996).
The U,B,V,I data (AB magnitudes) from HST and the J, H/K data of Cowie (1996) have been fitted with galaxy models from the library of Bruzual and Charlot (1993). The near infrared data were corrected to the same aperture as the optical data. For wavelengths beyond 2.5 $\mu$m, the Bruzual and Charlot models give predictions based on IRAS data for their standard stars, which appear to generate a spurious secondary peak at 12 $\mu$m (even for the youngest starburst models). We have therefore replaced the Bruzual and Charlot predictions with a Rayleigh-Jeans extrapolation beyond 2.5 $\mu$m. Excellent fits to the U to K data for our galaxies can be obtained using models with starburst of duration $10^{9}$ yrs, viewed at a range of subsequent times $\tau$ = 1 to 2.4 Gyr (values of $\tau$ are given in Table 1). In the case of the galaxies for which we have only photometric redshifts, the good fits of the models to the data provide support for the photometric redshifts. Almost equally good fits could be obtained with an exponentially decreasing star formation rate with a time-scale of 1 Gyr. No allowance is made for reddening at this stage. For one galaxy (12 36 43.9 +62 11 30), the 6.7 $\mu$m emission can be accounted for almost completely by starlight.
For the remaining 12 galaxies, there is a clear excess of infrared radiation. We have considered first the possibility that we are seeing the infrared ’cirrus’ in the galaxies, emission from starlight in the galaxies absorbed by interstellar grains and reemitted in the infrared. The cirrus models of Rowan-Robinson (1992) have been revised to incorporate very small grains and PAHs correctly (Efstathiou, Rowan-Robinson and Siebenmorgen 1997, in preparation). For 12 of the galaxies, there was no plausible cirrus model, because the resulting far infrared luminosity was always at least 3 times the total optical-uv luminosity of the galaxy. A typical value of $L_{fir}/L_{opt-uv}$ for cirrus emission is 0.2-0.3 (Rowan-Robinson et al 1987, Rowan-Robinson 1992). For one galaxy (12 36 48.1 +62 14 32), in which the cirrus model gave a far infrared luminosity comparable to that seen in the optical and uv, we accepted the cirrus model fit (see Fig 1).
Figure 1 shows fits of the standard starburst model of Efstathiou et al (1997) to the infrared spectral energy distributions of the remaining 11 ISO-detected HDF galaxies. The model is a development of the earlier starburst model of Rowan-Robinson and Efstathiou (1993), with a proper treatment of very small grains and PAHs, and gives an excellent fit to the spectra of M82 and the starburst galaxy NGC6090 studied by ISO (Acosta-Pulido et al 1996). Although in most cases we have only a single mid-infrared data point, it is satisfactory that in the case where we have detections at both 6.7 and 15 $\mu$m , the model fits both data points well. Where only upper limits are available at one of the ISO wavelengths, these are generally consistent with the predictions of the model (for objects 4 and 5 in Table 1, the upper limits at 15 $\mu$m may imply that the model parameters need adjustment, that an alternative model, eg a dusty AGN torus, may be needed, or that the 6.7 $\mu$m detections is unreliable). Even more impressive is the fit for 3 galaxies to the VLA detections by Fomalont et al (1996), for which we have assumed in the model the standard radio-far ir correlation ( S(60 $\mu$m)/S(1.4 GHz) = 90 ) and a radio spectral index of 0.8. This supports the idea that we really are seeing dust emission from starbursts with ISO. In most other cases the models are consistent with the 1.4 GHz upper limit, taken as 12.2 $\mu$Jy (Fomalont et al 1997) ( but for objects 4, 5 and 10, the radio limits lie significantly below the predictions of the starburst model). If we look at the morphologies of the galaxies in our sample, the 2 non-starburst galaxies are both ellipticals. Of the 11 galaxies whose seds we fit as starbursts, 8 are spirals, 2 are interacting pairs and 1 is a galaxy (possibly spiral) with a prominent nucleus (12 36 46.4 +62 14 06). Broad lines have been found in the optical spectrum of this galaxy and it is possible that the 6.7 mu emission could be from a dusty torus surrounding the AGN, rather than from a starburst. However if this interpretation were correct, the agreement of the radio flux with the prediction of the starburst model would be a coincidence. There are several other objects in which a dusty torus model would give an equally satisfactory fit to the observed infrared excess. However since the covering factor by dust in AGN is generally found to be $< 0.5$ (Rowan-Robinson 1995), it is unlikely that more than one object in our sample is AGN with its dust torus seen edge-on. The source-count models discussed by Oliver et al (1997) predict that the fraction of AGN expected in 6.7 and 15 $\mu$m samples should be small. We therefore assume that the infrared excess in the remaining 11 objects is from starbursts rather than from AGN with dusty tori.
Table 1 gives the inferred rest-frame values of $\nu L_{\nu}$ at 0.3, 0.8, 15 and 60 $\mu$m. It can be seen that $L_{60}/L_{0.8}$ ranges from 2 to 200, and $L_{60}/L_{0.3}$ ranges from 3 to 1000. Thus the implication of the ISO detections is that, at least for the detected galaxies, the bulk of the bolometric luminosity of the galaxies is emitted at far infrared wavelengths. The interpretation of this is similar to that for the starburst galaxies found in IRAS surveys: most massive star formation takes place in dense molecular clouds and is shrouded from view by a substantial optical depth in dust. What is seen in the optical and uv represents stars formed near the edges of clouds, so that the light from these stars can escape directly.
---- ---------------------- ------------------ -------- ----------------------------- ----------------------------- ---------------------------- ---------------------------- --------------------------------------- ------------
[**Galaxy**]{} [**redshift**]{} $\tau$ [**$L_{0.3}/L_{\odot}$**]{} [**$L_{0.8}/L_{\odot}$**]{} [**$L_{15}/L_{\odot}$**]{} [**$L_{60}/L_{\odot}$**]{} [**$(\dot{M}_{*,all}/M_{\odot})$**]{} notes
(ISOHDF) (Gyr) x$\phi^{-1}$
1 12 36 41.1 +62 11 29 (0.047) 1.02 2.8x$10^{8}$ 4.0x$10^{8}$ 2.1x$10^{8}$ 7.6x$10^{8}$ 0.20 S sb c
2 12 36 41.6 +62 11 42 0.585 1.14 2.8x$10^{9}$ 7.3x$10^{9}$ 1.1x$10^{11}$ 3.9x$10^{11}$ 101 I sb b,d
3 12 36 42.6 +62 12 10 0.454 1.14 6.9x$10^{9}$ 1.8x$10^{10}$ 1.7x$10^{10}$ 6.0x$10^{10}$ 16 S sb b
4 12 36 42.9 +62 13 09 (0.74) 1.14 5.5x$10^{9}$ 1.4x$10^{10}$ 5.3x$10^{11}$ 1.9x$10^{12}$ 495 S sb b
5 12 36 43.0 +62 11 52 (0.82) 2.4 3.4x$10^{9}$ 3.0x$10^{10}$ 8.9x$10^{11}$ 3.2x$10^{12}$ 840 S sb a
6 12 36 43.7 +62 12 55 0.558 1.14 1.4x$10^{10}$ 3.6x$10^{10}$ 8.6x$10^{10}$ 3.1x$10^{11}$ 80 I sb c,e
7 12 36 43.9 +62 11 30 1.01 3.5 1.75x$10^{10}$ 1.35x$10^{11}$ 2.0x$10^{10}$ 1.4x$10^{11}$ - E sl b,e
8 12 36 46.4 +62 14 06 0.960 1.28 2.2x$10^{10}$ 8.1x$10^{10}$ 1.1x$10^{12}$ 3.9x$10^{12}$ 1010 E sb a,e
9 12 36 48.1 +62 14 32 (0.023) 1.14 6.0x$10^{7}$ 1.5x$10^{8}$ 1.8x$10^{7}$ 1.26x$10^{8}$ - E cirr a,c
10 12 36 48.4 +62 12 15 (0.778) 1.02 4.5x$10^{9}$ 1.15x$10^{10}$ 6.8x$10^{11}$ 2.45x$10^{12}$ 640 S sb a
11 12 36 49.7 +62 13 15 0.475 1.28 1.75x$10^{9}$ 1.26x$10^{10}$ 9.5x$10^{10}$ 3.4x$10^{11}$ 88 S sb a,c,e
12 12 36 51.5 +62 13 57 0.557 1.14 4.7x$10^{9}$ 2.7x$10^{10}$ 5.2x$10^{10}$ 1.86x$10^{11}$ 48 S sb d
13 12 36 58.9 +62 12 48 0.320 1.02 2.4x$10^{9}$ 6.0x$10^{9}$ 9.0x$10^{9}$ 3.2x$10^{10}$ 8 S sb b
---- ---------------------- ------------------ -------- ----------------------------- ----------------------------- ---------------------------- ---------------------------- --------------------------------------- ------------
S - spiral
E - elliptical
I - interacting pair
sb - sed fitted with starburst model
cirr - sed fitted with cirrus model
sl - sed fitted with starlight
a - detected at 6.7 $\mu$m, from reliable and complete sub-sample
b - detected at 6.7 $\mu$m, from supplementary list
c - detected at 15 $\mu$m, from reliable and complete sub-sample
d - detected at 15 $\mu$m, from supplementary list
e - detected at 1.4 GHz (Fomalont et al 1997)
Star formation rate
===================
A number of authors have discussed how the star formation rate in a galaxy can be inferred from its optical, ultraviolet or far infrared luminosity. Scoville and Young (1983) estimated the star formation rate of O,B,A stars ($M > 1.6 M_{o}$) from the total far infrared luminosity of galaxies, implicitly assuming a burst of star formation lasts $10^{9}$ yrs, finding
$\dot{M}_{*,OBA}$ = 7.7x$10^{-11} L_{fir}/L_{o}$ . (1)
Thronson and Telesco (1986) used a Salpeter IMF to give rates of star formation of all stars and of OBA stars, averaged over the past 2x$10^{6}$ yrs, per unit far infrared luminosity:
$\dot{M}_{*,OBA}$ = 2.1x$10^{-10} L_{fir}/L_{o}$ . (2)
$\dot{M}_{*,all}$ = 6.5x$10^{-10} L_{fir}/L_{o}$ . (3)
They attribute the fact that (2) is a factor of 3 higher than (1) to the different assumptions about the duration of the burst. They also give the scaling factors for the star formation rate between the different IMFs and lower mass limits:
M/L (Miller-Scalo, 100, 0.1) : M/L (M-S, 100, 1.6) : M/L (Salpeter, 100, 0.1): M/L (Salpeter, 100, 1.6)
= 10.2 : 4.0 : 3.1 : 1
More recently Madau et al (1996) have calculated total star formation rates and heavy element production, $\dot{M}_{Z}$, from the ultraviolet luminosity densities at 2800 $\AA$, using a Salpeter IMF and the evolutionary models of Bruzual and Charlot (1993). The figures they give are equivalent to
$\dot{M}_{*,all}$ = 5.3x$10^{-10} L_{2800}/L_{o}$ . (4)
$\dot{M}_{*,all}$ = 42 $\dot{M}_{Z}$ (5)
To convert from 60 $\mu$m luminosity to star formation rate, we assume that a fraction $\epsilon$ ($\simeq$ 1) of the optical and uv energy emitted in a starburst is absorbed by dust and emitted in the far infrared, so that
$L_{bol,fir}$ = $\epsilon L_{bol,opt-uv}$. (6)
The bolometric correction at 2800 $\AA$ for the 1 Gyr starburst models of Bruzual and Charlot (1993), when viewed at early ages, is 3.5 and those at 15 and 60 $\mu$m for the dusty starburst model of Rowan-Robinson and Efstathiou (1993) are 6.0 and 1.7, respectively, so using (4) and (6):
$\dot{M}_{*,all} /[L_{60}/L_{o}]$ = 2.6 $\phi/\epsilon$ x$10^{-10}$ (7)
$\dot{M}_{*,all} /[L_{15}/L_{o}]$ = 9.3 $\phi/\epsilon$ x$10^{-10}$
where the factor $\phi$ incorporates (1) the correction from a Salpeter IMF to the true IMF (x3.3 if the Miller-Scalo IMF is the correct one), (2) a correction if the starburst event is forming only massive stars (x 1/3.1 if only O,B,A stars, $> 1.6 M_{o}$, are being formed). This estimate, which is now based on detailed starburst models for the optical-uv radiation and proper radiative transfer models for the far infrared emission, is a factor 1.9 higher than that of Scoville and Young (1983, eqn (1) above) and a factor 0.7 times that of Thronson and Telesco (1986, eqn (2) and (3) above).
Star formation rates for ISO-HDF galaxies
=========================================
Table 1 gives the inferred 15 and 60 $\mu$m luminosities, and star formation rates based on eqn (7), for the 11 HDF starburst galaxies detected by ISO. The star formation rates range (with one exception) from 8-1000 $\phi M_{o}$ per yr. The galaxies detected by ISO are forming stars at a prodigous rate compared with nearby normal spirals. Although star formation rates based only on 6.7 and 15 $\mu$m detections are bound to be rather uncertain, because most of the energy is emitted at much longer wavelengths, it is clear that the star formation rates deduced from the uv fluxes detected by HST are a severe underestimate for these galaxies.
It is of course of interest to ask whether these star formation rates can be typical of all the galaxies in the HDF. For 2 of the galaxies detected by ISO, the 6.7 and/or 15 $\mu$m flux is consistent with emission from starlight and/or cirrus and there is no evidence for a luminous starburst. For other bright galaxies in the HDF, the non-detection by ISO gives a significant upper limit on any excess far infrared emission. For these galaxies the estimates of star formation rate from the uv flux will be correct. However we can not rule out the possibility that for a significant fraction of the fainter HDF galaxies, particularly those with $z > 1$, the presence of a strong far infrared excess is the norm rather than the exception. When we see star formation within our Galaxy or in other nearby galaxies, the bulk of the massive stars (which produce all the heavy elements and most of the ultraviolet light) are formed within dense molecular clouds behind a high optical depth in dust. The starburst galaxies detected by IRAS emit most of their radiation at far infrared wavelengths. It is a reasonable expectation that as we look back to epochs when the bulk of the stars in a galaxy are formed, that this too will take place within dense clouds of molecules and dust and be primarily a far infrared phenomenon. Of course eventually we will see back to epochs when the very first stars form, when little or no heavy elements or dust are present, and star formation will be entirely an optical and uv phenomenon. However this transparent phase may last no more than a few percent of the main star formation phase, say $10^{7} - 10^{8}$ yrs, and be confined to very high redshifts ( $> 3-5$).
Infrared luminosity-density and the history of star formation
=============================================================
Madau et al (1996) have used the Canada-France Redshift Survey (Lilly et al 1996) and HDF data to calculate the history of star formation and heavy element generation, under the assumption that the uv gives a complete view of the star formation that is occurring. Integrating over the star formation density as a function of redshift, they conclude that all the heavy elements associated with the visible matter in galaxies can be generated. However if their calculated baryonic density is converted to a value for $\Omega$, a value of 0.0035 is obtained, only 7 $\%$ of the baryonic density of 0.05, for an assumed $H_{o}$ = 50, derived from cosmological nucleosynthesis of the light elements (Walker et al 1991). It is not unreasonable to assume that some of the remaining 93 $\%$ or baryons has participated in star formation and heavy element production. For example, the hot gas in clusters is known to have a heavy element abundance of at least 1/3rd of solar.
We first estimate the far infrared luminosity density for the luminosity function derived from IRAS 60 $\mu$m data. Oliver et al (1997) have shown that the 6.7 and 15 $\mu$m source-counts are consistent with the strongly (luminosity-)evolving starburst models which we have used to fit (1) the redshift distribution in IRAS redshift surveys (Saunders et al 1990, Oliver et al 1995), (2) the 60 $\mu$m source-counts (Pearson and Rowan-Robinson 1996), (3) the far infrared background, including the claimed detection using FIRAS data from COBE by Puget et al (1996) (Pearson and Rown-Robinson 1996, Rowan-Robinson and Pearson 1996), (4) the sub-mJy 1.4 GHz radio counts (Rowan-Robinson et al 1993, Hopkins et al 1996).
The solid curve in Fig 2 shows the luminosity-density at 60 $\mu$m as a function of redshift. For $z < 0.3$ this is directly derived from IRAS galaxy redshift surveys (the luminosity function given in line (23) of Table 3 of Saunders et al 1990). The extrapolation to higher redshift is the luminosity evolution model used in Pearson and Rowan-Robinson (1996), Rowan-Robinson et al (1993), and Oliver et al (1997) to fit the deep 60 $\mu$m and 1.4 GHz source-counts, for which
$L_{*}(z)$ = $(1+z)^{3.1}, z < 2$, (8)
$L_{*}(z)$ = $3^{3.1}, 2 < z < 5$.
Using eqn (7) we can convert this to a density of star formation, and integrate to derive a total mass-density in stars or in heavy elements. We find
$\Omega_{*}$ = 0.008 $h^{-2}_{50}$ $\phi$, $\Omega_{Z}$ =0.00019 $h^{-2}_{50}$. (9)
These values are not unreasonable. They require that twice as much star formation as has been inferred by Madau et al (1996) from the uv integrated light has taken place shrouded by dust. The total fraction of baryonic matter that has participated in star formation would be of order 20 $\%$, with about 1/3rd of the resulting heavy elements now residing in the luminous parts of galaxies. The remainder could be in baryonic objects in the halos of galaxies or in intergalactic gas (including the hot X-ray emitting gas in clusters). In fact evolutionary rates appreciably steeper than that assumed in eqn (8) can probably not be ruled out at this stage. If the star-forming galaxies we have detected with ISO are typical of the fainter HDF galaxies, then we may require that more than 50 $\%$ of baryons have participated in star-formation and heavy element production, presumably with a truncated IMF so that most of the baryons now reside in dark remnants. Similar conclusions are reached if we use the model for evolution of infrared galaxies of Franceschini et al (1994, 1997 in preparation), shown as a broken line in Fig 2.
We have also estimated the contribution to the 60 $\mu$m luminosity-density implied directly by the ISO-HDF starburst galaxies. There are 5 starburst galaxies in the redshift bin 0.4-0.7 and 3 in the redshift bin 0.7-1.0 (omitted the galaxy with broad lines). Estimating the volume of the universe sampled by the HDF survey we find contributions of 6.0 $\pm$2.0x$10^{8}$ and 2.6 $\pm$1.5x$10^{9} L_{o} Mpc^{-3}$ for the redshift ranges 0.4-0.7 and 0.7-1.0 respectively. These estimates take no account of sources fainter than the ISO limit and they are subject to any uncertainty in the ISO calibration (probably a factor of 50either way), as well as the considerable uncertainty associated with extrapolating from 6.7 and 15 $\mu$m to 60 $\mu$m, so must be seen as very preliminary. They appear to be significantly higher than the predictions for evolution of the form (8) (the solid line in Fig 3) by factors of 5 and 10 respectively. This may imply that the evolution of starburst galaxies is steeper than assumed in eqn (8) for 0 $<$ z $<$ 1. Alternatively our models may overestimate the 60 $\mu$m luminosities for at least some of the galaxies, for example because the 6.7 and 15 $\mu$m radiation comes from dust tori around AGN.
We have also shown in Fig 2 one of the more extreme of the models of Pei and Fall (1995, dotted line). This illustrates that the luminosity density estimated from the ISO-HDF galaxies is not at odds with current data on the number-density of quasar absorption-line clouds or the observed heavy element abundance at high redshift. However for this model $\Omega_*=0.032 h^{-1}_{50}\phi$, $\Omega_{\rm
Z}=0.00076 h^{-1}_{50}$, which would probably imply that only higher mass stars were being formed in the ISO-HDF galaxies.
Support for the idea that stronger evolution than (8) is required for the starburst galaxy population comes from the fit to the ISO counts by Oliver et al (1997). The Pearson and Rowan-Robinson (1996) model fails to predict a strong enough contribution to the counts by starburst galaxies. The Franceschini et al (1994) model, involving a strongly evolving starburst population in elliptical galaxies, appears to give a better fit to the ISO counts. This idea can be tested by the deep 90 $\mu$m surveys which we and others are carrying out with ISO. It will also be interesting to observe the HDF galaxies detected by ISO at submillimetre wavelengths, for example with SCUBA on the JCMT.
Conclusions
===========
We have modelled the spectral energy distributions of the 13 galaxies reliably associated with ISO sources detected at 6.7 and$/$or 15 $\mu$m. For 2 galaxies the emission detected by ISO is consistent with being starlight or normal infrared ’cirrus’ in the galaxies. For the remaining 11 galaxies there is a strong mid-infrared excess, which we interpret as emission from dust associated with a strong starburst. In 3 cases the starburst model appears to be confirmed by the good agreement of the the predicted radio flux with that detected by Fomalont et al (1996). Inferred rest-frame luminosities ($\nu L_{\nu}$) at 0.3, 0.8, 15 and 60 $\mu$m are given and $L_{60}/L_{0.3}$ ranges from 3 to 1000 for the 11 galaxies. Thus most of the the bolometric luminosity of the galaxies is predicted to emerge at far infrared wavelengths.
We give a new discussion of how the star formation rate can be deduced from the far infrared luminosity and derive star formation rates of 8-1000 $\phi M_{o}$ per yr, where $\phi$ takes account of the uncertainty in the initial mass function (=1 for Salpeter IMF). The HDF galaxies detected by ISO are clearly forming stars at a prodigous rate compared with nearby normal galaxies. We discuss the implications of our detections, and of the IRAS 60 $\mu$m luminosity function and evolution, for the history of star and heavy element formation in the universe. We conclude that at least 20 $\%$ of baryons must have participated in star formation.
Acknowledgements {#acknowledgements .unnumbered}
=================
This paper is based on observations with ISO, an ESA project, with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands and the United Kingdom) and with the participation of ISAS and NASA. We thank the referee, Harry Fergurson, for comments and suggestions which enabled us to improve this paper. This work was supported by PPARC (Grant no. GR/K97828) and by the EC TMR Network Programme (Contract no. FMRX-CT96-0068).
[99]{}
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=0.33333in
> TAUP 2686-2001
0.5in
[**Remarks on the Physical Meaning of\
Diffraction-Free Solutions of Maxwell Equations**]{}
E. Comay
School of Physics and Astronomy\
Raymond and Beverly Sackler Faculty of Exact Sciences\
Tel Aviv University\
Tel Aviv 69978\
Israel
0.5in 0.5in PACS No: 03.50.De, 41.20.Jb 0.2in Abstract:
It is proved that a source of electromagnetic radiation cannot emit a diffraction-free beam at the wave zone. A Bessel $J_0$ $\varphi $-invariant beam does not hold even at the intermediate zone. These results negate claims published recently in the literature.
An idea of creating a diffraction-free beam has been published\[1\]. The beam’s amplitude is cylindrically symmetric ($\varphi $-invariant) where the $r$-dependence is proportional to the Bessel function of the first kind $J_0(ar)$ and $a$ is a constant having the dimension of $L^{-1}$. Reference \[1\] has arisen a great interest in utilization of $J_0$ beams and has been cited more than 360 times\[2\]. An application of \[1\] shows the central peak of the assumed $J_0$ beam\[3\] and another one refers to its peculiar $z$-component wavelength\[4\]. Another publication related to \[1\] claims that a superluminal propagation of light in air has been detected\[5\]. Objections to \[5\] have been published\[6\]. The purpose of this work is to show that one [*cannot*]{} construct a diffraction-free electromagnetic beam at the wave zone and that the Bessel function $J_0(ar)$ is unsuitable for describing diffraction free $\varphi $-invariant wave at the intermediate zone too. This outcome proves that results of papers discussing this topic, in general, and those ascribing superluminal velocity to beams that take the form of Bessel function $J_0$, in particular, should be reevaluated. Units where the speed of light $c=1$ are used. The metric $g_{\alpha \beta}$ is diagonal and its entries are (1,-1,-1,-1). ${\bf u}_r $, ${\bf u}_\varphi $ and ${\bf u}_z$ denote unit vectors in cylindrical coordinates and ${\bf u}_x$, ${\bf u}_y$ and ${\bf u}_z$ are unit vectors in Cartesian coordinates.
A general analysis of diffraction-free solutions of Maxwell equations has been published\[7\]. Here the fields solving the problem are derived from a vector potential $\bf {A}$ that satisfies the wave equation together with the Lorentz-gauge requirement\[8\]. It turns out that this work is relevant to \[1\] and some of its results are analyzed here in detail. (Another work\[9\] is closely related to \[1\] and \[7\].) Let us start with the solution obtained in Example 1 (on p. 1557 of \[7\]). Using cylindrical coordinates and removing constant factors, the time dependent monochromatic electric field of this solution is obtained from the vector potential ${\bf E} = -\partial {\bf A}/\partial t$ $${\bf E} = \omega J_1(ar)e^{i(bz-\omega t)} {\bf u}_\varphi
\label{eq:E1}$$ where $J_1$ is the Bessel function of the first kind of order 1. The magnetic field is ${\bf B}= curl {\bf A}$ $${\bf B} = -b J_1(ar)e^{i(bz-\omega t)} {\bf u}_r -
ia J_0(ar)e^{i(bz-\omega t)} {\bf u}_z .
\label{eq:B1}$$
Ignoring constant factors, one finds that the magnetic field $(\!\!~\ref{eq:B1})$ is dual to the electric field of Example 2 of \[7\]. (The factor 2 in $U_r$ of example 2 is a misprint.) This outcome indicates that Examples 1 and 2 of \[7\] represent dual electromagnetic solutions where ${\bf E \rightarrow B,\;\; B \rightarrow -E}$ (see \[8\], p. 252).
Having the solution, let us examine the problem of a cylindrically shaped wave guide whose walls are made of a perfect conductor (see \[8\], p. 335). The length of the cylinder is much greater than both its diameter $2R$ and the wavelength $\lambda = 1/\omega $ (see fig. 1). The boundary conditions along the wave guide’s walls are (see \[8\], p. 335) $$E_\parallel = 0,\;\; B_\perp = 0.
\label{eq:BOUNDARY}$$ Thus, the solution $(\!\!~\ref{eq:E1})$ and $(\!\!~\ref{eq:B1})$ satisfies the boundary conditions provided $$J_1(aR) = 0.
\label{eq:BOUNDARY2}$$
Dynamical properties of the solution $(\!\!~\ref{eq:E1})$ and $(\!\!~\ref{eq:B1})$ are obtained from the energy-momentum tensor of the electromagnetic fields (see \[10\], p. 81 or \[8\], p. 605)) $$T_{F}^{\mu \nu } =
\frac {1}{4\pi }(F^{\mu \alpha }F^{\beta \nu }g_{\alpha \beta }
+\frac {1}{4}F^{\alpha \beta }F_{\alpha \beta }g^{\mu \nu })
\label{eq:TF}$$ where $F^{\mu \nu}$ denotes the tensor of the electromagnetic fields. Expression $(\!\!~\ref{eq:TF})$ is quadratic in the fields. Hence, one should use the real part of $(\!\!~\ref{eq:E1})$ and $(\!\!~\ref{eq:B1})$ in an evaluation of quantities belonging to it. Let us first examine the momentum density of the fields. This is the Poynting vector $${\bf S} = \frac {1}{4\pi }{\bf E} \times {\bf B}.
\label{eq:POYNTING}$$
The $z$-component of the momentum density and energy flux are obtained from the substitution of the appropriate real part of $(\!\!~\ref{eq:E1})$ and $(\!\!~\ref{eq:B1})$ $$S_z = \frac {b\omega }{4\pi }J_1^2(ar)cos^2(bz-\omega t).
\label{eq:PZ}$$ Expression $(\!\!~\ref{eq:PZ})$ is non-negative at all points, a property which is consistent with the beam’s expected flux of energy that travels away from a localized source.
The radial component of the momentum density is obtained analogously $$S_r = -\frac {a\omega }{8\pi }J_1(ar) J_0(ar) sin[2(kz-\omega t)].
\label{eq:PR}$$ Here one sees that, unlike the case of $(\!\!~\ref{eq:PZ})$, the sign of $(\!\!~\ref{eq:PR})$ alternates periodically in the time and $z$-coordinates. Moreover, for any fixed value of $t$ and $z$, it changes sign along the $r$-axis, because zeroes of the Bessel functions $J_0$ and $J_1$ do not coincide\[11\]. It follows that although the radial motion does not vanish locally, its mean value is null. This property indicates that the radial motion takes the type of a standing wave.
Now let us examine the interaction of the fields with the walls of the wave guide. Point $P$ at $x=R,\;y=z=0$ is used as a representation of the general case and cartesian coordinates are used. The $x$-component of the momentum current at $P$ is (see \[10\], p. 82 or \[8\], p. 605)) $$T_{xx} = \frac {1}{8\pi }
(E_y^2 + E_z^2 - E_x^2 + B_y^2 + B_z^2 - B_x^2).
\label{eq:TXX1}$$ Examining the fields $(\!\!~\ref{eq:E1})$ and $(\!\!~\ref{eq:B1})$ and the boundary value $(\!\!~\ref{eq:BOUNDARY2})$, one finds that only the $z$-component of the magnetic field makes a nonvanishing contribution. Thus, the momentum current at $P$ is $$T_{xx} = \frac {a^2}{8\pi }J_0^2(aR)sin^2(bz-\omega t).
\label{eq:TXX2}$$ This momentum current is absorbed by the walls, because the fields vanish in all space outside the inner part of the wave guide.
Another effect of the magnetic field $(\!\!~\ref{eq:B1})$ on the wave guide’s walls is the electric current induced in the $\varphi $-direction. Indeed, let us evaluate the line integral along the infinitesimal rectangular closed path of fig. 1. Using vector analysis, Maxwell equations and the boundary condition $(\!\!~\ref{eq:BOUNDARY2})$, one finds $$\oint {\bf B\cdot }d{\bf l} = \int curl {\bf B\cdot }d{\bf s} =
\int 4\pi {\bf j\cdot }d{\bf s}.
\label{eq:4PIJ}$$ Thus, a nonzero current ${\bf j}$ is induced on the walls, because only $B_z$ at the inner part of the wave guide makes a nonvanishing contribution to the line integral. This outcome proves that a time-dependent (and $z$-dependent) electric current flows along the $\varphi $-direction of the wave guide’s walls and that fields of this current are part of the solution $(\!\!~\ref{eq:E1})$ and $(\!\!~\ref{eq:B1})$. This electric current sustains the $B_z$ related standing wave in the radial direction. The walls also counteract against local electromagnetic pressure.
The dual solution of example 2 of \[7\] behaves analogously. Using the same global factor of $(\!\!~\ref{eq:E1})$ and $(\!\!~\ref{eq:B1})$, one finds for this case $${\bf B} = \omega J_1(ar)e^{i(bz-\omega t)} {\bf u}_\varphi
\label{eq:B2}$$ $${\bf E} = b J_1(ar)e^{i(bz-\omega t)} {\bf u}_r +
ia J_0(ar)e^{i(bz-\omega t)} {\bf u}_z .
\label{eq:E2}$$ Hence, the boundary conditions $(\!\!~\ref{eq:BOUNDARY})$ yield $$J_0(aR) = 0.
\label{eq:BOUNDARY3}$$ Since $J_0(ar)$ and $J_1(ar)$ have no common root\[11\], a nonvanishing radial electric field exists at the wave guide’s walls. It follows from Maxwell equation $div {\bf E} = 4\pi \rho $ that a time dependent and $z$-dependent charge density is built on the inner part of the wave guide’s walls. Thus, we have also in Example 2 a current that flows on the walls and affects the fields inside the wave guide.
Let us examine an analogous experimental setup. Here the source of the radiation at $z=-L$ is the same as that of the first experiment but the wave guide is removed. This situation is different from the wave guide case. Indeed, the fields of a closed electromagnetic system depend on charges and currents at the retarded space-time points (see \[10\], pp. 158-160 or \[8\], p. 225). Therefore, the wave guide’s solutions clearly do not hold for this case because here the current along the wave guide walls is missing.
Since in the second experiment the region at $z=0$ satisfies the wave zone requirements (see \[10\] p. 170 or \[8\], p. 392) $$L\gg \lambda ,\;\; L\gg 2R,
\label{eq:WAVEZONE}$$ one can use the wave zone solution. Let $\bf A$ denote the retarded vector potential at the wave zone. Thus, one finds the fields (see \[10\] p. 171) $${\bf B} = {\bf \dot{A}\times n},
\label{eq:BWZ}$$ $${\bf E} = ({\bf \dot{A}\times n}){\bf \times n}
\label{eq:EWZ}$$ where ${\bf n}$ is a unit vector in the radial direction.
It turns out that the solution for the free space experiment is inherently different from the one which fits the wave guide’s inner space. In particular, in the case of free space, fields at the wave zone are perpendicular to the radius vector from the source to the field point. On the other hand, the wave guide solution contains a $z$-component ($B_z$ or $E_z$) which is an inherent part of the solution. As shown above, the $B_z$ (or $E_z$) field is associated with the electric current induced on the wave guide’s walls. This conclusion obviously holds for any pattern of source elements put at the same spatial region as the one used here, because the analysis does not refer to the source’s details. Thus, the results disagree with the claim of \[9\].
One can use general arguments for proving that a diffraction-free electromagnetic beam that has a nonvanishing $z$-component for at least one of the fields, contains transverse standing wave. Indeed, the beam carries energy and therefore ${\bf S}$ of $(\!\!~\ref{eq:POYNTING})$ does not vanish. Hence, ${\bf E}$ is not parallel to ${\bf B}$ and, due to the $z$-component of the fields, ${\bf S}$ has a nonvanishing transverse component. Now, the diffraction-free property of the beam prevents energy from flowing transversally. Hence, the transverse component of ${\bf S}$ is a standing wave.
It can also be proved that all solutions of \[7\] have a nonvanishing $z$-component of at least one of the fields. Indeed, the vector potential $\bf A$ takes the form (see p. 1556 therein) $${\bf A} = \sum _{n} (\alpha _n {\bf M}_n + \beta _n {\bf N}_n),
\label{eq:A}$$ where $\alpha _n$ and $\beta _n$ are numerical coefficients of the expansion. Here $${\bf M}_n = curl [J_n(ar)e^{i(bz + n\varphi - \omega t)}{\bf u_z}]
\label{eq:MN}$$ and $${\bf N}_n = \frac {1}{k}curl {\bf M}_n
\label{eq:NN}$$ where $k$ is the wave number. Now ${\bf N}_n$ contains a $z$-component (see p. 1557 therein). Hence, if $\beta _n \neq 0$ then ${\bf E} = -\partial {\bf A}/\partial t = i\omega {\bf A}$ has a $z$-component too. In other cases all $\beta _n = 0$, which mean that for at least one $n$, $\alpha _n \neq 0$. Here the magnetic field ${\bf B} = curl {\bf A} = \alpha _n curl{\bf M} = k\alpha _n {\bf N}$, which means that $B_z \neq 0$ and the proof is completed.
It follows that the family of solutions of \[7\] involves standing waves associated with the $z$-components of the solutions. This diffraction-free family of solutions may fit cylindrical wave guides but are unsuitable for the case of a free space.
Example 4 of \[7\] (see p. 1558) is the last one which is analyzed here in detail. This example contains one component which is proportional to $J_0(ar)$ and is $\varphi $-invariant. Although it has a $\varphi $-dependent $z$-component term which is associated with a standing wave, it looks simpler to show another problem of this solution. The vector potential of this example is given in Cartesian coordinates $${\bf A} = -i\alpha [aJ_0(ar)\,{\bf u}_x -
i\frac {a^2}{b}J_1(ar)cos \varphi \,{\bf u}_z]e^{i(bz - \omega t)}.
\label{eq:EX4A}$$ Using ${\bf E} = -\partial {\bf A}/\partial t$, one finds $${\bf E} = \alpha \omega [aJ_0(ar)\,{\bf u}_x -
i\frac {a^2}{b}J_1(ar)cos \varphi \,{\bf u}_z]e^{i(bz - \omega t)}.
\label{eq:EX4E}$$ Let us examine the $z$-component of the Poynting vector which represents energy current flowing along the beam’s direction, namely, the quantity which is analogous to $(\!\!~\ref{eq:PZ})$ of Example 1. Examining $(\!\!~\ref{eq:EX4E})$, one finds that only $B_y$ is needed for this purpose. Thus, $(curl\,{\bf A})_y$ of $(\!\!~\ref{eq:EX4A})$ is $$B_y = \alpha [(ab - \frac {a^3}{2b})J_0(ar) +
\frac {a^3}{2b}cos 2\varphi \,J_2(ar)]
e^{i(bz - \omega t)}.
\label{eq:EX4BY}$$ Hence, the required $z$-component of the Poynting vector is obtained as the product of the real parts of $E_x$ of $(\!\!~\ref{eq:EX4E})$ and $B_y$ of $(\!\!~\ref{eq:EX4BY})$ $$S_z = \alpha ^2 \omega [(a^2 b - \frac {a^4}{2b})J_0^2(ar) +
\frac {a^4}{2b}cos 2\varphi \,J_0(ar)J_2(ar)]
cos^2(bz - \omega t).
\label{eq:EX4SZ}$$
Let us examine the $z$-component of the energy current near a point whose radial coordinate is $\bar {R}$ and $J_0(a\bar{R})=0$. In this neighbourhood $J_2$ is dominant\[12\] and the contribution of the $J_0^2(ar)$ term of $(\!\!~\ref{eq:EX4SZ})$ can be ignored. The rest of $(\!\!~\ref{eq:EX4SZ})$ is proportional to $J_0(ar)J_2(ar)cos\,2\varphi $. Now, let us examine the value of $S_z$ on a circle whose radius is $\bar {R} + \varepsilon $, where $\varepsilon $ is an appropriate small quantity. Due to the factor $cos\,2\varphi $, one realizes that $S_z$ takes different signs on this circle. Hence, in the solution of Example 4 of \[7\], energy flows in opposite $z$-directions in certain regions of space. This property of Example 4 is inconsistent with the notion of a beam, where electromagnetic energy flows [*away*]{} from a localized source.
It is clear from the analysis carried out above that, in free space, one cannot build a diffraction free [*beam*]{} from the family of Bessel functions of \[7\], because these functions are unsuitable at the wave zone.
Some conclusions can be drawn for the intermediate zone too. The diffraction free $\varphi $-invariant $J_0(ar)$ function proposed in \[1\] does [*not*]{} belong to the solutions of \[7\]. Indeed, in \[7\], there are only two truly $\varphi $-invariant solutions. They are the dual solutions of Examples 1 and 2 which are discussed above. As proved in this work, the $z$-component of the energy current is proportional to $J_1^2(ar)$. Hence, the flow of energy [*vanishes along the $z$-axis*]{}. It is also proved above that Example 4 of \[7\], where there is one $J_0$ term which is $\varphi $-invariant, does not describe a beam of electromagnetic radiation and its $z$-component is not $\varphi $-invariant. It follows that experiments using a $\varphi$-invariant setup and showing a strong peak at the center (like \[1,3,4\]) should not be interpreted by means of diffraction free solutions.
References:
- Email: [email protected]
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- Due to \[11\], the roots $(r > 0)$ of $J_n(r)$ and $J_{n + 1}(r)$ are simple, do not coincide and interlace. Hence, the recurrence formula $2J_1(r)/r=J_0(r)+J_2(r)$ proves that positive roots of $J_0(r)$ and $J_2(r)$ do not coincide.
Figure captions:\
Fig. 1:\
Electromagnetic radiation is emitted from a source into a cylindrical wave guide whose radius is $R$. The source is at $z=-L$ and $L\gg 2R$. $O$ denotes the origin of coordinates and the rectangle at point $P$ denotes a closed integration path (see text).
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Tyler Crain
- Christopher Natoli
- Vincent Gramoli
bibliography:
- 'reference.bib'
title: Evaluating the Red Belly Blockchain
---
Abstract {#abstract .unnumbered}
--------
In this paper, we present the most extensive evaluation of blockchain system to date. To achieve scalability across servers in more than 10 countries located on 4 different continents, we drastically revisited Byzantine fault tolerant blockchains and verification of signatures. The resulting blockchain, called the *Red Belly Blockchain (RBBC)*, commits more than a hundred thousand transactions issued by permissionless nodes. These transactions are grouped into blocks within few seconds through a partially synchronous consensus run by permissioned nodes. It prevents double spending by guaranteeing that a unique block is decided at any given index of the chain in a deterministic way by all participants.
We compared the performance of RBBC against traditional Byzantine fault tolerant alternatives and more recent randomized solutions. In the same geo-distributed environment with low-end machines, we noticed two interesting comparisons: *(i)* the RBBC throughput scales to hundreds of machines whereas the classic 3-step leader-based BFT state machine used by consortium blockchains cannot scale to 40 identically configured nodes; *(ii)* RBBC guarantees transaction finality in 3 seconds and experiences a third of the latency that randomized-based solutions like HoneyBadgerBFT can offer. This empirical evaluation demonstrates that blockchain scalability can be achieved without sacrificing security.
Introduction
============
Blockchain systems [@Nak08] aim at implementing a Byzantine fault tolerant replicated state machine (RSM) by totally ordering blocks or sets of transactions that are issued by requesters. Various replicated state machines have been proposed over the last decades to coordinate servers, but they typically apply to a small set of replicas. By contrast, blockchains aim at offering a peer-to-peer model where many geodistributed participants replicate the information and where many requesters can check their balance and issue cryptographically signed transactions. Permissioned (resp. permissionless) blockchains allow a pre-determined set of nodes (resp. all nodes) to be the deciders of new transaction blocks. The limitations of existing blockchains, be they permissioned or permissionless, are their performance: the verification of all transactions is computationally intensive while reaching consensus is communication intensive.
In this paper, we evaluate a fast blockchain called the *Red Belly Blockchain* (RBBC)[^1] that is deterministic and does not assume synchrony. RBBC offers a new sharding method that assigns, for each group of transactions, distinct groups of *proposer* and *verifier* nodes. *(i)* The sharding of proposers balances the communication load on multiple nodes, hence avoiding the congestion and slowdown induced by the least responsive node. As opposed to practical Byzantine consensus protocols that traditionally rely on a leader to propose a set of transactions, RBBC’s multiple proposers combine distinct sets of transactions into a super block to commit more transactions per consensus instance. *(ii)* The sharding of verifiers balances the computation load on different verifiers. As opposed to existing blockchains whose nodes typically verify the same transactions, each of our transaction signature is verified by at least $t+1$ and at most $2t+1$ *verifiers* (where $t$ is the maximum number of faulty nodes).
We conducted the most extensive evaluation of blockchain to date by evaluating *(i)* the peak throughput with the large bandwidth offered by hundreds of high end machines in a single data center, *(ii)* as compared to classic and randomized Byzantine tolerant blockchains, *(iii)* when attacked by a coalition of maliciously behaving machines and *(iv)* when deployed on low-end machines over 4 continents around the world. While well-documented blockchain experiments already involved a large number of virtual machines [@LNZ16; @GHM17], they typically spawn consensus participants in the same country, if not the same datacenter to make sure that message delays remain as low as possible.
Other blockchains even require that all message delays are lower than a bound imposed by the algorithm, an assumption called *synchrony* [@Nak08; @Woo15; @GHM17], which can be difficult to achieve in practice and may be exploited to double spend [@NG17; @EGJ18]. Some blockchain components that avoid the synchrony assumption cannot verify transaction signatures [@SBV17], allowing someone to withdraw from someone else’s account. Randomized alternatives that terminate with high probability may require additional messages to implement a common coin [@MSC16] that all participants can use. Until now, purely Byzantine fault tolerant blockchains have been notoriously unscalable [@Vuk15], often reaching their peak throughput with 4 nodes [@Buc16; @SBV17], hence tolerating at most one failure to maximize performance. This even led companies to recently trade security for crash fault tolerance [@ABB18].
For the sake of security, RBBC features the non Turing complete scripting language and the unspent transaction output (UTXO) model from Bitcoin [@Nak08]. Each transaction is cryptographically signed and verified using Elliptic Curve Digital Signature Algorithm (ECDSA) keys. In contrast with Bitcoin, transactions are only verified by sufficiently many verifiers to cope with $t$ Byzantine nodes. The underlying consensus protocol run by $n$ nodes is especially designed to run in a blockchain over the Internet [@CGLR18]. It is time optimal, resilience optimal in that it tolerates $t<\frac{n}{3}$ Byzantine nodes, and runs in a partially synchronous network. To evaluate the performance of RBBC, we deployed it on up to 1000 nodes on up to 14 datacenters in 4 different continents. For most workloads, we tested the performance of RBBC across continents and observed tens of thousands of transactions committed per second. The performance of our system peaks at 660,000 transactions per second, when run on 260 machines in a single datacenter with a low fault tolerance parameter $t$.
As cryptography is necessary but not sufficient to guarantee the security of a blockchain system, we evaluate the robustness of RBBC by implementing Byzantine attacks and assessing empirically the behavior of RBBC. In a distributed system, the misbehavior of some machines, that could be due to a simple misconfiguration, may affect the result of the entire computation, like the consensus decision. To observe the robustness of RBBC, we implemented alternative Red Belly Blockchain programs flipping the bits that they should send to slow down the consensus execution and sending wrong information to delay the broadcast of the information among correct nodes. We deployed these intentionally misbehaving codes and observed the Byzantine fault tolerance of RBBC and the impact on performance these misbehaviors could cause.
Finally, we compared the RBBC against a classic leader-based BFT [@CL02; @BSA14; @Kwo14] protocol and HoneyBadgerBFT [@MSC16]. The results indicate a latency among the lowest but significantly higher throughput. More specifically, in the same settings the throughput of RBBC peaks at hundreds of machines where the traditional Byzantine fault tolerant solutions would not scale to 40 machines and the latency of RBBC is a third of the HoneyBadgerBFT randomized alternative.
We start with the background on other blockchain systems (\[sec:rw\]) and present the design decision of the Red Belly Blockchain (\[sec:design\]). We then describe the settings of our experiments (\[sec:evaluation\]). We compare the Red Belly Blockchain to both leader-based and randomized blockchains when deployed on geo-distributed machines and illustrate the performance gained from its transaction signature verification and its proposal combination to decide large blocks (\[sec:geo1\]). We then evaluate the security of the Red Belly Blockchain by running Byzantine attacks and observing its resilience and how it maintains performance (\[sec:security\]). Finally, we present the performance on a thousand of virtual machines (\[ssec:thousand-vm\]) and we conclude (\[sec:conclusion\]).
Background {#sec:rw}
==========
Byzantine Fault Tolerance (BFT) requires a number of messages per consensus instance that grows quadratically with the number of nodes. Most previous work on Byzantine-fault tolerant blockchains [@Kwo14; @BSA14; @SBV17] would solve the traditional Byzantine consensus problem [@LSP82], deciding only one proposed value, regardless of the number of participants [@CL02; @BSA14; @MA06; @ABQ13; @AGK14]. Due to this scalability limitation [@Vuc15; @Buc16; @SBV17], we see various blockchain proposals reverting to a crash tolerant model to tolerate more but simpler failures, like Hyperledger Fabric [@ABB18]. Not tolerating Byzantine failures confines these blockchains to secure networks that are protected from intrusions by other means.
For the sake of scalability, one can instead solve a variant of the consensus problem that allows to combine proposals into a decision [@BKR94; @NCV05; @CGLR18]. Extra care is however needed before one can apply solutions to these problems to decide a “super” block combining multiple proposed blocks. For example, the related problems of Agreement on a Core Set or Asynchronous Common Subset (ACS) [@BKR94], Interactive Consistency (IC) [@LSP82] and Vector Consensus (VC) [@NCV05] require either $n-t$ or $t+1$ (at least $t+1$ as $n>3t$) proposed values to be decided. In blockchain, however, there may not even be $t+1$ compatible proposed blocks. This incompatibility arises when at least one transaction per block is not correctly signed or transactions of two distinct blocks conflict. Accepting to commit these invalid transactions could limit fairness and introduce starvation, yet at proposal time even correct proposers cannot anticipate these conflicts. Instead of trying to decide a minimal number of proposed blocks, RBBC propose sets of transactions and decides on a new block whose transactions are the union of correctly signed and non-conflicting proposed transactions.
Recent blockchains try to avoid the synchrony assumptions [@Kwo14; @MSC16; @SBV17; @ABB18]. The HoneyBadger Byzantine Fault Tolerance (HBBFT) [@MSC16] aims at solving ASC by building upon an asynchronous binary Byzantine consensus algorithm [@MMR14] that is probabilistic and assumes a fair scheduler [@MMR15]. As we show in \[sec:geo1\], even with a fair scheduler HBBFT is too costly for RBBC to build upon it because it requires a binary consensus [@MMR14] that requires a common coin. Some alternatives relax this fair scheduler assumption but they require more messages, which risks to increase the overhead [@MMR15]. To avoid both randomization and synchrony, various solutions [@CL02; @KADC07; @CMS07; @VCB09; @MJM09; @AGK14; @BSA14] assume partial synchrony [@DLS88]. Unfortunately, they all rely on a leader or a primary to propose to others and follow a three-step execution pattern that offers low latency but whose throughput cannot scale. The lack of scalability of this pattern was experimented in \[sec:evaluation\].
Recent blockchains, like Tendermint [@Kwo14] achieve security by building upon these BFT protocols. As they all inherit the same execution pattern, they all decide only one of the proposed set of transactions and do not scale to tens of nodes [@Vuc15; @Buc16; @SBV17]. A Byzantine fault tolerant ordering service was tested for Hyperledger Fabric, however, its best performance was achieved with $n=4$ nodes [@SBV17]. Hyperledger Fabric finally reverted to a crash fault tolerance version [@ABB18] to try to scale to a larger network. Despite the lack of Byzantine fault tolerance, its consensus was only evaluated across two data centres [@ABB18]. By contrast, RBBC features a Democratic BFT (DBFT) that extends a Byzantine consensus algorithm [@CGLR18] by creating a super block resulting from multiple proposed sets of transactions. The reason for choosing this consensus algorithm is that it does not follow the common leader-based pattern that has costly recoveries in the case of faulty or slow leaders [@CWA09; @BS09; @ABQ13].
Despite these differences, RBBC combines many of the optimizations proposed in the aforementioned BFT literature. Its leaderless DBFT algorithm stems from a provably correct algorithm [@CGLR18], its concurrent implementation leverages multicores similar to BFT-Smart [@BSA14]. It generalises the $n$ proposers of HoneyBadger BFT to any number [@MSC16]. In addition, RBBC could potentially benefit from other BFT optimizations. Some proposals [@SBV17] allow temporary inconsistencies and transaction rollback. Other proposals rely on trusted components [@VCB10; @MJM09; @BDK17]. In particular, RAM [@MJM09] suggests to use the Attested Append-Only Memory (A2M) [@CMS07] trusted service to scale, however, we are not aware of any implementation. Steward [@ADD06] organizes consensus into a hierarchy to scale to wide area networks and offers tens of updates per second on an emulated wide area network. RBBC extends this related work through the implementation of a replicated state machine tailored for blockchain.
The Design of the Red Belly Blockchain {#sec:design}
======================================
The Red Belly Blockchain was initially presented at MIT in July 2017 where a first version of the system could achieve 440,000 transactions per second on 100 machines. The performance was optimized and shown to achieve 660,000 transactions per second on 300 machines at Facebook and Visa Research in October 2017 as we explain in this paper. The details of these presentations are available online [@Gra17].
In short, the Red Belly Blockchain is a community blockchain [@VG18] with a dynamic set of consensus participants or proposers whose public keys are listed in a configuration block. These proposers receive from permissionless clients some balance, subscription and transaction requests. Proposers can answer balance requests based on the information they have about the current state of the blockchain and they keep a list of subcribers to send them updates about the balance of all accounts. Both proposers and subscribers are called *replicas* as they maintain a copy of the state of the blockchain either as the full blockchain or as a UTXO table. Proposers store transactions in a memory pool or *mempool* before proposing them to some consensus instance. Once a client receives an identical balance response from $t+1$ proposers, it knows the balance of its account. Once the consensus decides upon a combination of the proposed transactions that are correctly signed and not in conflict, this combination is wrapped into a block appended to the chain.
Optimized Democratic BFT {#ssec:dbft}
------------------------
The unprecedented performance of the Red Belly Blockchain is mainly due to a novel design that relies on a Byzantine consensus algorithm especially designed for blockchains and called *Democratic Byzantine Fault Tolerance (DBFT)* [@CGLR18] that does not assume synchrony. Most Byzantine consensus algorithms predate the blockchain era and were not designed to scale to a very large number of machines as it is needed in blockchains.
For this reason, there are few distinctions between DBFT and classic Byzantine fault tolerant algorithms.
First, DBFT does not rely on a leader to avoid any bottleneck effect at large scale. Instead it allows multiple proposers to propose disjoint sets of transactions that could all be inserted in the block decided at the end of the consensus.
Second, DBFT solves the consensus deterministically. Hence the Red Belly Blockchain never *forks*, a situation where multiple blocks are appended at the same index of the chain and that could be exploited by attackers to double-spend [@NG17]. In particular, it does neither require a common coin nor a fair scheduler. The interested reader can access the detailed proofs in the technical report [@CGLR17].
To reduce the bandwidth usage of the reliable broadcast of DBFT [@CGLR18], we included a SHA256 hash digest of the message instead of including the full proposal in the $\lit{echo}$ and $\lit{ready}$ messages. To increase throughput we grouped all valid and non-conflicting transactions obtained at the end of the consensus to create a new block.
Sharded Verification
--------------------
The Red Belly Blockchain also offers other advantages as it shards the verification of transaction signatures. Traditional blockchains either require all active participants to verify the signature of each individual transaction or assume the presence of trusted verifiers or endorsers. The Red Belly Blockchain leverages the computational resources of the participants by spreading the load of verifying transactions to different subsets of participants but without requiring trust. It only requires each transaction to be verified by at least $t+1$ participants but never more than $2t+1$ participants.
The Red Belly Blockchain is a full-fledge blockchain that supports UTXO transactions signed through Elliptic Curve Digital Signature Algorithm (ECDSA) and verified at run-time. All communications are encrypted through SSL, which does not impact performance significantly. It offers a model of open permissioned blockchain called *community blockchain* in that it relies on a dynamic set of participants whose public key are well identified to run the consensus but allows permissionless clients to issue transaction and balance requests. More details on how this community blockchain bypasses the predetermined set of participants requirement of consortium blockchain can be found in [@VG18].
The optimized deterministic leader-less DBFT consensus designed for blockchain and the sharded verification allows Red Belly Blockchain to be a secure blockchain that does not fork and whose performance scales with the amount of computational resources coming with hundreds of participants.
Experimental Settings {#sec:evaluation}
=====================
In this section, we evaluate RBBC on up to 1000 machines on Amazon EC2 located in up to 14 separate regions. To this end, we compare the performance of (1) RBBC with its sharding and its DBFT consensus. (2) RBBC where we replaced DBFT by the Honey Badger of BFT protocol (HBBFT) [@MSC16], for which we reused the publicly available cryptographic operations implementation and (3) RBBC where we replaced DBFT by a classic 3-step leader-based BFT algorithm *CONS1* [@CL02; @Kwo14; @BSA14].
We run three types of experiments: *(i)* with up to 300 deciders all deciding and generating the workload, allowing new proposals to be made as soon as the previous one is committed (\[sec:geo1\] and \[sec:datacenter\]); *(ii)* with requesters running on nodes separated from the permissioned nodes to measure their impact on performance and finally (\[ssec:remote-req\]); *(iii)* with up to 1000 nodes all runnings as replicas, some requesting and some deciding, but all updating their copy of all account balances (\[ssec:thousand-vm\]).
Leader-based and randomized BFT
-------------------------------
CONS1 is the classic 3-step leader-based Byzantine consensus implementation similar to PBFT [@CL02], the Tendermint consensus [@Kwo14], and including the concurrency optimizations of BFT-Smart [@BSA14]. To reduce network consumption CONS1 is implemented using digests in messages that follow the initial broadcast.
The HoneyBadger Byzantine Fault Tolerance (HBBFT) [@MSC16] aims at solving the ASC problem [@BKR94] by building upon an asynchronous binary Byzantine consensus algorithm [@MMR14] that is probabilistic and assumes a fair scheduler [@MMR15]. To evaluate HBBFT we used the source code provided by the authors of HBBFT.
Both CONS1 and HBBFT variants make use of a classic verification, as in traditional blockchain systems [@Nak08; @Woo15], that takes place at every decider upon delivery of the decided block from consensus. Unless otherwise stated, all nodes behave correctly. Apart from the sharded verification of RBBC, all algorithms run the same code for the state-machine component implementing the blockchain. Note that there exist BFT algorithms that terminate in less message steps than CONS1, but require additional assumptions like non-faulty clients [@AGK14; @KADC07] or $t<n/5$ [@MA06]. HBBFT uses a randomized consensus [@MMR14] and reliable broadcast using erasure codes.
Machine specification
---------------------
We run the blockchains on the 14 Amazon datacenters that we had at our disposal at the time of the experiment: North Virginia, Ohio, North California, Oregon, Canada, Ireland, Frankfurt, London, Tokyo, Seoul, Singapore, Sydney, Mumbai, São Paulo. We tested two different VMs: (1) *high-end* c4.8xlarge instances with an Intel Xeon E5-2666 v3 processor of 18 hyperthreaded cores, 60 GiB RAM and 10Gbps network performance when run in the same datacenter where storage is backed by Amazon’s Elastic Block Store (EBS) with 4 Gbps dedicated throughput; (2) *low-end* c4.xlarge instances with an Intel Xeon E5-2666 v3 processor of 4 vCPUs, 7.5 GiB RAM, and “moderate” network performance (as defined by Amazon). Storage is backed by EBS with 750Mbps dedicated throughput. To limit the bottleneck effect on the leader of PBFT, we always place the leader in the most central (w.r.t. latency) region, Oregon. When not specified, proposals contain 10,000 transactions and $t$ is set to the larger integer strictly lower than $\frac{n}{3}$.
Comparing geodistributed blockchains {#sec:geo1}
====================================
First, we report the performance when running 10 high-end VMs in each of the 14 regions for a total of 140 machines. At the time of this experiment Amazon was offering us only 14 availability zones: North Virginia, Ohio, North California, Oregon, Canada, Ireland, Frankfurt, London, Tokyo, Seoul, Singapore, Sydney, Mumbai, São Paulo. Each zone contains 10 high-end machines. As depicted on Table \[table:pings\], we computed the variation of communication latencies and throughput between these Amazon EC2 datacenters as measured using c4.xlarge instances. The minimum latency is $11$ms between London and Ireland, whereas the maximum latency is $332$ms observed between Sydney and São Paulo. Bandwidth between Ohio and Singapore is measured at approximately 64.9 Mbits/s (with variance between 6.5 Mbits/s and 20.4 Mbits/s).
![Impact of fault tolerance and verification on the RBBC throughput when $n=140$ geodistributed machines \[fig:consensus-fault-tolerance\]](fig/t-140)
Impact of verification {#ssec:verification-cost}
----------------------
To measure the impact of verification on performance, we varied the parameter $t$ from the minimum to its maximum value ($46 < \frac{140}{3}$) with sharded verification as depicted in Figure \[fig:consensus-fault-tolerance\] (left) and we compared all three blockchains with all nodes verifying all transactions (all) and with no verification (no validation) as depicted in Figure \[fig:consensus-fault-tolerance\] (right). The peak throughput of 151,000 *transactions per second* (tx/sec) is achieved with the fault-tolerance parameter $t=12$. When $t\leq 6$, performance is limited by the $(t-1)^{th}$ slowest node as the consensus waits for a higher number of $n-t$ proposers. When $t\geq 24$, performance is then limited by the growing number of $t+1$ necessary verifications. In Figure \[fig:consensus-fault-tolerance\] (right), the performance of all algorithms is higher without verification than with full verification. RBBC is the most affected dropping from 219,000tx/sec to 33,000tx/sec while HBBFT and CONS1 throughputs drop less but from a lower peak. As we will show in \[sec:proposals-gain\] and \[sec:geo2\], there are factors other than verification that have a larger impact on these algorithms.
![Throughput and latency comparison of the blockchain solutions with $n=140$ and $t=46$, and proposal sizes of $1,10,100,1000$ and $10000$\[fig:consensus-comparison-valid\]](fig/val-140 "fig:") ![Throughput and latency comparison of the blockchain solutions with $n=140$ and $t=46$, and proposal sizes of $1,10,100,1000$ and $10000$\[fig:consensus-comparison-valid\]](fig/lat-val-140 "fig:")
Combining proposals {#sec:proposals-gain}
-------------------
Figure \[fig:consensus-comparison-valid\] explored the effect of deciding the unions of proposals when running the blockchain. CONS1 has the lowest latency because in all executions the leader acts correctly, allowing it to terminate in only 3 message delays, where RBBC with DBFT requires 4 message delays. Probably due to its inherent concurrency, RBBC offers the best latency/throughput tradeoff: at 1000ms latency, RBBC offers 12,100tx/sec whereas at 1750ms latency, CONS1 offers only 5800tx/sec. Finally, the blockchain with HBBFT has the worst performance for several reasons: HBBFT relies on a consensus algorithm [@MMR14] whose termination is randomized and it uses erasure codes: the computation time needed for performing reliable broadcast using erasure codes on a single node with a proposal size of 1000 transactions takes over 200ms. Each node then has to do this for each proposal (i.e., 140 times in this experiment) increasing significantly the latency.
![The performance of CONS1 and RBBC with $t+1$ proposer nodes; the number following the algorithm name represents the number of transactions in the proposals; solid lines represent throughput, dashed lines represent latency\[fig:t+1scalability\]](fig/scali-100 "fig:") \[fig:t+1-100sca\] ![The performance of CONS1 and RBBC with $t+1$ proposer nodes; the number following the algorithm name represents the number of transactions in the proposals; solid lines represent throughput, dashed lines represent latency\[fig:t+1scalability\]](fig/scali-1000 "fig:") \[fig:t+1-1000sca\]
Low-end machines and distributed proposals {#sec:geo2}
------------------------------------------
We now experiment on up to 240 low-end VMs, whose CPU resource is closer to the one of cell phones, and evenly spread on 5 datacenters in the United States (Oregon, Northern California, and Ohio) and Europe (Ireland and Frankfurt). We examine the impact of having $t+1$ vs. $n$ proposers. Dedicating the 4 vCPUs of these low-end instances led to verify about 7800 serialized transactions per second with 97% of CPU time spent verifying signatures and with 3% spent deserializing and updating the UTXO table.
![Comparing throughput and latency of CONS1 and RBBC with $t+1$ proposer nodes on 100 geodistributed nodes; each point represents the number of transactions in the proposals, either 10, 100, 1000, 2500, 5000 or 10000 (HBBFT does not appear due to lower performance)\[fig:lat-tp-100\]](fig/lat-tp-100)
The impact of $t+1$ proposer nodes
----------------------------------
Figure \[fig:t+1scalability\] shows the throughput and latency of RBBC with $t+1$ proposers and CONS1 with different sizes of proposals. As CONS1 is limited to a single proposer (its leader) while RBBC supports multiple proposers, we tested whether CONS1 performance would be better with more transactions per proposal than RBBC.
With proposal size of 100, RBBC throughput increases from 1000 to 4000tx/sec while its latency increases from 750ms to 2seconds. The throughput increase stems from increasing CPU and bandwidth resources with more proposers. With larger proposal size (1000), performance increases faster (from 3000tx/sec to 9000tx/sec) with the number of nodes and flattens out earlier around 10,000tx/sec while latency increases from 2 to 8 seconds.
With proposal size of 100, CONS1 throughput decreases from 310tx/sec to 220tx/sec while latency increases from 320ms to 460ms. Unfortunately, this low latency does not help to increase throughput by increasing proposal size after a certain number of nodes. In particular, with proposal size of 5000 the throughput drops by 4 times (from 2800tx/sec to 700tx/sec). While CONS1 can broadcast message authentication codes (MACs) through UDP in local area networks, no such broadcast primitive is available in this wide area testnet.
Figure \[fig:lat-tp-100\] further examines the performance of CONS1 and RBBC with 100 nodes and proposal sizes of 1, 10, 100, 1000, 2500, and 5000. Here we see that the throughput of CONS1 reaches a limit of about 1100tx/sec while RBBC approaches 14,000tx/sec. CONS1 has a better minimum latency of 270ms compared to 640ms for RBBC for proposals of size $1$.
![The performance of HBBFT and RBBC with $n$ proposer nodes. The number following the algorithm name represents the number of transactions in the proposals; solid lines represent throughput, dashed lines represent latency \[fig:nscalability\]](fig/scali-n100 "fig:") \[fig:n-100sca\] ![The performance of HBBFT and RBBC with $n$ proposer nodes. The number following the algorithm name represents the number of transactions in the proposals; solid lines represent throughput, dashed lines represent latency \[fig:nscalability\]](fig/scali-n1000 "fig:") \[fig:n-1000sca\]
The impact of $n$ proposer nodes
--------------------------------
Figure \[fig:nscalability\] depicts the performance of RBBC and HBBFT with $n$ proposers, with proposal sizes of 100 and 1000 transactions. Unsurprisingly, with $n$ proposers the throughput of RBBC increases faster than with $t+1$ proposers. With a proposal size of 100, the throughput reaches 6000tx/sec at 80 nodes and slowly degrades, while latency starts at 740ms with 20 nodes and reaches 5160ms with 240 nodes. With a proposal size of 1000, the throughput reaches 10,000tx/sec at 40 nodes and remains mostly flat, latency starts at 2670ms with 20 nodes and reaches 25,100ms with 240 nodes. With larger node counts (around 200), the configurations with $t+1$ proposals achieve similar throughput, but with much lower latency. Thus when using nodes similar to the low-end instances, having $n$ proposers seems better suited for configurations of less than 100 nodes.
For HBBFT we observe that latencies increase superlinearly and throughput degrades as we increase the number of nodes. As mentioned before, this is primarily due to the computation needed for the erasure codes. Note that we only run HBBFT up to 100 nodes as afterwards we start seeing latencies approaching minutes.
Transaction verification count {#ssec:verification-count}
------------------------------
In the previous experiments we also recorded the average number of times a transaction is verified to examine the state of sharded verification, the results are shown in Figure \[fig:sca-val\]. The best case is $t+1$ verifications while the $2t+1$ is the worst case. We observe that with $t+1$ proposers the number of verifications stays close to the optimal, while with $n$ proposers the number of verifications remains around the middle of $t+1$ and $2t+1$. This is likely due to the increased load on the system causing verifications to occur in different orders at different nodes.
Experiment under Byzantine attacks {#sec:security}
==================================
We evaluate RBBC performance under 2 Byzantine attacks:
1. The payload of the reliable broadcast messages altered so that no proposal is delivered for reliable broadcast instances led by faulty nodes. The binary payloads of the binary consensus messages are flipped. The goal of this behavior is to reduce throughput and increase latency.
2. The Byzantine nodes form a coalition in order to maximize the bandwidth cost of the reliable broadcast using the digests described in \[ssec:dbft\]. As a result, for any reliable broadcast initiated by a Byzantine node, $t+1$ correct nodes will deliver the full message while the remaining $t$ will only deliver the digest of the message, meaning they will have to request the full message from $t+1$ different nodes from whom they receive $\lit{echo}$ messages.
As in \[sec:geo2\], experiments are run with 100 low-end machines using the same 5 datacenters from US and Europe and with $n$ proposers.
Figure \[fig:byz-tpt\] shows the impact of Byz1 on performance with $n$ proposers and proposal sizes of $100$. For RBBC, throughput drops from 5700tx/sec to 1900tx/sec, and latency increases from 920ms to 1750ms. The drop in throughput is partially due to having $t$ less proposals being accepted (the proposals sent by Byzantine nodes are invalid), and to the increase in latency. The increase in latency is due to the extra rounds needed to be executed by the binary consensus to terminate with $0$. The throughput of HBBFT drops from 350 to 256tx/sec due to the decrease in proposals, but interestingly the latency also decreases. This is due to the fact that since there are less proposals, less computation is needed for the erasure codes.
![Comparing throughput and latency of RBBC and HBBFT, with normal and Byzantine behavior on 100 geodistributed nodes; all $n$ nodes are making proposals of 100 transactions \[fig:byz-tpt\]](fig/byz-tpt)
![Comparing message complexity and latency of RBBFT and HBBFT with normal and Byzantine behaviors on 100 geodistributed nodes \[fig:byz-msg\]](fig/byz-msg)
Byz2 is a behavior designed against the digest compression of the reliable broadcast, with the goal of delaying the delivery of the message to $t$ of the correct nodes, and increasing the bandwidth used. HBBFT avoids this problem by using erasure codes, but has a higher bandwidth usage is the non-faulty case. Figure \[fig:byz-msg\] shows the impact of this behavior on bandwidth usage and latency for RBBC and HBBFT with $n$ proposers and proposal sizes of $100$. The bandwidth usage of RBBC increases from 538MB per multivalued consensus instance to 2622MB per multivalued consensus instance compared to HBBFT which uses 3600MB in all cases. Furthermore, the latency of RBBC increases from 920ms to 2300ms. Note that the bandwidth usage can further increase if additional delays are added to the network, in such cases the use of Erasure codes becomes beneficial.
\#Requesters Valid-tx/sec Read/sec R/W ratio Latency(ms) Valid-tx/block Invalid-tx/block
-- -------------- -------------- ---------- ----------- ------------- ---------------- ------------------
1,000 5,359 2,143 0.4 870 4,648 0
10,000 13,870 33,288 2.4 2,475 34,132 877
20,000 12,664 31,660 2.5 5,022 63,607 3,033
50,000 14,450 47,685 3.3 4,303 62,193 5,455
1,000 3,759 1,127 0.3 401 1,513 0
10,000 3,309 6,278 1.9 359 1,172 0
20,000 4,064 10,566 2.6 488 1,981 0
50,000 4,035 12,509 3.1 625 2,500 0
Single availability zone experiment {#sec:datacenter}
-----------------------------------
To really stress test RBBC, we tested the performance on 300 high-end VMs in the Oregon datacenter. We fixed $t$ to the largest fault tolerance parameter we can tolerate with $n=20$ nodes and increase the number of nodes from $20$ to $300$ permissioned nodes. While the setting is not realistic, it helps identifying potential performance bottlenecks. Note that Fig. \[fig:consensus-fault-tolerance\] depicts the impact of varying $t$ on performance. The results, shown in Figure \[fig:scalability\], indicates that the throughput scales up to $n=260$ nodes to reach 660,000tx/sec while the latency remains lower than 4 seconds. At $n=280$ throughput drops slightly. Other experiments not shown here indicated about $8$ verifications per transaction converging towards $7=t+1$ as $n$ increases. The performance is thus explained by the fact that the system is CPU-bound up to $n=260$, so that increasing $n$ adds CPU resources needed for the sharded verification and improves performance, after what the system becomes network-bound due to the consensus and performance flattens out.
![The performance (latency and throughput) of RBBC in a single datacenter\[fig:scalability\]](fig/scali-300)
![The number of times a transaction is verified in RBBC with proposal size of 100 transactions, with either $t+1$ or $n$ proposer nodes; the dashed lines $t+1$ and $2t+1$ represent the minimum and maximum number of possible verifications. \[fig:sca-val\]](fig/valid)
[ 1000clientexptable.tex ]{}
Impact of remote requesters {#ssec:remote-req}
---------------------------
For the following experiments we run the blockchain with requesters defined as follows. At the start of the benchmark each requester is assigned a random private key and a single UTXO contained within the genesis block with value 100,000 coins. The requester then loops over the following two steps until the benchmark completes: *(i)* For each UTXO currently assigned to the requester a new transaction is created using that UTXO as input. For the transaction’s output a UTXO is created using a randomly chosen account as the receiver with a value of 10 coins. Any change is included in a second UTXO sent back to the requester. Each transaction is then broadcast to the requester’s assigned proposers. *(ii)* The requester then repeatably performs the $\sf{request\_utxos}(\ms{account})$ operation until it receives at lest one new UTXO and then returns to step (i). Each requester is run in its own thread and maintains connections to $2t+1$ of the blockchain nodes, including the requester’s $t+1$ proposers (all CONS1 requesters have the same primary proposer).
For this experiment we ran RBBC and CONS1 using 100 c4.4xLarge server instances 25 c4.4xLarge requester instances. Both types of nodes are evenly distributed across the 5 datacenters from US and Europe. The c4.4xLarge instances use Intel Xeon E5-2666 v3 processors with 16 vCPUs, and 30 GiB RAM. The number of requesters vary from 1,000 to 50,000 and are evenly distributed across the requester nodes. For the proposal size $\beta$, we choose $1000$ for RBBFT as it gave the best throughput. For CONS1 we chose a proposal size of $2500$ as we found that larger sizes increased latency without increasing throughput and smaller sizes decreased throughput while only having a minor impact on latency. The experiments were run for 45 sec with a 15 sec warmup.
Table \[fig:tab-cli-exp\] shows the results. *Valid-tx/sec* is average number of valid transactions committed per second, *Read/sec* is the average number of $\sf{request\_utxos}(\ms{account})$ operations performed per second, *R/W ratio* is the ratio of the previous two values, *Latency* is the average amount of time between committed blocks, *Valid-tx/block* is the average number of valid transactions per block, and *Invalid-tx/block* is the average number of invalid transactions per block.
Similar to the previous experiments we see that RBBFT has the highest maximum throughput of $14,450$ tx/sec compared to $4,064$ with CONS1. RBBFT has the highest maximum latency between blocks of $5,022$ milliseconds compared to a maximum of $625$ milliseconds for CONS1. The higher throughput and latency is explained by the higher utilization of resources by the sharded proposers and reduced computation needed for sharded verification. In RBBFT increasing the number of requesters past 10,000 has little impact on the throughput as the system resources are already saturated by this point, as a result we see an increase in the R/W ratio as it takes longer for each individual node’s transaction to complete. A similar pattern is shown by CONS1, though this starts at 1,000 requesters as they are limited by the single primary proposer. Furthermore in RBBFT, increasing the number of requesters also increases the number of duplicate transactions occurring in blocks. This is due to the increased load in they system causing slower nodes to miss their proposals resulting in transactions being committed by secondary proposers.
Evaluation with 1000 machines {#ssec:thousand-vm}
=============================
To confirm that our blockchain scales to a large number of machines, we spawned 1000 VMs. To avoid wasting bandwidth, we segregated the roles: all 1000 VMs act as servers, keeping a local copy of the balances of all accounts. On these replicas, 10 requesters per 840 c4.large machines (60 VMs in each of 14 datacenters) send transactions and 160 c4.8xlarge machines (40 machines in each of the Ireland, London, Ohio and Oregon datacenters) decide upon each block.
Each of the 8400 requesters start with 100 UTXOs and each proposal contains up to 1000 transactions. Performance are depicted in Table \[fig:tab-cli-exp2\]: throughput is only around 30,000 tx/sec due to the difficulty of generating the workload: the replicas are located in 14 different datacenters and have to wait for a UTXO to request a transaction that consumes it (cf. \[ssec:remote-req\]). The asynchronous write latency measures the time a proposer acknowledges a transaction reception. The transaction commit time (latency) remains about 3 seconds despite the large traffic.
Conclusion {#sec:conclusion}
==========
In the most extensive experimentation of blockchain to date, we evaluated the Red Belly Blockchain, a deterministic blockchain system that does not need synchrony to be secure and performs well at large scale. Its main novelty is its novel sharding that minimizes both computation and communication wastes that allows to achieve unprecedented throughput with a low latency when deployed world-wide. The Red Belly Blockchain appears as a platform of choice for obtaining the security needed to move blockchain use-cases from innovation labs to production without sacrificing performance.
[^1]: “Red belly” is inspired by the name of a snake endemic to Sydney.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper proposes a new minimum description length procedure to detect multiple changepoints in time series data when some times are a priori thought more likely to be changepoints. This scenario arises with temperature time series homogenization pursuits, our focus here. Our Bayesian procedure constructs a natural prior distribution for the situation, and is shown to estimate the changepoint locations consistently, with an optimal convergence rate. Our methods substantially improve changepoint detection power when prior information is available. The methods are also tailored to bivariate data, allowing changes to occur in one or both component series.'
author:
-
bibliography:
- 'ChangepointTxTn.bib'
title: Multiple Changepoint Detection with Partial Information on Changepoint Times
---
[*Keywords:*]{} breakpoints, segmentation, structural breaks, empirical Bayes, time series, vector autoregression.
Introduction {#section:intro}
============
Changepoints, also called structural breaks or breakpoints, are times in a sequential record where the data abruptly shift in some manner (mean, variance, autocovariance, quantile, etc.). The primary goal of a retrospective multiple changepoint analysis, the case considered here, is to estimate the number of changepoints and their location times. Various approaches have been developed for independent data; good recent references include @Fryzlewicz_2014, @Pein_etal_2017, and the review paper @Niu_etal_2016 (and references therein). When the data are correlated, such as the monthly temperature records studied here, this feature can greatly impede changepoint detection; in fact, mean shifts can often be misattributed to positive correlation [@Lund_etal_2007].
One simple way to detect multiple changepoints is to combine an at most one changepoint (AMOC) technique (say a CUSUM or likelihood ratio test) with a binary segmentation procedure, e.g., @Shao_Zhang_2010 [@Aue_Horvath_2013; @Fryzlewicz_SubbaRao_2014]. Wild binary segmentation techniques usually improve upon ordinary binary segmentation methods [@Fryzlewicz_2014]. Since estimating the optimal multiple changepoint configuration can be formulated as a model selection problem, penalized likelihood methods such as BIC [@Yao_1988] and its modifications [@Zhang_Siegmund_2007; @Zhang_Siegmund_2012], and minimal description lengths (MDL) are also popular. In this paper, an MDL technique is developed that takes into account prior information on the changepoint numbers and locations. This scenario is shown to arise in the homogenization of temperature time series to account for gauge changes and station location moves.
The MDL principle [@Risanen_1989] from information theory has been successfully applied in statistical model selection problems [@Hansen_Yu_2001]. MDL penalties are the sum of penalties (i.e., description lengths, or code lengths) of all unknown model parameters. In the multiple changepoint literature, the seminal work of @Davis_etal_2006 develops an MDL penalty for piecewise autoregressive (AR) processes. Here, the penalty is constructed by following certain automatic rules that assign different penalties to different parameter types: bounded integer parameters, unbounded integer parameters, and real-valued parameters. Since MDL penalties are not just simple multiples of the number of model parameters, they are believed superior to AIC and BIC penalties (a belief supported by simulations), and are shown consistent for changepoint estimation under infill asymptotics [@Davis_etal_2006; @Davis_Yau_2013]. Following the automatic penalty rules, MDL methods have been extended to various time series structures, including GARCH processes [@Davis_etal_2008], periodic ARs [@Lu_etal_2010], autoregressive moving-averages [@Davis_Yau_2013], and threshold ARs [@Yau_etal_2015].
The main goal of this paper is to incorporate partial information on changepoint numbers and times into the MDL penalty, an aspect not readily handled by existing MDLs methods. Indeed, this will require us to revisit information theory. The motivating example involves the climate homogenization [@Caussinus_Mestre_2004; @Menne_WilliamsJr_2005] of monthly temperature records. Here, the aim is to detect abrupt mean shifts, which are often induced by artificial causes such as station relocations or gauge changes. Two types of [*a priori*]{} changepoint knowledge arise. First, metadata station history logs, which document the times of physical changes in the station, are sometimes available. Although metadata climate records are notoriously incomplete, and not all documented metadata times induce actual mean shifts in the series, climatologists believe that metadata times are more likely than non metadata times to be changepoints. Second, when multivariate records exist for the same station, changepoints may affect component records simultaneously. For example, with monthly maximum and minimum temperature averages (called Tmax and Tmin, respectively), moving a station to a drier location can both increase daytime highs and reduce nighttime lows. While changepoints in either Tmax or Tmin can occur by themselves, climatologists believe that it is more likely for changepoints to occur in both component series at the same time (called concurrent shifts).
While metadata is typically only used to verify climate changepoint conclusions in hindsight, Sections \[sec:simulation\] and \[sec:Tuscaloosa\] will show that use of metadata can improve detection power and time of estimation accuracy. This benefit is not limited to climatological pursuits; in other areas such as biology, economics, and engineering, domain expert knowledge is often available; e.g., knowledge from previous experiments on possible copy number variation locations, or the impact of certain political policy or regime changes on financial series.
Of course, Bayesian methods account for [*a priori*]{} knowledge via the construction of prior distributions. From a Bayesian model selection perspective, the optimal model (i.e., multiple changepoint configuration) is the one with the highest posterior probability [@Clyde_George_2004]. This maximum [*a posteriori*]{} (MAP) rule can be loosely viewed as a penalization method, where the posterior density is a penalized likelihood and the prior density is the penalty. Compared to frequentist approaches, one advantage of Bayesian posterior analysis is that it can also provide measure of uncertainty for model parameters and changepoint locations. Bayesian approaches have been proposed for retrospective multiple changepoint detection — see @Barry_Hartigan_1993 [@Chib_1998; @Fearnhead_2006; @Giron_etal_2007; @Zhang_Siegmund_2007; @Giordani_Kohn_2008; @Fearnhead_Vasileiou_2009; @Hannart_Naveau_2012]. However, theoretical studies of large sample performance of Bayesian methods are in general lacking; @Du_etal_2016 study asymptotic consistency of changepoint locations, but only for independent data. More importantly, since existing Bayesian changepoint approaches are typically derived under non-informative prior distributions, they rarely explicate how to incorporate real subjective prior knowledge. One (and maybe the only) exception is @Li_Lund_2015, who account for metadata in a univariate precipitation time series.
Changepoint detection on multivariate data has received significant attention in recent years, e.g., @Cho_Fryzlewicz_2015 [@Kirch_etal_2015; @Preuss_etal_2015; @Ma_Yau_2016]. In @Davis_etal_2006, the automatic MDL is applied to multivariate AR series, where changepoints affect all component series. However, for many applications, a changepoint may not affect all component series. The automatic MDL does not directly accommodate this case, probably because it is unclear whether a change affecting many components should receive the same penalty as one affecting only a few components. On the other hand, Bayesian approaches such as @Zhang_Siegmund_2012 [@Bardwell_Fearnhead_2017] can handle this problem, for independent data over both time and component. But since they are developed under non-informative prior distributions, they are not ready applicable to handle multivariate temperature homogenization, where concurrent changes in Tmax and Tmin should be encouraged.
In this paper, a new class of flexible MDL methods is proposed that incorporates domain experts’ [*a priori*]{} knowledge for multiple changepoint detection, in both univariate and multivariate time series. Multiple changepoint configurations are reformulated as vectors of zero/one indicators, thus permitting natural construction of subjective prior distributions, with straightforward hyper-parameter elicitation. To account for correlation in time and across components, AR processes for univariate data, and vector autoregressive (VAR) processes for multivariate data are employed. Our MDL method is termed a Bayesian MDL (BMDL) because it can be viewed as an empirical Bayes model selection approach. While our main focus is to improve and generalize conventional MDL changepoint detection approaches, to the best of our knowledge, this paper is the first Bayesian multiple changepoint work to establish asymptotic consistency with correlated observations. Under infill asymptotics, the estimated changepoint locations are shown to converge in probability to their true values at an optimal rate; moreover, estimation of the number of changepoints and model parameters such as regime means and AR coefficients are also consistent.
The rest of this paper is organized as follows. Section \[section:MDL\_review\] briefly reviews MDL principles. Section \[section:BMDL\] develops a BMDL penalty to detect mean shifts in univariate series. This work incorporates metadata, while allowing for a confounding seasonal mean cycle and AR errors. Frequentist large sample performance is studied. Section \[section:bivariateBMDL\] extends the BMDL to the multivariate setting, where Tmax and Tmin series are modeled jointly. Section \[sec:simulation\] presents simulation examples. Section \[sec:Tuscaloosa\] moves to an application to 114 years of monthly temperatures from Tuscaloosa, Alabama. Comments close the paper in Section \[sec:discussion\]. Technical results and proofs are delegated to an appendix.
A Brief Review of MDL {#section:MDL_review}
=====================
In information theory, a code length is the number of binary storage units required to transmit a random number or code. To reduce storage costs, one wants to assign shorter (longer) code lengths to common (rare) outcomes. Competing probability models can be compared by their code lengths; the true data generating distribution (i.e., the true model) should have the shortest expected code length. The MDL principle [@Risanen_1989] states that given the observed data, the model with the shortest code length is optimal.
For a discrete random variable $X$ with probability mass function $f(\cdot)$, @Shannon_1948 states that the encoding with code length $$\label{eq:basic_MDL}
\mathcal{L}(X) = -\log_2 \{f(X)\}$$ has the shortest expected code length. The existing MDL approach for multiple changepoint detection [@Davis_etal_2006] is developed under the automatic rules that the code length of a positive random integer $X$ bounded above by $N$ is $\log_2(N)$, and that of an unbounded positive random integer $X$ is $\log_2(X)$. The former rule implies a uniform distribution over the set $\{1, 2, \ldots, N \}$, which leads to the code length $\mathcal{L}(X) = -\log_2(1/N) =
\log_2(N)$, while the latter implies an improper power law distribution with the probability mass function $f(X) \propto 1/X$.
For a continuous random variable, say $X \in \mathbb{R}^k$ with density function $f(\cdot)$, after discretizing each dimension into equal cells of size $\delta$ (often viewed as the machine precision), one can mimic the discrete case to obtain $\mathcal{L}(X) = -\log_2 \{f(X)\delta^k\} =
-\log_2 f(X) - k\log_2(\delta)$. Because $k$ and $\delta$ do not vary with $X$, the term $-k\log_2(\delta)$ does not affect comparison between different outcomes of $X$ and is hence often omitted. Thus, the MDL for a continuous variable can also be expressed as in . In the rest of this paper, the natural logarithm is substituted for the base two logarithm — this does not affect model comparisons since $\log_2(x)/\log(x)$ is constant in $x$.
Now suppose that a dataset $\mathbf{X} = (X_1, \ldots, X_n)'$, believed to be generated from a certain parametric model $\mathcal{M}$ with density $f(\mathbf{X} \mid \theta, \mathcal{M})$, is to be transmitted along with a possibly unknown parameter $\theta \in \Theta$. As reviewed in @Hansen_Yu_2001, two types of MDL approaches, the two-part MDL and the mixture MDL, are commonly used.
Two-part MDLs
-------------
The two-part MDL, also called the two-stage MDL, considers the transmission of $\mathbf{X}$ and $\theta$ in two steps. If both the sender and receiver know $\theta$, the MDL of $\mathbf{X}$ is $\mathcal{L}(\mathbf{X} \mid \theta, \mathcal{M}) = -\log \{f(\mathbf{X}
\mid \theta, \mathcal{M})\}$. Here, notations such as $\mathcal{L}(\cdot
\mid \cdot)$ are analogous to the usual conditional distribution notations that emphasize dependence. Should $\theta$ also be unknown to the receiver, an additional cost of $\mathcal{L}(\theta \mid
\mathcal{M})$ is incurred in transmitting it. Hence, the two-part MDL is $$\mathcal{L}(\mathbf{X}, \theta \mid \mathcal{M}) =
\mathcal{L}(\mathbf{X} \mid \theta, \mathcal{M}) + \mathcal{L}(\theta
\mid \mathcal{M}).$$
Suppose that $\mathcal{L}(\mathbf{X}, \theta \mid \mathcal{M})$ is minimized at $\hat{\theta}$, an estimator of $\theta$ based on the data $\mathbf{X}$. If $\theta$ is a $k$-dimensional continuous parameter and $\hat{\theta}$ is a $\sqrt{n}$-consistent estimator, then one can set the machine precision to be $\delta = c/\sqrt{n}$, where $c$ is a positive constant. Under a uniform encoder $\pi(\theta \mid \mathcal{M})
\propto 1$, the code length needed to transmit $\theta$ (including $\hat{\theta}$) is hence $\mathcal{L}(\theta\mid \mathcal{M}) = -\log
\{\pi(\theta\mid \mathcal{M})\} -k \log(c/\sqrt{n}) = k\log(n) /
2 - k \log(c)$, which does not depend on $\theta$. Hence, the maximum likelihood estimator (MLE) minimizes $\mathcal{L}(\mathbf{X}, \theta
\mid \mathcal{M})$, and the two-part MDL coincides with the BIC [@Schwarz_1978]. In fact, $\hat{\theta}$ need not be the MLE; any $\sqrt{n}$-consistent estimator is justifiable. Again the constant term $k\log(c)$ can be dropped and the remaining code length $\mathcal{L}(\hat{\theta} \mid \mathcal{M}) = k \log(n) /2$ is adopted by @Davis_etal_2006 as the automatic MDL rule for a $k$-dimensional continuous parameter.
If there exists a discrete set of candidate models, to account for model uncertainty, the two-part MDL can be modified to include an additional code length for the model $\mathcal{M}$, i.e., $$\label{eq:MDL_two-part3}
\mathcal{L}(\mathbf{X}, \hat{\theta}, \mathcal{M})
= \mathcal{L}(\mathbf{X} \mid \hat{\theta}, \mathcal{M}) +
\mathcal{L}(\hat{\theta} \mid \mathcal{M}) +
\mathcal{L}(\mathcal{M}),$$ where $\hat{\theta}$ is model dependent, $\mathcal{L}(\mathcal{M}) =
-\log\{\pi(\mathcal{M})\}$, and $\pi(\mathcal{M})$ is the prior distribution over the model space. The model with the smallest MDL in is deemed optimal.
All existing automatic MDL methods for multiple changepoint detection are based on two-part MDLs. However, for a finite sample size $n$, the two-part MDL is problematic when the dimension of $\theta$ changes across models, as in the multiple changepoint case. Consider a setting of two competing models $\mathcal{M}_1$ and $\mathcal{M}_2$, whose parameters $\theta_j$ are $k_j$-dimensional continuous parameters, for $j = 1, 2$, and $k_1 \neq k_2$. Model $\mathcal{M}_1$ is favored if $\mathcal{L}(\mathbf{X}, \hat{\theta}_1, \mathcal{M}_1) -
\mathcal{L}(\mathbf{X}, \hat{\theta}_2, \mathcal{M}_2)$ is negative; otherwise, model $\mathcal{M}_2$ is favored. Note that the code length difference for the parameters $\mathcal{L}(\hat{\theta}_1 \mid
\mathcal{M}_1) - \mathcal{L}(\hat{\theta}_2 \mid \mathcal{M}_2)$ contains the term $(k_1 - k_2) \{\log(n) - 2\log(c)\}/2$. This term, and hence also $\mathcal{L}(\mathbf{X}, \hat{\theta}_1, \mathcal{M}_1) -
\mathcal{L}(\mathbf{X}, \hat{\theta}_2, \mathcal{M}_2)$, could be either positive or negative depending on $n$ and the arbitrary constant $c$. One cannot judge either model superior without knowledge of $c$. Of course, this issue does not conflict with the asymptotic consistency of BIC or automatic MDLs: as $n$ increases, $\log(n)$ dominates the constant $\log(c)$. Mixture MDLs, reviewed next, do not suffer from such a problem for a finite $n$.
Mixture MDLs
------------
By @Hansen_Yu_2001, the mixture MDL is defined to be based on the marginal likelihood $f(\mathbf{X} \mid \mathcal{M})$: $$\mathcal{L}(\mathbf{X} \mid \mathcal{M}) =
-\log \{f(\mathbf{X} \mid \mathcal{M})\}, \quad \text{where }
f(\mathbf{X} \mid \mathcal{M})
= \int_{\Theta} f(\mathbf{X} \mid \theta, \mathcal{M}) \pi(\theta \mid
\mathcal{M})d\theta$$ averages the likelihood $f(\mathbf{X} \mid \theta, \mathcal{M})$ over $\theta$ under its prior density $\pi(\theta \mid \mathcal{M})$. If this prior distribution depends on an unknown hyper-parameter $\psi$, then a two-part MDL can be used to account for the additional cost needed to transmit $\psi$. In this case, the overall mixture MDL, for any $\sqrt{n}$-consistent estimator of $\psi$, is $$\mathcal{L}(\mathbf{X},
\hat{\psi} \mid \mathcal{M}) = - \log \left\{ \int_{\Theta} f(\mathbf{X} \mid
\theta, \mathcal{M}) \pi(\theta \mid \hat{\psi}, \mathcal{M})d\theta \right\}
+ \mathcal{L}(\hat{\psi}\mid \mathcal{M}).$$
The mixture MDL for the model $\mathcal{M}$ is thus $\mathcal{L}(\mathbf{X}, \hat{\psi}, \mathcal{M}) =
\mathcal{L}(\mathbf{X}, \hat{\psi} \mid \mathcal{M}) + \mathcal{L}(
\mathcal{M})$, which is related to empirical Bayes (EB) approaches [@Carlin_Louis_2000]. If the prior probabilities of two models are the same, i.e., $\pi(\mathcal{M}_1) = \pi(\mathcal{M}_2)$, and the hyper-parameter $\psi$ is transmitted under the uniform encoder $\pi(\psi \mid \mathcal{M}_j) \propto 1$ for $j = 1,2$, then the difference of the two mixture MDLs, $\mathcal{L}(\mathbf{X},
\hat{\psi}_1, \mathcal{M}_1) - \mathcal{L} (\mathbf{X}, \hat{\psi}_2,
\mathcal{M}_2)$, equals the logarithm of their Bayes factor [@Kass_Raftery_1995] $\text{BF}_{\mathcal{M}_2: \mathcal{M}_1}$. Similarly, in EB settings, while the estimator $\hat{\psi}$ is often chosen to maximize the marginal likelihood $f(\mathbf{X} \mid \psi,
\mathcal{M})$, other consistent estimators (moments for example) can also be used.
Bayesian Minimum Description Lengths for a Univariate Time Series {#section:BMDL}
=================================================================
Consider a univariate time series $\mathbf{X}_{1:N}= (X_1, \ldots,
X_N)'$ with a seasonal mean cycle with fundamental period $T$. For monthly data, $T = 12$. A model with autoregressive errors describing this situation is $$\label{eq:likelihood1}
X_t = s_{v(t)} + \mu_{r(t)} + \epsilon_t, \quad
\epsilon_t = \sum_{j = 1}^p \phi_j \epsilon_{t-j} + Z_t.$$ Here, $v(t) = t - T \lfloor (t-1)/T \rfloor \in \{ 1, 2, \ldots, T\}$ is the season corresponding to time $t$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$. The seasonal means $\mathbf{s} = (s_1, \ldots, s_T)'$ are unknown. The errors $\{ \epsilon_
t \}_{t=1}^N$ are a causal zero mean AR process of known order $p$. The AR coefficients $\boldsymbol{\phi} = (\phi_1, \ldots, \phi_p)'$ and the white noise variance $\text{Var}(Z_t) = \sigma^2$ are assumed unknown. For likelihood computations, following @Davis_etal_2006, white noises are assumed iid normal. This can be justified as a quasi-likelihood approach; furthermore, in climate applications, monthly averaged temperatures are approximately normally distributed [@Wilks_2011].
Suppose a multiple changepoint configuration (i.e., a model) contains $m$ changepoints at the times $\tau_1 < \cdots < \tau_m \leq N$. These times partition the observations $\{ 1, \ldots, N \}$ into $m+1$ distinct regimes (segments), where the series’ overall mean (neglecting its seasonal component), $\mu_{r(t)}$, changes across regimes. To avoid trite work with edge effects of the autoregression, we assume that no changepoints occur during the first $p$ observations. For notation, set $\tau_0=1$ and $\tau_{m+1}=N+1$. The regime indicator $r(t)$ in satisfies $r(t)=r$ when $\tau_{r-1} \leq t <
\tau_r$. To ensure identifiability, $\mu_1$ is set to zero; hence, $E(X_t)=s_{v(t)}$ when $t$ lies in the first regime. The other regime means $\boldsymbol{\mu} = (\mu_2, \ldots, \mu_{m+1})'$ are unknown.
Following @Li_Lund_2015, the multiple changepoint configuration $(m; \boldsymbol{\tau})$ is reformulated as an $(N-p)$-dimensional vector of zero/one indicators: $\boldsymbol{\eta} = (\eta_{p+1}, \ldots,
\eta_N)'$. Here, $\eta_t=1$ indicates that time $t$ is a changepoint in this model; $\eta_t=0$ means that time $t$ is not a changepoint. The total number of changepoints in model $\boldsymbol\eta$ is thus $m=\sum_{t=p+1}^N \eta_t$.
Our idea is to apply the mixture MDL to the continuous parameter $\boldsymbol{\mu}$, whose dimension varies across models, and use the two-part MDL for the parameters $\mathbf{s}, \boldsymbol\sigma^2,
\boldsymbol\phi$, and the model $\boldsymbol\eta$. In the rest of this section, subsection \[subsection:prior\] introduces our priors on $\boldsymbol{\eta}$ and $\boldsymbol{\mu}$, subsection \[subsection:bmdl\_univariate\] derives the BMDL formula , subsection \[subsection:asymptotics\] studies asymptotic properties, and subsection \[subsection:computation\] concludes with computational strategies.
Prior specifications {#subsection:prior}
--------------------
Our prior distribution for the changepoint model $\boldsymbol{\eta}$ assumes that, in the absence of metadata, each time $t$ has an equal probability $\rho$ of being a changepoint, independently of all other times, i.e., $$\label{eq:eta_prior_bernoulli} \eta_t
\stackrel{\text{iid}}{\sim} \text{Bernoulli} (\rho), \quad t = p+1,
\ldots, N.$$ This independent Bernoulli prior has been used in previous Bayesian multiple changepoint detection works [@Chernoff_Zacks_1964; @Yao_1984; @Barry_Hartigan_1993]. From a hidden Markov perspective, this prior is equivalent to $\tau_r \mid
\tau_{r-1} \sim \text{Geometric}(\rho)$ for $r = 1, \ldots, m$ [@Fearnhead_Vasileiou_2009], and thus is a special case of the negative Binomial prior [@Hannart_Naveau_2012]. The uniform prior $\pi(\boldsymbol\eta) \propto 1$ adopted in @Du_etal_2016 is a special case of the Bernoulli prior with $\rho = 0.5$. For applications where knowledge beyond metadata is unavailable, an iid prior on $\{
\eta_t \}$ seems reasonable. In other applications, $\pi(\boldsymbol{\eta})$ is allowed to have different success probabilities in different regimes [@Chib_1998]; correlation across different changepoint times can also be achieved using Ising priors [@Li_Zhang_2010].
To account for uncertainty in the success probability $\rho$, a hyper-prior is placed on it. @Barry_Hartigan_1993 let $\rho$ have a uniform prior on the interval $(0, \rho_0)$, where $\rho_0 < 1$. For additional flexibility, we use the Beta distribution $$\label{eq:eta_prior_beta}
\rho \sim \text{Beta}(a, b),$$ where $a,b > 0$ are fixed hyper-parameters. The Beta-Binomial hierarchical priors in and are widely used in Bayesian model selection [@Scott_Berger_2010], and have been adopted to detect changepoints [@Giordani_Kohn_2008; @Li_Lund_2015]. Due to conjugacy, the marginal prior density of $\boldsymbol{\eta}$ has the following closed form, with $\beta(\cdot, \cdot)$ denoting the Beta function: $$\label{eq:prior_eta}
\pi(\boldsymbol\eta) = \int_0^1 \pi(\rho) \prod_{t = p + 1}^N \pi(\eta_t
\mid \rho) d\rho = \frac{\beta(a + m, b + N-p - m)}
{\beta(a, b)}.$$
For hyper-parameter choices, an objective Bayesian option [@Giron_etal_2007] is $a=b=1$. In this case, $\pi(\boldsymbol\eta)
= \left\{{N-p \choose m} (N-p+1)\right\}^{-1}$, which implies that marginally, the number of changepoints $m$ has a uniform prior on the set $\{0, 1, \ldots, N-p\}$, and all models containing the same number of changepoints have the same prior probabilities. The Beta-Binomial prior can be tuned to accommodate subjective knowledge from domain experts. For temperature homogenization, @Mitchell_1953 estimates an average of six station relocations and gauge changes per century in United States temperature series; this long-term rate is 0.005 changepoints per month and can be produced with $a=1$ and $b=199$; with these parameters, $E(\rho)=a/(a+b)=0.005$.
This prior is now modified to accommodate metadata. Suppose that during the times $\{p+1, \ldots, N\}$, there are $N^{(2)}$ documented times (times listed in the metadata) and $N^{(1)}=N-p-N^{(2)}$ undocumented times. For notation, all quantities superscripted with $(1)$ refer to undocumented times; quantities superscripted with $(2)$ refer to documented times. Following @Li_Lund_2015, we posit that the undocumented times have a Beta-Binomial$(a, b^{(1)})$ prior, and independently, the documented times have a Beta-Binomial$(a, b^{(2)})$ prior. To make the metadata times more likely to induce true mean shifts, we impose $b^{(1)} > b^{(2)}$ so that $$E\left(\rho^{(1)}\right) = \frac{a}{a + b^{(1)}} < \frac{a}{a + b^{(2)}} = E\left(\rho^{(2)}\right).$$ For monthly data, default values are $a=1, b^{(1)}=239$, and $b^{(2)}=47$, making $E(\rho^{(1)}) = 0.0042$ and $E(\rho^{(2)})=0.0208$; [*a priori*]{}, a documented time is roughly five times as likely to be a changepoint as an undocumented time. One may change the values of $b^{(1)}$ and $b^{(2)}$ to reflect different prior beliefs. The sensitivity analysis in @Li_Lund_2015 suggests that changepoint detection results are relatively stable under a range of $E(\rho^{(2)})/E(\rho^{(1)})$ values.
Following and writing Beta integrals via their Gamma function representations, a changepoint configuration $\boldsymbol{\eta}$ with $m^{(2)}$ documented changepoints and $m^{(1)}$ undocumented changepoints ($m=m^{(1)} + m^{(2)}$) has a marginal prior density (up to a normalizing constant) $$\pi(\boldsymbol{\eta}) \propto \prod_{k=1}^2
\Gamma\left(a + m^{(k)}\right) \Gamma\left(b^{(k)} + N^{(k)} - m^{(k)}\right).$$
For a changepoint model with $m > 0$ changepoints, priors for the $m$-dimensional regime means $\boldsymbol{\mu}$ are posited to have independent normal prior distributions: $$\label{eq:mu_prior_normal}
\boldsymbol{\mu} \mid \sigma^2, \boldsymbol{\eta}
\sim \text{N}(\mathbf{0}, \nu \sigma^2 \mathbf{I}_{m}).$$ Here, $\nu$ is a pre-specified non-negative parameter that is relatively large (making the variances of the regime means large multiples of the white noise variances). Similar to the sensitivity analysis in @Du_etal_2016, our experience suggests that model selection results are stable under a wide range of $\nu$ values. Our default takes $\nu = 5$.
In fact, $\pi(\boldsymbol{\mu})$ can be any zero mean continuous distribution. For example, if mean shifts are expected to be large, heavy-tailed distributions such as the Student-$t$ may be preferable. When $\boldsymbol{\mu}$ cannot be tractably integrated out, inferences can be based on Laplace approximations or posterior sampling with a reversible-jump MCMCs [@Green_1995]. Due to conjugacy under Gaussian likelihoods, the normal prior leads to closed form marginal likelihoods. Hence, for computational ease in the rest of this paper, normal regime mean priors are used.
The BMDL expression {#subsection:bmdl_univariate}
-------------------
To derive the BMDL expression in , the data likelihood is first obtained. This is then integrated over $\boldsymbol\mu$ to obtain the mixture MDL; finally, two-part MDLs are obtained for the rest of the parameters.
Given a changepoint model $\boldsymbol{\eta}$, the sampling distribution has the regression representation $$\label{eq:likelihood3}
\mathbf{X}_{1:N} = \mathbf{A}_{1:N} \mathbf{s} +
\mathbf{D}_{1:N}\boldsymbol\mu + \boldsymbol\epsilon_{1:N},$$ with $\mathbf{A}_{1:N}\in \mathbb{R}^{N \times T}$ and $\mathbf{D}_{1:N} \in \mathbb{R}^{N \times m}$ as seasonal and regime indicator matrices, respectively: $$\begin{aligned}
&\left[ \mathbf{A}_{1:N} \right]_{t,v} =
\mathbf{1}( \text{time } t \text{ is in season } v ), ~~~ v = 1, \ldots, T,\\
&\left[ \mathbf{D}_{1:N} \right]_{t,r-1} =
\mathbf{1}( \text{time } t \text{ is in regime } r ), ~~~ r = 2, \ldots, m + 1,\end{aligned}$$ where $\mathbf{1}(A)$ denotes the indicator of the event $A$. In , the subscript $1:N$, or in general ${t_1:t_2}$, signifies that only rows $t_1$ through $t_2$ are used in the quantities. The normal white noises $\{ Z_t \}$ in the AR process imply the distributional result $\boldsymbol\epsilon_{(p+1):N} - \sum_{j = 1}^p
\phi_j \boldsymbol\epsilon_{{(p+1-j):(N-j)}} \sim \text{N}(\mathbf{0},
\sigma^2 \mathbf{I}_{N-p})$, where $\mathbf{I}_k$ denotes the $k \times
k$ identity matrix. Now define $$\begin{aligned}
\label{eq:X_tilde}
\widetilde{\mathbf{X}}
&= \mathbf{X}_{(p+1):N}
- \sum_{j = 1}^p \phi_j \mathbf{X}_{(p+1-j):(N-j)}, \\ \label{eq:A_D_tilde}
\widetilde{\mathbf{A}}
&= \mathbf{A}_{(p+1):N}
- \sum_{j = 1}^p \phi_j \mathbf{A}_{(p+1-j):(N-j)}, \quad
\widetilde{\mathbf{D}}
= \mathbf{D}_{(p+1):N}
- \sum_{j = 1}^p \phi_j \mathbf{D}_{(p+1-j):(N-j)},\end{aligned}$$ and observe that $$\label{eq:likelihood5}
\widetilde{\mathbf{X}} - \widetilde{\mathbf{A}} \mathbf{s}
- \widetilde{\mathbf{D}} \boldsymbol\mu \sim
\text{N}(\mathbf{0}, \sigma^2 \mathbf{I}_{N-p}).$$ Note that all terms superscripted with $\sim$ depend on the unknown AR parameter $\boldsymbol{\phi}$. To avoid AR edge effects, a likelihood conditional on the initial observations $\mathbf{X}_{1:p}$ is used. In the change of variable computations, the Jacobian $|\partial
(\widetilde{\mathbf{X}} - \widetilde{\mathbf{A}} \mathbf{s} -
\widetilde{\mathbf{D}} \boldsymbol\mu) / \partial \mathbf{X}_{(p +
1):N}| = 1$ and the likelihood has the multivariate normal form $$f\left(\mathbf{X}_{(p+1):N} \mid
\boldsymbol\mu, \mathbf{s}, \sigma^2, \boldsymbol{\phi}, \boldsymbol{\eta}\right)
= \left(2\pi\sigma^2\right)^{-\frac{N-p}{2}} e^{-\frac{1}{2\sigma^2}
(\widetilde{\mathbf{X}} - \widetilde{\mathbf{A}} \mathbf{s} -
\widetilde{\mathbf{D}}\boldsymbol\mu)'
(\widetilde{\mathbf{X}} - \widetilde{\mathbf{A}} \mathbf{s} -
\widetilde{\mathbf{D}}\boldsymbol\mu)}.$$ Innovation forms of the likelihood [@Brockwell_Davis_1991] can be used if one wants a moving-average or long-memory component in $\{
\epsilon_t \}$.
We now obtain a BMDL for the changepoint model $\boldsymbol{\eta}$. If $m > 0$, we first use the mixture MDL on $\boldsymbol\mu$. The marginal likelihood, after integrating $\boldsymbol\mu$ out, has the closed form $$\begin{aligned}
& f (\mathbf{X}_{(p+1):N} \mid
\mathbf{s}, \sigma^2, \boldsymbol\phi, \boldsymbol\eta)
= \int_{\mathbb{R}^{m}} f\left(\mathbf{X}_{(p+1):N} \mid
\boldsymbol\mu, \mathbf{s}, \sigma^2, \boldsymbol\phi, \boldsymbol\eta
\right)
\pi(\boldsymbol\mu \mid \sigma^2, \boldsymbol\eta) d\boldsymbol\mu\\
=~& (2\pi\sigma^2)^{-\frac{N-p}{2}} \nu^{-\frac{m}{2}}
\left|\widetilde{\mathbf{D}}'
\widetilde{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right|^{-\frac{1}{2}}
e^{-\frac{1}{2\sigma^2} (\widetilde{\mathbf{X}} -
\widetilde{\mathbf{A}}\mathbf{s})'
\widetilde{\mathbf{B}}
(\widetilde{\mathbf{X}} - \widetilde{\mathbf{A}}\mathbf{s})
},\end{aligned}$$ where the notation has $$\label{eq:B_tilde}
\widetilde{\mathbf{B}} = \mathbf{I}_{N-p} - \widetilde{\mathbf{D}}
\left(\widetilde{\mathbf{D}}'
\widetilde{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right)^{-1}
\widetilde{\mathbf{D}}'.$$ If the parameters $\mathbf{s}$, $\sigma^2$, and $\boldsymbol{\phi}$ are known, the mixture MDL is simply $\mathcal{L}(\mathbf{X}_{(p+1):N} \mid
\mathbf{s}, \sigma^2, \boldsymbol\phi, \boldsymbol\eta) = -\log \{f
(\mathbf{X}_{(p+1):N} \mid \mathbf{s}, \sigma^2, \boldsymbol{\phi},
\boldsymbol{\eta})\}$.
The two-part MDL is used to quantify the cost of transmitting the parameters $\mathbf{s}$, $\sigma^2$, and $\boldsymbol{\phi}$. The optimal $\mathbf{s}$ and $\sigma^2$ that minimize the mixture MDL have closed forms: $$\begin{aligned}
\label{eq:s_hat}
\hat{\mathbf{s}}
& = \arg\min_{\mathbf{s}}
\mathcal{L}(\mathbf{X}_{(p+1):N} \mid \mathbf{s}, \sigma^2,
\boldsymbol\phi, \boldsymbol\eta)
= (\widetilde{\mathbf{A}}'\widetilde{\mathbf{B}}
\widetilde{\mathbf{A}})^{-1}
(\widetilde{\mathbf{A}}'\widetilde{\mathbf{B}}\widetilde{\mathbf{X}}),
\\ \label{eq:sigmasq_hat}
\hat{\sigma}^2
& = \arg\min_{\sigma^2}
\mathcal{L}(\mathbf{X}_{(p+1):N} \mid \hat{\mathbf{s}}, \sigma^2,
\boldsymbol\phi, \boldsymbol\eta)
= \frac{1}{N-p} \widetilde{\mathbf{X}}' \left\{
\widetilde{\mathbf{B}}
- \widetilde{\mathbf{B}} \widetilde{\mathbf{A}}
\left(\widetilde{\mathbf{A}}'
\widetilde{\mathbf{B}}\widetilde{\mathbf{A}}\right)^{-1}
\widetilde{\mathbf{A}}' \widetilde{\mathbf{B}}
\right\}\widetilde{\mathbf{X}}.\end{aligned}$$ These estimators depend on $\boldsymbol{\phi}$; however, the $\boldsymbol{\phi}$ that minimizes $\mathcal{L}(\mathbf{X}_{(p+1):N}
\mid \hat{\mathbf{s}}, \hat{\sigma}^2, \boldsymbol{\phi},
\boldsymbol{\eta})$ is intractable. In general, likelihood estimators for autoregressive models do not have closed forms. Hence, simple Yule-Walker moment estimators, which are asymptotically most efficient and $\sqrt{n}$-consistent under the true changepoint model, are used. There is actually little difference between moment and likelihood estimators for autoregressions [@Brockwell_Davis_1991].
In the linear model , the ordinary least squares residuals are $$\label{eq:Y}
\boldsymbol\epsilon_{1:N}^{\text{ols}} = (\mathbf{I}_N -
\mathcal{P}_{[\mathbf{A}_{1:N}|\mathbf{D}_{1:N}]})\mathbf{X}_{1:N},$$ where $[\mathbf{A}_{1:N}|\mathbf{D}_{1:N}]$ denotes the block matrix formed by $\mathbf{A}_{1:N}$ and $\mathbf{D}_{1:N}$, and $\mathcal{P}_{[\mathbf{A}_{1:N}|\mathbf{D}_{1:N}]}$ is the orthogonal projection matrix onto its column space. The sample autocovariance of the residuals at lag $h = 0, 1, \ldots, p$ are $\hat{\gamma}(h) = N^{-1} \sum_{t = h + 1}^N \epsilon_t^{\text{ols}}
\epsilon_{t-h}^{\text{ols}}$. The Yule-Walker estimator of $\boldsymbol{\phi}$ is $\hat{\boldsymbol{\phi}} =
\hat{\boldsymbol{\Gamma}}_p^{-1}\hat{\boldsymbol{\gamma}}_p$, where $\hat{\boldsymbol{\gamma}}_p = (\hat{\gamma}(1), \ldots,
\hat{\gamma}(p))'$ and $\hat{\boldsymbol{\Gamma}}_p$ is a $p \times p$ matrix whose $(i,j)$th entry is $\hat{\gamma}(|i-j|)$. This matrix is invertible whenever the data are non-constant [@Brockwell_Davis_1991]. Next, the Yule-Walker estimator $\hat{\boldsymbol{\phi}}$ is substituted for $\boldsymbol{\phi}$ in $\widetilde{\mathbf{X}}$, $\widetilde{\mathbf{A}}$, $\widetilde{\mathbf{D}}$, $\widetilde{\mathbf{B}}$, and $\hat{\sigma}^2$. The resulting quantities are denoted by $\widehat{\mathbf{X}}$, $\widehat{\mathbf{A}}$, $\widehat{\mathbf{D}}$, $\widehat{\mathbf{B}}$, and $\hat{\sigma}^2$, respectively. In particular, $\widehat{\mathbf{X}}$ contains estimated one-step-ahead prediction residuals (innovations).
By , the BMDL for transmitting the data $\mathbf{X}_{(p+1):N}$, the model $\boldsymbol{\eta}$, and its parameters is $$\begin{aligned}
\nonumber
\text{BMDL}(\boldsymbol\eta)
& = \mathcal{L}(\mathbf{X}_{(p+1):N} \mid \hat{\mathbf{s}}, \hat{\sigma}^2,
\hat{\boldsymbol\phi}, \boldsymbol\eta)
+ \mathcal{L}(\hat{\mathbf{s}}, \hat{\sigma}^2,\hat{\boldsymbol\phi} \mid \boldsymbol\eta)
+ \mathcal{L}(\boldsymbol\eta)\\ \label{eq:BMDL_sum}
& = -\log \left\{f(\mathbf{X}_{(p+1):N} \mid \hat{\mathbf{s}}, \hat{\sigma}^2,
\hat{\boldsymbol\phi}, \boldsymbol\eta)\right\} -\log \left\{\pi(\boldsymbol\eta)\right\}.\end{aligned}$$ The second equality holds because under a uniform encoder $\pi(\mathbf{s}, \sigma^2, \boldsymbol{\phi}) \propto 1$, the two-part MDL $\mathcal{L}(\hat{\mathbf{s}}, \hat{\sigma}^2,\hat{\boldsymbol\phi}
\mid \boldsymbol\eta) = (T + 1 + p)\log(N-p)/2$ is constant across models and hence can be omitted. Therefore, for a model with $m > 0$ changepoints, its BMDL is (up to a constant) $$\begin{aligned}
\label{eq:BMDL_univariate}
\text{BMDL}(\boldsymbol\eta) = ~&
\frac{N-p}{2}\log \left( \hat{\sigma}^2 \right) + \frac{m}{2}\log(\nu)
+ \frac{1}{2}\log\left( \left|\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right| \right)\\ \nonumber
& -\sum_{k=1}^2\log\left\{\Gamma\left(a + m^{(k)}\right)
\Gamma\left(b^{(k)} + N^{(k)} - m^{(k)}\right) \right\}.\end{aligned}$$
For a model with no changepoints ($m=0$), denoted by $\boldsymbol{\eta}_{\o}$, the above procedure needs modification. Since $\boldsymbol{\eta}_{\o}$ does not involve $\boldsymbol{\mu}$, the mixture MDL step can be skipped. As $\mathbf{D}$ has no columns, $\widetilde{\mathbf{B}}$ in is reduced to $\mathbf{I}_{N-p}$, and hence still holds. With the convention that the determinant of a $0 \times 0$ matrix is unity, $\log\left( \left|\widehat{\mathbf{D}}' \widehat{\mathbf{D}}+
\mathbf{I}_{m}/ \nu\right| \right) = 0$. Therefore, also holds for $\boldsymbol{\eta}_{\o}$. This resolves the issue of evaluating $\log(m)$ at $m = 0$ with some existing MDL methods.
Asymptotic properties of the BMDL {#subsection:asymptotics}
---------------------------------
Infill asymptotics, which assume regime lengths tend to infinity with the sample size $N$, have been widely adopted to study consistency of multiple changepoint detection procedures [@Davis_etal_2006; @Davis_Yau_2013; @Du_etal_2016] Under infill asymptotics, a relative changepoint configuration with $m$ changepoints is denoted by $\boldsymbol{\lambda} = (\lambda_1, \ldots, \lambda_{m})'$, where $0 <
\lambda_1 < \cdots < \lambda_m < 1$. Here, time is scaled to $[0,1]$ by mapping time $t$ to $t/N$. For the edges, set $\lambda_0=0$ and $\lambda_{m+1}=1$. For a given $N$, the $r$th changepoint location $\tau_r$ can be recovered from $\boldsymbol{\lambda}$ via $\tau_r =
\lfloor \lambda_r N \rfloor$. The length of the $r$th regime, $N_r =
\lfloor \lambda_r N \rfloor - \lfloor \lambda_{r-1} N \rfloor$, satisfies $\lim_{N\rightarrow \infty} N_r / N = \lambda_r -
\lambda_{r-1}$, for $r = 1, \ldots, m+1$. For any $\boldsymbol{\lambda}$, no changepoints occur in time $\{1, \ldots, p
\}$ when $N$ is large.
Suppose that the true relative changepoint configuration is $\boldsymbol{\lambda}^0 = (\lambda_1^0, \ldots, \lambda_{m^0}^0)'$, where true parameter values are superscripted with zero. Our goal is to identify $\boldsymbol{\lambda}^0$ over many candidate models. In fact, for a (fixed) large integer $M$, all relative changepoint configurations in $$\boldsymbol{\Lambda} = \{
\boldsymbol{\lambda}: 0 \leq m \leq M,
\min_{r = 1, 2, \ldots, m + 1} \lambda_r - \lambda_{r-1} \geq d \}$$ are considered, where $d$ is a small positive constant, smaller than $\lambda_r^0 - \lambda_{r-1}^0$ for all $r =1, \ldots, m^0 + 1$. We assume that $m^0 \leq M$, such that $\boldsymbol{\lambda}^0 \in
\boldsymbol{\Lambda}$. Also, $M \leq 1/d$.
Under the same assumptions, the automatic MDL for piece-wise AR processes [@Davis_etal_2006] has been shown to consistently estimate relative changepoint locations and model parameters [@Davis_Yau_2013]. The following two theorems show that the BMDL also achieve the same large sample consistency.
\[thm:lambda\_convergence\] Given the observed time series of length $N$, denote the estimated relative changepoint model as $$\label{eq:lambda_hat}
\hat{\boldsymbol\lambda}_N = \arg \min_{\boldsymbol\lambda \in \boldsymbol\Lambda}
~ \text{BMDL}(\boldsymbol\lambda),$$ with $\hat{m}_N = |\hat{\boldsymbol\lambda}_N|$ changepoints. As $N \rightarrow \infty$, $$\label{eq:lambda_convergence}
\hat{m}_N \stackrel{P}{\longrightarrow} m^0 \quad \text{and} \quad
\hat{\boldsymbol\lambda}_N \stackrel{P}{\longrightarrow} \boldsymbol\lambda^0.$$ Furthermore, under large $N$, the convergence rate for each $r = 1, \ldots, m^0$ is $$\label{eq:lambda_convergence_rate}
\left| \hat{\lambda}_r - \lambda^0_r \right| = O_P\left(\frac{1}{N}\right).$$
\[thm:parameter\_convergence\] Suppose that under the true model $\boldsymbol\lambda^0$, the true model parameters are $\boldsymbol\mu^0, \mathbf{s}^0, (\sigma^{2})^0$, and $\boldsymbol\phi^0$. Under the estimated relative changepoint model $\hat{\boldsymbol\lambda}_N$ , the BMDL estimator for $\boldsymbol\phi$, denoted by $\hat{\boldsymbol\phi}_N$, is given by the Yule-Walker estimator described in Section \[subsection:bmdl\_univariate\]; the BMDL estimator for $\mathbf{s}$ and $\sigma^2$, denoted by $\hat{\mathbf{s}}_N$ and $\hat{\sigma}^{2}_N$, are given by and after replacing all terms containing $\boldsymbol\phi$ by $\hat{\boldsymbol\phi}_N$, respectively; the BMDL estimator for $\boldsymbol\mu$ is taken as its conditional posterior mean $$\label{eq:mu_hat}
\hat{\boldsymbol\mu}_N =
E\left(\boldsymbol\mu \mid \hat{\mathbf{s}}_N, \hat{\sigma}^{2}_N, \hat{\boldsymbol\lambda}_N,
\hat{\boldsymbol\lambda}_N, \mathbf{X}_{1:N}\right)
= \left(\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right)^{-1}
\widehat{\mathbf{D}}' \left( \widehat{\mathbf{X}} - \widehat{\mathbf{A}}\hat{\mathbf{s}}_N \right).$$ Then as $N\rightarrow \infty$, all estimators converge to their true values in probability, i.e., $$\label{eq:parameter_convergence}
\hat{\boldsymbol\mu}_N \stackrel{P}{\longrightarrow} \boldsymbol\mu^0, \quad
\hat{\mathbf{s}}_N \stackrel{P}{\longrightarrow} \mathbf{s}^0, \quad
\hat{\sigma}^{2}_N \stackrel{P}{\longrightarrow} (\sigma^{2})^0, \quad
\hat{\boldsymbol\phi}_N \stackrel{P}{\longrightarrow} \boldsymbol\phi^0.$$
Proofs of Theorem \[thm:lambda\_convergence\] and \[thm:parameter\_convergence\] are given in the Appendix Section \[PROOFthm:lambda\_convergence\] and \[PROOFthm:parameter\_convergence\], respectively. The convergence rate $O_P(1/N)$ in is viewed as the optimal rate in the multiple changepoint detection literature [@Niu_etal_2016]. From a Bayesian model selection perspective, a model selection criterion is consistent if the ratio of posterior probabilities between the true model $\boldsymbol{\lambda}^0$ and any other model $\boldsymbol{\lambda} \in \boldsymbol{\Lambda}$ tends to infinity [@Clyde_George_2004]. This is equivalent to the BMDL difference $\text{BMDL}\left(\boldsymbol\lambda\right)
-\text{BMDL}\left(\boldsymbol\lambda^0\right) \longrightarrow \infty$, which is shown to hold in Proposition \[prop:pairwise\_BMDL\_OpN\] and \[prop:pairwise\_BMDL\_OplogN\] in the Appendix.
To better understand our BMDL penalty, we compare it to the MDL obtained when the automatic code length rules in @Davis_etal_2006 are applied to our multiple mean shift problem: $$\label{eq:MDL_univariate}
\text{MDL}(\boldsymbol\eta) =
\frac{N-p}{2}\log \left( \hat{\sigma}^2_{\nu = \infty} \right)
+ \frac{1}{2}\sum_{j=2}^{m+1}\log(N_r) + \log(m + 1) + (m+1) \log(N-p).$$ The first term in approximates the negative logarithm of the maximum likelihood, where the Yule-Walker estimator of $\sigma^2$ is used, which equals with $\nu =
\infty$ after $\boldsymbol\phi$ is replaced by $\hat{\boldsymbol\phi}$. This estimator is denoted by $\hat{\sigma}^2_{\nu = \infty}$ here. The other terms in are the two-part MDLs for the regime means $\mu_2, \ldots, \mu_{m+1}$, the number of changepoints $m$ (the original penalty of $\log(m)$ is undefined for the null model with $m=0$; the ad-hoc fix to this simply uses $m+1$ in the logarithm), and the regime lengths $N_1, \ldots, N_{m+1}$, respectively. The two-part MDLs of the global parameters $\mathbf{s}$, $\sigma^2$, and $\boldsymbol\phi$ are constants and hence omitted. An MDL for the AR order $p$ is not needed as $p$ is assumed known. Under a given relative changepoint model $\boldsymbol\lambda$, increases linearly with $N$. The following theorem states that the difference between the BMDL in and the automatic MDL in is asymptotically bounded.
\[prop:BMDL\_MDL\] For any relative changepoint model $\boldsymbol{\lambda} \in \boldsymbol{\Lambda}$, as $N \rightarrow \infty$, up to an additive constant, $$\text{BMDL}(\boldsymbol\lambda) - \text{MDL}(\boldsymbol\lambda) = O_P(1).$$
A proof of Theorem \[prop:BMDL\_MDL\] is obtained by comparing the large sample performance of the corresponding terms in and via order estimates derived in the Appendix. In the BMDL expression , all but the last term arise from the mixture MDL. The term $(N-p)\log \left( \hat{\sigma}^2 \right)/2$ measures the model’s goodness-of-fit. By Lemma \[lemma:sigmasq1\] in the Appendix, $\hat{\sigma}^2 =\hat{\sigma}^2_{\nu = \infty} + O_P(1/N)$; hence, the difference between the first terms in and obeys $$\frac{N-p}{2}\log \left( \hat{\sigma}^2 \right) - \frac{N-p}{2}\log
\left( \hat{\sigma}^2_{\nu = \infty} \right) = O_P(1).$$ In , the second term is $O_P(1)$, while the third term, by Lemma \[lemma:det\] in the Appendix, satisfies $$\frac{1}{2}
\log\left( \left|\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right| \right)
= \frac{1}{2}\sum_{j=2}^{m+1}\log(N_r) + O_P\left(1\right),$$ which interestingly suggests that the mixture MDL in contains a built in penalty on $\boldsymbol\mu$ that performs similarly to the two-part MDL penalty on $\boldsymbol\mu$ in . The last term in is the penalty on the changepoint configuration $\boldsymbol{\lambda}$. With or without metadata, Lemma \[lemma:Gamma\_pairdiff\] in the Appendix suggests that this term is asymptotically $m\log(N) + O_P(1)$, which only differs from the last term in by $O_P(1)$ plus a constant.
An implication of Theorem \[prop:BMDL\_MDL\] is that the model selection consistency results in Theorem \[thm:lambda\_convergence\] also hold for the automatic MDL , which gives alternate confirmation of the asymptotic results in @Davis_etal_2006 and @Davis_Yau_2013. In addition, without metadata, the BMDL and the automatic MDL perform similarly for large samples. Section \[sec:simulation\] confirms this result via simulation examples, also demonstrating that when metadata is available and incorporated, the BMDL significantly increases changepoint detection power and precision under finite samples.
BMDL optimization {#subsection:computation}
-----------------
The optimal changepoint model $\hat{\boldsymbol{\eta}}$ is selected as the one with the smallest BMDL score. However, exhaustively searching the changepoint configuration space is formidable since the total number of admissible models, $2^{N-p}$, is extremely large. To overcome this, genetic algorithms are used as optimization tools in @Davis_etal_2006 [@Lu_etal_2010]. Genetic algorithms efficiently explore the model space, only evaluating the penalized likelihood at a relatively small number of promising models.
The following connection to empirical Bayes (EB) methods allow us to borrow MCMC model search algorithms that are commonly used in Bayesian model selection. The BMDL under model $\boldsymbol\eta$ represented in is equivalent to the negative logarithm of an EB estimator of the posterior probability of $\boldsymbol{\eta}$: $$p_{\text{EB}}(\boldsymbol\eta \mid \mathbf{X}_{(p+1):N})
\propto~ \pi(\boldsymbol\eta)
\int_{\mathbb{R}^{m}} f\left(\mathbf{X}_{(p+1):N} \mid
\boldsymbol\mu, \hat{\mathbf{s}}, \hat{\sigma}^2, \hat{\boldsymbol\phi},
\boldsymbol\eta \right)
\pi(\boldsymbol\mu \mid \hat{\sigma}^2, \boldsymbol\eta)
d\boldsymbol\mu.$$
As our BMDL formula is tractable, Bayesian stochastic model search algorithms can be used; see @Garcia-Donato_Martinez-Beneito_2013 and the references therein. Here, we modify the Metropolis-Hastings algorithm in @George_McCulloch_1997 by intertwining two types of proposals: a component-wise flipping at a random location and a simple random swapping between a changepoint and a non-changepoint. This algorithm is described in detail in @Li_Lund_2015 and can be implemented by the R package [BayesMDL]{} (<https://github.com/yingboli/BayesMDL>).
Extensions to Multivariate Time Series {#section:bivariateBMDL}
======================================
Mimicking the univariate setup, this section develops a BMDL for multivariate time series. While the details are illustrated for bivariate series, similar extensions apply to multivariate series of more than two components. The BMDL penalty constructed here allows changepoints to occur in one or both component series. Furthermore, it can accommodate domain experts’ knowledge that encourage concurrent changes, i.e., changes affecting both series at the same time.
In temperature homogenization, to model Tmax and Tmin series jointly, both series are concatenated via $\mathbf{X}_{1:N} =
(\mathbf{X}_{1:N,1}', \mathbf{X}_{1:N,2}')'$ $\in \mathbb{R}^{2N}$, where $\mathbf{X}_{1:N,i} = (X_{1,i}, \ldots, X_{N,i})'$ is the record for Tmax ($i=1$) or Tmin ($i=2$). Again, each time in $\{ p+1, \ldots, N
\}$ is allowed to be a changepoint in either the Tmax or Tmin series, or both. A multiple changepoint configuration is denoted by $\boldsymbol{\eta} = (\boldsymbol{\eta}_1', \boldsymbol{\eta}_2')'$, where $\boldsymbol{\eta}_i = (\eta_{p+1,i}, \ldots, \eta_{N,i})' \in
\{0, 1\}^{N-p}$ is defined as in the univariate case. Given a bivariate changepoint model $\boldsymbol{\eta}$, series $i$ has $m_i =
\sum_{t=p+1}^N \eta_{t,i}$ changepoints. As in the univariate case, the seasonal means are denoted by $\mathbf{s}_i = (s_{1,i}, \ldots,
s_{T,i})' \in \mathbb{R}^T$; regime means are denoted by $\boldsymbol{\mu}_i = (\mu_{2,i}, \ldots, \mu_{m_i+1,i})' \in
\mathbb{R}^{m_i}$. The seasonal and regime indicator matrices $\mathbf{A}_{1:N,i} \in \mathbb{R}^{N \times T}$ and $\mathbf{D}_{1:N,
i} \in \mathbb{R}^{N \times m_i}$ are constructed analogously to their univariate counterparts.
The regression representation holds for the bivariate case, with $\mathbf{s} = (\mathbf{s}_1', \mathbf{s}_2')'$, $\boldsymbol{\mu} = (\boldsymbol{\mu}_1', \boldsymbol{\mu}_2')'$, $\boldsymbol{\epsilon}_{1:N} = (\boldsymbol\epsilon_{1:N,1}',
\boldsymbol\epsilon_{1:N,2}')'$ denoting the concatenated seasonal means, regime means, and regression errors, respectively. The seasonal and regime indicator matrices have the block diagonal forms $\mathbf{A}_{1:N} = \text{diag}\left(\mathbf{A}_{1:N, 1},
\mathbf{A}_{1:N, 2} \right)$ and $\mathbf{D}_{1:N} =
\text{diag}\left(\mathbf{D}_{1:N, 1}, \mathbf{D}_{1:N, 2} \right)$. Note that the seasonal indicators for Tmax and Tmin coincide, i.e., $\mathbf{A}_{1:N, 1} = \mathbf{A}_{1:N, 2}$, while $\mathbf{D}_{1:N,1}$ and $\mathbf{D}_{1:N,2}$ differ unless all changepoints are concurrent.
As Tmax and Tmin temperature series tend to fluctuate about the seasonal mean in tandem (positive correlation), the errors $\{
\boldsymbol{\epsilon}_t = (\epsilon_{t, 1}, \epsilon_{t, 2})' \}$ need to be correlated across components. For this, a vector autoregressive model (VAR) of order $p$ is employed: $$\boldsymbol\epsilon_t = \sum_{j=1}^p
\boldsymbol\Phi_j \boldsymbol\epsilon_{t-j}
+ \mathbf{Z}_t,
\quad
\mbox{Cov}(\mathbf{Z}_t) = \boldsymbol\Sigma,$$ where $\boldsymbol{\Phi}_1, \ldots, \boldsymbol{\Phi}_p$ are $2 \times
2$ VAR coefficient matrices. The VAR model allows for correlation in time and between components.
As holds after replacing $\sigma^2\mathbf{I}_{N-p}$ with $\boldsymbol{\Sigma} \otimes
\mathbf{I}_{N-p}$, the likelihood of $\mathbf{X}_{(p+1):N}$, conditional on the initial observations $\mathbf{X}_{1:p}$, is (up to a multiplicative constant) $$f(\mathbf{X}_{(p+1):N} \mid \mathbf{s}, \boldsymbol\mu, \boldsymbol\Sigma,
\boldsymbol{\Phi}_{1:p}, \boldsymbol\eta)
\propto
\left| \boldsymbol\Sigma \right|^{-\frac{N-p}{2}}
e^{
-\frac{1}{2} (\widetilde{\mathbf{X}}
- \widetilde{\mathbf{A}} \mathbf{s} - \widetilde{\mathbf{D}} \boldsymbol\mu)'
(\boldsymbol\Sigma^{-1} \otimes \mathbf{I}_{N-p}) (\widetilde{\mathbf{X}}
- \widetilde{\mathbf{A}}\mathbf{s}- \widetilde{\mathbf{D}} \boldsymbol\mu)
}.$$ Here, $\otimes$ denotes a Kronecker product and the terms $\widetilde{\mathbf{X}}, \widetilde{\mathbf{A}}, \widetilde{\mathbf{D}}$ are modified by replacing $\phi_j$ with $\boldsymbol{\Phi}_j \otimes
\mathbf{I}_{N-p}$ in and , for $j
= 1, \ldots, p$.
Prior specifications {#prior-specifications}
--------------------
For $t = p + 1, \ldots, N$, the indicator $\boldsymbol{\eta}_{t} = (\eta_{t,1}, \eta_{t,2})'$ takes values in one of the four categories: $(1,1)'$, mean shifts in both Tmax and Tmin; $(1,0)'$, a mean shift in Tmax but not in Tmin; $(0,1)'$, a mean shift in Tmin but not in Tmax; and $(0,0)'$, no mean shifts. As a natural extension of the Beta-Binomial prior, a Dirichlet-Multinomial prior is put on $\boldsymbol{\eta}_{t}$: $$\boldsymbol\eta_{t} \mid \boldsymbol\rho
\stackrel{\text{iid}}{\sim} \text{Multinomial}(1; \boldsymbol{\rho}),
\quad
\boldsymbol{\rho} \sim \text{Dirichlet}(\boldsymbol{\alpha}),$$ where $\boldsymbol{\rho} = (\rho_1, \ldots, \rho_4)'$ are the probabilities of the four categories satisfying $\sum_{\ell=1}^4
\rho_\ell = 1$, and $\boldsymbol{\alpha} = (\alpha_1, \ldots,
\alpha_4)'$ are the Dirichlet parameters with $\alpha_\ell > 0$ for each $\ell = 1, \ldots, 4$. Suppose that the changepoint configuration $\boldsymbol{\eta}$ has $m_\ell$ times in category $\ell$. Due to Dirichlet-multinomial conjugacy, the marginal prior of $\boldsymbol{\eta}$ has a closed form after integrating out $\boldsymbol{\rho}^{(1)}$ and $\boldsymbol{\rho}^{(2)}$: $$\pi(\boldsymbol\eta) \propto
\prod_{k = 1}^2\prod_{\ell=1}^4
\Gamma\left(\alpha_\ell^{(k)} + m_\ell^{(k)}\right).$$ Again, superscripts $(1)$ and $(2)$ refer to non-metadata and metadata related terms, respectively.
The choice of the hyper-parameter $\boldsymbol{\alpha}$ should reflect our belief that concurrent changepoints are more likely to occur than when the component series are independent. The ratios between the prior expectations satisfy $E(\rho_1) : E(\rho_2) : E(\rho_3) : E(\rho_4) =
\alpha_1 : \alpha_2 : \alpha_3 : \alpha_4$. If changepoints in the Tmax and Tmin series at time $t$ are independent events, then $\rho_1 =
P(\eta_{t,1} = 1, \eta_{t,2} = 1) = P(\eta_{t,1} = 1) P(\eta_{t,2} = 1)
= (\rho_1 + \rho_2)(\rho_1 + \rho_3)$. To encourage concurrent shifts, $\boldsymbol{\alpha}$ is hence chosen such that $$E(\rho_1) =
\frac{\alpha_1}{\sum_{\ell=1}^4 \alpha_{\ell}} > \frac{\alpha_1 +
\alpha_2} {\sum_{\ell=1}^4 \alpha_{\ell}} ~ \frac{\alpha_1 +
\alpha_3}{\sum_{\ell=1}^4 \alpha_{\ell}} = E(\rho_1 +\rho_2) E(\rho_1 +
\rho_3).$$ In addition, the prior probability of not obtaining a changepoint at a time is set to its counterpart in the univariate case, i.e., $\alpha_4/
\sum_{\ell=1}^4 \alpha_{\ell} = b / (a+b)$. After consulting climatologists, default hyper-parameters are set to $\boldsymbol{\alpha}^{(1)} = \left(3/7, 2/7, 2/7, 239\right)'$ and $\boldsymbol{\alpha}^{(2)} = \left(3/7, 2/7, 2/7, 47\right)'$ for monthly data.
To obtain the mixture MDL in a closed form, for a bivariate model with $m = m_1+m_2 > 0$ changepoints, the regime means $\boldsymbol{\mu}$ again are taken to have independent normal priors $$\boldsymbol\mu \mid \boldsymbol\Sigma, \boldsymbol\eta
\sim \text{N}(\mathbf{0}, \boldsymbol\Omega), \quad
\boldsymbol\Omega = \nu ~ \text{diag}
\left(\underbrace{\sigma_1^2, \ldots, \sigma_1^2}_{m_1},
\underbrace{\sigma_2^2, \ldots, \sigma_2^2}_{m_2}\right),$$ where $\sigma_1^2$ and $\sigma_2^2$ are the diagonal entries of the white noise covariance $\boldsymbol{\Sigma}$.
The bivariate BMDL
------------------
For a model $\boldsymbol\eta$ with $m > 0$, the marginal likelihood, after integrating $\boldsymbol{\mu}$ out, has a closed form: $$f(\mathbf{X}_{(p+1):N} \mid \mathbf{s}, \boldsymbol\Sigma,
\boldsymbol\Phi_{1:p}, \boldsymbol\eta)\
\propto
\left| \boldsymbol\Sigma \right|^{-\frac{N-p}{2}} \left| \boldsymbol\Omega \right|^{-\frac{1}{2}}
\left| \widetilde{\mathbf{D}}' (\boldsymbol\Sigma^{-1}\otimes \mathbf{I}_{N-p})
\widetilde{\mathbf{D}} + \boldsymbol\Omega^{-1} \right|^{-\frac{1}{2}}
e^{-\frac{1}{2} (\widetilde{\mathbf{X}} -
\widetilde{\mathbf{A}}\mathbf{s})'
\widetilde{\mathbf{B}}
(\widetilde{\mathbf{X}} - \widetilde{\mathbf{A}}\mathbf{s})},$$ where $\widetilde{\mathbf{B}}$ is modified to $$\widetilde{\mathbf{B}} =
(\boldsymbol\Sigma^{-1}\otimes \mathbf{I}_{N-p}) \times
\left[\mathbf{I}_{2(N-p)} - \widetilde{\mathbf{D}}
\left\{\widetilde{\mathbf{D}}' (\boldsymbol\Sigma^{-1}\otimes \mathbf{I}_{N-p})
\widetilde{\mathbf{D}}+\boldsymbol\Omega^{-1}\right\}^{-1}
\widetilde{\mathbf{D}}'(\boldsymbol\Sigma^{-1}\otimes \mathbf{I}_{N-p})
\right].$$ The maximum marginal likelihood estimator $\tilde{\mathbf{s}}$ is unaltered from . However, after plugging $\hat{\mathbf{s}}$ back into the likelihood, the maximum likelihood estimators of $\boldsymbol{\Sigma}$ and $\boldsymbol{\Phi}_1, \ldots,
\boldsymbol{\Phi}_p$ do not have closed forms. Again, Yule-Walker estimators are used.
To find Yule-Walker estimators for the time series regression , generalized least squares residuals of the mean fit, denoted by $\boldsymbol\epsilon_{1:N}^\text{gls} =
((\boldsymbol\epsilon_{1:N, 1}^{\text{gls}})',
(\boldsymbol\epsilon_{1:N, 2}^\text{gls})')' \in \mathbb{R}^{2N}$, are computed via $$\boldsymbol\epsilon_{1:N}^\text{gls} = \left[\mathbf{I}_{2N} - \mathbf{G}
\left\{ \mathbf{G}'\left(\hat{\boldsymbol{\Gamma}}^\text{ols}(0)^{-1} \otimes
\mathbf{I}_N\right)\mathbf{G} \right\}^{-1}
\mathbf{G}'\left(\hat{\boldsymbol{\Gamma}}^\text{ols}(0)^{-1} \otimes
\mathbf{I}_N\right)\right]
\mathbf{X}_{1:N},$$ where $$\mathbf{G} = \left[ \begin{array}{cccc}
\mathbf{A}_{1:N, 1} & \mathbf{D}_{1:N,1} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{A}_{1:N,2} & \mathbf{D}_{1:N,2} \\
\end{array} \right].$$ Here, $\hat{\boldsymbol{\Gamma}}^\text{ols}(0) = N^{-1}\sum_{t=1}^N
\boldsymbol\epsilon_t^{\text{ols}}
(\boldsymbol\epsilon_t^{\text{ols}})^\prime$ is a $2 \times 2$ covariance matrix of the ordinary (unweighted) least squares residuals $\boldsymbol\epsilon_t^{\text{ols}} = (\epsilon_{t, 1}^{\text{ols}},
\epsilon_{t, 2}^{\text{ols}})^\prime$, where $\epsilon_{t,
1}^{\text{ols}}$ and $\epsilon_{t, 2}^{\text{ols}}$ are computed analogously to with the design matrices $[\mathbf{A}_{1:N,1}| \mathbf{D}_{1:N,1}]$ and $[\mathbf{A}_{1:N,2} |
\mathbf{D}_{1:N,2}]$, respectively. The sample autocovariances at lag $h
= 0, 1, \ldots, p$ of the generalized least squares residuals $\boldsymbol\epsilon_t^\text{gls} = (\epsilon_{t, 1}^\text{gls},
\epsilon_{t, 2}^\text{gls})^\prime, t =1, \ldots, N$ are computed as $\hat{\boldsymbol{\Gamma}}(h) = N^{-1}\sum_{t = h+1}^N
\boldsymbol\epsilon_t^\text{gls}
(\boldsymbol\epsilon_{t-h}^{\text{gls}})^\prime$. The Yule-Walker estimators thus obey $$\left(
\hat{\boldsymbol{\Phi}}_1, \ldots, \hat{\boldsymbol{\Phi}}_p \right) =
\left( \widehat{\boldsymbol\Gamma}(1), \ldots,
\widehat{\boldsymbol\Gamma}(p) \right)
\left[
\begin{array}{cccc}
\widehat{\boldsymbol\Gamma}(0) & \widehat{\boldsymbol\Gamma}(1)
& \cdots & \widehat{\boldsymbol\Gamma}(p-1)\\
\widehat{\boldsymbol\Gamma}(1)' & \widehat{\boldsymbol\Gamma}(0)
& \cdots & \widehat{\boldsymbol\Gamma}(p-2)\\
\vdots & \vdots
& \ddots & \vdots \\
\widehat{\boldsymbol\Gamma}(p-1)' & \widehat{\boldsymbol\Gamma}(p-2)'
& \cdots & \widehat{\boldsymbol\Gamma}(0)\\
\end{array}
\right]^{-1}$$ and $\widehat{\boldsymbol{\Sigma}} = \hat{\boldsymbol{\Gamma}}(0) -
\sum_{j = 1}^p \hat{\boldsymbol{\Phi}}_j \hat{\boldsymbol{\Gamma}}(j)'$.
After plugging $\widehat{\boldsymbol\Sigma}$ and $\widehat{\boldsymbol\Phi}_1, \ldots, \widehat{\boldsymbol\Phi}_p$ back into the marginal likelihood, the terms $\widetilde{\mathbf{X}},
\widetilde{\mathbf{A}}, \widetilde{\mathbf{D}}, \widetilde{\mathbf{B}}$, and $\boldsymbol\Omega$, which depend on $\boldsymbol\Sigma$ and $\boldsymbol\Phi_1, \cdots, \boldsymbol\Phi_p$, are denoted by $\widehat{\mathbf{X}}, \widehat{\mathbf{A}}, \widehat{\mathbf{D}},
\widehat{\mathbf{B}}, \widehat{\boldsymbol\Omega}$, respectively. Hence, the Bayesian MDL for $\boldsymbol{\eta}$ is (up to a constant) $$\begin{aligned}
\text{BMDL}(\boldsymbol\eta)
=~& \frac{N-p}{2}\log \left( \left| \widehat{\boldsymbol\Sigma} \right| \right)
+ \frac{1}{2}\sum_{i=1}^2 m_i \log (\nu\hat{\sigma}_i^2)
+ \frac{1}{2}\log \left( \left| \widehat{\mathbf{D}}'
(\widehat{\boldsymbol\Sigma}^{-1}\otimes \mathbf{I}_{N-p})
\widehat{\mathbf{D}} + \widehat{\boldsymbol\Omega}^{-1} \right|\right)\\
& + \frac{1}{2}\widehat{\mathbf{X}}' \left\{
\widehat{\mathbf{B}}
- \widehat{\mathbf{B}} \widehat{\mathbf{A}}
\left(\widehat{\mathbf{A}}'
\widehat{\mathbf{B}}\widehat{\mathbf{A}}\right)^{-1}
\widehat{\mathbf{A}}' \widehat{\mathbf{B}}
\right\}\widehat{\mathbf{X}} - \sum_{k = 1}^2\sum_{\ell=1}^4
\log \left\{ \Gamma\left(\alpha_{\ell}^{(k)} + m_{\ell}^{(k)}\right) \right\}.\end{aligned}$$ Under the null model $\boldsymbol\eta_{\o}$, since $\widehat{\mathbf{B}}
= \widehat{\boldsymbol\Sigma}^{-1}\otimes \mathbf{I}_{N-p}$, with the convention that the determinant of a $0 \times 0$ matrix is unity, the above BMDL still holds.
Simulation Studies {#sec:simulation}
==================
This section studies changepoint detection performance under finite samples via simulation. Our simulation parameters are selected to roughly resemble the bivariate Tuscaloosa data, which will be studied in Section \[sec:Tuscaloosa\]. Specifically, the bivariate error series $\{ \boldsymbol\epsilon_t \}$ is chosen to follow a zero mean Gaussian VAR model with $p=3$. The VAR parameters are taken as $$\boldsymbol\Phi_1 = \left(
\begin{array}{cc}
0.2 & 0.02 \\
0.02 & 0.2
\end{array} \right),
\boldsymbol\Phi_2 = \left(
\begin{array}{cc}
0.1 & 0.01 \\
0.01 & 0.1
\end{array} \right),
\boldsymbol\Phi_3 = \left(
\begin{array}{cc}
0.05 & 0.005 \\
0.005 & 0.05
\end{array} \right),
\boldsymbol\Sigma = \left(
\begin{array}{cc}
9 & 2 \\
2 & 9
\end{array} \right).$$ In each of 1000 independent runs, 50 year monthly Tmax and Tmin series ($N = 600$) are simulated with $m = 3$ changepoints in each series. For the Tmax series, mean shifts are placed at the times $150, 300$, and $450$. The regime means have form $\boldsymbol{\mu}_1 = (0, \Delta, 2
\Delta, 3 \Delta)'$ where $\Delta > 0$ will be varied. For the Tmin series, mean shifts are placed at times $150, 300$, and $375$. The regime means are $\boldsymbol{\mu}_2 = (0, -\Delta, \Delta, 0)'$. Here, Tmax has monotonic “up, up, up” shifts of equal shift magnitudes; Tmin shifts in a “down, up, down” fashion and the second shift is twice as large as the other two shifts. The shifts at times 150 and 300 are concurrent in both series.
Seasonal means are set to $\mathbf{s} = (0, 3, 10, 18, 26, 33, 36, 36,
31, 20, 8, 2)'$ in both series. Seasonal mean parameters are not critical, but the $\Delta$ parameter controlling the mean shift size is. Our detection powers will be reported under different signal to noise ratios, measured by $\kappa = \Delta / \sigma$. Our study examines $\kappa \in \{ 1, 1.5, 2 \}$, with $\sigma = 3$ agreeing with the diagonal elements of $\boldsymbol\Sigma$. For metadata, a record containing four documented changes at the times $75, 150, 250$, and $550$ is posited. Among the documented times, only time 150 is a true changepoint.
A simulated series with $\kappa = 1.5$ is shown in Figure \[fg:simulation\_sample1\]. Figure \[fg:simulation\_sample2\] in the Appendix shows the same series after subtraction of sample monthly means.
Univariate simulations
----------------------
First, the Tmax and Tmin series are analyzed separately, each fitted by univariate BMDL methods with default parameters, once with the fictitious metadata and once without metadata. We also compare various methods without metadata, including BMDL under the objective Bayes parameters $a=b=1$ (denoted by oBMDL), the automatic MDL (denoted by MDL), and the BIC, which up to a constant, is $$\text{BIC} (\boldsymbol\eta) = \frac{N-p}{2}\log \left(
\hat{\sigma}^2_{\nu = \infty} \right) + m \log(N-p).$$ In each fit, an MCMC chain of 100,000 iterations is generated. The optimal multiple changepoint model is taken as the one that minimizes the objective function.
For Tmax series, Table \[tb:simulation\_Tmax\_uni\] reports empirical detection percentages, including true positive rates at the exact times of changepoints and average false positive rates at non-changepoint times, along with estimated number of changepoints $\hat{m}$ and its standard error. When metadata is ignored, since the three shifts are of equal size $\Delta$, their detection rates are similar. False detection rates are very low; even when $\kappa = 1$, on average, a non-changepoint is detected 0.43% of the time or less.
Among different methods without metadata, detection rates of true changepoints are similar, while BIC flags slightly more false positives than MDL-based methods (BMDL, oBMDL, and MDL). When $\kappa = 1$, the number of changepoints $m=3$ is underestimated by the MDL-based methods and better estimated by BIC penalties; when $\kappa = 1.5$ and $2$, $m$ is better estimated by the MDL-based methods, and overestimated by BIC. Overall, BIC tends to favor models with more changepoints than the MDL-based methods. As suggested by Theorem \[prop:BMDL\_MDL\], the BMDL performs similarly to the automatic MDL.
Interestingly, without metadata, the BMDL under the default parameters $a = 1$ and $b = 239$ and the objective choices $a=b=1$ perform similarly. Figure \[fg:BMDL\_penalties\] in the Appendix reveals that as functions of $m$, the code lengths $\mathcal{L}(\boldsymbol\eta) =
-\log\{\pi(\boldsymbol\eta)\}$ under BMDL and oBMDL have similar shapes, with a nearly constant difference over the region where $m$ is small. In this case, if knowledge of changepoint frequency is not available, a BMDL can still be used with objective parameters.
Metadata use substantially increases detection power for the BMDL. In Figure \[fg:simulation\_Tmax\], the true documented change at time 150 is detected $75.7\%$ of the time when metadata is used, more than twice as high ($36.3\%$) when metadata is eschewed. Moreover, times near the changepoint at time 150 are less likely to be erroneously flagged as changepoints. Our prior belief that metadata times are more likely to be changepoints is especially important when the mean shift is small: when $\kappa = 1$, using metadata increases the detection rate of the time 150 changepoint from $15.4\%$ to $58.8\%$. For false positives, Figure \[fg:simulation\_Tmax\] shows that using metadata does not increase false detection rates at the documented times 75, 250, and 550 (where no shifts occur). This suggests that the prior distribution does not “overwhelm" the data. Table \[tb:simulation\_Tmax\_uni\] shows that average false positive rates even drop after using metadata.
For Tmin series, the non-monotonic shift aspect (down, up, down) that troubles some at most one change (AMOC) binary segmentation approaches [@Li_Lund_2012] is well handled by all multiple changepoint detection methods examined. Table \[tb:simulation\_Tmin\_uni\] shows that when metadata is ignored, the larger shift at time 300 is more easily detected than the two smaller shifts at times 150 and 375. When metadata is used, the detection rate of the time 150 shift becomes comparable to the detection rate of time 300 shift, which is twice as large in size, but is not a metadata time. False positive rates are uniformly low, and $m$ is well-estimated by MDL-based methods when $\kappa$ is not too small. Again, without metadata, the MDL-based methods are similar, while BIC tends to favor models with larger $m$.
Bivariate simulations
---------------------
Since the BMDL is flexible enough to handle non-concurrent shifts for bivariate series, we now apply it to Tmax and Tmin series jointly. Each bivariate series is fitted by an MCMC chain of 50,000 iterations, once without metadata, and once with metadata. Metadata impacts are similar to the univariate case, increasing detection of true mean shifts at metadata times and also slightly decreasing average false positive rates (see Tables \[tb:simulation\_Tmax\_bi\] and \[tb:simulation\_Tmin\_bi\]). Figure \[fg:simulation\_txtn\] shows bivariate detection rates with metadata when $\kappa = 1.5$. For the non-concurrent shifts times 375 and 450, detection rates for the component series are very different; in most runs, concurrent shifts are not erroneously signaled.
While concurrent shifts are not always the case, they are believed to be more likely in our parameter elicitation settings. Compared to the univariate BMDL, the bivariate method enhances detection power of concurrent changepoints. When $\kappa = 1.5$, at time 150, where Tmax (Tmin) shifts $\Delta$ ($-\Delta$), the bivariate BMDL increases the univariate detection rates from both series from about $77\%$ to above $81\%$. At time 300, where Tmax (Tmin) shifts by $\Delta$ ($2\Delta$), the detection rate increases from $41.1\%$ to $82.2\%$ for Tmax, while it remains roughly the same for Tmin. Tables \[tb:simulation\_Tmax\_bi\] and \[tb:simulation\_Tmin\_bi\] show that detection power gains under the bivariate approach are greater for small signals $\kappa = 1$, without metadata. An interesting phenomenon is observed: bivariate BMDL improves univariate methods more when the concurrent shifts move the series in opposite directions than in the same direction (detection rates at time 300 do not increase for Tmin). Because Tmax and Tmin are positively correlated series, concurrent shifts in the same direction may be misattributed to positively correlated errors; this cannot happen for shifts in opposite directions.
Overall, while bivariate detection does not induce more false positives, it tends to flag more false positives at locations where the mean in the other series shifts. Figure \[fg:simulation\_txtn\] shows that at time 375, a changepoint time in Tmin but not in Tmax, a false detection rate of $7.3\%$ for Tmax is obtained. At time 450, a changepoint in Tmax but not Tmin, a false detection rate of $15.2\%$ is obtained for Tmin. These false positive rates slightly degrade inferences at nearby changepoints; for example, at time 450 for Tmax and time 375 for Tmin, detection rates are $34.2\%$ and $33.0\%$, respectively, slightly lower than the $37.9\%$ and $38.2\%$ reported in the univariate case. Finally, Tables \[tb:simulation\_Tmax\_bi\] and \[tb:simulation\_Tmin\_bi\] show that the bivariate approach tends to overestimate $m$, which differs from the univariate case.
The Tuscaloosa Data {#sec:Tuscaloosa}
===================
Monthly Tmax and Tmin series from Tuscaloosa, Alabama (the target station) over the 114 year period January, 1901 – December, 2014 are plotted in Figure \[fg:Tuscaloosa1\]. [@Lu_etal_2010] study annually averaged values of this series from 1901-2000. The Tuscaloosa metadata lists station relocations in November 1921, March 1939, June 1956, and May 1987; November 1956 and May 1987 are listed as instrument change times.
In this section, the Tmax and Tmin series will be analyzed from both univariate and bivariate perspectives via the penalization methods of Section \[sec:simulation\]. All parameters are set to default values; the AR order $p = 2$ is judged as appropriate: by Figure \[fg:Tuscaloosa\_acf\] in the Appendix, almost all sample autocorrelations of residuals fitted with $p=2$ lie inside the pointwise $95\%$ confidence bands.
To ensure convergence in the MCMC search algorithm, for each fit, 50 Markov chains are generated from different starting points, each containing 1,000,000 (univariate) or 100,000 (bivariate) iterations. Among all changepoint models visited by the 50 Markov chains, the one with the smallest BMDL is reported as the optimal model.
Univariate fits
---------------
The top half of Table \[tb:Tuscaloosa\_analysis\] displays estimated changepoints for the univariate fits. When metadata is ignored, all methods (BMDL, oBMDL, MDL, and BIC) estimate the same optimal changepoint configuration: Tmax has two estimated changepoints and Tmin has three; of these, only January 1990 is a concurrent change. Another changepoint is approximately concurrent: March 1957 for Tmax and July 1957 for Tmin. The 1918 changepoint flagged for Tmin is close to the station relocation in November 1921; the station relocation in June 1956 and the equipment change in November 1956 are near the two estimated changepoints in 1957. The metadata time in May 1987 is about three years from the concurrent changepoints flagged in January 1990. Of course, when metadata is ignored, estimated changepoint times may not coincide (exactly) with metadata times.
Repeating the above analysis with metadata, two changepoints are found in Tmax and three in Tmin. All estimated changepoint times now coincide with metadata times. Only the May 1987 changepoint is concurrent. Between Tmax and Tmin, the two estimated changepoints in 1956 (i.e., the two metadata times in 1956) are just a few months apart. As parameter estimates are similar with or without metadata, only estimates for the optimal changepoint model with metadata are reported. For Tmax, estimated regime means are (standard errors in parentheses) $\hat{\mu}_2
= -1.50~(0.24)$ and $\hat{\mu}_3 = 0.66~(0.25)$ (recall that $\mu_1 =
0$); estimated AR(2) coefficients are $\hat{\phi}_1 = 0.21, \hat{\phi}_2
= 0.05$, and $\hat{\sigma}^2 = 11.59$. For Tmin, the estimated parameters are $\hat{\mu}_2 = 1.76~(0.21), \hat{\mu}_3 = -1.06~(0.22),
\hat{\mu}_4 = 2.35~(0.24), \hat{\phi}_1 = 0.18, \hat{\phi}_2 = 0.05$, and $\hat{\sigma}^2 = 10.81$. The concurrent May 1987 changepoint shifts both series to warmer regimes.
Bivariate fits
--------------
Both Tmax and Tmin series are now analyzed in tandem with our methods. Three changepoints are detected in both series, with or without metadata, and all are concurrent (see the bottom half of Table \[tb:Tuscaloosa\_analysis\]). Figure \[fg:Tuscaloosa1\] illustrates the optimal bivariate BMDL changepoint configuration. When metadata is used, all estimated changepoint times migrate to metadata times. Comparing to the univariate results, the bivariate approach yields the same changepoint configuration for Tmin; for Tmax, a new changepoint in November 1921 is flagged and the November 1956 changepoint moves to June 1956, thus becoming a concurrent change. For this changepoint configuration, the estimated VAR parameters are $$\widehat{\boldsymbol{\Phi}}_1 =
\left(
\begin{array}{cc}
0.21 & -0.01 \\
-0.02 & 0.20
\end{array}
\right),
\quad
\widehat{\boldsymbol{\Phi}}_2 =
\left(
\begin{array}{cc}
0.06 & -0.02 \\
-0.04 & 0.08
\end{array} \right),
\quad
\widehat{\boldsymbol{\Sigma}} =
\left(
\begin{array}{cc}
11.56 & 8.13 \\
8.13 & 10.81
\end{array}
\right).$$
In temperature homogenization problems, the goal is often to detect (and then adjust for) “artificial" changes. Naturally occurring climate shifts should be left in the record if possible. Because of this, analyses often consider target minus reference series, where a reference series is a record from a nearby station that shares similar weather with the target station. A changepoint detection analysis using bivariate BMDL is performed on target minus reference data, and is included in the Appendix Section \[subsec:target\_minus\_reference\].
Discussion {#sec:discussion}
==========
This paper developed a flexible MDL-based multiple changepoint detection approach to accommodate [*a priori*]{} information on changepoint times via prior distributional specifications. Motivated by climate homogenization problems, our Bayesian MDL (BMDL) method incorporates subjective knowledge such as metadata in mean shift detection for univariate autoregressive processes with seasonal means, and then extends these ideas to bivariate VAR settings while encouraging concurrent changes in the component series. Both theoretical and simulation studies show that without metadata, our BMDL performs similarly to the state-of-art automatic MDL method; with metadata, the BMDL’s detection power significantly improves under finite samples. Our BMDL has several practical advantages, including simple parameter elicitation, asymptotic consistency, and efficient MCMC computation.
The approach can be extended to accommodate more flexible time series structures, including periodic autoregressions [@Hewa_etal_2017], moving-averages, and multivariate data with more than two series. The methods could also be tailored to categorical data. For count data, the likelihood could be Poisson-based. With a conjugate Gamma prior on means, the resulting marginal likelihoods will again have closed forms. There is no technical difficulty in allowing a background linear trend, or even piecewise linear trends. This said, linear trends can be mistaken for multiple mean shifts should trends be present and ignored in the analysis [@Li_Lund_2015]. In addition, with straightforward modification, the BMDL can handle changes in variance or autocovariance.
Non-MCMC stochastic search methods could also be used. Genetic algorithms, popular in multiple changepoint MDL analyses, are also capable of minimizing the BMDL. Pre-screening methods such as @Chan_etal_2014 [@Yau_Zhao_2016] can speed up model search algorithms. In simple settings when no global parameters exist (i.e., independent observations, no seasonal cycle, error variance known), dynamic programming based techniques such as the PELT [@Killick_etal_2012] can further accelerate computational speed.
Supplementary Materials {#supplementary-materials .unnumbered}
-----------------------
[**Appendix**]{}: includes more theoretical results and theorem proofs in Section \[sec:appendix\_proofs\], and additional simulation and read data examples in Section \[sec:appendix\_examples\].
Acknowledgement {#acknowledgement .unnumbered}
---------------
The authors thank Matthew Menne, Jared Rennie, Claude Williams Jr., and Bin Yu for helpful discussions. The climate application was posed at SAMSI’s 2014 climate homogeneity summit in Boulder, Colorado. Robert Lund and Anuradha Hewaarachchi thank NSF Grant DMS 1407480 for partial support. Clemson University is acknowledged for generous allotment of computation time on its Palmetto cluster.
[Appendix for “Multiple Changepoint Detection with Partial Information on Changepoint Times” ]{}
Theoretical Results and Proofs {#sec:appendix_proofs}
==============================
In this Appendix, the asymptotic limits of the Yule-Walker estimator $\hat{\boldsymbol\phi}$ and white noise variance $\hat{\sigma}^2$ under a given changepoint model $\boldsymbol\lambda$ are investigated in Sections \[subsection:YW\_phi\] and \[subsection:YW\_sigmasq\], respectively. In Section \[sec:asym\_pairwise\_BMDL\], the BMDL difference between the true model $\boldsymbol\lambda^0$ and other models is studied, showing that $\boldsymbol\lambda^0$ achieves the smallest BMDL in the limit. Last, the proofs of Theorem \[thm:lambda\_convergence\] and Theorem \[thm:parameter\_convergence\] are given in Sections \[PROOFthm:lambda\_convergence\] and \[PROOFthm:parameter\_convergence\], respectively.
Asymptotic behavior of the Yule-Walker estimator of the autoregression coefficients $\hat{\boldsymbol\phi}$ {#subsection:YW_phi}
-----------------------------------------------------------------------------------------------------------
For a sample size $N$, the observations obey the true changepoint model $\boldsymbol\lambda^0$ in : $$\mathbf{X}= \mathbf{A} \mathbf{s} +
\mathbf{D}^0\boldsymbol\mu^0 + \boldsymbol\epsilon.$$ Here, $\boldsymbol\epsilon$ is a zero-mean causal AR$(p)$ series. Wherever there is no ambiguity, we simplify the notations $\boldsymbol\mu^0,
\mathbf{s}^0, (\sigma^{2})^0, \boldsymbol\phi^0$ to $\boldsymbol\mu,
\mathbf{s}, \sigma^2, \boldsymbol\phi$, respectively, and omit subscripts such as $1:N$ on the data vector and other quantities.
For any relative changepoint model $\boldsymbol\lambda$, suppose that $\boldsymbol\eta$ is the corresponding changepoint configuration under the sample size $N$. From , the ordinary least squares residual vector is $$\label{eq:Y2}
\boldsymbol\epsilon^{\text{ols}} =
(\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}|\mathbf{D}]})\mathbf{X}
= (\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}|\mathbf{D}]})
(\mathbf{A} \mathbf{s} +
\mathbf{D}^0\boldsymbol\mu+ \boldsymbol\epsilon)
=(\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}|\mathbf{D}]})
(\mathbf{D}^0\boldsymbol\mu + \boldsymbol\epsilon).$$ Here, $[\mathbf{A}|\mathbf{D}]$ is the block matrix formed by $\mathbf{A}$ and $\mathbf{D}$, $\mathcal{P}_{\mathbf{A}}$ is the orthogonal projection onto the columns of the matrix $\mathbf{A}$. The regime indicator matrix $\mathbf{D}$ depends on $\boldsymbol\lambda$ and may not equal $\mathbf{D}^0$.
\[lemma:Y\] For each relative changepoint configuration $\boldsymbol\lambda \in
\boldsymbol\Lambda$ and $t \in \{ 1, \ldots, N \}$, when $N$ is large, each entry of $\boldsymbol\epsilon^{\text{ols}}$ can be expressed as $$\label{eq:delta_t_W_t}
\epsilon_t^{\text{ols}}=\delta_t + W_t, \quad \text{where} \quad
\delta_t = \mu_{r^0(t)} - \bar{\mu}_{r(t)}
\quad \mbox{and} \quad
W_t = \epsilon_t - \bar{\epsilon}_{r(t)} - \bar{\epsilon}_{v(t)} + \bar{\epsilon}.$$ Here, the functions $r^0(t)$ and $r(t)$ are the regimes that time $t$ is in under the models $\boldsymbol\lambda^0$ and $\boldsymbol\lambda$, respectively. In regime $\ell$ of the changepoint configuration $\boldsymbol\lambda$, $\bar{\mu}_{\ell} = N_{\ell}^{-1}
\sum_{t \in {\cal R}_\ell} \mu_t$ is the average of the true mean parameters, $N_\ell$ is the number of time points in this regime, and ${\cal R}_\ell$ is the set of all time points in this regime. Likewise, $\bar{\epsilon}_\ell$ is the average of errors in regime $\ell$, $\bar{\epsilon}_{v}$ is the average of errors during season $v$, and $\bar{\epsilon}$ is the average of all errors.
Because of , our main objective is to study the projection residual $\mathbf{I}_N -
\mathcal{P}_{[\mathbf{A}|\mathbf{D}]}$ under large $N$. Since the two column spaces spanned by $(\mathbf{I}_N - \mathcal{P}_{\mathbf{D}})
\mathbf{A}$ and $\mathbf{D}$ are perpendicular, Theorem B.45 in @Christensen_2002 [pp. 411] gives $\mathcal{P}_{[(\mathbf{I}_N -
\mathcal{P}_{\mathbf{D}}) \mathbf{A}|\mathbf{D}]} =
\mathcal{P}_{(\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}} +
\mathcal{P}_{\mathbf{D}}$. Projection properties give $$\label{eq:Q_AD}
\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}|\mathbf{D}]}
= \mathbf{I}_N - \mathcal{P}_{[(\mathbf{I}_N - \mathcal{P}_{\mathbf{D}})\mathbf{A}|\mathbf{D}]}
= \mathbf{I}_N - \mathcal{P}_{(\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}}
- \mathcal{P}_{\mathbf{D}}.$$ The term $\mathcal{P}_{(\mathbf{I}_N - \mathcal{P}_{\mathbf{D}})
\mathbf{A}}$ can be expanded as $$\label{eq:P_QD_A}
\mathcal{P}_{(\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}}
= (\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}
\left\{\mathbf{A}' (\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}\right\}^{-1}
\mathbf{A}' (\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}).$$ For any $n \in \mathbb{N}$, let $\mathbf{0}_n$ be the $n$-dimensional vector containing all zero entries, $\mathbf{1}_n$ be the $n$-dimensional vector whose entries are all unity, and $\mathbf{J}_n$ be the $n \times n$ matrix whose entries are all unity, i.e., $\mathbf{J}_n = \mathbf{1}_n \mathbf{1}_n'$.
For $v \in \{ 1, \ldots, T \}$, suppose there are $k(v,\ell)$ time points in regime $\ell$ that are also in season $v$. Since $N_\ell$ increases linearly with $N$, so does $k(v,\ell)$. Moreover, when $N$ is large, inside each regime, the seasonal counts $k(v,\ell)$ are equal except for edge effects, i.e., $k(v,\ell)/N_\ell \approx 1/T$ for all seasons $v$. We will ignore these edge effects in the ensuing calculations. Proceeding under this simplification, the $v$th column in $\mathbf{A}$, denoted by $\mathbf{A}_v$, under the projection $\mathcal{P}_{\mathbf{D}}$, becomes $$\label{eq:PD_Av}
\mathcal{P}_{\mathbf{D}}\mathbf{A}_v
= \left(\mathbf{0}_{N_1}', \frac{k(v, 2)}{N_2}\mathbf{1}_{N_2}', \ldots,
\frac{k(v, m+1)}{N_{m+1}}\mathbf{1}_{N_{m+1}}' \right)'
= \left(\mathbf{0}_{N_1}', \frac{1}{T}\mathbf{1}_{N - N_1}' \right)'.$$
We can now obtain an expression for $\mathbf{A}' (\mathbf{I}_N -
\mathcal{P}_{\mathbf{D}}) \mathbf{A}$. To do this, note that for $u,w
\in \{1, 2, \ldots, T\}$, $$[\mathbf{A}' (\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}]_{u,w}
= \mathbf{A}_u' \mathbf{A}_w
-(\mathcal{P}_{\mathbf{D}}\mathbf{A}_u)'
(\mathcal{P}_{\mathbf{D}}\mathbf{A}_w)
= \begin{cases}
\frac{N}{T^2}(T - (1-\lambda_1)), & \text{if } u = w,\\
-\frac{N}{T^2}(1-\lambda_1), & \text{if } u \neq w,
\end{cases}$$ and it follows that $\mathbf{A}' (\mathbf{I}_N -
\mathcal{P}_{\mathbf{D}}) \mathbf{A} = NT^{-2}\{T \mathbf{I}_T - (1 -
\lambda_1)\mathbf{J}_T\}$. The inverse of this matrix can be verified as $$\left\{\mathbf{A}' (\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}\right\}^{-1}
= \frac{1}{N}\left(T \mathbf{I}_T +
\frac{1-\lambda_1}{\lambda_1}\mathbf{J}_T\right).$$ Plugging this inverse into and denoting $\mathcal{Q}_{\mathbf{D}} = \mathbf{I}_N - \mathcal{P}_{\mathbf{D}}$ produce $$\begin{aligned}
\label{eq:P_QD_A2}
\mathcal{P}_{( \mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}}
&= \frac{1}{N} (\mathcal{Q}_{\mathbf{D}} \mathbf{A})
\left(T \mathbf{I}_T +
\frac{1-\lambda_1}{\lambda_1}\mathbf{J}_T\right)
(\mathcal{Q}_{\mathbf{D}} \mathbf{A})'\\ \nonumber
&= \frac{T}{N} (\mathcal{Q}_{\mathbf{D}} \mathbf{A})
(\mathcal{Q}_{\mathbf{D}} \mathbf{A})'
+ \frac{1-\lambda_1}{N\lambda_1}
(\mathcal{Q}_{\mathbf{D}} \mathbf{A} \mathbf{1}_T)
(\mathcal{Q}_{\mathbf{D}} \mathbf{A} \mathbf{1}_T)'.\end{aligned}$$ For simplicity, we assume that regime $\ell$ starts with season one, ends with season $T$, and contains $n_\ell$ full cycles. Using $n = N/T=
\sum_{r=1}^{m+1} n_r$ and gives $$\mathcal{Q}_{\mathbf{D}} \mathbf{A} = \left(
\begin{array}{c}
\mathbf{1}_{n_1} \otimes \mathbf{I}_T \\ \hdashline
\mathbf{1}_{n - n_1} \otimes \left(\mathbf{I}_T - \frac{1}{T}\mathbf{J}_T\right)
\end{array}
\right), \quad
\mathcal{Q}_{\mathbf{D}}\mathbf{A}\mathbf{1}_T
= \left(
\begin{array}{c}
\mathbf{1}_{N_1} \\ \hdashline
\mathbf{0}_{N - N_1}
\end{array}
\right).$$ Hence, quadratic forms of these matrices are $$(\mathcal{Q}_{\mathbf{D}} \mathbf{A})
(\mathcal{Q}_{\mathbf{D}} \mathbf{A})'
= \left(
\begin{array}{c:c}
\mathbf{J}_{n_1} \otimes \mathbf{I}_T
& \mathbf{J}_{n_1\times (n - n_1)}
\otimes \left(\mathbf{I}_T - \frac{1}{T}\mathbf{J}_T\right) \\ \hdashline
\mathbf{J}_{(n - n_1) \times n_1}
\otimes \left(\mathbf{I}_T - \frac{1}{T}\mathbf{J}_T\right)
& \mathbf{J}_{n - n_1}
\otimes \left(\mathbf{I}_T - \frac{1}{T}\mathbf{J}_T\right)
\end{array}
\right),$$ and $$(\mathcal{Q}_{\mathbf{D}} \mathbf{A} \mathbf{1}_T)
(\mathcal{Q}_{\mathbf{D}} \mathbf{A} \mathbf{1}_T)'
= \left(
\begin{array}{c:c}
\mathbf{J}_{N_1} & \mathbf{0}\\ \hdashline
\mathbf{0} & \mathbf{0}
\end{array}
\right).$$ Plugging these into produces $$\mathcal{P}_{(\mathbf{I}_N - \mathcal{P}_{\mathbf{D}}) \mathbf{A}}
= \frac{1}{N_1} \left(
\begin{array}{c:c}
\mathbf{J}_{N_1} & \mathbf{0}\\ \hdashline
\mathbf{0} & \mathbf{0}
\end{array}
\right) + \frac{T}{N} \mathbf{J}_{n} \otimes \mathbf{I}_T
- \frac{1}{N}\mathbf{J}_{N}.$$ Since $\mathcal{P}_{\mathbf{D}}$ is block-diagonal of form $$\mathcal{P}_{\mathbf{D}} = \text{diag}\left(\mathbf{0}_{N_1 \times N_1},
\frac{\mathbf{J}_{N_2}}{N_2}, \ldots, \frac{\mathbf{J}_{N_{m+1}}}{N_{m+1}}\right),$$ we have $$\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}|\mathbf{D}]}
= \mathbf{I}_N - \text{diag}\left(\frac{\mathbf{J}_{N_1}}{N_1},
\frac{\mathbf{J}_{N_2}}{N_2}, \ldots, \frac{\mathbf{J}_{N_{m+1}}}{N_{m+1}}\right)
- \frac{T}{N}\mathbf{J}_{n} \otimes \mathbf{I}_T
+ \frac{1}{N}\mathbf{J}_{N}.$$ Therefore, for $t \in \{ 1, 2, \ldots, N \}$, the $t$th entries of the vectors in are $$\begin{aligned}
W_t &= [(\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}|\mathbf{D}]})\boldsymbol\epsilon]_t
= \epsilon_t - \bar{\epsilon}_{r(t)}
- \bar{\epsilon}_{v(t)} + \bar{\epsilon}, \\
\delta_t &= [(\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}|\mathbf{D}]})\mathbf{D}^0\boldsymbol\mu ]_t
= \mu_{r^0(t)} - \bar{\mu}_{r(t)}.\end{aligned}$$
For any changepoint configuration $\boldsymbol\lambda \in
\boldsymbol\Lambda$, as $N$ tends to infinity, $N^{-1}\sum_{t=h+1}^N
\delta_t \delta_{t-h}$ converges to a constant that does not depend on the lag $h \in \{ 0, 1, \ldots, p \}$. This is because for any lag $h$, $\delta_t = \delta_{t-h}$ for all $t \in \{ 1, \ldots, N \}$, except for at most $(m + m^0)h \leq (m + m^0)p$ times near the changepoints in $\boldsymbol\lambda$ and $\boldsymbol\lambda^0$. Hence, as $N
\rightarrow \infty$, $N^{-1}\sum_{t=h+1}^N\delta_t \delta_{t-h}$ converges to its limit at rate $O\left(1/N\right)$. We denote this limit as $$\label{eq:delta_sq_average}
\delta^2 \stackrel{\text{def}}{=}
\lim_{N\rightarrow \infty} \frac{1}{N}\sum_{t=1}^N \delta_t^2
= \lim_{N\rightarrow \infty} \frac{1}{N}\sum_{t=1}^N \left(
\mu_{r^0(t)} - \bar{\mu}_{r(t)}\right)^2,$$ which is non-negative and depends on $\boldsymbol\lambda$, but not on $N$. It is not hard to see that $\delta_t = 0$ for all $t \in \{ 1,
\ldots, N \}$ if and only if $\boldsymbol{\lambda}$ contains all relative changepoints in $\boldsymbol{\lambda}^0$ (denoted by $\boldsymbol\lambda
\supset \boldsymbol\lambda^0$). Therefore, $\delta^2=0$ only for models $\boldsymbol\lambda$ such that $\boldsymbol\lambda \supset
\boldsymbol\lambda^0$, including $\boldsymbol{\lambda}^0$ itself.
\[lemma:gamma\_h\] Under any relative changepoint configuration $\boldsymbol\lambda \in \boldsymbol\Lambda$ (which may or may not be the true changepoint configuration), for $h \in \{ 0, 1, \ldots,
p \}$, as $N \rightarrow \infty$, the lag $h$ sample autocovariance $$\hat{\gamma}(h) = \frac{1}{N} \sum_{t = h + 1}^N
\epsilon_t^{\text{ols}} \epsilon_{t-h}^{\text{ols}}$$ obeys $$\label{eq:gamma_h}
\hat{\gamma}(h)
= \gamma(h) +
\delta^2 + O_P\left(\frac{1}{\sqrt{N}}\right),$$ where $\gamma(h)$ is the true lag $h$ autocovariance for the AR$(p)$ series $\boldsymbol\epsilon$.
Since the AR$(p)$ errors are assumed causal, we may write $$\epsilon_t = \sum_{j = 0}^{\infty} \psi_{j} Z_{t-j}$$ for some weights $\{ \psi_j \}_{j=0}^\infty$, where $\sum_{j =
0}^{\infty} |\psi_j| < \infty$. Since $W_t = \epsilon_t -
\bar{\epsilon}_{r(t)}- \bar{\epsilon}_{v(t)} + \bar{\epsilon}$, one can write $W_t$ as a linear combination of all $Z_t$s up to and before time $N$: $$W_t = \sum_{j=-\infty}^{\infty} \psi_j^{(t)} Z_{t-j},$$ where $$\label{eq:psi_j^t}
\psi_j^{(t)}= \psi_{j} -
\frac{\sum_{k: r(k) = r(t)}\psi_{k-t+j}}{N_{r(t)}}
- \frac{\sum_{l: v(l) = v(t)}\psi_{l-t+j}}{N/T}
+ \frac{\sum_{u=1}^N \psi_{u-t+j}}{N}.$$ Since $\psi_j =0$ when $j < 0$, $\psi_j^{(t)} = 0$ if $j < t-N$.
The asymptotic limit of the sample autocovariances can now be derived: $$\begin{aligned}
\nonumber
\hat{\gamma}(h) & = \frac{1}{N} \sum_{t = h + 1}^N \epsilon_t^{\text{ols}} \epsilon_{t-h}^{\text{ols}}
= \frac{1}{N} \sum_{t = h + 1}^N (W_t + \delta_t)(W_{t-h} + \delta_{t-h})\\
\label{eq:autocov1}
& = \frac{1}{N} \sum_{t = h + 1}^N
(W_t W_{t-h} + \delta_{t-h} W_t + \delta_{t} W_{t-h} + \delta_t \delta_{t-h}).\end{aligned}$$ Arguing as in Proposition 7.3.5 of @Brockwell_Davis_1991 [pp. 232] gives $$\frac{1}{N} \sum_{t = h + 1}^N W_t W_{t-h}
= \frac{1}{N}\sum_{t=h+1}^N \sum_{j=-\infty}^{\infty} \psi_j^{(t)}\psi_{j-h}^{(t-h)} Z_{t-j}^2
+ O_P\left(\frac{1}{\sqrt{N}}\right).$$ In , since $\sum_{j=0}^\infty |\psi_j| < \infty$, and $N_{r(t)} = O(N)$ for all $t \in \{ 1, \ldots, N \}$, it is not difficult to show that there exists a positive finite constant $c$ such that, $$\sup_{t, j} \left|\psi_j^{(t)} -\psi_{j}\right| \leq \frac{c}{N}.$$ Therefore, for each $t$ and $h$, $\left\{\psi_j^{(t)}\psi_{j-h}^{(t-h)}
\right\}_{j=-\infty}^{\infty}$ is absolutely convergent, and $$\left| \sum_{j=-\infty}^{\infty} \psi_j^{(t)}\psi_{j-h}^{(t-h)}
- \sum_{j=-\infty}^{\infty} \psi_j\psi_{j-h} \right| = O\left( \frac{1}{N} \right).$$ Since $\{ Z_t \}$ is iid with variance $\sigma^2$, the weak law of large numbers (WLLN) for linear processes [@Brockwell_Davis_1991 pp. 208, Proposition 6.3.10] gives $$\begin{aligned}
\frac{1}{N} \sum_{t = h + 1}^N W_t W_{t-h}
&= \frac{1}{N}\sum_{t=h+1}^N
\sum_{j=-\infty}^{\infty} \psi_j^{(t)}\psi_{j-h}^{(t-h)}\sigma^2 + O_P\left(\frac{1}{\sqrt{N}}\right)\\
&= \frac{1}{N}\sum_{t=h+1}^N
\sum_{j=-\infty}^{\infty} \psi_{j}\psi_{j-h}\sigma^2 + O_P\left(\frac{1}{\sqrt{N}}\right).\end{aligned}$$ Now using that $\gamma(h) = \sigma^2
\sum_{j=-\infty}^\infty \psi_j \psi_{j-h}$ gives $$\frac{1}{N} \sum_{t = h + 1}^N W_t W_{t-h}
=\frac{N-h}{N}\gamma(h)+ O_P\left(\frac{1}{\sqrt{N}}\right)
=\gamma(h)+ O_P\left(\frac{1}{\sqrt{N}}\right).$$ This identifies the limit of the first term in the bottom line of . By , it is not hard to show that for each $t$, $\left\{\psi_j^{(t)} \right\}_{j = -\infty}^{\infty}$ is absolutely convergent. For the second and third terms in , apply the WLLN again to see that these terms converge to zero in probability at rate $O_P(1/\sqrt{N})$. Hence, as $N \rightarrow \infty$, $$\hat{\gamma}(h)
= \gamma(h) + \frac{1}{N}\sum_{t=h+1}^N \delta_t \delta_{t-h}
+ O_P\left(\frac{1}{\sqrt{N}}\right)
= \gamma(h) + \delta^2 + O_P\left(\frac{1}{\sqrt{N}}\right).$$
Since the Yule-Walker estimator $\hat{\boldsymbol\phi}$ is formulated based on $\hat{\gamma}(h)$’s, the following asymptotic result follows from Lemma \[lemma:gamma\_h\].
\[prop:YW\_phi\] Under any relative changepoint configuration $\boldsymbol\lambda \in \boldsymbol\Lambda$, the Yule-Walker estimator $\hat{\boldsymbol\phi} =
\hat{\boldsymbol\Gamma}_p^{-1} \hat{\boldsymbol\gamma}_p$ obeys $$\label{eq:YW_phi_limit}
\hat{\boldsymbol\phi} =
\left(\boldsymbol\Gamma_p + \delta^2 \mathbf{J}_p \right)^{-1}
\left(\boldsymbol\gamma_p + \delta^2 \mathbf{1}_p \right)
+ O_P\left(\frac{1}{\sqrt{N}}\right),$$ where ${\boldsymbol\gamma}_p = ({\gamma}(1), \ldots, {\gamma}(p))'$ and ${\boldsymbol\Gamma}_p$ is a $p \times p$ matrix with $(i,j)$th entry ${\gamma}(|i-j|)$.
Asymptotic behavior of estimators of ${\sigma}^2$ {#subsection:YW_sigmasq}
-------------------------------------------------
In the BMDL and (automatic) MDL formulas, estimators for $\sigma^2$ are $$\begin{aligned}
\label{eq:sigmasq_hat_BMDL}
\hat{\sigma}^2
& = \frac{1}{N-p}\widehat{\mathbf{X}}' \left\{
\widehat{\mathbf{B}}
- \widehat{\mathbf{B}} \widehat{\mathbf{A}}
\left(\widehat{\mathbf{A}}' \widehat{\mathbf{B}}\widehat{\mathbf{A}}\right)^{-1}
\widehat{\mathbf{A}}' \widehat{\mathbf{B}}
\right\}\widehat{\mathbf{X}},\\ \label{eq:sigmasq_hat_MDL}
\hat{\sigma}^2_{\nu = \infty}
& =
\frac{1}{N-p}\widehat{\mathbf{X}}'
\left(\mathbf{I}_N - \mathcal{P}_{[\widehat{\mathbf{A}}|\widehat{\mathbf{D}}]}\right)
\widehat{\mathbf{X}},\end{aligned}$$ respectively. The following lemma suggests that under any model $\boldsymbol\lambda$, these two estimators are asymptotically the same as the Yule-Walker estimator of $\sigma^2$, i.e., $$\label{eq:sigmasq_hat_YW}
\hat{\sigma}^2_{\text{YW}} = \hat\gamma(0) - \hat{\boldsymbol\gamma}_p'
\hat{\boldsymbol\Gamma}_p^{-1}\hat{\boldsymbol\gamma}_p.$$
\[lemma:sigmasq1\] Under any changepoint configuration $\boldsymbol\lambda \in \boldsymbol\Lambda$, as $N
\rightarrow \infty$, $$\begin{aligned}
\label{eq:sigmasq1}
\hat{\sigma}^2
&= \hat{\sigma}^2_{\nu = \infty} + O_P\left( \frac{1}{N} \right), \\ \label{eq:sigmasq2}
\hat{\sigma}^2_{\nu = \infty}
&= \hat{\sigma}^2_{\text{YW}} + O_P\left( \frac{1}{N} \right).\end{aligned}$$
Under the null model $\boldsymbol\lambda_{\o}$ ($m = 0$), the column space of $\mathbf{D}$ is the null space and both $\hat{\sigma}^2$ and $\hat{\sigma}^2_{\nu = \infty}$ are $\frac{1}{N-p}\widehat{\mathbf{X}}' \left(\mathbf{I}_N -
\mathcal{P}_{\widehat{\mathbf{A}}}\right) \widehat{\mathbf{X}}$. Since $\hat{\sigma}^2_{\text{YW}} = \frac{1}{N}\widehat{\mathbf{X}}'
\left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\widehat{\mathbf{X}}$, the conclusion holds. The rest of the proof is for any model $\boldsymbol\lambda$ that contains $m \geq 1$ relative changepoints.
We first establish . Since $\hat{\boldsymbol\phi}$ has the limit in , it is not hard to show that as $N$ tends to infinity, $\widehat{\mathbf{D}}' \widehat{\mathbf{D}}/N$ and $\widehat{\mathbf{D}}' \widehat{\mathbf{X}}/N$ converge in probability to a $m \times m$ positive definite matrix and an $m$-dimensional vector, respectively. In the prior of $\boldsymbol\mu$, the parameter $\nu$ is a constant; hence,
$$\begin{aligned}
\frac{1}{N}\widehat{\mathbf{X}}'\widehat{\mathbf{B}}\widehat{\mathbf{X}}
&= \frac{\widehat{\mathbf{X}}'\widehat{\mathbf{X}}}{N}
- \frac{\widehat{\mathbf{X}}' \widehat{\mathbf{D}}}{N}
\left(\frac{\widehat{\mathbf{D}}' \widehat{\mathbf{D}}}{N}
+\frac{\mathbf{I}_{m}}{N \nu}\right)^{-1}
\frac{\widehat{\mathbf{D}}' \widehat{\mathbf{X}}}{N}\\
&= \frac{\widehat{\mathbf{X}}'\widehat{\mathbf{X}}}{N}
- \frac{\widehat{\mathbf{X}}' \widehat{\mathbf{D}}}{N}
\left(\frac{\widehat{\mathbf{D}}' \widehat{\mathbf{D}}}{N}\right)^{-1}
\frac{\widehat{\mathbf{D}}' \widehat{\mathbf{X}}}{N}
+ O_P\left( \frac{1}{N} \right) \\
&= \frac{1}{N}\widehat{\mathbf{X}}'
\left(\mathbf{I}_{N-p} - \mathcal{P}_{\widehat{\mathbf{D}}}\right)
\widehat{\mathbf{X}} + O_P\left( \frac{1}{N} \right).\end{aligned}$$
Similar arguments give $$\frac{1}{N}\widehat{\mathbf{X}}'\widehat{\mathbf{B}}\widehat{\mathbf{A}}
= \frac{1}{N}\widehat{\mathbf{X}}'
\left(\mathbf{I}_{N-p} - \mathcal{P}_{\widehat{\mathbf{D}}}\right)
\widehat{\mathbf{A}} + O_P\left( \frac{1}{N} \right),\quad
\frac{1}{N}\widehat{\mathbf{A}}'\widehat{\mathbf{B}}\widehat{\mathbf{A}}
= \frac{1}{N}\widehat{\mathbf{A}}'
\left(\mathbf{I}_{N-p} - \mathcal{P}_{\widehat{\mathbf{D}}}\right)
\widehat{\mathbf{A}} + O_P\left( \frac{1}{N} \right).$$ Hence, the left hand side of has the limit $$\begin{aligned}
\label{eq:X_Q_Ahat_Dhat_X2}
\hat{\sigma}^2
=& \frac{1}{N-p}\widehat{\mathbf{X}}' \left\{
\widehat{\mathbf{B}}
- \widehat{\mathbf{B}} \widehat{\mathbf{A}}
\left(\widehat{\mathbf{A}}' \widehat{\mathbf{B}}\widehat{\mathbf{A}}\right)^{-1}
\widehat{\mathbf{A}}' \widehat{\mathbf{B}}
\right\}\widehat{\mathbf{X}}\\ \nonumber
=& \frac{1}{N-p}\widehat{\mathbf{X}}' \left\{
\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{D}}}
- \mathcal{P}_{\left(\mathbf{I}_{N} - \mathcal{P}_{\widehat{\mathbf{D}}}\right) \widehat{\mathbf{A}}}
\right\}\widehat{\mathbf{X}} + O_P\left( \frac{1}{N} \right) \\ \nonumber
=& \frac{1}{N-p}\widehat{\mathbf{X}}'
\left(\mathbf{I}_{N-p} - \mathcal{P}_{[\widehat{\mathbf{A}}|\widehat{\mathbf{D}}]}\right)
\widehat{\mathbf{X}} + O_P\left( \frac{1}{N} \right) \\
=& \hat{\sigma}^2_{\nu = \infty}+ O_P\left( \frac{1}{N} \right),\end{aligned}$$ where the second to last equality follows from .
We now show that for any $\boldsymbol\lambda$ with $m\geq 1$, holds. For notational simplicity, for any $j \in \{ 0, 1, \ldots, p \}$, matrices formed from the rows of $\mathbf{A}$ and $\mathbf{D}$ are denoted by $$\mathbf{A}_j \stackrel{\text{def}}{=} \mathbf{A}_{(p+1-j):(N-j)},\quad
\mathbf{D}_j \stackrel{\text{def}}{=} \mathbf{D}_{(p+1-j):(N-j)}.$$ Since both $\widehat{\mathbf{A}}$ and $\mathbf{A}_j$ are $(N-p)\times T$ matrices and each column in $\widehat{\mathbf{A}}$ can be written as a linear combination of the columns in $\mathbf{A}_j$, the corresponding column spaces agree: $C(\widehat{\mathbf{A}}) = C(\mathbf{A}_j)$. Therefore, $\mathcal{P}_{\widehat{\mathbf{A}}} =
\mathcal{P}_{\mathbf{A}_j}$ for all $j$. Now define $$\label{eq:D_hat_Dj}
\boldsymbol\Delta_j =
\mathbf{D}_j - \frac{\widehat{\mathbf{D}}}
{1 - \hat{\phi}_1 - \hat{\phi}_2 - \cdots - \hat{\phi}_p}.$$ The denominator in cannot be zero since $1 -
\sum_{k=1}^p \hat{\phi}_k \neq 0$ for Yule-Walker estimates when $N$ is large [@Brockwell_Davis_1991].
Since there are at most $2m(p+h)$ non-zero entries in $\boldsymbol\Delta_j$, and none of these entries depend on $N$, $\boldsymbol\Delta_j' \boldsymbol\Delta_j = O_P(1)$. In addition, for any $N$-dimensional vectors $\boldsymbol\alpha$ whose entries do not depend on $N$, $\boldsymbol\alpha' \boldsymbol\Delta_j = O_P(1)$. Using , we have $$\begin{aligned}
\frac{\widehat{\mathbf{D}}' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\widehat{\mathbf{D}}}{N\left(1 - \sum_{k=1}^p \hat{\phi}_k\right)^2}
&= \frac{1}{N} (\mathbf{D}_j - \boldsymbol\Delta_j)'
\left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
(\mathbf{D}_j - \boldsymbol\Delta_j)
= \frac{\mathbf{D}_j' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\mathbf{D}_j}{N} + O_P\left(\frac{1}{N}\right),\\
\frac{\boldsymbol\alpha' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\widehat{\mathbf{D}}}{N\left(1 - \sum_{k=1}^p \hat{\phi}_k\right)}
&= \frac{1}{N} \boldsymbol\alpha' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
(\mathbf{D}_j - \boldsymbol\Delta_j)
= \frac{\boldsymbol\alpha' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\mathbf{D}_j}{N} + O_P\left(\frac{1}{N}\right).\end{aligned}$$ Therefore, for any $\boldsymbol\alpha, \boldsymbol\beta \in
\mathbb{R}^N$ whose entries do not depend on $N$, $$\begin{aligned}
\frac{1}{N}\boldsymbol\alpha'
\mathcal{P}_{\left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\widehat{\mathbf{D}}}\boldsymbol\beta
&= \frac{\boldsymbol\alpha' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\widehat{\mathbf{D}}}{N\left(1 - \sum_{k=1}^p \hat{\phi}_k\right)}
\left\{ \frac{\widehat{\mathbf{D}}' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\widehat{\mathbf{D}}}{N\left(1 - \sum_{k=1}^p \hat{\phi}_k\right)^2} \right\}^{-1}
\frac{\widehat{\mathbf{D}}' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\boldsymbol\beta}{N\left(1 - \sum_{k=1}^p \hat{\phi}_k\right)}\\
&= \frac{1}{N}\boldsymbol\alpha' \left\{
\left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)\mathbf{D}_j
\left( \mathbf{D}_j' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)\mathbf{D}_j \right)^{-1}
\mathbf{D}_j' \left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\right\}\boldsymbol\beta+ O_P\left(\frac{1}{N}\right)\\
&= \frac{1}{N}\boldsymbol\alpha'
\mathcal{P}_{\left(\mathbf{I}_N - \mathcal{P}_{\widehat{\mathbf{A}}}\right)
\mathbf{D}_j}\boldsymbol\beta+ O_P\left(\frac{1}{N}\right).\end{aligned}$$ Hence, from , $$\label{eq:P_hatA_hatD}
\frac{1}{N}\boldsymbol\alpha'
\mathcal{P}_{[\widehat{\mathbf{A}}|\widehat{\mathbf{D}}]}\boldsymbol\beta
= \frac{1}{N}\boldsymbol\alpha'
\mathcal{P}_{[\mathbf{A}_j|\mathbf{D}_j]}\boldsymbol\beta
+ O_P\left(\frac{1}{N}\right).$$
Since $\widehat{\mathbf{X}} = \mathbf{X}_{(p+1):N} - \sum_{j = 1}^p \hat{\phi}_j
\mathbf{X}_{(p+1-j):(N-j)}$, for any $j, k \in \{0, 1, \ldots, p\}$, shows that $$\begin{aligned}
& \frac{1}{N}\mathbf{X}_{(p+1-j):(N-j)}'
\left(\mathbf{I}_N - \mathcal{P}_{[\widehat{\mathbf{A}}|\widehat{\mathbf{D}}]}\right)
\mathbf{X}_{(p+1-k):(N-k)}\\
=& ~\frac{1}{N}\left\{\left(\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}_j|\mathbf{D}_j]}\right)
\mathbf{X}_{(p+1-j):(N-j)}\right\}'
\left\{\left(\mathbf{I}_N - \mathcal{P}_{[\mathbf{A}_k|\mathbf{D}_k]}\right)
\mathbf{X}_{(p+1-k):(N-k)}\right\} + O_P\left(\frac{1}{N}\right)\\
=& ~\frac{1}{N}\left(\boldsymbol\epsilon_{(p+1-j):(N-j)}^{\text{ols}}\right)' \boldsymbol\epsilon_{(p+1-k):(N-k)}^{\text{ols}}
+ O_P\left(\frac{1}{N}\right).\end{aligned}$$ Therefore, the left hand side of is $$\begin{aligned}
& \frac{1}{N}\widehat{\mathbf{X}}'
\left(\mathbf{I}_N - \mathcal{P}_{[\widehat{\mathbf{A}}|\widehat{\mathbf{D}}]}\right)
\widehat{\mathbf{X}} \\
=& ~\frac{1}{N}\left\{ \boldsymbol\epsilon_{(p+1):N}^{\text{ols}}
- \sum_{j = 1}^p \hat{\phi}_j \boldsymbol\epsilon_{(p+1-j):(N-j)}^{\text{ols}} \right\}'
\left\{ \boldsymbol\epsilon_{(p+1):N}^{\text{ols}}
- \sum_{k = 1}^p \hat{\phi}_k \boldsymbol\epsilon_{(p+1-k):(N-k)}^{\text{ols}} \right\}
+ O_P\left( \frac{1}{N} \right)\\
=& ~\hat{\gamma}(0) - 2\sum_{j=1}^p \hat{\phi}_j \hat{\gamma}(j)
+ \sum_{j=1}^p \sum_{k=1}^p \hat{\phi}_j \hat{\phi}_k \hat{\gamma}(|j-k|)
+ O_P\left( \frac{1}{N} \right)\\
=& ~\hat\gamma(0) - 2\hat{\boldsymbol\gamma}_p' \hat{\boldsymbol\phi}
+\hat{\boldsymbol\phi} '
\hat{\boldsymbol\Gamma}_p\hat{\boldsymbol\phi} + O_P\left( \frac{1}{N} \right)\\
=& ~\hat\gamma(0) - \hat{\boldsymbol\gamma}_p'
\hat{\boldsymbol\Gamma}_p^{-1}\hat{\boldsymbol\gamma}_p
+ O_P\left( \frac{1}{N} \right),\end{aligned}$$ which is the right hand side of .
Under any model $\boldsymbol\lambda$, Lemma \[lemma:gamma\_h\] shows that the Yule-Walker estimator $\hat{\sigma}^2_{\text{YW}}$ converges to $$\label{eq:f}
f(\delta^2)\stackrel{\text{def}}{=}
~\gamma(0) + \delta^2
- \left(\boldsymbol\gamma_p + \delta^2 \mathbf{1}_p \right)'
\left(\boldsymbol\Gamma_p + \delta^2 \mathbf{J}_p \right)^{-1}
\left(\boldsymbol\gamma_p + \delta^2 \mathbf{1}_p \right),$$ at rate $O_P(1/\sqrt{N})$. We define the limit in (\[eq:f\]) as $f(\delta^2)$, emphasizing dependence on $\delta^2$. By Lemma \[lemma:sigmasq1\], the asymptotic behavior of the BMDL estimator $\hat{\sigma}^2$ can be summarized in the following proposition.
\[prop:sigmasq2\] Under any relative changepoint configuration $\boldsymbol\lambda \in \boldsymbol\Lambda$, the BMDL estimator of the white noise variance $\hat{\sigma}^2$ obeys $$\label{eq:sigmasq3}
\hat{\sigma}^2= f(\delta^2) + O_P\left( \frac{1}{\sqrt{N}} \right),$$ where $f(\delta^2)$ is defined in . Furthermore, $f(\delta^2)$ strictly increases in $\delta^2$.
We show that $f(\delta^2)$ strictly increases in $\delta^2$. According to (2.22) in @Harville_2008 [pp. 428], for any matrices $\mathbf{R}\in \mathbb{R}^{r
\times r}, \mathbf{S}\in \mathbb{R}^{r \times l}, \mathbf{T}\in
\mathbb{R}^{l \times l}, \mathbf{U}\in \mathbb{R}^{l \times r}$ with $\mathbf{R}, \mathbf{U}$ non-singular, $(\mathbf{R} +
\mathbf{S}\mathbf{T}\mathbf{U})^{-1} = \mathbf{R}^{-1} -
\mathbf{R}^{-1}\mathbf{S} (\mathbf{T}^{-1} +
\mathbf{U}\mathbf{R}^{-1}\mathbf{S})^{-1} \mathbf{U}\mathbf{R}^{-1}$. Hence, for $\delta^2>0$, $$\label{eq:inv_matrix}
\left(\boldsymbol\Gamma_p + \delta^2 \mathbf{J}_p \right)^{-1}
= \left(\boldsymbol\Gamma_p + \mathbf{1}_p
\delta^2 \mathbf{1}_p' \right)^{-1}
= \boldsymbol\Gamma_p^{-1} - \boldsymbol\Gamma_p^{-1}\mathbf{1}_p
\left(\frac{1}{\delta^2} + \mathbf{1}_p'
\boldsymbol\Gamma_p^{-1}\mathbf{1}_p\right)^{-1}
\mathbf{1}_p'\boldsymbol\Gamma_p^{-1}.$$ For notational simplicity, denote the following scalars by $$\label{eq:ab}
a \stackrel{\text{def}}{=}
\mathbf{1}_p' \boldsymbol\Gamma_p^{-1}\mathbf{1}_p, \quad
b \stackrel{\text{def}}{=}
\mathbf{1}_p' \boldsymbol\Gamma_p^{-1}\boldsymbol\gamma_p=\sum_{k=1}^p \phi_k.$$ Then $f(\delta^2)$ can be expanded as $$f(\delta^2)
= \gamma(0) + \delta^2 -
\boldsymbol\gamma_p' \boldsymbol\Gamma_p^{-1}\boldsymbol\gamma_p
- 2b\delta^2 - a(\delta^2)^2 + \frac{b^2}{\frac{1}{\delta^2}+a}
+ \frac{2ab\delta^2}{\frac{1}{\delta^2}+a} + \frac{a^2(\delta^2)^2}{\frac{1}{\delta^2}+a}.$$ Differentiation of $f(\delta^2)$ with respect to $\delta^2$ gives $$f'(\delta^2)
= 1-2b-2a\delta^2 + \frac{b^2 \frac{1}{(\delta^2)^2}}{\left(\frac{1}{\delta^2}+a\right)^2}
+ \frac{2ab\left( \frac{2}{\delta^2} + a\right)}{\left(\frac{1}{\delta^2}+a\right)^2}
+ \frac{a^2\left( 3 +
2a\delta^2\right)}{\left(\frac{1}{\delta^2}+a\right)^2} =
\frac{(b-1)^2}{(1 + a\delta^2)^2} > 0.$$ The strict inequality follows from causality of the AR($p$) errors, which implies that $b = \sum_{k=1}^p \phi_k > 1$. Therefore, $f(\delta^2)$ is strictly increasing in $\delta^2$ and $f(0)=\sigma^2$.
Asymptotic behavior of the BMDL in {#sec:asym_pairwise_BMDL}
-----------------------------------
Recall that under the relative changepoint model $\boldsymbol\lambda$, its BMDL is $$\begin{aligned}
\text{BMDL}(\boldsymbol\lambda) = ~&
\frac{N-p}{2}\log \left( \hat{\sigma}^2 \right) + \frac{m}{2}\log(\nu)
+ \frac{1}{2}\log\left( \left|\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right| \right)\\ \nonumber
& -\sum_{k=1}^2\log\left\{\Gamma\left(a + m^{(k)}\right)
\Gamma\left(b^{(k)} + N^{(k)} - m^{(k)}\right) \right\}.\end{aligned}$$ The next two lemmas quantify the asymptotic behavior of the third and forth terms in the above BMDL formula, respectively.
\[lemma:det\] Under any changepoint model $\boldsymbol\lambda \in \boldsymbol\Lambda$ with $m>0$, $$\label{eq:det}
\frac{1}{2}\log\left( \left|\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right| \right)
= \frac{1}{2}\sum_{j=2}^{m+1}\log(N_r) -m \log \left( 1- \sum_{k=1}^p \hat{\phi}_k \right)
+ O_P\left(\frac{1}{N}\right).$$
By and the corresponding results in the proof of Lemma \[lemma:sigmasq1\], as $N \rightarrow \infty$,
$$\frac{\widehat{\mathbf{D}}' \widehat{\mathbf{D}}}{N}+\frac{\mathbf{I}_{m}}{N\nu}
= \frac{\widehat{\mathbf{D}}' \widehat{\mathbf{D}}}{N} + O\left(\frac{1}{N}\right)
= \frac{\mathbf{D}' \mathbf{D}}{N\left( 1- \sum_{k=1}^p \hat{\phi}_k \right)^2}
+ O_P\left(\frac{1}{N}\right).$$ The determinant of the $m \times m$ matrix (of finite dimension) is then $$\begin{aligned}
\log\left( \left|\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right| \right)
& = m \log(N) + \log\left(\left| \frac{\widehat{\mathbf{D}}' \widehat{\mathbf{D}}}{N}
+\frac{\mathbf{I}_{m}}{N\nu} \right| \right)\\
& = m \log(N) + \log\left(\frac{\left|\mathbf{D}' \mathbf{D}\right|}
{N^m\left( 1- \sum_{k=1}^p \hat{\phi}_k \right)^{2m}} \right)+ O_P\left(\frac{1}{N}\right)\\
& = \log\left( \left|\mathbf{D}' \mathbf{D}\right|\right)
-2m \log \left( 1- \sum_{k=1}^p \hat{\phi}_k \right) + O_P\left(\frac{1}{N}\right)\\
& = \log\left( \prod_{j=2}^{m+1} N_r\right)
-2m \log \left( 1- \sum_{k=1}^p \hat{\phi}_k \right) + O_P\left(\frac{1}{N}\right),\end{aligned}$$ and follows immediately.
Since $N_r = O(N)$ for all $r \in \{ 2, \ldots, m+1\}$, Lemma \[lemma:det\] implies that for any changepoint model $\boldsymbol\lambda$, $$\label{eq:det2}
\frac{1}{2}\log\left( \left|\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right| \right)
= \frac{m}{2}\log(N) + O_P\left(1\right).$$
\[lemma:Gamma\_pairdiff\] Suppose that both the number of documented and undocumented times increases linearly with $N$, i.e., $N^{(k)} = O(N)$, for $k = 1, 2$. Then under any two changepoint models $\boldsymbol\lambda_1, \boldsymbol\lambda_2 \in \boldsymbol\Lambda$, whose total number of changepoints are $m_1, m_2$, respectively, the pairwise difference of the last term in the BMDL formula is $$\begin{aligned}
\nonumber
& -\sum_{k=1}^2\left[\log\left\{\Gamma\left(a + m_1^{(k)}\right)
\Gamma\left(b^{(k)} + N^{(k)} - m_1^{(k)}\right) \right\}
-\log\left\{\Gamma\left(a + m_2^{(k)}\right)
\Gamma\left(b^{(k)} + N^{(k)} - m_2^{(k)}\right) \right\} \right]\\ \label{eq:Gamma_pairdiff}
& = (m_1 - m_2) \log(N) + O_P(1).\end{aligned}$$
The left hand side of can be simplified to $$\label{eq:Gamma_pairdiff2}
\sum_{k=1}^2\log \left\{\frac{\Gamma\left(a + m_2^{(k)}\right)
\Gamma\left(b^{(k)} + N^{(k)} - m_2^{(k)}\right)}
{\Gamma\left(a + m_1^{(k)}\right)
\Gamma\left(b^{(k)} + N^{(k)} - m_1^{(k)} \right)}\right\}.$$ Stirling’s formula quantifies the asymptotic limit of the following Gamma function ratio: $$\frac{\Gamma\left(b^{(k)} + N^{(k)} - m_2^{(k)}\right)}
{\Gamma\left(b^{(k)} + N^{(k)} - m_1^{(k)} \right)}
\approx~
e^{m_2^{(k)} - m_1^{(k)}}~\frac{\left(b^{(k)} + N^{(k)} - m_2^{(k)}-1\right)^{b^{(k)} + N^{(k)} - m_2^{(k)} - 1/2}}
{\left(b^{(k)} + N^{(k)} - m_1^{(k)}-1 \right)^{b^{(k)} + N^{(k)} - m_1^{(k)} - 1/2}}
\approx~
\left(\frac{N}{e}\right)^{m_1^{(k)} - m_2^{(k)}}.$$ Therefore, equals $(m_1 - m_2) \log N +
O_P(1)$.
The asymptotic behavior of the BMDL is established in the following two propositions. They consider the pairwise difference of BMDLs between the true model $\boldsymbol\lambda^0$ and another changepoint model $\boldsymbol\lambda$. Proposition \[prop:pairwise\_BMDL\_OpN\] considers the case where the model $\boldsymbol\lambda$ does not contain all relative changepoints in $\boldsymbol\lambda^0$, i.e., $\boldsymbol{\lambda} \not\supset \boldsymbol{\lambda}^0$, whereas Proposition \[prop:pairwise\_BMDL\_OplogN\] considers the case where $\boldsymbol{\lambda} \supset \boldsymbol{\lambda}^0$, i.e., $\boldsymbol\lambda$ contains all relative changepoints in $\boldsymbol\lambda^0$, and also may have some redundant changepoints.
\[prop:pairwise\_BMDL\_OpN\] For any relative changepoint configuration $\boldsymbol{\lambda} \in
\boldsymbol{\Lambda}$, if $\boldsymbol{\lambda} \not\supset
\boldsymbol{\lambda}^0$, then as $N \rightarrow \infty$, $$\text{BMDL}\left(\boldsymbol\lambda\right)
> \text{BMDL}\left(\boldsymbol\lambda^0\right), \quad
\text{BMDL}\left(\boldsymbol\lambda\right)
-\text{BMDL}\left(\boldsymbol\lambda^0\right) = O_P(N).$$
In this proof, when necessary, subscripts $\boldsymbol\lambda$ and $\boldsymbol\lambda^0$ are used to distinguish the same terms under different models. By and , the difference between BMDLs in the (non-true) model $\boldsymbol\lambda$ and the true model $\boldsymbol\lambda^0$ is asymptotically $$\begin{aligned}
\label{eq:diff_MDL}
\text{BMDL}\left(\boldsymbol\lambda\right)
-\text{BMDL}\left(\boldsymbol\lambda^0\right)
=& ~\frac{N-p}{2}\log \left(\frac{\hat{\sigma}^2_{\boldsymbol\lambda}}
{\hat{\sigma}^2_{\boldsymbol\lambda^0}}\right)
+ \frac{3(m-m^0)}{2} \log (N) + O_P\left(1\right) \\ \label{eq:diff_MDL2}
=& ~\frac{N-p}{2}\log \left\{\frac{f(\delta^2_{\boldsymbol\lambda})+ O_P\left(\frac{1}{\sqrt{N}}\right)}
{f(0)+ O_P\left(\frac{1}{\sqrt{N}}\right)}\right\}
+ \frac{3(m-m^0)}{2} \log (N) + O_P\left(1\right) . \end{aligned}$$ Here, the last equality is justified via Proposition \[prop:sigmasq2\]. For the model $\boldsymbol{\lambda} \not\supset
\boldsymbol{\lambda}^0$, its corresponding $\delta^2_{\boldsymbol\lambda} > 0$. By Proposition \[prop:sigmasq2\], $f(\delta^2)$ strictly increases in $\delta^2$, which shows that the logarithm term in has a strictly positive limit. Therefore, when $N$ is large, the first term in is positive, of order $O_P(N)$, and dominates the other terms in .
\[prop:pairwise\_BMDL\_OplogN\] For any relative changepoint configuration $\boldsymbol{\lambda} \in
\boldsymbol{\Lambda}$, if $\boldsymbol{\lambda} \supset
\boldsymbol{\lambda}^0$, then as $N \rightarrow \infty$, $$\text{BMDL}\left(\boldsymbol\lambda\right)
> \text{BMDL}\left(\boldsymbol\lambda^0\right), \quad
\text{BMDL}\left(\boldsymbol\lambda\right)
-\text{BMDL}\left(\boldsymbol\lambda^0\right) = O_P(\log N).$$
In the case where $\boldsymbol{\lambda} \supset
\boldsymbol{\lambda}^0$, still holds. Moreover, since $\boldsymbol{\lambda}$ also contains redundant changepoints, $m
> m^0$. Hence, for large $N$, the second term in is positive and of order $O_P(\log N)$. To prove Proposition \[prop:pairwise\_BMDL\_OplogN\], we need to show that the first term in is bounded in probability. A sufficient condition for this is that $$\label{eq:sigmasq3}
\hat{\sigma}^2_{\boldsymbol\lambda} = \hat{\sigma}^2_{\boldsymbol\lambda^0}
+ O_P\left(\frac{1}{N}\right).$$
To establish , we first focus on the model $\boldsymbol{\lambda}$. For notational simplicity, the subscript $\boldsymbol\lambda$ is omitted when there is no ambiguity. Under any model $\boldsymbol{\lambda} \supset \boldsymbol{\lambda}^0$, its corresponding $\delta_t$ in is zero for all $t
\in \{ 1, \ldots, N \}$; hence, by Lemma \[lemma:Y\], the lag-$h$ sample autocovariance $\hat{\gamma}(h)$ in for all $h \in \{ 0, 1, \ldots, p\}$ can be written as $$\begin{aligned}
\nonumber
\hat{\gamma}(h)
& = \frac{1}{N} \sum_{t = h + 1}^N W_t W_{t-h}
= \frac{1}{N} \sum_{t = h + 1}^N
\left(\epsilon_t - \bar{\epsilon}_{r(t)} - \bar{\epsilon}_{v(t)} + \bar{\epsilon}\right)
\left(\epsilon_{t-h} - \bar{\epsilon}_{r(t-h)} - \bar{\epsilon}_{v(t-h)} + \bar{\epsilon}\right)\\
\label{eq:autocov2}
& = \frac{1}{N} \sum_{t = h + 1}^N \left\{
\epsilon_t \epsilon_{t-h}
- \epsilon_t \left( \bar{\epsilon}_{r(t-h)} + \bar{\epsilon}_{v(t-h)} - \bar{\epsilon} \right)
- \epsilon_{t-h} \left( \bar{\epsilon}_{r(t)} + \bar{\epsilon}_{v(t)} - \bar{\epsilon} \right)
\right.\\ \nonumber
& \left.~~~~~~~~~~~~~~~~
+ \left( \bar{\epsilon}_{r(t-h)} + \bar{\epsilon}_{v(t-h)} - \bar{\epsilon} \right)
\left( \bar{\epsilon}_{r(t)} + \bar{\epsilon}_{v(t)} - \bar{\epsilon} \right)
\right\}. \end{aligned}$$
Recall that $\bar{\epsilon}_{r(\cdot)}, \bar{\epsilon}_{v(\cdot)},
\bar{\epsilon}$ are averages of AR$(p)$ errors. These averages are taken over error blocks whose size is proportional to $N$. By the WLLN for linear processes, these averages all converge to zero in probability with order $O_P(1/\sqrt{N})$. Since the fourth term in is a sum of their two-way interactions and quadratic forms, it is also $O_P(1/N)$. The second term in can be expanded as $$\begin{aligned}
\frac{1}{N} \sum_{t = h + 1}^N
\epsilon_t \left( \bar{\epsilon}_{r(t-h)} + \bar{\epsilon}_{v(t-h)} - \bar{\epsilon} \right)
&= \frac{1}{N} \left\{
\sum_{r = 1}^{m + 1} \sum_{t = 1}^{N_r}\epsilon_{r, t} \bar{\epsilon}_{r}
+ \sum_{v = 1}^{T} \sum_{t = 1}^{N/T}\epsilon_{v, t} \bar{\epsilon}_{v}
+ \sum_{t = 1}^N \epsilon_t \bar{\epsilon} + O_P(1)
\right\}\\
&= \frac{1}{N} \left\{
\sum_{r = 1}^{m + 1} N_r \bar{\epsilon}_{r}^2
+ \sum_{v = 1}^{T} \left(\frac{N}{T}\right) \bar{\epsilon}_{v}^2
+ N \bar{\epsilon}^2
\right\} + O_P\left( \frac{1}{N} \right)\\
&= O_P\left( \frac{1}{N} \right), \end{aligned}$$ where $\epsilon_{r, t}$ denotes the error during time $t$ in the $r$th regime, $\epsilon_{v, t}$ denotes the error during time $t$ in the $v$th month, and $\bar{\epsilon}_{r}$ and $\bar{\epsilon}_{v}$ are the error averages for the $r$th regime and $v$th month, respectively. Similarly, we can show that the third term in is also $O_P(1/N)$. Therefore, under any model $\boldsymbol{\lambda} \supset
\boldsymbol{\lambda}^0$, including $\boldsymbol{\lambda}^0$ itself, becomes $$\hat{\gamma}(h) = \frac{1}{N} \sum_{t = h + 1}^N
\epsilon_t \epsilon_{t-h} + O_P\left( \frac{1}{N} \right),$$ which shows that $\hat{\gamma}(h)$ under the two models $\boldsymbol{\lambda}$ and $\boldsymbol{\lambda}^0$ only changes by $O_P(1/N)$. By , $\hat{\sigma}^2_{\text{YW}}$ under the two models $\boldsymbol{\lambda}$ and $\boldsymbol{\lambda}^0$ also can only differ by $O_P(1/N)$. By Lemma \[lemma:sigmasq1\], the BMDL estimator $\hat{\sigma}^2 = \hat{\sigma}^2_{\text{YW}} + O_P(1/N)$, which establishes . Thus, $\hat{\sigma}^2$ under the two models $\boldsymbol{\lambda}$ and $\boldsymbol{\lambda}^0$ only differ by $O_P(1/N)$.
A proof of Theorem \[thm:lambda\_convergence\] {#PROOFthm:lambda_convergence}
----------------------------------------------
To prove Theorem \[thm:lambda\_convergence\], we first establish the asymptotic consistency of $\hat{\boldsymbol\lambda}_N$ in the case where $m^0$ is known. Here, $\boldsymbol\Lambda_m$ denotes a subset of $\boldsymbol\Lambda$ formed by models that have $m$ relative changepoints.
\[prop:lambda\_convergence\_known\_m0\] If the true number of changepoints $m^0$ is known, then as $N \rightarrow \infty $, the estimator $\hat{\boldsymbol\lambda}_N = \arg\min_{\boldsymbol\lambda
\in \boldsymbol\Lambda_{m^0}} \text{BMDL}(\boldsymbol\lambda)$ satisfies that $\hat{\boldsymbol\lambda}_N \stackrel{P}{\longrightarrow}
\boldsymbol\lambda^0$.
Since $\boldsymbol\Lambda_{m^0}$ is a closed and bounded subset of $\mathbb{R}^{m^0}$, it is sequentially compact [@Browder_1996 pp. 144, Theorem 6.67]. Therefore, there exist a subsequence $\{N_k\}$ and a $\boldsymbol\lambda^* \in \boldsymbol\Lambda_{m^0}$ such that $\hat{\boldsymbol\lambda}_N \stackrel {P}{\longrightarrow}
\boldsymbol\lambda^*$ along the subsequence $\{N_k\}$. For notational simplicity, we simply replace $N_k$ by $N$. To prove Proposition \[prop:lambda\_convergence\_known\_m0\], we need only show that $\boldsymbol\lambda^* = \boldsymbol\lambda^0$.
Using proof by contradiction, suppose that $\boldsymbol\lambda^* \neq
\boldsymbol\lambda^0$. Since $m^0$ is known, $\hat{m} = m^0$. By and , $$\text{BMDL}(\hat{\boldsymbol\lambda}_N)
- \text{BMDL}\left( \boldsymbol\lambda^* \right)
= \frac{N-p}{2}\log \left( \frac{\hat{\sigma}^2_{\hat{\boldsymbol\lambda}_N}}
{\hat{\sigma}^2_{\boldsymbol\lambda^*}} \right)
= \frac{N-p}{2}\log \left\{ \frac{f(\delta^2_{\hat{\boldsymbol\lambda}_N})
+ O_P\left( \frac{1}{\sqrt{N}} \right)}
{f(\delta^2_{\boldsymbol\lambda^*}) + O_P\left( \frac{1}{\sqrt{N}} \right)}
\right\} = o_P(N).$$ Here, the last equality follows from the fact that as $N \rightarrow
\infty$, $\hat{\boldsymbol\lambda}_N \longrightarrow
\boldsymbol\lambda^*$, and hence $\delta^2_{\hat{\boldsymbol\lambda}_N}
\longrightarrow \delta^2_{\boldsymbol\lambda^*}$. Since $\boldsymbol\lambda^* \neq \boldsymbol\lambda^0$ but the number of changepoints in these two models are the same, we have $\boldsymbol\lambda^* \not\supset \boldsymbol\lambda^0$. By Proposition \[prop:pairwise\_BMDL\_OpN\], the BMDL difference $$\text{BMDL}(\boldsymbol\lambda^*) - \text{BMDL}(\boldsymbol\lambda^0) = O_P(N)$$ and is positive. Therefore, for large $N$, $$\begin{aligned}
\text{BMDL}(\hat{\boldsymbol\lambda}_N)
- \text{BMDL}(\boldsymbol\lambda^0)
& = \left\{ \text{BMDL}(\hat{\boldsymbol\lambda}_N)
- \text{BMDL}(\boldsymbol\lambda^*) \right\}
+ \left\{ \text{BMDL}(\boldsymbol\lambda^*)
- \text{BMDL}(\boldsymbol\lambda^0) \right\} \\
& = o_P(N) + O_P(N) = O_P(N),\end{aligned}$$ and is positive, which contradicts $\hat{\boldsymbol\lambda}_N$ being the BMDL minimizer.
Next, under the assumption that $m^0$ is unknown, we first establish the following convergence rate lemma on estimated changepoint locations $\hat{\lambda}_j$.
\[lemma:lambda\_convergence\] Suppose that $m^0$ is unknown. For a $\omega \leq 0$ and each $\lambda_r^0$, $r \in \{ 1,
\ldots, m^0 \}$, there exists a $\hat{\lambda}_j$ in $\hat{\boldsymbol\lambda}_N$ such that
$$\label{eq:lambda_convergence_rate2}
\left| \hat{\lambda}_j - \lambda^0_r \right| = O_P(N^{\omega - 1}).$$
Following the contradiction argument in the proof of Proposition \[prop:lambda\_convergence\_known\_m0\], if $m^0$ is unknown, then the estimated relative changepoint model $\hat{\boldsymbol\lambda}_N$ given by converges to a limit $\boldsymbol\lambda^*$ that contains all changepoints in $\boldsymbol\lambda^0$, i.e., $\boldsymbol\lambda^* \supset
\boldsymbol\lambda^0$. This means that for each $\lambda_r^0$, $r = 1,
\ldots, m^0$, there exists a $\hat{\lambda}_j$ in $\hat{\boldsymbol\lambda}_N$ such that $\hat{\lambda}_j \stackrel {P}
{\longrightarrow} \lambda_r^0$, i.e., $\left| \hat{\lambda}_j -
\lambda^0_r \right| = o_P(1)$. Denote that $\left| \hat{\lambda}_j -
\lambda^0_r \right| = O_P(N^{\omega_j-1})$, then, $\omega_j < 1$ for all $r$. Since the total number of changepoints is bounded, set $\omega = \max_{1\leq r \leq m^0} \omega_r$. Then holds with $\omega < 1$.
To show that $\omega \leq 0$, proof by contradiction will be used. If $\omega > 0$, then there exists an $r \in \{1, \ldots, m^0\}$ such that $\omega_r > 0$. Suppose $\hat{\lambda}_j$ is the estimated changepoint that converges to $\lambda^0_r$. For a sufficiently large $N$, based on the BMDL estimator $\hat{\boldsymbol\lambda}_N$, a new model $\tilde{\boldsymbol\lambda}_N$ is created by putting the changepoint $\hat{\lambda}_j$ in $\hat{\boldsymbol\lambda}_N$ with $\lambda^0_r$, i.e., $$\tilde{\boldsymbol\lambda}_N
= \left(\hat{\lambda}_1, \ldots, \hat{\lambda}_{j-1}, \lambda^0_r, \hat{\lambda}_{j+1}, \ldots,
\hat{\lambda}_{\hat{m}}\right)'.$$ A contradiction would arise if $\text{BMDL}(\tilde{\boldsymbol\lambda}_N) <
\text{BMDL}(\hat{\boldsymbol\lambda}_N)$ since $\hat{\boldsymbol\lambda}_N$ minimizes the BMDL.
We first investigate the difference in $\hat{\gamma}(h)$ under the two models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$, for each $h \in \{ 0, 1, \ldots,
p\}$. Following the same argument in Proposition \[prop:pairwise\_BMDL\_OplogN\], $$\label{eq:autocov3}
\frac{1}{N} \sum_{t = h + 1}^N W_t W_{t-h} =
\frac{1}{N} \sum_{t = h + 1}^N
\epsilon_t \epsilon_{t-h} + O_P\left( \frac{1}{N} \right),$$ only depends on the observed data up to an $O_P(1/N)$ error. Hence, its difference under the models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ is $O_P(1/N)$.
For the other terms in , we need only focus on the summation over $t$ satisfying $\lfloor\hat{\lambda}_{j-1} N \rfloor \leq
t \leq \lfloor \hat{\lambda}_{j+1} N \rfloor - 1$, depicted in Figure \[fig:diagram1\]. This is because $(W_t, \delta_t)$ for all $t$ elsewhere are identical in the models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$. For notational simplicity, lengths of time intervals on the rescaled timeline are denoted by $$l_{r} = \lambda^0_{r} - \lambda^0_{r-1}, \quad
l_{r+1} = \lambda^0_{r+1} - \lambda^0_{r}.$$ We first consider the case where $\hat{\lambda}_{j-1}$ is to the left of $\lambda^0_{r-1}$ and $\hat{\lambda}_{j+1}$ is to the right of $\lambda^0_{r+1}$. Without loss of generality, we assume that $\hat{\lambda}_{j}$ is to the left of $\lambda^0_{r}$. The vanishing length between these estimated changepoints and their limits are denoted by $$\Delta l_{r-1} = \lambda^0_{r-1} - \hat{\lambda}_{j-1}, \quad
\Delta l_r = \lambda^0_r - \hat{\lambda}_j, \quad
\Delta l_{r+1} = \hat{\lambda}_{j+1} - \lambda^0_{r+1},$$ all of which converge to zero at rates no slower than $O_P(N^{\omega - 1})$.
(-1, 0) – (11, 0); at (-3, 1) [true regime mean]{}; at (-3, 0) [timeline]{}; at (-3, -1.50) [length]{}; (0.5, -0.1) – (0.5, 0.1); at (0.5, -0.4) [$\hat{\lambda}_{j-1}$]{}; (1.5, -0.1) – (1.5, 1.2); at (1.5, -0.4) [$\lambda_{r-1}^0$]{}; (3.5, -0.1) – (3.5, 0.1); at (3.5, -0.4) [$\hat{\lambda}_{j}$]{}; (5, -0.1) – (5, 1.2); at (5, -0.4) [$\lambda^0_r$]{}; (8.5, -0.1) – (8.5, 1.2); at (8.5, -0.4) [$\lambda_{r+1}^0$]{}; (10, -0.1) – (10, 0.1); at (10, -0.4) [$\hat{\lambda}_{j+1}$]{}; (0.6, 1) – (1.4, 1); (-1.2, 1) – (-0.4, 1); at (0.1, 1) [$\mu_{r-1}$]{}; (1.6, 1) – (2.4, 1); (4.1, 1) – (4.9, 1); at (3.25, 1) [$\mu_{r}$]{}; (5.1, 1) – (5.9, 1); (7.6, 1) – (8.4, 1); at (6.75, 1) [$\mu_{r+1}$]{}; (10.6, 1) – (11.4, 1); (8.6, 1) – (9.4, 1); at (10, 1) [$\mu_{r + 2}$]{}; (0.5, -1.0) – (1.45, -1.0); at (1, -1.3) [$\Delta l_{r-1}$]{}; (3.5, -1.0) – (5, -1.0); at (4.25, -1.3) [$\Delta l_r$]{}; (8.55, -1.0) – (10, -1.0); at (9.25, -1.3) [$\Delta l_{r+1}$]{}; (1.5, -1.7) – (4.95, -1.7); at (3.25, -2) [$l_{r}$]{}; (5.05, -1.7) – (8.5, -1.7); at (6.75, -2) [$l_{r+1}$]{}; (-1, 2) – (11, 2);
Under the model $\hat{\boldsymbol\lambda}_N$, $\delta_t$ in can be written as $$\label{eq:delta_t_lambdahat_proof1}
\delta_{\hat{\boldsymbol\lambda}_N, t} =
\begin{cases}
\mu_{r-1} - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r (l_r - \Delta l_r)}{\Delta l_{r-1} + l_r - \Delta l_r},
& \text{ if } \lfloor \hat{\lambda}_{j-1} N \rfloor \leq t \leq \lfloor \lambda^0_{r-1} N \rfloor - 1,\\
\mu_r - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r (l_r - \Delta l_r)}{\Delta l_{r-1} + l_r - \Delta l_r},
& \text{ if } \lfloor \lambda^0_{r-1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_j N \rfloor - 1,\\
\mu_r - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \hat{\lambda}_j N \rfloor \leq t \leq \lfloor \lambda^0_r N \rfloor - 1,\\
\mu_{r+1} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \lambda^0_r N \rfloor \leq t \leq \lfloor \lambda^0_{r+1} N \rfloor - 1,\\
\mu_{r+2} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \lambda^0_{r+1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+1} N \rfloor - 1;
\end{cases}$$ whereas under the model $\tilde{\boldsymbol\lambda}_N$, $$\label{eq:delta_t_lambdatilde_proof1}
\delta_{\tilde{\boldsymbol\lambda}_N, t} =
\begin{cases}
\mu_{r-1} - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r l_r }{\Delta l_{r-1} + l_r},
& \text{ if } \lfloor \hat{\lambda}_{j-1} N \rfloor \leq t \leq \lfloor \lambda^0_{r-1} N \rfloor - 1,\\
\mu_r - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r l_r }{\Delta l_{r-1} + l_r},
& \text{ if } \lfloor \lambda^0_{r-1} N \rfloor \leq t \leq \lfloor \lambda^0_r N \rfloor - 1,\\
\mu_{r+1} - \frac{\mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{ l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \lambda^0_r N \rfloor \leq t \leq \lfloor \lambda^0_{r+1} N \rfloor - 1,\\
\mu_{r+2} - \frac{ \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \lambda^0_{r+1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+1} N \rfloor - 1.\\
\end{cases}$$ When $N$ is large, $\delta_t = \delta_{t-h}$ for all but a finite number of times $t$; hence, for the second term (a similar argument applies to the third term) in , $$\begin{aligned}
\label{eq:autocov4}
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor -1}
\delta_{t-h} W_t
=~ &
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j} N \rfloor - 1}
\delta_{t} W_t
+ \frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j} N \rfloor}^{\lfloor \lambda^0_{r} N \rfloor - 1}
\delta_{t} W_t
+ \frac{1}{N} \sum_{t = \lfloor \lambda^0_{r} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor - 1}
\delta_{t} W_t
+O_P\left( \frac{1}{N} \right).\end{aligned}$$ By and , under the two models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$, the difference of $\delta_t$ is piecewise constant: $$\label{eq:delta_t_diff_proof1}
\delta_{\hat{\boldsymbol\lambda}_N, t} - \delta_{\tilde{\boldsymbol\lambda}_N, t}
= \begin{cases}
\frac{(\mu_r - \mu_{r-1}) \Delta l_{r-1} \Delta l_r}
{(\Delta l_{r-1} + l_r)(\Delta l_{r-1} + l_r - \Delta l_r)}
= O_P\left( N^{2\omega - 2}\right),
& \text{ if } \lfloor \hat{\lambda}_{j-1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_j N \rfloor - 1,\\
\frac{(\mu_r - \mu_{r+1}) l_r l_{r+1} + O_P(\Delta l)}
{(\Delta l_{r-1} + l_r)(\Delta l_r + l_{r+1} + \Delta l_{r+1})}
= O_P\left( 1\right),
& \text{ if } \lfloor \hat{\lambda}_j N \rfloor \leq t \leq \lfloor \lambda^0_{r} N \rfloor - 1,\\
\frac{(\mu_{r+1} - \mu_r)\Delta l_r l_{r+1} + (\mu_{r+2} - \mu_r)\Delta l_r \Delta l_{r+1}}
{(l_{r+1} + \Delta l_{r+1})(\Delta l_r + l_{r+1} + \Delta l_{r+1})}
= O_P\left( N^{\omega - 1}\right),
& \text{ if } \lfloor \lambda^0_{r} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+1} N \rfloor - 1.
\end{cases}$$ To study the sum of $W_t$ in over the above intervals, apply the WLLN for linear processes to see that $\bar{\epsilon}_{r(t)}, \bar{\epsilon}_{v(t)}, \bar{\epsilon}$ all converge to zero at the rate $O_P(1/\sqrt{N})$ for any $t$. Hence, for an interval $t \in [a,b]$ whose length $b-a = O_P(N^{\xi})$ with $\xi \in (0, 1]$, the sums of $\epsilon_t$ and $W_t$ over this interval satisfy
$$\sum_{t = a}^{b} \epsilon_t
= (b-a) \left(\frac{\sum_{t = a}^{b} \epsilon_t}{b-a}\right)
= O_P(N^{\xi})O_P\left( \frac{1}{\sqrt{N^{\xi}}} \right) = O_P(N^{\frac{\xi}{2}})$$
and $$\begin{aligned}
\label{eq:sum_Wt}
\sum_{t = a}^{b} W_t
=~&
\sum_{t = a}^{b}
\left( \epsilon_t - \bar{\epsilon}_{r(t)} - \bar{\epsilon}_{v(t)} + \bar{\epsilon} \right)
= \sum_{t = a}^{b} \epsilon_t
+ (b-a) O_P \left(\frac{1}{\sqrt{N}}\right) \\ \nonumber
=~& O_P(N^{\frac{\xi}{2}}) + O_P(N^{\xi - \frac{1}{2}}) = O_P(N^{\frac{\xi}{2}}), \end{aligned}$$ where the last equality follows from $\xi \leq 1$. For the three interval sums in , the corresponding convergence rates $\xi$ of their lengths are $1, \omega, 1$, respectively. Hence, in , when decomposed as three sums in these intervals, differences under the models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ are thus $$\begin{aligned}
& \frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor - 1}
\left(\delta_{\hat{\boldsymbol\lambda}_N, t} - \delta_{\tilde{\boldsymbol\lambda}_N, t}\right) W_t
\\
=~& \frac{1}{N} \left\{
O_P\left( N^{2\omega - 2} \right)~ O_P\left( N^{\frac{1}{2}} \right)
+ O_P\left( 1 \right)~ O_P\left( N^{\frac{\omega}{2}} \right)
+ O_P\left( N^{\omega - 1} \right)~ O_P\left( N^{\frac{1}{2}} \right)
\right\} + O_P\left( N^{-1}\right)\\
=~& O_P\left( N^{\frac{\omega}{2}-1}\right),\end{aligned}$$ where the last equality follows from $\omega \leq 1$. Therefore, the second (and third) term differences in under the two models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ is $O_P\left( N^{\frac{\omega}{2}-1}\right)$.
For the last term in , we similarly have $$\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor -1}
\delta_{t-h} \delta_t
= \frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor -1}
\delta_t^2 + O_P\left( \frac{1}{N} \right).$$ Under the model $\hat{\boldsymbol\lambda}_N$, $$\begin{aligned}
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor -1}
\delta_{\hat{\boldsymbol\lambda}_N, t}^2
=~& \left(\mu_{r-1} - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r (l_r - \Delta l_r)}
{\Delta l_{r-1} + l_r - \Delta l_r}\right)^2 \Delta l_{r-1}\\
& +\left(\mu_r - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r (l_r - \Delta l_r)}
{\Delta l_{r-1} + l_r - \Delta l_r}\right)^2 (l_r - \Delta l_r)\\
& + \left( \mu_r - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}} \right)^2 \Delta l_r \\
& + \left( \mu_{r+1} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}} \right)^2 l_{r+1} \\
& + \left( \mu_{r+2} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}} \right)^2 \Delta l_{r+1} \\
=~& \frac{\left(\mu_r - \mu_{r-1}\right)^2 (l_r - \Delta l_r) \Delta l_{r-1}}
{\Delta l_{r-1} + l_r - \Delta l_r}\\
& + \frac{(\mu_{r+1} - \mu_r)^2 \Delta l_r l_{r+1} + (\mu_{r+2} - \mu_r)^2 \Delta l_r \Delta l_{r+1}
+ (\mu_{r+2} - \mu_{r+1})^2 \Delta l_{r+1} l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}}.\end{aligned}$$ On the other hand, under the model $\tilde{\boldsymbol\lambda}_N$, $$\begin{aligned}
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor -1}
\delta_{\tilde{\boldsymbol\lambda}_N, t}^2
=~& \left(\mu_{r-1} - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r l_r}
{\Delta l_{r-1} + l_r}\right)^2 \Delta l_{r-1}
+\left(\mu_r - \frac{\mu_{r-1} \Delta l_{r-1} + \mu_r l_r }
{\Delta l_{r-1} + l_r}\right)^2 l_r \\ \nonumber
& + \left( \mu_{r+1} - \frac{\mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{ l_{r+1} + \Delta l_{r+1}} \right)^2 l_{r+1}
+ \left( \mu_{r+2} - \frac{\mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1} }
{ l_{r+1} + \Delta l_{r+1}} \right)^2 \Delta l_{r+1}\\
=~& \frac{\left(\mu_r - \mu_{r-1}\right)^2 l_r \Delta l_{r-1}}
{\Delta l_{r-1} + l_r }
+ \frac{\left(\mu_{r+2} - \mu_{r+1}\right)^2 l_{r+1} \Delta l_{r+1}}
{\Delta l_{r+1} + l_{r+1} }. \end{aligned}$$ The difference of the last term in under the two models, up to an $O_P(1/N)$ error, is thus $$\begin{aligned}
\nonumber
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor -1}
\left( \delta_{\hat{\boldsymbol\lambda}_N, t}^2 -
\delta_{\tilde{\boldsymbol\lambda}_N, t}^2 \right)
=~& -\frac{(\mu_r - \mu_{r-1})^2 \Delta l_{r-1}^2 \Delta l_r}
{(\Delta l_{r-1} + l_r - \Delta l_r)(\Delta l_{r-1} + l_r)}
- \frac{(\mu_{r+2} - \mu_{r+1})^2 \Delta l_r l_{r+1} \Delta l_{r+1}}
{(\Delta l_r + l_{r+1} + \Delta l_{r+1}) (\Delta l_{r+1} + l_{r+1})} \\
& + \frac{(\mu_{r+1} - \mu_r)^2 \Delta l_r l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}}
+ \frac{(\mu_{r+2} - \mu_r)^2 \Delta l_r \Delta l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}}\\ \nonumber
=~& (\mu_{r+1} - \mu_r)^2 \Delta l_r + o_P(\Delta l_r) = O_P\left( N^{\omega -1} \right).\end{aligned}$$
Therefore, if $\omega > 0$, the change in $\hat{\gamma}(h)$ in under models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ is
$$\begin{aligned}
\nonumber
\hat{\gamma}(h)_{\hat{\boldsymbol\lambda}_N} - \hat{\gamma}(h)_{\tilde{\boldsymbol\lambda}_N}
& = O_P(N^{-1}) + O_P(N^{\frac{\omega}{2}-1}) + O_P(N^{\omega -1}) = O_P(N^{\omega -1})\\
\label{eq:autocov5}
& = (\mu_{r+1} - \mu_r)^2 \Delta l_r, \end{aligned}$$
which is positive and constant for all $h \in \{ 0, 1, \ldots, p\}$.
Following similar reasoning, if $\hat{\lambda}_{j}$ is to the right of $\lambda^0_{r}$, still holds. This conclusion does not change if $\hat{\lambda}_{j-1}$ is to the right of $\lambda^0_{r-1}$ (or $\hat{\lambda}_{j+1}$ is to the left of $\lambda^0_{r+1}$): simply take $\Delta l_{r-1} = 0$ (or $\Delta l_{r+1} = 0$) and all above derivations hold unaltered.
Next, we will show that for sufficiently large $N$, model $\tilde{\boldsymbol\lambda}_N$ has a smaller BMDL than model $\hat{\boldsymbol\lambda}_N$. From the proof of Proposition \[prop:sigmasq2\] showing that $f(\delta^2)$ is strictly increasing in $\delta^2$, the difference of the BMDL estimator $\hat{\sigma}^2$ under $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ satisfies $$\hat{\sigma}^2_{\hat{\boldsymbol\lambda}_N}
- \hat{\sigma}^2_{\tilde{\boldsymbol\lambda}_N}
= O_P(N^{\omega - 1}),$$ and is positive. This limit can be denoted by $C N^{\omega - 1}$, where $C > 0$. Recall that the model $\tilde{\boldsymbol\lambda}_N$ contains the same number of changepoints as the model $\hat{\boldsymbol\lambda}_N$; therefore,
$$\begin{aligned}
\nonumber
\text{BMDL}(\hat{\boldsymbol\lambda}_N)
-\text{BMDL}(\tilde{\boldsymbol\lambda}_N)
&= ~\frac{N-p}{2}\log \left(\frac{\hat{\sigma}^2_{\hat{\boldsymbol\lambda}_N}}
{\hat{\sigma}^2_{\tilde{\boldsymbol\lambda}_N}}\right)
+ O_P\left(1\right) \\ \nonumber
&= ~\frac{N}{2}\log \left(1 + \frac{C}
{\hat{\sigma}^2_{\tilde{\boldsymbol\lambda}_N} N^{1 - \omega} }\right)
+ O_P\left(1\right) \\ \nonumber
&= \frac{N^{\omega}}{2} \log \left(1 + \frac{C}
{\hat{\sigma}^2_{\tilde{\boldsymbol\lambda}_N} N^{1 - \omega} }\right)^{N^{1-\omega}}
+ O_P\left(1\right) \\ \nonumber
&= \frac{N^{\omega}}{2} \frac{C}
{\hat{\sigma}^2_{\tilde{\boldsymbol\lambda}_N}} + O_P\left(1\right),\end{aligned}$$
where the last equality holds because $\lim_{n\rightarrow \infty}(1 + \frac{x}{n})^n \rightarrow e^x$ and $\omega \leq 0$. Hence, $\text{BMDL}(\hat{\boldsymbol\lambda}_N) -
\text{BMDL}(\tilde{\boldsymbol\lambda}_N)$ diverges to infinity at rate $O_P(N^{\omega})$ should $\omega > 0$. Here, a contradiction arises since $\hat{\boldsymbol\lambda}_N$ minimizes the BMDL.
In Theorem \[thm:lambda\_convergence\], the convergence rate in comes from Lemma \[lemma:lambda\_convergence\]. Now the proof of is given.
\[A proof of in Theorem \[thm:lambda\_convergence\]\] In the proof of Lemma \[lemma:lambda\_convergence\], $\boldsymbol\lambda^* \supset
\boldsymbol\lambda^0$. To verify , we need only show that $\boldsymbol\lambda^* = \boldsymbol\lambda^0$; in other words, there are no changepoints in $\boldsymbol\lambda^*$ that are not in $\boldsymbol\lambda^0$.
Proof by contradiction will again be used. Suppose that for a large $N$, the BMDL estimator $\hat{\boldsymbol\lambda}_N$ contains more than $m^0$ changepoints. More specifically, suppose that during the $(r+1)$th regime in the true model $\boldsymbol\lambda^0$, there are redundant changepoints estimated in $\hat{\boldsymbol\lambda}_N$, i.e., for some integer $d > 1$, $$\hat{\lambda}_j \stackrel {P} {\longrightarrow} \lambda_r^0, \quad
\hat{\lambda}_{j+d} \stackrel {P} {\longrightarrow} \lambda_{r+1}^0,$$ where $\hat{\lambda}_j$ can be to the left or right of $\lambda_r^0$, and $\hat{\lambda}_{j+d}$ can be to the left or right of $\lambda_{r+1}^0$. Since the estimated changepoints $\hat{\lambda}_{j+1}, \ldots, \hat{\lambda}_{j+d-1}$ are redundant, a new relative multiple changepoint model $$\tilde{\boldsymbol\lambda}_N
= \left(\hat{\lambda}_1, \ldots, \hat{\lambda}_j, \hat{\lambda}_{j+d},
\ldots, \hat{\lambda}_{\hat{m}}\right)'$$ is created by removing the redundant changepoints $\hat{\lambda}_{j+1},
\ldots, \hat{\lambda}_{j+d-1}$ from $\hat{\boldsymbol\lambda}_N$. A contradiction would arise if $\text{BMDL}(\hat{\boldsymbol\lambda}_N)>
\text{BMDL}( \tilde{\boldsymbol\lambda}_N)$ for large $N$ since $\hat{\boldsymbol\lambda}_N$ minimizes the BMDL.
(-1, 0) – (11, 0); at (-3, 1) [true regime mean]{}; at (-3, 0) [timeline]{}; at (-3, -1.50) [length]{}; (0.5, -0.1) – (0.5, 0.1); at (0.5, -0.4) [$\hat{\lambda}_j$]{}; (1.5, -0.1) – (1.5, 1.2); at (1.5, -0.4) [$\lambda_r^0$]{}; (3.5, -0.1) – (3.5, 0.1); at (3.5, -0.4) [$\hat{\lambda}_{j + 1}$]{}; (6.5, -0.1) – (6.5, 0.1); at (6.5, -0.4) [$\hat{\lambda}_{j + d - 1}$]{}; (8.5, -0.1) – (8.5, 1.2); at (8.5, -0.4) [$\lambda_{r+1}^0$]{}; (10, -0.1) – (10, 0.1); at (10, -0.4) [$\hat{\lambda}_{j+d}$]{}; (0.6, 1) – (1.4, 1); (-0.8, 1) – (0, 1); at (0.3, 1) [$\mu_{r}$]{}; (1.6, 1) – (4, 1); (6, 1) – (8.4, 1); at (5, 1) [$\mu_{r+1}$]{}; (10.6, 1) – (11.4, 1); (8.6, 1) – (9.4, 1); at (10, 1) [$\mu_{r + 2}$]{}; (0.5, -1.0) – (1.45, -1.0); at (1, -1.3) [$\Delta l_r$]{}; (1.55, -1.0) – (3.5, -1.0); at (2.5, -1.3) [$a_1$]{}; (6.5, -1.0) – (8.45, -1.0); at (7.5, -1.3) [$a_d$]{}; (8.55, -1.0) – (10, -1.0); at (9.25, -1.3) [$\Delta l_{r+1}$]{}; (1.5, -1.7) – (8.5, -1.7); at (5, -2) [$l_{r+1}$]{}; (-1, 2) – (11, 2);
Similar to the proof of Lemma \[lemma:lambda\_convergence\], the difference of $\hat{\gamma}(h)$ under the two models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ will be investigated, for each $h \in \{ 0, 1, \ldots, p \}$. By , the first term in is the same under $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$, up to a $O_P(1/N)$ difference.
For the other terms in , we need only focus on the summation over $t$ in the interval $\lfloor \hat{\lambda}_j N \rfloor
\leq t \leq \lfloor \hat{\lambda}_{j+d} N \rfloor - 1$, illustrated in Figure \[fig:diagram2\], since $(W_t, \delta_t)$ are the same for all other $t$ in $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$. For simplicity, lengths of time intervals on the rescaled timeline are denoted by $$l_{r+1} = \lambda^0_{r+1} - \lambda^0_{r}, \quad
a_1 = \hat{\lambda}_{j+1} - \lambda^0_{r}, \quad
a_d = \lambda^0_{r} - \hat{\lambda}_{j+d-1}.$$ If $\hat{\lambda}_j$ is to the left of $\lambda^0_r$ and $\hat{\lambda}_{j+d}$ is to the right of $\lambda^0_{r+1}$ (see Figure \[fig:diagram2\]), then the vanishing length between them and their limits are denoted by $$\Delta l_r = \lambda^0_r - \hat{\lambda}_j, \quad
\Delta l_{r+1} = \hat{\lambda}_{j+d} - \lambda^0_{r+1}.$$
Under the model $\hat{\boldsymbol\lambda}_N$, $\delta_t$ in can be written as $$\label{eq:delta_t_lambdahat}
\delta_{\hat{\boldsymbol\lambda}_N, t} =
\begin{cases}
\mu_r - \frac{\mu_r \Delta l_r + \mu_{r+1} a_1}{\Delta l_r + a_1},
& \text{ if } \lfloor \hat{\lambda}_j N \rfloor \leq t \leq \lfloor \lambda^0_r N \rfloor - 1,\\
\mu_{r+1} - \frac{\mu_r \Delta l_r + \mu_{r+1} a_1}{\Delta l_r + a_1},
& \text{ if } \lfloor \lambda^0_r N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+1} N \rfloor - 1,\\
0,
& \text{ if } \lfloor \hat{\lambda}_{j+1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+d-1} N \rfloor - 1,\\
\mu_{r+1}-\frac{\mu_{r+2} \Delta l_{r+1} + \mu_{r+1} a_d}{\Delta l_{r+1} + a_d},
& \text{ if } \lfloor \hat{\lambda}_{j+d-1} N \rfloor \leq t \leq \lfloor \lambda^0_{r+1} N \rfloor - 1,\\
\mu_{r+2}-\frac{\mu_{r+2} \Delta l_{r+1} + \mu_{r+1} a_d}{\Delta l_{r+1} + a_d},
& \text{ if } \lfloor \lambda^0_{r+1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+d} N \rfloor - 1.
\end{cases}$$ On the other hand, under the model $\tilde{\boldsymbol\lambda}_N$, $$\label{eq:delta_t_lambdatilde}
\delta_{\tilde{\boldsymbol\lambda}_N, t} =
\begin{cases}
\mu_r - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \hat{\lambda}_j N \rfloor \leq t \leq \lfloor \lambda^0_r N \rfloor - 1,\\
\mu_{r+1} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \lambda^0_r N \rfloor \leq t \leq \lfloor \lambda^0_{r+1} N \rfloor - 1,\\
\mu_{r+2} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}},
& \text{ if } \lfloor \lambda^0_{r+1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+d} N \rfloor - 1.
\end{cases}$$
When $N$ is large, $\delta_t = \delta_{t-h}$ for all but a finite number of times $t$; hence, for the second term (and similarly, the third term) in , $$\label{eq:autocov6}
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_j N \rfloor}^{\lfloor \hat{\lambda}_{j+d} N \rfloor -1}
\delta_{t-h} W_t
= \frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_j N \rfloor}^{\lfloor \hat{\lambda}_{j+1} N \rfloor - 1}
\delta_{t} W_t
+ \frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j+1} N \rfloor}^{\lfloor \hat{\lambda}_{j+d-1} N \rfloor - 1}
\delta_{t} W_t
+ \frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j+d-1} N \rfloor}^{\lfloor \hat{\lambda}_{j+d} N \rfloor - 1}
\delta_{t} W_t
+O_P\left( \frac{1}{N} \right).$$ By and , under the models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$, the difference of $\delta_t$ is piecewise constant, i.e., $$\begin{aligned}
\label{eq:delta_t_diff}
& \delta_{\hat{\boldsymbol\lambda}_N, t} - \delta_{\tilde{\boldsymbol\lambda}_N, t}\\ \nonumber
=~&
\begin{cases}
\frac{\left(\mu_{r+1} - \mu_r\right) \Delta l_r (l_{r+1} - a_1)
+ (\mu_{r+2} - \mu_{r+1})\Delta l_{r+1} a_1 + O_P(\Delta l^2) }
{\left(\Delta l_r + a_1\right) \left( \Delta l_r + l_{r+1} + \Delta l_{r+1} \right)}
= O_P\left( N^{\omega - 1}\right),
& \text{ if } \lfloor \hat{\lambda}_{j} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+1} N \rfloor - 1,\\
\frac{\left(\mu_{r+1} - \mu_r\right) \Delta l_r
+ (\mu_{r+2} - \mu_{r+1})\Delta l_{r+1} }
{\Delta l_r + l_{r+1} + \Delta l_{r+1}}
= O_P\left( N^{\omega - 1}\right),
& \text{ if } \lfloor \hat{\lambda}_{j+1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+d-1} N \rfloor - 1,\\
\frac{\left(\mu_{r+1} - \mu_{r+2}\right) \Delta l_{r+1} (l_{r+1} - a_d)
+ (\mu_r - \mu_{r+1})\Delta l_r a_d + O_P(\Delta l^2)}
{(\Delta l_{r+1} + a_d)(\Delta l_r + l_{r+1} + \Delta l_{r+1})}
= O_P\left( N^{\omega - 1}\right),
& \text{ if } \lfloor \hat{\lambda}_{j+d-1} N \rfloor \leq t \leq \lfloor \hat{\lambda}_{j+d} N \rfloor - 1.
\end{cases}\end{aligned}$$ For the three time intervals in , their lengths are $\Delta l_r + a_1 = O_P(N^{\xi_1})$, $l_{r+1} - a_1 - a_d = O_P(1)$, and $a_d + \Delta l_{r+1} = O_P(N^{\xi_d})$, respectively, with $\xi_1, \xi_d \in [\omega, 1]$. For , when decomposed as three sums in these intervals, by , its difference under the models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ is $$\begin{aligned}
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j} N \rfloor}^{\lfloor \hat{\lambda}_{j+d} N \rfloor - 1}
\left(\delta_{\hat{\boldsymbol\lambda}_N, t} - \delta_{\tilde{\boldsymbol\lambda}_N, t}\right) W_t
&= \frac{1}{N} O_P\left( N^{\omega - 1} \right) \left\{
O_P\left( N^{\frac{\xi_1}{2}} \right)
+ O_P\left( N^{\frac{1}{2}} \right) + O_P\left( N^{\frac{\xi_d}{2}} \right)
\right\} + O_P\left( N^{-1}\right)\\
&= O_P\left( N^{\omega -\frac{3}{2}}\right) + O_P\left( N^{-1}\right). \end{aligned}$$ By Lemma \[lemma:lambda\_convergence\], $\omega < 0$; hence, for the second term (and similarly for the third term) in , its difference under the two models converges to zero at rate $O_P(1/N)$.
For the fourth term in , since $$\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j} N \rfloor}^{\lfloor \hat{\lambda}_{j+d} N \rfloor - 1}
\delta_{t-h} \delta_t
= \frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j} N \rfloor}^{\lfloor \hat{\lambda}_{j+d} N \rfloor - 1}
\delta_t^2 + O_P\left( \frac{1}{N} \right),$$ under the model $\hat{\boldsymbol\lambda}_N$, it can be written as $$\begin{aligned}
\nonumber
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j} N \rfloor}^{\lfloor \hat{\lambda}_{j+d} N \rfloor - 1}
\delta_{\hat{\boldsymbol\lambda}_N, t}^2
=~& \left(\mu_r - \frac{\mu_r \Delta l_r + \mu_{r+1} a_1}{\Delta l_r + a_1}\right)^2 \Delta l_r
+ \left( \mu_{r+1} - \frac{\mu_r \Delta l_r + \mu_{r+1} a_1}{\Delta l_r + a_1} \right)^2 a_1\\ \nonumber
& + \left( \mu_{r+1} - \frac{\mu_{r+2} \Delta l_{r+1} + \mu_{r+1} a_d}{\Delta l_{r+2} + a_d}
\right)^2 a_d
+ \left( \mu_{r+2} - \frac{\mu_{r+2} \Delta l_{r+1} + \mu_{r+1} a_d}{\Delta l_{r+2} + a_d}
\right)^2 \Delta l_{r+1}\\ \label{eq:deltasq_lambdahat}
=~& \frac{(\mu_{r+1} - \mu_r)^2 a_1 \Delta l_r}{a_1 + \Delta l_r}
+ \frac{(\mu_{r+2} - \mu_{r+1})^2 a_d \Delta l_{r+1}}{a_d + \Delta l_{r+1}},\end{aligned}$$ whereas under the model $\tilde{\boldsymbol\lambda}_N$, $$\begin{aligned}
\nonumber
\frac{1}{N} \sum_{t = \lfloor \hat{\lambda}_{j} N \rfloor}^{\lfloor \hat{\lambda}_{j+d} N \rfloor - 1}
\delta_{\tilde{\boldsymbol\lambda}_N, t}^2
=~& \left( \mu_r - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}}\right)^2 \Delta l_r\\ \nonumber
& + \left( \mu_{r+1} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}} \right)^2 l_{r+1}\\ \nonumber
& + \left( \mu_{r+2} - \frac{\mu_r \Delta l_r + \mu_{r+1} l_{r+1} + \mu_{r+2} \Delta l_{r+1}}
{\Delta l_r + l_{r+1} + \Delta l_{r+1}} \right)^2 \Delta l_{r+1}\\ \label{eq:deltasq_lambdatilde}
=~& \frac{(\mu_{r+1} - \mu_r)^2 l_{r+1} \Delta l_r
+(\mu_{r+2} - \mu_r)^2 \Delta l_r \Delta l_{r+1}
+(\mu_{r+2} - \mu_{r+1})^2 l_{r+1} \Delta l_{r+1} }{\Delta l_r + l_{r+1} + \Delta l_{r+1}}.\end{aligned}$$ Since $\Delta l_r = O_P\left(N^{\omega-1}\right)$ and $\Delta l_{r+1} =
O_P\left(N^{\omega-1}\right)$, where $\omega \leq 0$, both and converge to zero at rate $O_P(N^{\omega-1})$. Hence, the difference of the fourth term in under the two models converges to zero at rate $O_P(1/N)$.
The difference in $\hat{\gamma}(h)$ in under the two models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$ thus satisfies $$\hat{\gamma}_{\hat{\boldsymbol\lambda}_N}(h) = \hat{\gamma}_{\tilde{\boldsymbol\lambda}_N}(h)
+ O_P\left( \frac{1}{N} \right),$$ which holds for all $h \in \{ 0, 1, \ldots, p \}$. By Lemma \[lemma:sigmasq1\], a similar result holds for the BMDL estimators of $\sigma^2$ under the two models $\hat{\boldsymbol\lambda}_N$ and $\tilde{\boldsymbol\lambda}_N$: $$\label{eq:diff_sigmasq_lambdahat_lambdatilde}
\hat{\sigma}^2_{\hat{\boldsymbol\lambda}_N} =
\hat{\sigma}^2_{\tilde{\boldsymbol\lambda}_N} + O_P\left( \frac{1}{N} \right).$$ Note that if $\hat{\lambda}_j$ is to the right of $\lambda^0_r$ (or $\hat{\lambda}_{j+d}$ is to the left of $\lambda^0_{r+1}$), then we simply let $\Delta l_r = 0$ (or $\Delta l_{r+1} = 0$), so that all above derivations, including , , and more importantly, hold as stated.
The difference between $\text{BMDL}(\hat{\boldsymbol\lambda}_N)$ and $\text{BMDL}(\tilde{\boldsymbol\lambda}_N)$ will be studied. Recall that $\hat{\boldsymbol\lambda}_N$ has $d - 1$ more changepoints than $\tilde{\boldsymbol\lambda}_N$. By and , the BMDL difference is $$\begin{aligned}
\text{BMDL}(\hat{\boldsymbol\lambda}_N)
-\text{BMDL}(\tilde{\boldsymbol\lambda}_N)
=~& \frac{N-p}{2}\log \left(\frac{\hat{\sigma}^2_{\hat{\boldsymbol\lambda}_N} }
{\hat{\sigma}^2_{\tilde{\boldsymbol\lambda}_N} }\right)
+ \frac{3(d - 1)}{2} \log (N) + O_P\left(1\right) \\ \label{eq:diff_MDL2}
=~& O_P\left(1\right)
+ \frac{3(d-1)}{2} \log (N) + O_P\left(1\right) \\
=~& O_P(\log N), \end{aligned}$$ and is positive. Here, the second equality follows from . This contradicts that $\hat{\boldsymbol\lambda}_N$ minimizes the BMDL.
Proof of Theorem \[thm:parameter\_convergence\] {#PROOFthm:parameter_convergence}
-----------------------------------------------
\[A proof of Theorem \[thm:parameter\_convergence\]\] By Theorem \[thm:lambda\_convergence\], as $N$ tends to infinity, $\hat{\boldsymbol\lambda}_N \stackrel{P}{\longrightarrow}
\boldsymbol\lambda^0$, and hence $\delta^2_{\hat{\boldsymbol\lambda}_N}
\stackrel{P}{\longrightarrow} 0$. Therefore, by Proposition \[prop:YW\_phi\], the BMDL estimator $$\hat{\boldsymbol\phi}_N =
\left(\boldsymbol\Gamma_p + \delta^2_{\hat{\boldsymbol\lambda}_N} \mathbf{J}_p \right)^{-1}
\left(\boldsymbol\gamma_p + \delta^2_{\hat{\boldsymbol\lambda}_N} \mathbf{1}_p \right)
+ O_P\left(\frac{1}{\sqrt{N}}\right)
\stackrel{P}{\longrightarrow}
\boldsymbol\Gamma_p^{-1} \boldsymbol\gamma_p = \boldsymbol\phi^0.$$ By , when $\delta = 0$, $f(0) = \gamma(0) -
\boldsymbol\gamma_p'\boldsymbol\Gamma_p^{-1} \boldsymbol\gamma_p =
\left(\sigma^2\right)^0$, i.e., the true value of $\sigma^2$. Since $f(\delta^2)$ is continuous in $\delta^2$, Proposition \[prop:sigmasq2\] shows that as $N \rightarrow \infty$, the BMDL estimator $$\hat{\sigma}^2_N \stackrel{P}{\longrightarrow} f(0) = \left(\sigma^2\right)^0.$$
For sufficiently large $N$, since $\hat{\boldsymbol\lambda}_N$ is close to the true model $\boldsymbol\lambda^0$, the regime indicator matrix $\mathbf{D}$ under $\hat{\boldsymbol\lambda}_N$ is close to its counterpart $\mathbf{D}^0$ under the true model. Therefore, implies that $$\label{eq:likelihood4}
\widehat{\mathbf{X}}= \widehat{\mathbf{A}} \mathbf{s} +
\widehat{\mathbf{D}}\boldsymbol\mu + \hat{\mathbf{z}},$$ where $\hat{\mathbf{z}} = (\hat{z}_{p+1}, \ldots,
\hat{z}_N)'$, and $\hat{z}_t = \epsilon_t - \sum_{j=1}^p
\hat{\phi}_j \epsilon_{t-j}$. Since $\hat{\mathbf{z}}$ is a series of white noises [@Brockwell_Davis_1991 pp. 240], can be viewed as a linear model with unknown coefficients $(\mathbf{s},
\boldsymbol\mu)$.
Following the proof of Lemma \[lemma:sigmasq1\], the BMDL estimators for $\mathbf{s}$ and $\boldsymbol\mu$ have the following limits: $$\begin{aligned}
\hat{\mathbf{s}}_N
& = (\widehat{\mathbf{A}}'\widehat{\mathbf{B}}
\widehat{\mathbf{A}})^{-1}
(\widehat{\mathbf{A}}'\widehat{\mathbf{B}}\widehat{\mathbf{X}})\\
& = \left\{ \widehat{\mathbf{A}}'
\left(\mathbf{I}_{N-p} - \mathcal{P}_{\widehat{\mathbf{D}}}\right)
\widehat{\mathbf{A}} \right\}^{-1}
\left\{ \widehat{\mathbf{A}}'
\left(\mathbf{I}_{N-p} - \mathcal{P}_{\widehat{\mathbf{D}}}\right)
\widehat{\mathbf{X}} \right\} + O_P\left(\frac{1}{N}\right),\\
\hat{\boldsymbol\mu}_N
& = \left(\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}+\frac{\mathbf{I}_{m}}{\nu}\right)^{-1}
\widehat{\mathbf{D}}' \left( \widehat{\mathbf{X}} - \widehat{\mathbf{A}}\hat{\mathbf{s}}_N \right)\\
& = \left(\widehat{\mathbf{D}}'
\widehat{\mathbf{D}}\right)^{-1}
\widehat{\mathbf{D}}' \left( \widehat{\mathbf{X}} - \widehat{\mathbf{A}}\hat{\mathbf{s}}_N \right)
+ O_P\left(\frac{1}{N}\right).\end{aligned}$$ After rewriting as $$\begin{aligned}
\widehat{\mathbf{X}}
= &\left\{\left(\mathbf{I}_{N-p} - \mathcal{P}_{\widehat{\mathbf{D}}}\right)
\widehat{\mathbf{A}}\right\} \mathbf{s}
+ \left\{\widehat{\mathbf{D}}\boldsymbol\mu
+ \mathcal{P}_{\widehat{\mathbf{D}}} \widehat{\mathbf{A}}\mathbf{s} \right\}
+ \hat{\mathbf{z}}\\
= & \left\{\left(\mathbf{I}_{N-p} - \mathcal{P}_{\widehat{\mathbf{D}}}\right)
\widehat{\mathbf{A}}\right\} \mathbf{s}
+ \widehat{\mathbf{D}}\left\{ \boldsymbol\mu +
\left(\widehat{\mathbf{D}}' \widehat{\mathbf{D}}\right)^{-1}\widehat{\mathbf{D}}'
\widehat{\mathbf{A}} \mathbf{s}\right\} + \hat{\mathbf{z}},\end{aligned}$$ it is not hard to see that $\hat{\mathbf{s}}_N$ and $\hat{\boldsymbol\mu}_N$ are the least square estimators of this linear model. Since least square estimators are asymptotically consistent: $\hat{\mathbf{s}}_N \stackrel{P}{\longrightarrow} \mathbf{s}^0$ and $\hat{\boldsymbol\mu}_N \stackrel{P}{\longrightarrow} \boldsymbol\mu^0$.
Additional Simulations and Real Examples {#sec:appendix_examples}
========================================
Simulation Examples
-------------------
Additional figures related to our simulation examples in Section \[sec:simulation\] are included here.
Tuscaloosa Data Analysis: Target Minus Reference {#subsec:target_minus_reference}
------------------------------------------------
A reference series is a record from a station near the target station that is subtracted from the target series. The idea is that two nearby stations should experience similar weather; hence, any trends or seasonal cycles should be lessened (if not altogether removed) in the target minus reference subtraction. Changepoints caused by artificial reasons, rather than by real climate changes, are easier to detect (visually) in target minus reference comparisons. Following @Lu_etal_2010, our reference series is obtained by averaging three nearby stations: Aberdeen, MS; Greensboro, AL; and Selma, AL. By averaging multiple reference series (this is called a composite reference), impacts of mean shifts in any of the individual stations in the composite reference are lessened.
Figure \[fg:Tuscaloosa3\] shows the optimal changepoint configuration for the target minus reference series and contains 12 concurrent changes: June 1914, January 1919, July 1933, July 1937, August 1937, October 1938, December 1938, June 1946, July 1946, November 1956, May 1987, and October 1996. Among them, the 1956 and 1987 changepoints are in the metadata; the two changepoints in 1938 are close to the 1939 station relocation. The changepoints in 1919, 1933, and 1990 are also flagged by @Lu_etal_2010. One of the shifts, November 1956, moves the Tmax series warmer and the Tmin series colder.
The October and December 1938 changepoints are likely due to typos in the data record. Specifically, the October and November 1938 Tmin values in the target minus reference series appear to be abnormally high. While the data have been quality checked, some errors persist. This conjecture is made because the three reference stations lie in various directions from Tuscaloosa; climatologically, series to the north and west of Tuscaloosa should be cooler and those to the south and east should be warmer. In this case, Tuscaloosa was significantly warmer than all three references. Similar statements apply to the two “outlier” changepoints in 1937, and the two changepoints in 1946, where the Tmin records for Tuscaloosa are lower than those for all three reference stations. It is interesting that our method picked up outliers.
It is natural to flag more changepoints in the target minus reference series than the target series alone. An ideal reference series should have the same trend and seasonal cycles as the target series and be free of artificial mean shifts. This said, we do not assume that the target minus reference comparison completely removes the monthly mean cycle; indeed, [@Liu_etal_2016] shows that this is seldom the case. Reference series selection is a problem currently studied by climatologists. As our reference series averages three neighbor stations, mean shifts in any of the reference records may induce shifts in the target minus reference series. For example, the estimated changepoint in 1914 is close to the 1915 metadata time listed in the Aberdeen reference. This said, averaging three neighbors should help mitigate the effects of changepoints in any individual reference series.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Numerical analysis is conducted for a generalized particle method for a Poisson equation. Unique solvability is derived for the discretized Poisson equation by introducing a connectivity condition for particle distributions. Moreover, by introducing discrete Sobolev norms and a semi-regularity of a family of discrete parameters, stability is obtained for the discretized Poisson equation based on the norms.'
address: '${}^\dagger$Tohoku Forum for Creativity, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577 Japan'
author:
- 'Yusuke Imoto${}^\dagger$'
title: |
Stability of a generalized particle method for a Poisson equation by\
discrete Sobolev norms
---
Introduction
============
Numerical analysis is conducted for the generalized particle method introduced in [@imoto2016phd]. This method is a generalization of conventional particle methods such as smoothed particle hydrodynamics [@liu2010smoothed] and the moving particle semi-implicit [@koshizuka1996moving]. For the generalized particle method, we have established numerical analyses involving the truncation error estimates of approximate operators [@imoto2016teintpV; @imoto2017tederiV] and the error estimates for the Poisson and heat equations based on the maximum norm [@imoto2016phd]. Therefore, as the next step of this study, we focus on the numerical analyses of the generalized particle method using discrete Sobolev norms.
This study considers a Poisson equation with a source term given by a divergence form. This formulation is selected because it has several practical applications. For example, a pressure Poisson equation, which is appeared in formulations of particle methods for the incompressible Navier–Stokes equations [@shao2003incompressible], uses a source term including a divergence of a predictor of velocity.
A connectivity condition for particle distributions and a semi-regularity of a family of discrete parameters are introduced for analyzing the discretized Poisson equation. By virtue of the connectivity condition, a unique solvability of the discretized Poisson equation is derived. Moreover, by demonstrating certain properties of the discrete Sobolev norms, such as integration by parts, the stability of the discretized Poisson equation is obtained with semi-regularity.
Formulation {#sec:formulation}
===========
Let $\Dm\subset\dRd\,(\Dim\in\dN)$ be a bounded domain with smooth boundary $\Bd$. Let $\FsCz{\DmOl}$ be the space of real continuous functions defined in $\DmOl$. For $k\in\dN$, let $\FsC{k}{\DmOl}$ be the space of functions in $\FsCz{\DmOl}$ with derivatives up to the $k$th order. We define a function space, $\FsSol$, as $$\FsSol\deq\Set{\Func\in\FsCz{\DmOl}}{\Func(\x)=0\,(\x\in\Bd)}.$$ Then, we consider the following Poisson equation with a homogeneous boundary condition: $$\mbox{Find}~\Sol\in\FsSol\cap\FsC{2}{\DmOl} \st -\Lap\Sol = \Div\DmFunc,
\label{Poisson}$$ where $\DmFunc\in(\FsSol\cap\FsC{1}{\DmOl})^\Dim$ is given.
We introduce approximate operators in the generalized particle method. Let $\H$ be a fixed positive number. For $\Dm$ and $\H$, we define $\DmH\subset\dRd$ and $\BdH$ by $$\begin{aligned}
\DmH &\deq \left\{\x\in\dRd \midd \ex \y\in\Dm \st |\x-\y|<\H\right\}, \\
\BdH &\deq \DmH\setminus\Dm. \end{aligned}$$ For $\N\in\dN$, we define a particle distribution, $\PtSet$, and a particle volume set, $\PvSet$, as $$\begin{aligned}
\PtSet \deq \Set{\Pt{\i}{}\in\Dm}{\i=1,2,\dots,\N,\,\Pt{\i}{}\neq\Pt{\j}{}\,(\i\neq\j)},
\label{def:PtSet}\\
\PvSet \deq \Set{\Pv{\i}>0}{\i=1,2,\dots,\N,\,\sum_{\i=1}^\N\Pv{\i}=\Meas{\DmH}}.
\label{def:PvSet}\end{aligned}$$ Here, $\Meas{\DmH}$ denotes the volume of $\DmH$. We refer to $\Pt{\i}{}\in\PtSet$ and $\Pv{\i}\in\PvSet$ as a particle and particle volume, respectively. We define a function set, $\FsWeightFunc$, as $$\FsWeightFunc \deq \Set{\w:[0,\infty)\ra\dR}{\w(r)>0\,(0<r<1),\w(r)=0\,(r\geq1),\,\int_{\dRd}\w(|\x|)\dx=1}.$$ We refer to $\w\in\FsWeightFunc$ as a reference weight function. We define an influence radius, $\h$, as a positive number that satisfies $\min\SetNd{|\Pt{\i}{}-\Pt{\j}{}|}{\i\neq\j}<\h<\H$. For reference weight function $\w\in\FsWeightFunc$ and influence radius $\h$, we define a weight function, $\wh:[0,\infty)\ra\dR$, as $$\wh(r) \deq \frac{1}{\h^{\Dim}}\w\left(\frac{r}{\h}\right).
\label{def:wh}$$ Then, for discrete parameters $(\PtSet, \PvSet, \h)$ and reference weight function $\w\in\FsWeightFunc$, we define an approximate divergence operator, $\DivAppPlus$, for $\psi:\PtSet\ra\dRd$ as $$\DivAppPlus\psi_{\i} \deq \Dim\sum_{\j\neq\i}\Pv{\j} \frac{\psi_{\j}+\psi_{\i}}{|\Pt{\j}{}-\Pt{\i}{}|}\cdot\frac{\Pt{\j}{}-\Pt{\i}{}}{|\Pt{\j}{}-\Pt{\i}{}|} \wh(|\Pt{\j}{}-\Pt{\i}{}|)
\label{def:DivAppPlus}$$ and an approximate Laplace operator, $\LapApp{}$, for $\phi:\PtSet\ra\dR$ as $$\LapApp{} \phi_{\i}\deq 2\Dim\sum_{\j\neq\i}\Pv{\i} \frac{\phi_{\j}-\phi_{\i}}{|\Pt{\j}{}-\Pt{\i}{}|^2} \wh(|\Pt{\j}{}-\Pt{\i}{}|),
\label{def:LapApp}$$ where $\psi_{\i}\deq\psi(\Pt{\i}{})$ and $\phi_{\i}\deq\phi(\Pt{\i}{})$.
We define an index set, $\IndexSet{S}\,(S\subset\dRd)$, and a function space, $\FsSolApp$, as $$\begin{aligned}
&\IndexSet{S}\deq \BrM{\i=1,2,\dots,\N\midd \Pt{\i}{}\in \PtSet\cap S},\\
&\FsSolApp\deq\Set{\Func:\PtSet\ra\dR}{\Func(\Pt{\i}{})=0\,(\i\in\IndexSet{\BdH})}. \end{aligned}$$ Then, we consider the following generalized particle method for the Poisson equation: $$\mbox{Find}~\SolApp\in\FsSolApp \st -\LapApp{}\SolApp_{\i} = \DivAppPlus\DmFuncEx_{\i}\quad\i\in\IndexSet{\Dm},
\label{Poisson_disc}$$ where $\DmFuncEx\in\FsSolApp^\Dim$ such that $\DmFuncEx_{\i}=\DmFunc_{\i}\,(\i\in\IndexSet{\Dm})$, $\DmFuncEx_{\i}=0\,(\i\in\IndexSet{\BdH})$.
In particle methods, approximate operators of the first derivative, including the difference in function values, have also been proposed, such as in [@liu2010smoothed]. To distinguish between our operators and their operators, we used the notation with the plus symbol, e.g., $\DivAppPlus$ in .
We can introduce approximate operators and using the weighted averages of approximations based on the finite volume method, as shown in Appendix \[appendix:deriv\_app\_diver\].
Connectivity and semi-regularity
================================
We introduce a connectivity condition for particle distribution $\PtSet$ and a semi-regular condition for a family of discrete parameters.
\[def\_connectivity\] For influence radius $\h$, we consider that particle distribution $\PtSet$ satisfies $\h$-connectivity if for all $\i\in\IndexSet{\Dm}$, there exists an integer, $\IndexSetSeqLast\,(\leq\N)$, and a sequence, $\{\IndexSetSeq{\i}{\k}\}_{\k=1}^{\IndexSetSeqLast}\subset\{1,2,\dots,\N\}$, such that $$\IndexSetSeq{\i}{1}=\i,\quad|\Pt{\IndexSetSeq{\i}{\k}}{}-\Pt{\IndexSetSeq{\i}{\k+1}}{}|<\h\,(1\leq\k<\IndexSetSeqLast),\quad\IndexSetSeq{\i}{\k}\in\IndexSet{\Dm}\,(1\leq\k<\IndexSetSeqLast),\quad\IndexSetSeq{\i}{\IndexSetSeqLast}\in\IndexSet{\BdH}.
\label{h_connectivity}$$
A family, $\{(\PtSet, \PvSet, \h)\}$, is *semi-regular* if there exists a positive constant, ${c_0}$, such that for all elements of the family, $$\max_{\i=1,2,\dots,\N}\BrM{\sum_{\j\neq\i}\Pv{\j}\wh(|\Pt{\j}{}-\Pt{\i}{}|)}\leq {c_0}.
\label{def:semi-regular}$$ We refer to the constant, ${c_0}$, as a semi-regular constant.
Consider graph $G$, whose vertex set is particle distribution $\PtSet$ and whose edges are a pair, $(\Pt{\i}{}, \Pt{\j}{})$, which satisfies $0<|\Pt{\j}{}-\Pt{\i}{}|<\h$, e.g., see Figure \[fig:connectivity\]. By Definition \[def\_connectivity\], particle distribution $\PtSet$ satisfies $\h$-connectivity is equivalent to all vertices of $G$ on $\Dm$ have a path to a vertex of $G$ on $\BdH$.
Stability analysis
==================
First, we show the unique solvability for the discrete Poisson equation .
\[theorem:unique\_solvability\] If particle distribution $\PtSet$ satisfies $\h$-connectivity, then discrete Poisson equation has a unique solution.
Let $\Np$ be the number of particles included in $\Dm$. We renumber the index of particles so that $\Pt{\i}{}\in\Dm\,(\i=1,2,\dots,\Np)$. Let ${a_{\i\j}}\,(i,j=1,2,\dots,N)$ be $$\begin{aligned}
{a_{\i\j}}&\deq
\begin{cases}
\ds 0, \quad & i=j,\\
\ds 2 \Dim \frac{\wh(|\Pt{\j}{}-\Pt{\i}{}|)}{|\Pt{\j}{}-\Pt{\i}{}|^2}, \quad & \i\neq\j.
\end{cases}\end{aligned}$$ We define matrix ${A}\in\dR^{\Np\times\Np}$ as $${A}_{\i\j}\deq
\begin{cases}
\ds \sum_{k=1}^{\N} \frac{\Pv{k}}{\Pv{\i}}{a_{\i\k}}, \quad & i=j,\\
\ds -{a_{\i\j}}, \quad & \i\neq\j,
\end{cases}$$ Then, discrete Poisson equation is equivalent to $$\mbox{Find}~{y}\in\dR^{\Np} \st {A}{D}{y}={b},$$ where ${D}\deq \Diag(\Pv{\i})$, ${b}_{\i}\deq\DivAppPlus\DmFuncEx_{\i}\,(\i=1,2,\dots,\Np)$, and ${y}_{\i}\deq \SolApp_{\i}\,(\i=1,2,\dots,\Np)$. As $\Pv{\i}>0\,(\i=1,2,\dots,\N)$, diagonal matrix ${D}$ is a regular matrix. Therefore, it is sufficient to prove that ${A}$ is a regular matrix. As ${A}$ is symmetric, we will prove that ${A}$ is a positive definite matrix. For all ${\alpha}\in\dR^{\Np}\setminus\{0\}$, we have $$\begin{aligned}
\sum_{i,j=1}^{\Np} {\alpha}_{\i}{\alpha}_{\j} {A}_{\i\j}
&= 2\sum_{1\leq i<j\leq\Np} {\alpha}_{\i}{\alpha}_{\j} {A}_{\i\j} + \sum_{\i=1}^{\Np} {\alpha}_{\i}^2 {A}_{\i\i}
\nn\\
& = -2\sum_{1\leq i<j\leq\Np}{\alpha}_{\i}{\alpha}_{\j} {a_{\i\j}} + \sum_{\i=1}^{\Np} {\alpha}_{\i}^2 \sum_{k=1}^{\N} \frac{\Pv{k}}{\Pv{\i}}{a_{\i\k}}
\nn\\
& = \sum_{1\leq i<j\leq\Np}\frac{\BrS{\Pv{\j} {\alpha}_{\i}-\Pv{\i}{\alpha}_{\j}}^2}{\Pv{\i} \Pv{\j}}{a_{\i\j}} + \sum_{\i=1}^{\Np} {\alpha}_{\i}^2 \sum_{k=\Np+1}^{\N} \frac{\Pv{k}}{\Pv{\i}}{a_{\i\k}}.
\label{prf_poisson_unique_01}\end{aligned}$$ As ${a_{\i\j}}$ is nonnegative, is nonnegative. For $a\in\dR^{\Np}\setminus\{0\}$, we set $\i$ such that ${\alpha}_{\i}\neq 0$. Because of particle distribution $\PtSet$ with $\h$-connectivity, we can consider a sequence, $\{\IndexSetSeq{\i}{\k}\}_{\k=1}^{\IndexSetSeqLast}$, such that . As all terms of the last equation in are nonnegative, we have $$\begin{aligned}
\sum_{i,j=1}^{\Np} {\alpha}_{\i}{\alpha}_{\j} {A}_{\i\j}
& \geq \sum_{k=1}^{\IndexSetSeqLast-1}\frac{\BrS{\Pv{\IndexSetSeq{\i}{\k+1}} {\alpha}_{\IndexSetSeq{\i}{\k}}-\Pv{\IndexSetSeq{\i}{\k}}{\alpha}_{\IndexSetSeq{\i}{\k+1}}}^2}{\Pv{\IndexSetSeq{\i}{\k}}\Pv{\IndexSetSeq{\i}{\k+1}}}{a_{\IndexSetSeq{\i}{\k}\IndexSetSeq{\i}{\k+1}}}+\frac{\Pv{\IndexSetSeq{\i}{\IndexSetSeqLast}} }{\Pv{\IndexSetSeq{\i}{\IndexSetSeqLast-1}}}{\alpha}_{\IndexSetSeq{\i}{\IndexSetSeqLast}}^2 {a_{\IndexSetSeq{\i}{\IndexSetSeqLast-1} \IndexSetSeq{\i}{\IndexSetSeqLast}}}.
\label{proof:unieque:positive_definite_subseq}\end{aligned}$$ As $|\Pt{\IndexSetSeq{\i}{\k}}{}-\Pt{\IndexSetSeq{\i}{\k+1}}{}|<\h$, the value of ${a_{\IndexSetSeq{\i}{\k}\IndexSetSeq{\i}{\k+1}}}\,(\k=1,2,\dots,\IndexSetSeqLast-1)$ is positive. Thus, if the right hand side of equals zero, then ${\alpha}_{\IndexSetSeq{\i}{\k}}=0\,(\k=1,2,\dots,\IndexSetSeqLast)$. As this is inconsistent with ${\alpha}_{\i}\,(={\alpha}_{\IndexSetSeq{\i}{1}})\neq 0$, the right hand side of is positive. Therefore, matrix ${A}$ is a positive definite matrix.
Next, we introduce a few notations and show certain lemmas. Hereafter, assume that particle distribution $\PtSet$ satisfies $\h$-connectivity. For $S\subset\dRd$ and $n\in\dN$, we define a discrete inner product, $\InnerProdDisc{\cdot}{\cdot}{S}:\FsSolApp^n\times\FsSolApp^n\ra\dR$, a discrete $L^2$ norm, $\NormDiscL{\cdot}{2}{S}:\FsSolApp^n\ra\dR$, and a discrete $H^1_0$ norm, $\NormDiscHz{\cdot}{1}{S}:\FsSolApp^n\ra\dR$, as $$\begin{aligned}
&\InnerProdDisc{\phi}{\psi}{S} \deq \sum_{\i\in\IndexSet{S}}\Pvi\,\phi_{\i}\cdot\psi_{\i},\\
&\NormDiscL{\phi}{2}{S} \deq \InnerProdDisc{\phi}{\phi}{S}^{1/2}=\BrS{\sum_{\i\in\IndexSet{S}}\Pvi\,\phi_{\i}^2}^{1/2},\\
&\NormDiscHz{\phi}{1}{S} \deq \BrS{\Dim \sum_{\i\in\IndexSet{S}}\Pv{\i} \sum_{\j\neq\i}\Pv{\j} \frac{|\phi_{\j}-\phi_{\i}|^2}{|\Pt{\j}{}-\Pt{\i}{}|^2}\wh(|\Pt{\j}{}-\Pt{\i}{}|)}^{1/2}.\end{aligned}$$ For $\phi:\PtSet\ra\dR$, we define an approximate gradient operator, $\GradApp{}$, by $$\GradApp{} \phi_{\i} \deq \Dim\sum_{\j\neq\i}\Pv{\j}\frac{\phi_{\j}-\phi_{\i}}{|\Pt{\j}{}-\Pt{\i}{}|}\frac{\Pt{\j}{}-\Pt{\i}{}}{|\Pt{\j}{}-\Pt{\i}{}|}\wh(|\Pt{\j}{}-\Pt{\i}{}|)
\label{def:GradApp}$$ Then, we obtain the following lemma:
\[lem:propaties\_disc\_norm\] For $\phi\in\FsSolApp$ and $\psi\in\FsSolApp^\Dim$, we have $$\InnerProdDisc{\DivAppPlus\psi}{\phi}{\Dm}
=-\InnerProdDisc{\psi}{\GradApp{}\phi}{\Dm},
\label{lem:propaties_disc_norm:integral_of_parts}$$ $$-\InnerProdDisc{\LapApp{}\phi}{\phi}{\Dm}=\NormDiscHz{\phi}{1}{\DmH}^2\geq\NormDiscHz{\phi}{1}{\Dm}^2.
\label{lem:propaties_disc_norm:equiv_norm}$$
First, we prove . Let ${I_{\i \j}}$ be $${I_{\i \j}}\deq
\begin{cases}
0,\quad&\i=\j,\\
\ds\Dim\frac{\Pt{\j}{}-\Pt{\i}{}}{|\Pt{\j}{}-\Pt{\i}{}|^2} \wh(|\Pt{\j}{}-\Pt{\i}{}|),\quad&\i\neq\j.
\end{cases}
\label{def:InteractCofGrad}$$ As $\phi\in\FsSolApp$, $\psi\in\FsSolApp^\Dim$, and ${I_{\i \j}}=-{I_{\j \i}}$, we have $$\begin{aligned}
\InnerProdDisc{\DivAppPlus\psi}{\phi}{\Dm}
&=\sum_{\i\in\IndexSet{\Dm}}\Pv{\i}\phi_{\i}\sum_{\j=1}^\N\Pv{\j}\BrS{\psi_{\j}+\psi_{\i}}\cdot{I_{\i \j}}\\
&=\sum_{\i=1}^\N\sum_{\j=1}^\N\Pv{\i}\Pv{\j}\phi_{\i}\BrS{\psi_{\j}+\psi_{\i}}\cdot{I_{\i \j}}\\
&=\dfrac{1}{2}\sum_{\i=1}^\N\sum_{\j=1}^\N\Pv{\i}\Pv{\j}(\phi_{\i}-\phi_{\j})\BrS{\psi_{\j}+\psi_{\i}}\cdot{I_{\i \j}}\\
&=\sum_{\i=1}^\N\Pv{\i}\psi_{\i}\cdot\sum_{\j=1}^\N\Pv{\j}(\phi_{\i}-\phi_{\j}){I_{\i \j}}\\
&=-\InnerProdDisc{\psi}{\GradApp{}\phi}{\Dm}.
\end{aligned}$$ Next, we prove . Let ${J_{\i \j}}\,(\geq0)$ be $${J_{\i \j}}\deq
\begin{cases}
0,\quad&\i=\j,\\
\ds\Dim\frac{\wh(|\Pt{\j}{}-\Pt{\i}{}|)}{|\Pt{\j}{}-\Pt{\i}{}|^2} ,\quad&\i\neq\j.
\end{cases}$$ As $\phi\in\FsSolApp$ and ${J_{\i \j}}={J_{\j \i}}$, we have $$\begin{aligned}
-\InnerProdDisc{\LapApp{}\phi}{\phi}{\Dm}
&=2\sum_{\i\in\IndexSet{\Dm}}\Pv{\i}\phi_{\i}\sum_{\j=1}^\N\Pv{\j}\BrS{\phi_{\i}-\phi_{\j}}{J_{\i \j}}\\
&=2\sum_{\i=1}^\N\sum_{\j=1}^\N\Pv{\i}\Pv{\j}\phi_{\i}\BrS{\phi_{\i}-\phi_{\j}}{J_{\i \j}}\\
&=\sum_{\i=1}^\N\sum_{\j=1}^\N\Pv{\i}\Pv{\j}\BrS{\phi_{\i}-\phi_{\j}}^2{J_{\i \j}}\\
&=\NormDiscHz{\phi}{1}{\DmH}^2\\
&=\NormDiscHz{\phi}{1}{\Dm}^2 +
\sum_{\i\in\IndexSet{\BdH}}\Pv{\i} \sum_{\j=1}^\N\Pv{\j}\phi_{\j}^2{J_{\i \j}}\\
&\geq\NormDiscHz{\phi}{1}{\Dm}^2.
\end{aligned}$$
\[lem:inequarity\_disc\_norm\] Assume that a family, $\{(\PtSet, \PvSet, \h)\}$, is semi-regular. Then, we have $$\NormDiscL{\GradApp{}\phi}{2}{\Dm}^2
\leq \Dim\,{c_0}\NormDiscHz{\phi}{1}{\Dm}^2.$$ Here, ${c_0}$ is the semi-regular constant in .
By the Cauchy–Schwarz inequality, we have $$\begin{aligned}
\NormDiscL{\GradApp{}\phi}{2}{\Dm}^2&= \sum_{\i\in\IndexSet{\Dm}}\Pvi\BrA{\GradApp{}\phi_{\i}}^2\\
&\leq \Dim^2\sum_{\i\in\IndexSet{\Dm}}\Pvi\BrS{\sum_{\j\neq\i}\Pv{\j}\frac{|\phi_{\j}-\phi_{\i}|}{|\Pt{\j}{}-\Pt{\i}{}|}\wh(|\Pt{\j}{}-\Pt{\i}{}|)}^2\\
&\leq \Dim^2\sum_{\i\in\IndexSet{\Dm}}\Pvi\BrS{\sum_{\j\neq\i}\Pv{\j}\frac{|\phi_{\j}-\phi_{\i}|^2}{|\Pt{\j}{}-\Pt{\i}{}|^2}\wh(|\Pt{\j}{}-\Pt{\i}{}|)}\BrS{\sum_{\j\neq\i}\Pv{\j}\wh(|\Pt{\j}{}-\Pt{\i}{}|)}.
\end{aligned}$$ As family $\{(\PtSet, \PvSet, \h)\}$ is semi-regular, we obtain $$\NormDiscL{\GradApp{}\phi}{2}{\Dm}^2\leq\Dim\,{c_0}\NormDiscHz{\phi}{1}{\Dm}^2.$$
Then, we establish the following stability of the generalized particle method for Poisson equation .
\[thm:stability\] Assume that particle distribution $\PtSet$ satisfies $\h$-connectivity and family $\{(\PtSet, \PvSet, \h)\}$ is semi-regular. Then, there exists constant $c$ such that $$\NormDiscHz{\SolApp}{1}{\Dm}\leq {c}\NormDiscL{\DmFunc}{2}{\Dm}.
\label{thm:stability:ineq}$$
By the Cauchy–Schwarz inequality, , and Lemmas \[lem:propaties\_disc\_norm\] and \[lem:inequarity\_disc\_norm\], we have $$\begin{aligned}
\NormDiscHz{\SolApp}{1}{\Dm}^2
&\leq\BrA{-\InnerProdDisc{\LapApp{}\SolApp}{\SolApp}{\Dm}}
\\
&=\BrA{\InnerProdDisc{\DivAppPlus\widehat{f}}{\SolApp}{\Dm}}
\\
&=\BrA{-\InnerProdDisc{\widehat{f}}{\GradApp{}\SolApp}{\Dm}}
\\
&\leq \NormDiscL{\DmFunc}{2}{\Dm}\NormDiscL{\GradApp{}\SolApp}{2}{\Dm}
\\
&\leq \sqrt{\Dim\,{c_0}}\NormDiscL{\DmFunc}{2}{\Dm}\NormDiscHz{\SolApp}{1}{\Dm}. \end{aligned}$$ Consequently, we obtain .
For function space $\FsSolApp^n\,(n\in\dN)$, the discrete $L^2$ norm, $\NormDiscL{\cdot}{2}{\Dm}$, satisfies the conditions of the norm. Moreover, if and only if particle distribution $\PtSet$ satisfies $\h$-connectivity, then the discrete $H^1_0$ norm, $\NormDiscHz{\cdot}{1}{\Dm}$, satisfies the conditions of the norm.
Concluding remarks {#sec6}
==================
We have analyzed the stability of a generalized particle method for a Poisson equation with a source term given by a divergence form. We have obtained a unique solvability of the discretized Poisson equation by introducing a connectivity condition for particle distributions, which is referred to as $h$-connectivity. Moreover, we have established the stability of the discretized Poisson equation based on the semi-regularity of a family of discrete parameters and discrete Sobolev norms with properties such as integration by parts.
In future, we will analyze the error estimates of the discretized Poisson equation by showing properties such as the discrete Poincaré inequality. Moreover, we will extend these results to the incompressible viscous flow problem.
Derivation of approximate operators {#appendix:deriv_app_diver}
===================================
Assume a two-dimensional or three-dimensional space, $\Dim=2,3$. Assume a particle distribution on a square lattice with spacing $\Dx$. For $\i,\j=1,2,\dots,\N$, let ${\sigma_{\i}}=(\Pt{\i}{}-\Dx/2,\Pt{\i}{}+\Dx/2)^\Dim$, ${\gamma_{\i\j}}\deq{\ol{\sigma}_{\i}}\cap{\ol{\sigma}_{\j}}$, and ${\lambda_{\i}}\deq\SetNd{\k=1,2,\dots,\N}{\Meas{{\gamma_{\i\k}}}\neq0,\,\k\neq\i}$. As $\Meas{{\sigma_{\i}}}=\Dx^\Dim$ and $\Meas{{\gamma_{\i\j}}}=\Dx^{\Dim-1}\,(\j\in{\lambda_{\i}})$, by the divergence theorem, we can approximate the divergence of $\psi:\DmH\ra\dRd$ on $\Pt{\i}{}\in\PtSet$ as $$\begin{aligned}
\Div\psi_{\i}
&\approx\dfrac{1}{\Meas{{\sigma_{\i}}}}\int_{{\sigma_{\i}}}\Div\psi(x)\dx\\
&=\dfrac{1}{\Dx^\Dim}\int_{\partial{\sigma_{\i}}}\psi(x)\cdot n\,ds\\
&\approx\dfrac{1}{\Dx^\Dim}\sum_{\j\in{\lambda_{\i}}}\Meas{{\gamma_{\i\j}}}\psi\BrS{\dfrac{\Pt{\i}{}+\Pt{\j}{}}{2}}\cdot\dfrac{\Pt{\j}{}-\Pt{\i}{}}{|\Pt{\j}{}-\Pt{\i}{}|}\\
&\approx\dfrac{1}{\Dx^\Dim}\sum_{\j\in{\lambda_{\i}}}\Meas{{\gamma_{\i\j}}}\dfrac{\psi_{\j}+\psi_{\i}}{2}\cdot\dfrac{\Pt{\j}{}-\Pt{\i}{}}{|\Pt{\j}{}-\Pt{\i}{}|}\\
&=\dfrac{1}{2}\sum_{\j\in{\lambda_{\i}}}\dfrac{\psi_{\j}+\psi_{\i}}{|\Pt{\j}{}-\Pt{\i}{}|}\cdot\dfrac{\Pt{\j}{}-\Pt{\i}{}}{|\Pt{\j}{}-\Pt{\i}{}|},
\label{appendix:approx_diver}\end{aligned}$$ where $n$ is the outward normal vector on the boundary $\partial{\sigma_{\i}}$. Further, using the central difference, we can approximate the Laplacian of $\phi:\DmH\ra\dR$ on $\Pt{\i}{}\in\PtSet$ as $$\begin{aligned}
\Lap\phi_{\i}
&\approx\dfrac{1}{\Meas{{\sigma_{\i}}}}\int_{{\sigma_{\i}}}\Lap\phi(x)\dx\nn\\
&=\dfrac{1}{\Dx^\Dim}\int_{\partial{\sigma_{\i}}}\Grad\phi(x)\cdot n\,ds\\
&\approx\dfrac{1}{\Dx^\Dim}\sum_{\j\in{\lambda_{\i}}}\Meas{{\gamma_{\i\j}}}\Grad\phi\BrS{\dfrac{\Pt{\i}{}+\Pt{\j}{}}{2}}\cdot\dfrac{\Pt{\j}{}-\Pt{\i}{}}{|\Pt{\j}{}-\Pt{\i}{}|}\\
&\approx\dfrac{1}{\Dx^\Dim}\sum_{\j\in{\lambda_{\i}}}\Meas{{\gamma_{\i\j}}}\dfrac{\phi_{\j}-\phi_{\i}}{|\Pt{\j}{}-\Pt{\i}{}|}\\
&=\sum_{\j\in{\lambda_{\i}}}\dfrac{\phi_{\j}-\phi_{\i}}{|\Pt{\j}{}-\Pt{\i}{}|^2}
\label{appendix:approx_lap}\end{aligned}$$ By noting that the number of elements of ${\lambda_{\i}}$ is $2\Dim$, we can derive approximate operators and as the weighted averages of and , respectively. As the approximation procedures in and are same as that of the finite volume method based on Voronoi decomposition, we can regard approximate operators and as approximations based on the finite volume method.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by JSPS KAKENHI Grant Number 17K17585 and by JSPS A3 Foresight Program.
[1]{} Y. Imoto, Error estimates of generalized particle methods for the Poisson and heat equations, Ph. D thesis, Kyushu University, 2016.
M. B. Liu and G. R. Liu, Smoothed Particle Hydrodynamics (SPH): an overview and recent developments, Arch. Comput. Methods Eng., [**17**]{} (2010), 25–76.
S. Koshizuka and Y. Oka, Moving-particle semi-implicit method for fragmentation of incompressible fluid, Nucl. Sci. Eng., [**123**]{} (1996), 421–434.
Y. Imoto and D. Tagami, A truncation error estimate of the interpolant of a particle method based on the Voronoi decomposition, JSIAM Letters, [**8**]{} (2016), 29–32.
Y. Imoto and D. Tagami, Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition, JSIAM Letters, [**9**]{} (2017), 69–72.
S. Shao and L. Edmond, Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface, Adv. Water Resources, [**26**]{} (2003), 787–800.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The famous Hanna Neumann Conjecture (now the Friedman–Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups $H$ and $K$ of a non-abelian free group. It is an interesting question to ‘quantify’ this bound with respect to the rank of $H\vee K$, the subgroup generated by $H$ and $K$. We describe a set of realizable values $\big(\!{\operatorname{rk}}(H\vee K),{\operatorname{rk}}(H\cap K)\big)$ for arbitrary $H$, $K$, and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for $H$ and $K$ with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of $H\vee K$, $H\cap K$ are not realizable, thus resolving the remaining open case $m=4$ of Guzman’s “Group-Theoretic Conjecture” in the affirmative. This in turn implies the validity of the corresponding “Geometric Conjecture” on hyperbolic $3$–manifolds with a $6$–free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when ${\operatorname{rk}}(H)=2$.'
address: |
Department of Mathematics\
Louisiana State University\
Baton Rouge\
LA 70803\
USA
author:
- Ignat Soroko
title: Realizable ranks of joins and intersections of subgroups in free groups
---
=1
Introduction
============
Let $F$ be a free group and $H,K\le F$ finitely generated subgroups. Denote $H\vee K$ the subgroup of $F$ generated by $H$ and $K$. Define the *reduced rank* of $H$ by $${\operatorname{rr}}(H)=\max(0,{\operatorname{rk}}(H)-1).$$ The famous Hanna Neumann Conjecture (now the Friedman–Mineyev theorem [@Mi1], [@DM], [@Fr], [@DF]; see also a recent proof of Jaikin-Zapirain [@Jai]) states that $${\operatorname{rr}}(H\cap K)\le {\operatorname{rr}}(H){\operatorname{rr}}(K).$$
It is an interesting problem to try to ‘quantify’ the possible ranks of $H\cap K$ with respect to the rank of the join $H\vee K$ of $H$ and $K$ (i.e. the subgroup generated by $H$ and $K$). Ideally, one wishes to determine the set of all realizable values for tuples $$\big(\!{\operatorname{rk}}(H\vee K),\,{\operatorname{rk}}(H\cap K)\big)$$ for any given values of ${\operatorname{rk}}(H)$ and ${\operatorname{rk}}(K)$. It seems plausible, by the analogy with the linear algebra identity for vector spaces, $\dim(U\cap V)+\dim(U+V)=\dim(U)+\dim(V)$, that the bigger ${\operatorname{rk}}(H\cap K)$ is, the smaller ${\operatorname{rk}}(H\vee K)$ should be. Several partial results and conjectures have been made in this direction.
In [@IM] Imrich and Müller have proved the following:
If $H,K$ are finitely generated subgroups of $F$ and either $H$ or $K$ is of finite index in $H\vee K$ then $$\label{eq1}
{\operatorname{rr}}(H\cap K){\operatorname{rr}}(H\vee K)\le {\operatorname{rr}}(H){\operatorname{rr}}(K).$$
In general, without the finite index assumption, this inequality does not hold, see [@IM Ex. 3]. Moreover, Hunt [@Hun] has shown that the ratio of the left-hand side of to its right-hand side can be made arbitrarily large. Recently, Sergei Ivanov [@Iva17 p.826] has posed the following open question:
Does inequality hold true if ${\operatorname{rr}}(H\cap K)$ is the maximal possible in the Friedman–Mineyev theorem, i.e. if ${\operatorname{rr}}(H\cap K)={\operatorname{rr}}(H){\operatorname{rr}}(K)>0$? Equivalently, does this assumption imply that ${\operatorname{rr}}(H\vee K)=1$?
Another circle of questions about the relationship between ${\operatorname{rk}}(H\cap K)$ and ${\operatorname{rk}}(H\vee K)$ was motivated by the study of hyperbolic $3$–manifolds. Continuing the program started by Agol, Culler and Shalen [@ACS], [@CS], Guzman [@Gu] formulated the following “Group-Theoretic Conjecture” (GTC):
If two subgroups $H,K\le F$ both have ranks equal to $m\ge 2$, and ${\operatorname{rk}}(H\cap K)\ge m$, then ${\operatorname{rk}}(H\vee K)\le m$.
She proved that this conjecture, if true, implies the following “Geometric Conjecture” (GC), with $k=m+2$ (recall that a group is called *$k$–free* if all of its $k$–generator subgroups are free):
Let $M$ be a closed, orientable, hyperbolic $3$–manifold. If $\pi_1(M)$ is $k$–free for $k\ge3$ then there exists a point $P$ in $M$ such that the set of all elements of $\pi_1(M,P)$ represented by loops of length less than $\log(2k-1)$ is contained in a (free) subgroup of $\pi_1(M)$ of rank $\le k-3$.
The case $m=2$ of the GTC appeared earlier as a question in a preprint of Culler and Shalen and was subsequently resolved in the affirmative by Kent [@Ke2] and, independently, by Louder–McReynolds [@LMR]. This was used in [@CS Th.1.4] to prove a statement equivalent to the GC for $k=4$ and to obtain a lower bound on the volume of a closed orientable hyperbolic $3$–manifold with a $4$–free fundamental group. (The case $k=3$ of the GC and the corresponding lower bound on the volume for the case of $3$–free fundamental groups was proved earlier in [@ACS Cor.9.3].) Using results and techniques from [@Ke2], Guzman proves the GTC for $m=3$ and hence the GC for $k=5$. However, Hunt [@Hun] has shown by example that the GTC is no longer true for $m=5$. Below we will show that the GTC is false for all values $m\ge 6$, but holds true for $m=4$.
It must be noted that, very recently, Guzman and Shalen [@GuSha] proved the Geometric Conjecture in full generality, without dependence on the Group-Theoretic Conjecture.
It is an easy consequence of the Hopfian property of finite rank free groups that the only possibility for ${\operatorname{rk}}(H\vee K)$ to equal the maximal possible value ${\operatorname{rk}}(H)+{\operatorname{rk}}(K)$ is to have $H\vee K\cong H*K$. Thus, in this case ${\operatorname{rk}}(H\cap K)$ must equal $0$.
In [@Ke2], Kent proved the following inequality:
Let $H$ and $K$ be nontrivial finitely generated subgroups of $F$ with reduced ranks $h={\operatorname{rr}}(H)$, $k={\operatorname{rr}}(K)$, and $k\ge h$. Assume also that $H\cap K\ne 1$. Then $${\operatorname{rr}}(H\cap K)\le 2hk-h{\operatorname{rr}}(H\vee K).$$
Recently, Sergei Ivanov has improved the above estimate of Kent, see [@Iva (4.2)] (where he mentions that inequality was also obtained independently by Dicks):
Let $H$ and $K$ be nontrivial finitely generated subgroups of $F$ with reduced ranks $h={\operatorname{rr}}(H)$, $k={\operatorname{rr}}(K)$. Then $$\label{eq2}
{\operatorname{rr}}(H\cap K)\le \frac12\big(h+k-{\operatorname{rr}}(H\vee K)\big)\big(h+k-{\operatorname{rr}}(H\vee K)+1\big).$$
The last result suggests that it may be convenient to describe the locus of possible values $\big(\!{\operatorname{rk}}{(H\vee K)},{\operatorname{rk}}{(H\cap K)}\big)$ in terms of the difference between ${\operatorname{rr}}(H\vee K)$ and its largest possible value $h+k+1$. If we denote $i=h+k+1-{\operatorname{rr}}(H\vee K)$ then inequality reads: $
{\operatorname{rr}}(H\cap K)\le\frac{i(i-1)}{2}.
$
The results mentioned so far do not guarantee that if certain numbers $(v,c)$ satisfy the respective inequalities, then there exist subgroups $H$, $K$ realizing them as $(v,c)=\big(\!{\operatorname{rk}}(H\vee K),{\operatorname{rk}}(H\cap K)\big)$. Hence these results are, in effect, describing the regions of tuples which are *non-realizable*. By contrast, in [@Ke1], Kent exhibited for arbitrary $h,k\ge2$ a family of subgroups $H=H(h,k,m)$ and $K=K(k)$ such that ${\operatorname{rk}}(H)=h$, ${\operatorname{rk}}(K)=k$, ${\operatorname{rk}}(H\vee K)=2$ and ${\operatorname{rk}}(H\cap K)$ takes on all possible ranks $m=0,\dots,(h-1)(k-1)+1$ allowed by the Friedman–Mineyev theorem. This answered a question of Myasnikov [@BMS (AUX1)].
Our first contribution to this theme is the following addition to the region of known realizable values of $\big(\!{\operatorname{rk}}(H\vee K),{\operatorname{rk}}(H\cap K)\big)$.
\[thm:real\] Let $F$ be a free group and let integers $h,k,c,v$ satisfy $2\le h\le k$, $2\le v\le h+k$, $0\le c\le (h-1)(k-1)+1$. Define a sequence $a_i$ as follows: $$\begin{aligned}
a_0&=0\,;\\
a_i&=\Big\lfloor \frac {i^2}{4}\Big\rfloor +1,\quad \text{for\quad $i=1,\dots, 2(h-1)\,;$}\\
a_i&=(h-1)(i-h+1)+1,\quad \text{for\quad $i=2(h-1),\dots,h+k-2$.}\end{aligned}$$ If we denote $i=h+k-v$, then for any $c\le a_i$ there exist subgroups $H,K\le F$ such that ${\operatorname{rk}}(H)=h$, ${\operatorname{rk}}(K)=k$, ${\operatorname{rk}}(H\cap K)=c$ and ${\operatorname{rk}}(H\vee K)=v$.
The sequence $a_i$ from this theorem is a splicing of a discrete quadratic function and a linear function. The linear part exists only if $h<k$. Written in terms of reduced ranks, the quadratic part implies: ${{\operatorname{rr}}(H\cap K)\le \big\lfloor \frac{i^2}{4}\big\rfloor}$, which is smaller than Ivanov’s upper bound $\frac{i(i-1)}{2}$ above, and the gap between the two becomes unbounded as $i$ grows. If $h=2$, the quadratic part trivializes and the sequence $a_i$ becomes especially simple: $a_i=i$ for all $i$, see Figure \[fig:2.10\].
The realizable values from Theorem \[thm:real\] allow us to establish the following.
Guzman’s “Group-Theoretic Conjecture” does not hold for any $m\ge 5$.
Figure \[fig:intro\] depicts all the regions described above for the case ${\operatorname{rk}}(H)={\operatorname{rk}}(K)=6$.
(6.5cm,0.75cm) node [${\operatorname{rk}}(H\cap K)$]{}; (-1.3cm,-2cm) node \[rotate=90\] [${\operatorname{rk}}(H\vee K)$]{}; (-0.5cm,-4.82cm) node [[$(i=0)$]{}]{}; (-0.5cm,-4.35cm) node [[$(i=1)$]{}]{}; (-0.5cm,-3.88cm) node [[$(i=2)$]{}]{}; (-0.5cm,-3.45cm) node [[$(i=3)$]{}]{}; (-0.5cm,-0.25cm) node [[$(i=10)$]{}]{}; (-0.5cm,-1.85cm) node [[$\vdots$]{}]{};
(0cm,0cm,::::::::::::::::::::::::::::,::::::::::::::::::::::::::::,:::::::::::::::::::::::;;;;;,:::::::::::::::::::;;;;;;;;;,:::::::::::::::;;;;;;;;;;;;;,::::::::::::;;;;;;;;;;;;;;;;,:::::::::;;;;;;;;;;;;;;;;;;;,:::::::;;;;;;;;;;;;;;;;;;;;;,:::::;;;;;;;;;;;;;;;;;;;;;;;,::::;;;;;;;;;;;;;;;;;;;;;;;;,:::;;;;;;;;;;;;;;;;;;;;;;;;;,::;;;;;;;;;;;;;;;;;;;;;;;;;;) (0cm,0cm,::<0>:1:2:3:4:5:6:7:8:9:<10>:<[1]{}1>:<12>:<13>:<14>:<15>:<16>:<17>:<18>:<19>:<20>:<21>:<[2]{}2>:<23>:<24>:<25>:<26>,:2;;;;;;;;;;;;;;;;;;;;;;;;;;;,:3;;;;;;;;;;;;;;;;;;;;;;,:4;;;;;;;;;;;;;;;;;;,:5;;;;;;;;;;;;;;,:6;;;;;;;;;;;,:7;;;;;;;;,:8;;;;;;,:9;;;;,:<10>;;;,:<[1]{}1>;;,:<12>;) (0cm,0cm,::::::::::::::::::::::::::::,:;;;;;;;;;;;;;;;;;;;;;;;;;;;,:;;;;;;;;;;;;;;;;;;;;;;,:;;;;;;;;;;;;;;;;;;,:;;;;;;;;;;;;;;,:;;;;;;;;;;;,:;;;;;;;;,:;;;;;;,:;;;;,:;;;,:;;,:;) (0cm,0cm,::::::::::::::::::::::::::::,::::::::::::::::::::::::::::,::::::::::::::::::::::::::::,::::::::::::::::::::::::::::,:::::::::::::::::::::::;;;;;,:::::::::::::::::;;;;;;;;;;;,::::::::::::;;;;;;;;;;;;;;;;,::::::::;;;;;;;;;;;;;;;;;;;;,:::::;;;;;;;;;;;;;;;;;;;;;;;,::::;;;;;;;;;;;;;;;;;;;;;;;;,:::;;;;;;;;;;;;;;;;;;;;;;;;;,::;;;;;;;;;;;;;;;;;;;;;;;;;;)
(13pt,-143pt) – ++(351pt,0) – ++ (0,143pt) – ++(-351pt,0) – ++(0,-143pt); (26pt,-143pt) – ++(0,13pt) – ++(338pt,0); (13pt,-13pt) – ++(351pt,0); (364pt,-13pt) – ++(-169pt,0) – ++(0,-13pt) – ++(-52pt,0)– ++(0,-13pt) – ++(-26pt,0) – ++(0,-13pt) – ++(-13pt,0)– ++(0,-13pt)– ++(-13pt,0)– ++(0,-13pt)– ++(-13pt,0)– ++(0,-26pt) – ++(-13pt,0) – ++(0,-39pt); (323pt,-20pt) node [(1)]{}; (39pt,-130pt) – ++(325pt,0) – ++(0,65pt) – ++(-65pt,0) – ++(0,-13pt) – ++(-65pt,0) – ++(0,-13pt) – ++(-65pt,0) – ++(0,-13pt) – ++(-65pt,0) – ++(0,-13pt) – ++(-65pt,0) – ++(0,-13pt) – ++(-13pt,0); (364pt,-65pt) – ++ (0,26pt) – ++(-52pt,0) – ++(0,-13pt) – ++(-78pt,0) – ++(0,-13pt) – ++(-65pt,0) – ++(0,-13pt) – ++(-52pt,0) – ++(0,-13pt) – ++(-39pt,0) – ++(0,-13pt) – ++(-26pt,0) – ++(0,-13pt); (299pt,-13pt) – ++(0,-13pt) – ++(-52pt,0) – ++(0,-13pt) – ++(-52pt,0)– ++(0,-13pt) – ++(-39pt,0)– ++(0,-13pt) – ++(-39pt,0)– ++(0,-13pt) – ++(-26pt,0)– ++(0,-13pt) – ++(-26pt,0)– ++(0,-13pt); (364pt,-65pt) – ++(-273pt,0) – ++(0,-78pt); (235pt,-71.5pt) node [*Guzman’s GTC $(m=6)$*]{}; (364pt,-13pt) – ++(-13pt,0) – ++(0,-130pt); (357.75pt,-64pt) node \[rotate=90\] [*Ivanov’s question*]{}; (71.5pt,-97.5pt) node [$*$]{}; (110.5pt,-84.5pt) node [$*$]{}; (162.5pt,-71.5pt) node [$*$]{}; (227.5pt,-58.5pt) node [$*$]{}; (305.5pt,-45.5pt) node [$*$]{}; (182pt,-136.5pt) node [*Hopfian property*]{}; (260pt,-110.5pt) node [*Kent’09*]{}; (172pt,-84.5pt) node [*Ivanov’18*]{}; (182pt,-6.5pt) node [*Kent’05*]{}; (47pt,-58.5pt) node [*Th.1.1*]{};
We conjecture that the set of realizable values from Theorem \[thm:real\] is complete.
\[cnj:real\] Let $F$ be a free group and let integers $h,k,v,c$ satisfy: $2\le h\le k$, $2\le v\le h+k$ and $0\le c\le (h-1)(k-1)+1$. Then there exist subgroups $H,K\le F$ such that ${\operatorname{rk}}(H)=h$, ${\operatorname{rk}}(K)=k$, ${\operatorname{rk}}(H\vee K)=v$, and ${\operatorname{rk}}(H\cap K)=c$ if and only if $c\le a_i$ for $i=h+k-v$, where $a_i$ is the sequence defined in Theorem \[thm:real\].
The author tested this conjecture on a computer (searching for a possible counterexample) using a Monte-Carlo type algorithm of Bassino, Nicaud and Weil [@BNW], which randomly generates core graphs on a given number of vertices with the uniform distribution for subgroups of the given size (of their core graph) in a free group. The author learned about this algorithm from the preprint of Hunt [@Hun], and also used the computer algebra system GAP [@GAP] (with the package FGA [@FGA] for methods dealing with free groups) to implement it. Testing about $5\cdot 10^8$ pairs of random core graphs did not produce any values outside of the conjectured locus.
Note that Conjecture \[cnj:real\] subsumes the open question of Ivanov above.
We prove this conjecture for the special case of $h={\operatorname{rk}}(H)=2$ (when the sequence $a_i$ from Theorem \[thm:real\] becomes linear: $a_i=i$ for all $i$):
\[thm:h2\] Let $F$ be a free group and let integers $k,v,c$ satisfy: $k\ge 2$, $2\le v\le k+2$ and $0\le c\le k$. Then there exist subgroups $H,K\le F$ such that ${\operatorname{rk}}(H)=2$, ${\operatorname{rk}}(K)=k$, ${\operatorname{rk}}(H\vee K)=v$, and ${\operatorname{rk}}(H\cap K)=c$ if and only if $c+v\le k+2$.
The diagram in Figure \[fig:2.10\] shows all realizable values for ${\operatorname{rk}}(H)=2$, ${\operatorname{rk}}(K)=10$.
[(6.5cm,0.75cm) node [$c={\operatorname{rk}}(H\cap K)$]{}; (-1.3cm,-2.3cm) node \[rotate=90\] [$v={\operatorname{rk}}(H\vee K)$]{}; (-0.4cm,-4.82cm) node [[$(i=0)$]{}]{}; (-0.4cm,-4.35cm) node [[$(i=1)$]{}]{}; (-0.4cm,-0.25cm) node [[$(i=10)$]{}]{}; (-0.4cm,-2.30cm) node [[$\vdots$]{}]{}; ]{} [ (0cm,0cm,:::::::::::::::::::::::::::,::::::::::::,:::::::::::;,::::::::::;;,:::::::::;;;,::::::::;;;;,:::::::;;;;;,::::::;;;;;;,:::::;;;;;;;,::::;;;;;;;;,:::;;;;;;;;;,::;;;;;;;;;;) (0cm,0cm,::0:1:2:3:4:5:6:7:8:9:<10>,:2;;;;;;;;;;;,:3;;;;;;;;;;,:4;;;;;;;;;,:5;;;;;;;;,:6;;;;;;;,:7;;;;;;,:8;;;;;,:9;;;;,:<10>;;;,:<[1]{}1>;;,:<12>;) ]{}
Our next contribution is the resolution of a few extremal cases which are not covered by the realizable values from Theorem \[thm:real\] and are not eliminated by Ivanov’s inequality (see the cells marked with an asterisk in Figure \[fig:intro\]).
\[thm:guzman4\] Let $F$ be a free group. Then there do not exist subgroups $H,K\le F$ such that ${\operatorname{rk}}(H)$, ${\operatorname{rk}}(K)\ge 2$, ${\operatorname{rk}}(H\vee K)={\operatorname{rk}}(H)+{\operatorname{rk}}(K)-i$ for some $i\ge 3$, and ${\operatorname{rk}}(H\cap K)=\frac{i(i-1)}{2}+1$.
As an immediate consequence we conclude that Guzman’s “Group-Theoretic Conjecture” holds true for the remaining unresolved case of $m=4$:
Let $F$ be a free group. If two subgroups $H,K\le F$ both have ranks equal to $4$, and ${{\operatorname{rk}}(H\cap K)\ge 4}$, then ${\operatorname{rk}}(H\vee K)\le 4$.
Invoking the implication theorem from [@Gu], we obtain a proof of the “Geometric Conjecture” for $k=6$:
Let $M$ be a closed, orientable, hyperbolic $3$–manifold. If $\pi_1(M)$ is $6$–free then there exists a point $P$ in $M$ such that the set of all elements of $\pi_1(M,P)$ represented by loops of length less than $\log(11)$ is contained in a free subgroup of $\pi_1(M)$ of rank at most $3$.
Our paper is organized as follows.
Section \[sec:graphs\] contains the basic definitions of graph related concepts necessary for our needs. We introduce Stallings’ core graphs and describe the construction of the topological pushout of two core graphs, following Kent [@Ke2]. The topological pushout is an intermediate object between the join of two core graphs ${\Gamma}_H$, ${\Gamma}_K$ and the core graph ${\Gamma}_{H\vee K}$ of the join of the two subgroups, and the rank of the topological pushout serves as an upper bound for the rank of the join.
In Section \[sec:real\] we show how to obtain all the values of $\big(\!{\operatorname{rk}}(H\vee K),{\operatorname{rk}}(H\cap K)\big)$ from Theorem \[thm:real\] by properly adding new generators to the family of examples exhibited by Kent in [@Ke1].
In Section \[sec:dicks\] we study the combinatorial structure of the topological pushout in terms of graphs introduced by Dicks in [@Di] in the context of the Amalgamated Graph Conjecture. Our approach is motivated by the construction of graphs $\Upsilon$ and $Z$ in section 6 of [@Di]. In Section \[sec:cn\] we establish a technical condition on Dicks graphs which specifies when the rank of the topological pushout is the maximal possible.
Finally, in Section \[sec:guzman4\], we use the results obtained so far to prove Theorems \[thm:h2\], \[thm:guzman4\] and the consequences of the latter, the Guzman’s GTC for the remaining case of $m=4$, and hence the GC for $k=6$. The key observation is that in the situation described in Theorems \[thm:h2\], \[thm:guzman4\], the components of the corresponding Dicks graphs $\Omega_{abc}$ are ‘incompatible’ in the sense that one component contains a highly connected subgraph (a complete bipartite graph $K_{i,i-1}$ in Theorem \[thm:guzman4\] and $K_{2,m}$ in Theorem \[thm:h2\]), while others are singleton vertices. Dicks’ duality implies the existence of an isomorphic copy of the highly connected subgraph in $\Omega$, which must be ‘spread’ along two or more subgraphs $\Omega_{ab}$, $\Omega_{ac}$, $\Omega_{bc}$. This forces the rank of the join to be less than required, which makes the corresponding tuples from Theorems \[thm:h2\], \[thm:guzman4\] non-realizable.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author is very grateful to Jing Tao for bringing Guzman’s “Group-Theoretic Conjecture” to his attention and for supporting him with the research assistantship from her grants (NSF grant DMS 1611758 and NSF Career grant DMS 1651963). The author extends his gratitude to Noel Brady for his interest in this project and numerous constructive discussions and to Sergei Ivanov who suggested to the author that Warren Dicks’ methods may prove useful for this project. The author would also like to thank the anonymous referee for numerous suggestions which greatly improved the clarity of the exposition, and Pallavi Dani, Warren Dicks, Max Forester, Autumn Kent, and Lars Louder for their valuable remarks. Last but not least the author would like to thank Till Tantau and Matthew Fayers for creating LaTeX packages [Ti*k*Z]{} [@TikZ] and [genyoungtabtikz]{} [@gytt], respectively, which proved to be extremely useful in typesetting this article.
Graphs {#sec:graphs}
======
In this paper we will deal with two types of graphs: directed labeled graphs and undirected ones. Among the former are Stallings’ core graphs that are used to represent finitely generated subgroups of a free group, see [@Sta], [@KM]. Among the latter ones, are bipartite graphs introduced by Dicks [@Di] to study the structure of the intersection of two subgroups in a free group; they will be useful for the description of the topological pushout in the sense of Kent [@Ke2]. We now remind the reader of the relevant definitions adapted to our needs from [@Bog], [@KM], [@Di].
Basic definitions
-----------------
A *graph* ${\Gamma}$ is a pair of sets $V({\Gamma})$, $E({\Gamma})$, where $V({\Gamma})$ is a nonempty set of *vertices* of ${\Gamma}$ and $E({\Gamma})$ is a set of *(directed) edges* of ${\Gamma}$ equipped with the three maps: $o\colon E({\Gamma})\to V({\Gamma})$, $t\colon E({\Gamma})\to V({\Gamma})$ and $\overline{\phantom{e}}\colon E({\Gamma})\to E({\Gamma})$ called the *origin* map, the *terminus* map and the map of taking the *inverse* of an edge, respectively, with the following properties: for each $e\in E({\Gamma})$, $\overline{\overline{e}}=e$, $\overline e\ne e$ and $o(\overline e)=t(e)$.
A *morphism* between two graphs ${\Gamma}$ and $\Delta$ is a map $\pi\colon {\Gamma}\to\Delta$ that sends vertices to vertices and edges to edges and has the property that $o(\pi(e))=\pi(o(e))$, $t(\pi(e))=\pi(t(e))$ and $\pi(\overline e)=\overline{\pi(e)}$ for any edge $e\in E({\Gamma})$.
Each graph ${\Gamma}$ admits a geometric realization as a $1$–dimensional CW complex $X_{\Gamma}$, with vertices of ${\Gamma}$ being the $0$–cells of $X_{\Gamma}$, and each pair of mutually inverse edges $e,\overline e$ of ${\Gamma}$ corresponding to the two opposite orientations of the same open $1$–cell of $X_{\Gamma}$.
A graph ${\Gamma}$ is called *directed* (or *oriented*) if in each pair of its mutually inverse edges $e, \overline e$ one edge is chosen, which is called *positively oriented*, and the other is called *negatively oriented*. The set of all positively (negatively) oriented edges is denoted $E^+({\Gamma})$ (respectively, $E^-({\Gamma})$). A morphism of directed graphs $\pi\colon{\Gamma}\to\Delta$ is required to send $E^+({\Gamma})$ to $E^+(\Delta)$.
Let ${\mathcal A}$ be a finite alphabet, and ${\mathcal A}^{-1}$ be the set of formal inverses of ${\mathcal A}$. A *directed ${\mathcal A}$–labeled graph* (or just a *directed labeled graph*, if ${\mathcal A}$ is obvious from the context) is a directed graph ${\Gamma}$ with a labeling $\mu\colon E({\Gamma})\to {\mathcal A}\sqcup {\mathcal A}^{-1}$ such that $\mu(E^+({\Gamma}))\subseteq {\mathcal A}$ and $\mu(\overline e)=\mu(e)^{-1}$ for each $e\in E({\Gamma})$. A morphism of directed ${\mathcal A}$–labeled graphs $\pi\colon{\Gamma}\to\Delta$ is required to preserve the labeling, i.e. $\mu(\pi(e))=\mu(e)$ for each $e\in E({\Gamma})$.
The *star* of a vertex $v$ of $V({\Gamma})$ is the set of all edges $e$ in $E({\Gamma})$ such that $o(e)=v$. The star of $v$ can be thought of as the link of $v$ in the geometric realization of ${\Gamma}$ in our context. The *valence* ${\operatorname{val}}(v)$ of $v$ is the cardinality of the star of $v$. If $k={\operatorname{val}}(v)$ we call the vertex $v$ *$k$-valent*.
A morphism $\pi\colon {\Gamma}\to\Delta$ is called an *immersion* if its restriction to the star of each vertex of ${\Gamma}$ is injective.
A *path* $p$ in ${\Gamma}$ is a sequence of edges $p=e_1,\dots,e_k$ of $E({\Gamma})$ such that for each $i=2,\dots,k$, we have $o(e_i)=t(e_{i-1})$. The length of $p$ is set to be $k$ (with the case $k=0$ possible). In this situation we call vertex $x=o(e_1)$ the *origin of $p$* and $y=t(e_k)$ the *terminus of $p$*. We also say that $p$ is a path *from $x$ to $y$*, and use notation $x{-}y$ to denote any such path.
If ${\Gamma}$ is a directed labeled graph, then any path $p=e_1,\dots,e_k$ has a naturally defined label $\mu(p)=\mu(e_1)\dots\mu(e_k)$, which is a word in the alphabet ${\mathcal A}\sqcup{\mathcal A}^{-1}$. (If $k=0$ then $\mu(p)=1$, the empty word.)
The notion of the fundamental group of a graph is a combinatorial analog of the notion of the fundamental group of the geometric realization of the graph, see [@Bog Ch. 2.4]. In what follows, we will not distinguish graphs and their geometric realizations and will use these notions interchangeably. All directed labeled graphs will be labeled by the set ${\mathcal A}=\{a,b,c\}$ of free generators of a rank $3$ free group $F(a,b,c)$.
With each graph ${\Gamma}$ we can associate an *undirected graph* ${\Gamma}_u$ which has the same set of vertices $V({\Gamma})$ but whose set of *undirected edges* is obtained by identifying each pair $\{e,\overline e\}$ of mutually inverse directed edges of ${\Gamma}$ into a single equivalence class. Such an undirected edge has two vertices that are *incident* to it, namely $\{o(e),t(e)\}=\{o(\overline e),t(\overline e)\}$, and these two vertices may coincide if $e$ is a loop. We also say that such vertices are *adjacent* to each other. We will abuse the notation and denote an undirected edge $\{e,\overline e\}$ simply by $e$, and the set of all undirected edges of ${\Gamma}_u$ also by $E({\Gamma}_u)$. A *path* in an undirected graph is a sequence $p=x_1e_1x_2e_2\dots x_ke_kx_{k+1}$ of pairwise distinct vertices $x_i$ and undirected edges $e_i$ such that for all $i$, the vertices $x_i,x_{i+1}$ are incident to the edge $e_i$. We say that $p$ is a path *from $x_1$ to $x_{k+1}$*, and we will also denote it as $x_1{-}x_{k+1}$. The length of it is $k$, as above, with $k=0$ possible. If we denote a subpath $e_1x_2e_2\dots x_ke_k$ as $q$, we can also write: $p=x_1qx_{k+1}$. A *cycle* in an undirected graph ${\Gamma}$ is a union of a path $x_0e_0x_1e_1\dots x_k$ with an edge $e_k$ which is incident to both $x_k$ and $x_0$. We denote a cycle also as $x_0e_0x_1e_1\dots x_ke_kx_0$ and consider its length to be equal $k$.
Core graphs represent subgroups
-------------------------------
Let $F=F(a,b)$ be a free group of rank $2$ and let $X$ be a finite graph viewed as a $1$–dimensional CW complex such that $\pi_1(X)$ is isomorphic to $F$. Traditionally $X$ is identified with a wedge of two circles, but we will implement Dicks’ approach [@Di] and take $X$ to be the graph with two $0$–cells $u$, $v$ and three $1$–cells $a$, $b$, $c$ all originating at $u$ and terminating at $v$, see Figure \[fig:xgraph\]. If we take $u$ as the basepoint for $X$, this realizes $F$ as a subgroup of a rank $3$ free group $F(a,b,c)$ on free generators $\{a,b,c\}$ with the inclusion $\theta\colon F(a,b)\xhookrightarrow{\quad} F(a,b,c)$ given by $a\mapsto ca^{-1}$, $b\mapsto cb^{-1}$. (Choosing $v$ as the basepoint of $X$ yields an inclusion given by $a\mapsto a^{-1}c$ and $b\mapsto b^{-1}c$.)
(5.5,-0.5) to \[out=90, in=90\] (6.5,-0.5); (5.5,-0.5) to \[out=0, in=180\] (6.5,-0.5); (5.5,-0.5) to \[out=-90, in=-90\] (6.5,-0.5); (5.5,-0.5) circle (1.5pt); (5.5,-0.5) circle (2.5pt); (6.5,-0.5) circle (1.5pt); (4.5,-0.6) node [$X$:]{}; (5.3,-0.6) node [$u$]{}; (6.7,-0.6) node [$v$]{}; (6.3,0) node [$a$]{}; (6.25,-0.35) node [$b$]{}; (6.3,-0.95) node [$c$]{};
For any subgroup $H\le F$ there is a covering $\widetilde X_H\to X$ corresponding to $H$. If we fix the vertex $u$ (or $v$, for that matter) as the basepoint of $X$, there is a choice of the basepoint $x_H$ in $\widetilde X_H$ such that $\pi_1(\widetilde X_H, x_H)$ is identical to $H$. (Such choice is not unique if $\widetilde X_H$ has a nontrivial deck transformation.) Let ${\Gamma}_H$ be the smallest subgraph of $\widetilde X_H$ containing $x_H$ that carries $\pi_1(\widetilde X_H,x_H)=H$. We call $({\Gamma}_H,x_H)$ the *core graph* for $H$. The vertices of ${\Gamma}_H$ fall into two classes: ‘sources’ (preimages of $u\in X$) and ‘sinks’ (preimages of $v\in X$). Every edge of ${\Gamma}_H$ is oriented from a source to a sink and inherits a unique label $a$, $b$, or $c$ induced by the covering map $\widetilde X_H\to X$. Notice that this labeling is *proper* in the sense that for every vertex $x$ of ${\Gamma}_H$ and each letter $\eta\in\{a,b,c\}$ there is at most one edge in ${\Gamma}_H$ with the origin $x$ labeled $\eta$ and there is at most one edge in ${\Gamma}_H$ with the terminus $x$ labeled $\eta$. Notice also that any vertex of ${\Gamma}_H$ is at most $3$–valent, and the only vertex that may have valence $1$ is the basepoint $x_H$.
In what follows, we will call the edges of ${\Gamma}_H$ which map to the edge $a$ (respectively, $b$, $c$) of $X$, as $a$–edges (resp., $b$–edges, $c$–edges), and paint them in diagrams with red (resp., blue, green) color. (We also depict $a$–edges with a single arrow, $b$–edges with a double arrow, and $c$–edges with a solid triangular arrow in all diagrams.)
Intersection of subgroups is represented by pullback
----------------------------------------------------
Let $H$, $K\le F$ be two finitely generated subgroups of $F$ and $({\Gamma}_H, x_H)$, $({\Gamma}_K,x_K)$ be the corresponding core graphs, with the natural maps $p_H\colon({\Gamma}_H, x_H)\to (X,u)$, $p_K\colon({\Gamma}_K, x_K)\to (X,u)$, which are injective on links of vertices, i.e. are immersions. Let $G_{H\cap K}$ be the pullback of these maps, defined as follows. The vertex set of $G_{H\cap K}$ is $V({\Gamma}_H)\times V({\Gamma}_K)$ and there is an oriented edge labeled $\eta$ ($\eta\in\{a,b,c\}$) from the vertex $(p,q)$ to the vertex $(r,s)$ in $G_{H\cap K}$ if and only if there is an edge labeled $\eta$ from $p$ to $r$ in ${\Gamma}_H$ and an edge labeled $\eta$ from $q$ to $s$ in ${\Gamma}_K$. The natural projections $V({\Gamma}_H)\times V({\Gamma}_K)\to V({\Gamma}_H)$ and $V({\Gamma}_H)\times V({\Gamma}_K)\to V({\Gamma}_K)$ give rise to immersions $\Pi_H\colon G_{H\cap K}\to {\Gamma}_H$, $\Pi_K\colon G_{H\cap K}\to {\Gamma}_K$, and the fundamental group of the component of $G_{H\cap K}$ containing the basepoint $(x_H,x_K)$ is equal to $H\cap K$, see [@Sta Th. 5.5]. Denote by ${\Gamma}_{H\cap K}$ the minimal subgraph of $G_{H\cap K}$ that contains $(x_H,x_K)$ and carries the fundamental group of the connected component of $(x_H,x_K)$. Then $\big({\Gamma}_{H\cap K},(x_H,x_K)\big)$ is the core graph for $H\cap K$. The construction guarantees that the two compositions of immersions $\big({\Gamma}_{H\cap K},(x_H,x_K)\big)\xrightarrow{\,\Pi_H\,}({\Gamma}_H,x_H)\xrightarrow{\,p_H\,} (X,u)$ and $\big({\Gamma}_{H\cap K},(x_H,x_K)\big)\xrightarrow{\,\Pi_K\,}({\Gamma}_K,x_K)\xrightarrow{\,p_K\,} (X,u)$ commute and thus define the canonical immersion $\big({\Gamma}_{H\cap K},(x_H,x_K)\big)\to (X,u)$.
\[ex:main\] Figure \[fig:mainpic\] shows the core graphs ${\Gamma}_H$, ${\Gamma}_K$, ${\Gamma}_{H\cap K}$ for the subgroups $H,K\le \theta(F)\le F(a,b,c)$ given by $H=\theta\big(\langle a,bab^{-1}\rangle\big)=\langle ca^{-1}, cb^{-1}ca^{-1}bc^{-1}\rangle$, $K=\theta\big(\langle b^{-1}a,ba\rangle\big)=\langle ba^{-1},cb^{-1}ca^{-1}\rangle$, and for their intersection $H\cap K=\theta\big(\langle bab^{-1}a\rangle\big)=\langle cb^{-1}ca^{-1}ba^{-1}\rangle$.
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Now we show that without loss of generality we can assume that the core graphs ${\Gamma}_H$, ${\Gamma}_K$ and ${\Gamma}_{H\cap K}$ do not have vertices of valence $1$.
\[lem:noextremal\] For any $H,K\le \theta(F)\le F(a,b,c)$ such that $H\cap K\ne 1$, we can find an element $g\in F(a,b,c)$ such that all the core graphs ${\Gamma}_{H^g}$, ${\Gamma}_{K^g}$, ${\Gamma}_{H^g\cap K^g}$ for the conjugated subgroups $H^g$, $K^g$, $H^g\cap K^g$, respectively, do not have vertices of valence $1$.
The only vertices in ${\Gamma}_H$, ${\Gamma}_K$, ${\Gamma}_{H\cap K}$ that may have valence $1$ are the basepoints $x_H$, $x_K$, $(x_H, x_K)$, respectively. If $(x_H,x_K)$ has valence $2$ or more, then its projections $x_H$ and $x_K$ also have valence $2$ or more, and there is nothing to prove.
Now suppose that the basepoint $p=(x_H,x_K)$ of ${\Gamma}_{H\cap K}$ has valence $1$. Since $H\cap K\ne 1$, the graph ${\Gamma}_{H\cap K}$ must have a closed circuit, and the vertex $p$ does not belong to it. Therefore there exists a vertex of valence $3$ in ${\Gamma}_{H\cap K}$. Let $p{-}q$ be the shortest path in ${\Gamma}_{H\cap K}$ to a valence $3$ vertex $q$, and let $w\in F(a,b,c)$ be the label on this path. Then projections $\Pi_H(p{-}q)$ and $\Pi_K(p{-}q)$ are immersed paths in ${\Gamma}_H$, ${\Gamma}_K$, respectively, with the same label $w$ on them. Since $q$ is a valence $3$ vertex in ${\Gamma}_{H\cap K}$, the vertices $q_H=\Pi_H(q)$ and $q_K=\Pi_K(q)$ also have valence $3$. However, one may have other vertices of valence $3$ on the paths $\Pi_H(p{-}q)=x_H{-}q_H$ and $\Pi_K(p{-}q)=x_K{-}q_K$. Let $q'_H$ be a vertex on the path $x_H{-}q_H$ defined as follows: $q'_H=x_H$, if $x_H$ has valence greater than $1$ in ${\Gamma}_H$, and $q'_H$ is the vertex of valence $3$ on the path $x_H{-}q_H$ with the least distance along this path from $x_H$, otherwise. Define $q'_K$ similarly in ${\Gamma}_K$. Then conjugating by $g=w^{-1}$ inside $F(a,b,c)$ yields a triple of subgroups $H^g, K^g, H^g\cap K^g$ such that their core graphs ${\Gamma}_{H^g}$, ${\Gamma}_{K^g}$, ${\Gamma}_{H^g\cap K^g}$ differ from ${\Gamma}_{H}$, ${\Gamma}_{K}$, ${\Gamma}_{H\cap K}$ by moving their basepoint to vertices $q_H$, $q_K$, and $q$, respectively, and deleting the hanging trees $p{-}q$ from ${\Gamma}_{H\cap K}$ and $x_H{-}q'_H$, $x_K{-}q'_K$ from ${\Gamma}_H$, ${\Gamma}_K$, respectively. (Vertices $q$, $q'_H$, $q'_K$ themselves are not deleted.) This gives us a triple of the core graphs ${\Gamma}_{H^g}$, ${\Gamma}_{K^g}$, ${\Gamma}_{H^g\cap K^g}$ for the subgroups $H^g$, $K^g$, $H^g\cap K^g$ with no vertices of valence $1$. See Figure \[fig:extremal\] for an illustration.
\[rem:novalence1\] Since ${\operatorname{rk}}H^g={\operatorname{rk}}H$, ${\operatorname{rk}}K^g={\operatorname{rk}}K$, ${\operatorname{rk}}H^g\cap K^g={\operatorname{rk}}H\cap K$, for the purposes of this paper we may assume (and will do so from now on) without loss of generality that the groups $H$, $K$, $H\cap K$ have the core graphs which do not have vertices of valence $1$. It may happen that after the procedure described in Lemma \[lem:noextremal\] the basepoints of ${\Gamma}_H$, ${\Gamma}_K$ and ${\Gamma}_{H\cap K}$ all map to the vertex $v\in X$, instead of $u$, but that does not limit generality, since we may have chosen vertex $v\in X$ as the basepoint of $X$ from the very beginning.
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Join of subgroups and the topological pushout {#ssec:tp}
---------------------------------------------
As was shown by Stallings [@Sta], the core graph ${\Gamma}_{H\vee K}$ for the join of two subgroups is obtained by joining the core graphs for ${\Gamma}_H$ and ${\Gamma}_K$ at their respective basepoints and performing a sequence of identifications of edges with the same labels called *foldings*: $${\Gamma}_H\vee {\Gamma}_K\xrightarrow{\mathrm{\ foldings\ }}{\Gamma}_{H\vee K}$$ In general, the number of foldings required to produce ${\Gamma}_{H\vee K}$ and the rank of ${\Gamma}_{H\vee K}$ are hard to estimate directly from the information about ${\Gamma}_H$, ${\Gamma}_K$, without actually performing the required sequence of foldings. In [@Ke2], Kent works with an intermediate object, the *topological pushout* ${\mathcal T}$ of ${\Gamma}_H$ and ${\Gamma}_K$, which fits into the diagram: $${\Gamma}_H\vee{\Gamma}_K \xrightarrow{\mathrm{\ foldings\ }} {\mathcal T}\xrightarrow{\mathrm{\ foldings\ }} {\Gamma}_{H\vee K}$$ and whose rank is much easier to estimate than the rank of ${\Gamma}_{H\vee K}$. Since the folding operation is surjective at the level of fundamental groups [@Sta Cor. 4.4], we also have $${\operatorname{rk}}{\mathcal T}\ge {\operatorname{rk}}{\Gamma}_{H\vee K}.$$
${\mathcal T}$ is defined as follows.
Let $x\in{\Gamma}_H$ and $y\in{\Gamma}_K$ be either two vertices or two edges of ${\Gamma}_H$ and ${\Gamma}_K$. The graph ${\mathcal T}$ is the quotient of the disjoint union ${\Gamma}_H\sqcup{\Gamma}_K$ by the equivalence relation generated by the following relation: $x\sim y$ if $x\in \Pi_H\big((\Pi_K|_{{\Gamma}_{H\cap K}})^{-1}(y)\big)$ or $y\in \Pi_K\big((\Pi_H|_{{\Gamma}_{H\cap K}})^{-1}(x)\big)$. In other words, $x\sim y$ if and only if there is an element $z$ of ${\Gamma}_{H\cap K}$ such that $x$ and $y$ are the images under $\Pi_H$, $\Pi_K$, respectively, of $z$. Recall that, by construction, the vertices of ${\Gamma}_{H\cap K}$ can be identified with a certain subset of $V({\Gamma}_H)\times V({\Gamma}_K)$ and the same is true for edges. Thus two elements (i.e. two vertices or two edges) $a,b$ of ${\Gamma}_H\sqcup{\Gamma}_K$ map to the same element in ${\mathcal T}$ if and only if there is a sequence of elements $(x_1,y_1),\dots, (x_n,y_n)$ in ${\Gamma}_{H\cap K}$ with $x_i\in {\Gamma}_H$, $y_i\in {\Gamma}_K$ such that $a$ is either $x_1$ or $y_1$, $b$ is either $x_n$ or $y_n$ and for each $i$ either $x_i=x_{i+1}$ or $y_i=y_{i+1}$.
Equivalently, ${\mathcal T}$ can be obtained from the join of ${\Gamma}_H$ and ${\Gamma}_K$ over their respective basepoints $x_H$ and $x_K$, followed by a sequence of foldings along the edges of ${\Gamma}_{H\cap K}$ only. I.e. we may choose a circuit (i.e. a closed path) $\gamma$ in ${\Gamma}_{H\cap K}$ that starts at the basepoint $(x_H,x_K)$ and traverses each edge of ${\Gamma}_{H\cap K}$ at least once, and perform a sequence of foldings, identifying $\Pi_H(z)\in{\Gamma}_H$ with $\Pi_K(z)\in{\Gamma}_K$ for $z$ running consecutively through all vertices and edges along $\gamma$. Since ${\Gamma}_{H\cap K}$ is connected, a simple inductive argument shows that the result of this sequence of foldings is exactly the topological pushout ${\mathcal T}$ of ${\Gamma}_H$ and ${\Gamma}_K$.
Figure \[fig:mainpic\] shows the topological pushout for the groups $H,K$ of Example \[ex:main\]. We see that the topological pushout ${\mathcal T}$ may be different from ${\Gamma}_{H\vee K}$. In Figure \[fig:mainpic\], the graph ${\Gamma}_{H\vee K}$ is equal to $X$ and it is obtained from ${\mathcal T}$ by identifying (folding) two $c$–edges.
Proof of Theorem \[thm:real\] {#sec:real}
=============================
For the purposes of this section it will be convenient to assume that $F$ is a free group of countable rank so that we have a freedom to add new generators if necessary, without the need of explicitly embedding them into the free group of rank $2$. Also, for that purpose, we fix the wedge of countably many circles as the base CW complex for $F$.
We will call a tuple of values $(h,k;v,c)$ *realizable*, if there exist finitely generated subgroups $H,K$ of $F$ with ${\operatorname{rk}}H=h$, ${\operatorname{rk}}K=k$, ${\operatorname{rk}}(H\vee K)=v$, and ${\operatorname{rk}}(H\cap K)=c$. If the values $h,k$ (and sometimes also $v$) are clear from the context, we will also call the tuple $(v,c)$ (respectively, the number $c$) *realizable*, and say that it belongs to *page* $(h,k)$.
Excluding trivial cases, we may assume that ${\operatorname{rk}}H\ge 2$ and ${\operatorname{rk}}K\ge 2$, so that ${\operatorname{rk}}(H\vee K)\ge 2$. The upper bound for ${\operatorname{rk}}(H\vee K)$ is obviously $h+k$. On the other hand, the limits for ${\operatorname{rk}}(H\cap K)$ are $0$ and $(h-1)(k-1)+1$, as is stipulated by the Friedman–Mineyev theorem.
It turns out that the set of all known realizable values $(h,k;v,c)$ can be described for any fixed $(h,k)$ by a finite sequence of nonnegative integers $(a_i)$, such that for any given $h,k,v$ all (known) realizable values of $c$ are described as the range $0\le c\le a_i$, where $i=h+k-v$:
Let $F$ be a free group and let integers $h,k,c,v$ satisfy $2\le h\le k$, $2\le v\le h+k$, $0\le c\le (h-1)(k-1)+1$. Define a sequence $(a_i)$ as follows: $$\begin{aligned}
a_0&=0\,;\\
a_i&=\Big\lfloor \frac {i^2}{4}\Big\rfloor +1,\quad \text{for\quad $i=1,\dots, 2(h-1)\,;$}\\
a_i&=(h-1)(i-h+1)+1,\quad \text{for\quad $i=2(h-1),\dots,h+k-2$.}\end{aligned}$$ If we denote $i=h+k-v$, then for any $c\le a_i$ there exist subgroups $H,K\le F$ such that ${\operatorname{rk}}(H)=h$, ${\operatorname{rk}}(K)=k$, ${\operatorname{rk}}(H\cap K)=c$, and ${\operatorname{rk}}(H\vee K)=v$.
The diagram in Figure \[fig:57\] shows the realizable values from Theorem \[thm:real\] for $h={\operatorname{rk}}H=5$, $k={\operatorname{rk}}K=7$. They correspond to the sequence $$(a_i)=(0,1,2,3,5,7,10,13,17,21,25).$$
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We see that the sequence $(a_i)$ from the Theorem \[thm:real\] is a union of a discrete quadratic function and a linear function, with the linear part present only when $h<k$. The value $a_0=0$ reflects the fact that if ${\operatorname{rk}}(H\vee K)={\operatorname{rk}}H + {\operatorname{rk}}K$ then ${\operatorname{rk}}(H\cap K)=0$. (This is a consequence of the property of finitely generated free groups being Hopfian.) On the other hand, the value $v=2$ corresponds to $i=h+k-2$, and $a_{h+k-2}$ equals $(h-1)(k-1)+1$. This reflects the fact that all possible values for ${\operatorname{rk}}(H\cap K)=0,\dots,(h-1)(k-1)+1$ are realizable when ${\operatorname{rk}}(H\vee K)=2$, as was shown by Kent in [@Ke1].
In what follows, we will call a finite sequence $(a_i)$, ($i=0,\dots,n$), *greater* than a sequence $(b_i)$, ($i=0,\dots,n$), if for each $i$, we have $a_i\ge b_i$. In this case we also say that the sequence $(b_i)$ is *smaller* than the sequence $(a_i)$.
We will obtain the required set of realizable values for page $(h',k')$ inductively from the realizable values for page $(h,k)$ with $h\le h'$, $k\le k'$, by using the following operations:
Ia. Adding a new generator to $H$. This operation copies all realizable values from page $(h,k)$ to page $(h+1,k)$ as follows: $$(h,k;v,c)\longmapsto (h+1,k;v+1,c)$$
Ib. Adding a new generator to $K$. This operation copies all realizable values from page $(h,k)$ to page $(h,k+1)$ as follows: $$(h,k;v,c)\longmapsto (h,k+1;v+1,c)$$
II\. Adding the same new generator to both $H$ and $K$. We get: $$(h,k;v,c)\longmapsto (h+1,k+1;v+1,c+1)$$
III\. Populating the first row of any page $(h,k)$ with values $$v=2,\quad c=0,\,\dots,\,(h-1)(k-1)+1$$ corresponding to the explicit examples produced by Kent [@Ke1].
To prove the claimed effect on ranks under operations Ia, Ib and II, we notice that adding a new generator to a subgroup $H$ amounts to attaching the loop corresponding to this generator to the core graph ${\Gamma}_H$, at its basepoint. (We can do that in view of the assumption in the opening paragraph of the current section.) This makes the effect of operations Ia and Ib obvious, while for operation II we recall the construction of the core graph for $H\cap K$ from Section \[sec:graphs\]. It is clear that if we attach a loop labeled with the same new generator to both core graphs ${\Gamma}_H$, ${\Gamma}_K$ at their respective basepoints, then the core graph of their intersection ${\Gamma}_{H\cap K}$ also gets the loop labeled with the same generator attached to its basepoint.
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We start with page $(h,k)=(2,2)$, which corresponds to the sequence $(a_n)=(0,1,2)$, (as was shown in [@Ke2 p. 307], see Figure \[fig:22\]), and determine a sequence of operations Ia, Ib, II, III that leads to the greatest sequence $(a_n)$, for the required values $(h,k)$.
We first show that if $h<k$ then applying (Ia + III) followed by (Ib + III) produces a greater sequence $(a_n)$ of realizable $c$–values for page $(h+1,k+1)$ than applying first (Ib + III) and then (Ia + III). Indeed, let $(a_n)$ be the sequence for $(h,k)$ with $n$ ranging from $0$ to $h+k-2$. Denote $(a_n')$ the result of applying (Ia + III) to $(a_n)$ and $(a_n'')$ the result of applying (Ib + III) to $(a_n')$. Similarly, denote $(b_n')$ the result of applying (Ib + III) to $(a_n)$ and $(b_n'')$ the result of applying (Ia + III) to $(b_n')$: $$\begin{gathered}
(a_n)\xrightarrow{\text{Ia+III}}(a_n')\xrightarrow{\text{Ib+III}}(a_n''),\\
(a_n)\xrightarrow{\text{Ib+III}}(b_n')\xrightarrow{\text{Ia+III}}(b_n'').\end{gathered}$$ We want to show that $a_n''\ge b_n''$ for all $n$. Since operation Ia copies all values of $(a_i)$ to the new page, we have: $a_i'=a_i$ if $i<(h+1)+k-2$, and since operation III populates the first row (corresponding to the last value of $a_i$) with a sequence of $h(k-1)+1$ values, we get $a'_{(h+1)+k-2}=h(k-1)+1$. Similarly, $a_i''=a_i$ for $i<(h+1)+(k+1)-3$, $a''_{(h+1)+(k+1)-3}=h(k-1)+1$, and $a''_{(h+1)+(k+1)-2}=hk+1$. Performing these operations in the opposite order, i.e. applying (Ib + III) to $(a_n)$ first followed by (Ia + III), we get in a similar fashion: $b''_i=a_i$ for $i<(h+1)+(k+1)-3$, $b''_{(h+1)+(k+1)-3}=hk-k+1$, and $b''_{(h+1)+(k+1)-2}=hk+1$. The sequences $(a_n'')$ and $(b_n'')$ agree for all values of $n$ except the penultimate one, $n=(h+1)+(k+1)-3$. Comparing them and taking into account that $h<k$, we see that $hk-h+1>hk-k+1$, i.e. $a_n''\ge b_n''$ for all $n$. This means that if $h<k$, applying (Ia + III) followed by (Ib + III) produces a greater sequence $(a_n)$ than if applying these operations in the opposite order.
The above formulas also show that in the case when $h=k$ the result of applying operations (Ia + III) and (Ib + III) does not depend on their order.
Now let’s examine operation (II + III). It copies realizable values from page $(h,k)$ to page $(h+1,k+1)$ as shown in the lower part of Figure \[fig:23\]. Let $(a_n)$ and $(a_n''')$ be the corresponding sequences of realizable values for pages $(h,k)$ and $(h+1,k+1)$, respectively, and let $(a''_n)$ be the sequence obtained from $(a_n)$ by the composition of operations (Ia + III) and (Ib + III), as before. We claim that $(a_n''')$ is smaller than $(a_n'')$. Indeed, we saw in the previous paragraph that $a_i''=a_i$ for $i<h+k-1$, $a''_{h+k-1}=hk-h+1$ and $a''_{h+k}=hk+1$. From Figure \[fig:23\] we see that $a'''_{i}=a_{i-1}+1$ for $i=1,\dots,h+k-1$ and $a'''_{h+k}=hk+1$. Thus to prove that $a''_i\ge a'''_i$ for all $i$, we claim the following:
1. $a_{i}\ge a_{i-1}+1$ for $i=1,\dots,h+k-2$, and
2. $hk-h+1\ge a_{h+k-2}+1$.
We will prove these using the following fact: *for any page $(h,k)$ the last value of the sequence $(a_n)$, i.e. the term $a_{h+k-2}$, equals $(h-1)(k-1)+1$*, which is the content of operation III. (Recall that operation III is applied every time we apply Ia, Ib or II.) Thus, inequality (ii) is established: $a_{h+k-2}=(h-1)(k-1)+1\le hk-h$ since $k\ge2$. To prove (i) we observe that all operations Ia, Ib, II preserve the difference between consecutive elements $a_{j}-a_{j-1}$, so all that needs to be proved is that the top value $a_{h+k-2}=(h-1)(k-1)+1$ is always at least $1$ bigger than the previous value of $a_{h+k-3}$. Assuming that (i) holds true for page $(h,k)$, we examine how it changes under the application of operations (Ia + III), (Ib + III) and (II + III). In the first case, operation Ia makes the next-to-last value of the sequence $(a_n')$ to be $(h-1)(k-1)+1$, while the last one is $h(k-1)+1$, with the difference $k-1$ between the two. In the second case, arguing in a similar fashion, we get that the difference between the last two values of $(b_n')$ equals $h-1$. And in the case of operation (II + III), we get that the difference equals $(hk+1) - \big((h-1)(k-1)+2\big)=h+k-2$. We see that in all three cases this difference is at least $1$, which proves claim (i). Thus operation (II + III) creates a sequence $(a_n''')$ which is smaller than the sequence $(a_n'')$ created by (Ia + III) followed by (Ib + III).
in [2,...,10]{} in [2,...,5]{} [ (,) +(-.5,-.5) rectangle ++(.5,.5); (,7-) node[(,)]{}; ]{}
(2,5)+(0.35,0.35) circle (1.5pt); (3,5)+(0.35,0.35) circle (1.5pt); (3,4)+(0.35,0.35) circle (1.5pt); (4,4)+(0.35,0.35) circle (1.5pt); (4,3)+(0.35,0.35) circle (1.5pt); (5,3)+(0.35,0.35) circle (1.5pt); (5,2)+(0.35,0.35) circle (1.5pt); (6,2)+(0.35,0.35) circle (1.5pt); (7,2)+(0.35,0.35) circle (1.5pt); (8,2)+(0.35,0.35) circle (1.5pt); (9,2)+(0.35,0.35) circle (1.5pt); (10,2)+(0.35,0.35) circle (1.5pt);
(2.35,5.35)–(3.35,5.35); (3.35,5.35)–(3.35,4.35); (3.35,4.35)–(4.35,4.35); (4.35,4.35)–(4.35,3.35); (4.35,3.35)–(5.35,3.35); (5.35,3.35)–(5.35,2.35); (5.35,2.35)–(6.35,2.35); (6.35,2.35)–(7.35,2.35); (7.35,2.35)–(8.35,2.35); (8.35,2.35)–(9.35,2.35); (9.35,2.35)–(10.35,2.35);
(3.35,5.35)–(4.35,5.35); (4.35,5.35)–(5.35,5.35); (5.35,5.35)–(6.35,5.35); (6.35,5.35)–(7.35,5.35); (7.35,5.35)–(8.35,5.35); (8.35,5.35)–(8.35,4.35); (8.35,5.35)–(9.35,5.35); (9.35,5.35)–(9.35,4.35); (8.35,4.35)–(9.35,4.35); (9.35,4.35)–(10.35,4.35); (10.35,4.35)–(10.35,3.35); (10.35,3.35)–(10.35,2.35);
(4,5)+(0.35,0.35) circle (1.5pt); (5,5)+(0.35,0.35) circle (1.5pt); (6,5)+(0.35,0.35) circle (1.5pt); (7,5)+(0.35,0.35) circle (1.5pt); (8,5)+(0.35,0.35) circle (1.5pt); (8,4)+(0.35,0.35) circle (1.5pt); (9,5)+(0.35,0.35) circle (1.5pt); (9,4)+(0.35,0.35) circle (1.5pt); (10,4)+(0.35,0.35) circle (1.5pt); (10,3)+(0.35,0.35) circle (1.5pt);
The above analysis shows that to obtain the greatest sequence $(a_n)$ of realizable values for page $(h,k)$, $h\le k$, one can discard operation (II + III) completely and apply only operations (Ia + III) and (Ib + III), starting with page $(h_0,k_0)=(2,2)$. All the ways to get from page $(2,2)$ to page $(h,k)$ by applying the said operations can be encoded by broken lines running in a rectangular table from entry $(2,2)$ to entry $(h,k)$ with horizontal and vertical segments corresponding to operations (Ib + III) and (Ia + III), respectively, see Figure \[fig:pages\]. Every time when operation (Ib + III) followed by (Ia + III) is applied to a page $(h',k')$ with $h'<k'$, we can interchange the order of these operations thus producing a bigger sequence $(a_n)$. By repeatedly doing this interchange, we obtain an optimal sequence which can be described as follows:
> *Start with $(h_0,k_0)=(2,2)$. Alternate operations (Ib + III) and (Ia + III) to reach page $(h,h)$. If $h<k$, keep applying (Ib + III) to reach page $(h,k)$*:
$$\begin{gathered}
(2,2)\xrightarrow{\text{Ib+III}}(2,3)\xrightarrow{\text{Ia+III}}(3,3)\xrightarrow{\text{Ib+III}}\dots
\xrightarrow{\text{Ia+III}}(h,h)\xrightarrow{\text{Ib+III}}\\
\xrightarrow{\text{Ib+III}}(h,h+1)\xrightarrow{\text{Ib+III}}(h,h+2)\xrightarrow{\text{Ib+III}}\dots
\xrightarrow{\text{Ib+III}}(h,k).\end{gathered}$$
Note that $a_0=0,a_1=1,a_2=2$ for $(h_0,k_0)=(2,2)$, and each of operations (Ia + III), (Ib + III) augments the existing sequence $a_0,\dots,a_n$ with a new value $a_{n+1}$, which is computed according to operation III as follows: $$\begin{aligned}
(\ell,\ell)&\xrightarrow{\text{Ib+III}}(\ell,\ell+1): &a_{n+1}=\ell(\ell-1)+1,\quad &\text{with }n=2(\ell-1);\\
(\ell,\ell+1)&\xrightarrow{\text{Ia+III}}(\ell+1,\ell+1): &a_{n+1}=\ell^2+1,\quad &\text{with }n=2\ell-1,$$ for $\ell=2,\dots,h-1$, and $$(h,h+j)\xrightarrow{\text{Ib+III}}(h,h+j+1):\quad a_{n+1}=(h-1)(h+j)+1,\,\text{with }n=2(h-1)+j,$$ for $j=0,\dots,k-h-1$, if $h<k$.
Now we observe that the values $a_{n+1}$ for $0\le n\le 2h-3$ can be written concisely as $a_{n+1}=\big\lfloor\big(\frac{n+1}2\big)^2\big\rfloor+1$. Indeed, if $n=2(\ell-1)$, then $\big\lfloor\big(\frac{n+1}2\big)^2\big\rfloor=\big\lfloor(\ell-\frac12)^2\big\rfloor=\big\lfloor\ell^2-\ell+\frac14\big\rfloor=\ell(\ell-1)$, and if $n=2\ell-1$, then $\big\lfloor\big(\frac{n+1}2\big)^2\big\rfloor=\ell^2$. This proves that the above sequence of operations produces the sequence $a_i$ described in the Theorem, which finishes the proof.
Now we have counterexamples to GTC for all $m\ge 5$.
Guzman’s “Group-Theoretic Conjecture” does not hold for any $m\ge 5$.
Theorem \[thm:real\] guarantees the existence of subgroups $H,K\le F$ such that $h={\operatorname{rk}}(H)=m$, $k={\operatorname{rk}}(K)=m$, $c={\operatorname{rk}}(H\cap K)=m$ and $v={\operatorname{rk}}(H\vee K)=m+1$, if $m\ge 5$. Indeed, $i=h+k-v=m-1$ in this case, which is less than $2(h-1)=2(m-1)$. So the value of $a_i$ in Theorem \[thm:real\] is equal to $\big\lfloor \frac{i^2}4\big\rfloor+1 = \big\lfloor\frac{(m-1)^2}4\big\rfloor+1$, which is bigger than or equal to $c=m$ for all $m\ge 5$. See Figure \[fig:intro\] for an illustration of the case $m=6$, which shows the existence of rank $6$ subgroups $H,K$ with ${\operatorname{rk}}(H\vee K)=7$ and ${\operatorname{rk}}(H\cap K)=6,7$.
The structure of the topological pushout {#sec:dicks}
========================================
In this section we study the combinatorial structure of the topological pushout using graphs introduced by Dicks in [@Di]. This will allow us to obtain an upper bound on the rank of the topological pushout of two core graphs and hence on the rank of the join of the corresponding subgroups.
The Dicks graphs
----------------
Let ${\Gamma}_H$, ${\Gamma}_K$ be the core graphs for subgroups $H,K\le F$ and the core graph ${\Gamma}_{H\cap K}$ is constructed as the pullback of graph immersions, as in Section \[sec:graphs\]: $$\begin{CD}
{\Gamma}_{H\cap K} @>\Pi_K>> {\Gamma}_K\\
@VV\Pi_H V @VVp_K V\\
{\Gamma}_H @>p_H>> X
\end{CD}$$ Each element $z$ (a vertex or an edge) of ${\Gamma}_H$ and ${\Gamma}_K$ inherits its *type* from the mapping to $X$, that is, an element of $V(X)\sqcup E(X)=\{u,v,a,b,c\}$ to which $z$ maps.
We are now going to define five bipartite undirected graphs $\Omega_u, \Omega_v, \Omega_a, \Omega_b, \Omega_c$, adapting the construction of [@Di] to our needs.
First, we define $\Omega_u$ as follows. $$\begin{aligned}
V(\Omega_u)&=\{z\in V({\Gamma}_H) \mid p_H(z)=u\} \sqcup \{z'\in V({\Gamma}_K) \mid p_K(z')=u\},\\
E(\Omega_u)&=\{(z,z')\in V({\Gamma}_{H\cap K})\mid p_H(z)=u \text{ and } p_K(z')=u\}.\end{aligned}$$ In other words, two vertices $z\in V({\Gamma}_H)$, $z'\in V({\Gamma}_K)$ from $V(\Omega_u)$ are connected with a single undirected edge if the vertex $(z,z')$ of $G_{H\cap K}$ actually belongs to ${\Gamma}_{H\cap K}$.
The graph $\Omega_v$ is defined analogously to $\Omega_u$ with the obvious modification ($u\rightsquigarrow v$). Denote also $\Omega=\Omega_u\sqcup\Omega_v$.
We define $\Omega_a$ similarly, by dealing with edges instead of vertices: $$\begin{aligned}
V(\Omega_a)&=\{e\in E({\Gamma}_H) \mid p_H(e)=a\} \sqcup \{e'\in E({\Gamma}_K) \mid p_K(e')=a\},\\
E(\Omega_a)&=\{(e,e')\in E({\Gamma}_{H\cap K})\mid p_H(e)=a \text{ and } p_K(e')=a\}\end{aligned}$$
The graphs $\Omega_b$, $\Omega_c$ are defined analogously with the obvious modifications.
The bipartite structure on the defined graphs is given by grouping all vertices/edges of graph ${\Gamma}_H$ in one part, and those of ${\Gamma}_K$ in the other.
Figure \[fig:dicks-uvabc\] shows the graphs $\Omega_u, \Omega_v, \Omega_a, \Omega_b, \Omega_c$ for the core graphs of the subgroups $H,K$ from Example \[ex:main\].
(0,0) circle (3pt); (2,0) circle (3pt); (0,3) circle (3pt); (2,3) circle (3pt); (0,0)–(2,3); (2,0)–(0,3); (2,0)–(2,3); (0,-0.5) node [$1$]{}; (2,-0.5) node [$3$]{}; (0,3.5) node [$5$]{}; (2,3.5) node [$7$]{}; (-1,1.5) node [$\Omega_u$:]{};
(0,0) circle (3pt); (2,0) circle (3pt); (0,3) circle (3pt); (2,3) circle (3pt); (0,0)–(2,3); (2,0)–(0,3); (0,0)–(0,3); (0,-0.5) node [$2$]{}; (2,-0.5) node [$4$]{}; (0,3.5) node [$6$]{}; (2,3.5) node [$8$]{}; (-1,1.5) node [$\Omega_v$:]{};
(0,0) circle (3pt); (2,0) circle (3pt); (1,3) circle (3pt); (0,0)–(1,3); (2,0)–(1,3);
(0,-0.5) node [(1,2)]{}; (2,-0.5) node [(3,4)]{}; (1,3.5) node [(7,6)]{}; (-1,1.5) node [$\Omega_a$:]{};
(1,0) circle (3pt); (0,3) circle (3pt); (2,3) circle (3pt); (1,0)–(0,3); (2,3)–(1,0);
(1,-0.5) node [(3,2)]{}; (0,3.5) node [(5,8)]{}; (2,3.5) node [(7,6)]{}; (-1,1.5) node [$\Omega_b$:]{};
(0,0) circle (3pt); (2,0) circle (3pt); (0,3) circle (3pt); (2,3) circle (3pt); (0,0)–(0,3); (2,0)–(2,3);
(0,-0.5) node [(1,2)]{}; (0,3.5) node [(7,8)]{}; (2,-0.5) node [(3,4)]{}; (2,3.5) node [(5,6)]{}; (-1,1.5) node [$\Omega_c$:]{};
Notice that the operations of taking the origin and the terminus of an edge induce embeddings $\tilde o$, $\tilde t$ of the graph $\Omega_a\sqcup \Omega_b\sqcup\Omega_c$ into $\Omega_u, \Omega_v$, respectively. Let’s show that $\tilde o|_{\Omega_a}\colon\Omega_a\to \Omega_u$ is an embedding. If $e,e'\in V(\Omega_a)$, with $e\ne e'$, then they correspond to $a$–edges of ${\Gamma}_H\sqcup{\Gamma}_K$. If $e\in E({\Gamma}_H)$, $e'\in E({\Gamma}_K)$, their origins are different. If they both belong to the same graph then their origins are also different, since ${\Gamma}_H$ and ${\Gamma}_K$ are folded, i.e. at every vertex of ${\Gamma}_H$, ${\Gamma}_K$ there is at most one $a$–edge having this vertex as the origin. This proves that the map $\tilde o|_{\Omega_a}\colon V(\Omega_a)\to V(\Omega_u)$ is injective. Now, if $e,e'\in V(\Omega_a)$ are connected with an edge, this means that there is an $a$–edge $(e,e')$ in $E({\Gamma}_{H\cap K})$ whose projections under $\Pi_H, \Pi_K$ are $e,e'$. In particular, the origin of $(e,e')$ projects to the origins of $e,e'$, and we see that the images of $e,e'$ in $V(\Omega_u)$ are connected with an edge as well. This proves that $\tilde o|_{\Omega_a}\colon\Omega_a\to\Omega_u$ is an injective graph homomorphism, i.e. is an isomorphism onto its image. Denote $$\begin{aligned}
\label{eq:dicks2}
\Omega_{u,a}&={\operatorname{im}}(\Omega_a\xhookrightarrow{\quad\tilde o\quad}\Omega_u), & \Omega_{v,a}&={\operatorname{im}}(\Omega_a\xhookrightarrow{\quad\tilde t\quad}\Omega_v), \nonumber\\
\Omega_{u,b}&={\operatorname{im}}(\Omega_b\xhookrightarrow{\quad\tilde o\quad}\Omega_u), & \Omega_{v,b}&={\operatorname{im}}(\Omega_b\xhookrightarrow{\quad\tilde t\quad}\Omega_v), \\
\Omega_{u,c}&={\operatorname{im}}(\Omega_c\xhookrightarrow{\quad\tilde o\quad}\Omega_u), & \Omega_{v,c}&={\operatorname{im}}(\Omega_c\xhookrightarrow{\quad\tilde t\quad}\Omega_v) \nonumber\end{aligned}$$ the images of these embeddings, and set $$\begin{aligned}
\Omega_{u,ab}&=\Omega_{u,a}\cap\Omega_{u,b}, & \Omega_{v,ab}&=\Omega_{v,a}\cap\Omega_{v,b}, & \Omega_{ab}&=\Omega_{u,ab}\sqcup\Omega_{v,ab},\\
\Omega_{u,bc}&=\Omega_{u,b}\cap\Omega_{u,c}, & \Omega_{v,bc}&=\Omega_{v,b}\cap\Omega_{v,c}, & \Omega_{bc}&=\Omega_{u,bc}\sqcup\Omega_{v,bc},\\
\Omega_{u,ac}&=\Omega_{u,a}\cap\Omega_{u,c}, & \Omega_{v,ac}&=\Omega_{v,a}\cap\Omega_{v,c}, & \Omega_{ac}&=\Omega_{u,ac}\sqcup\Omega_{v,ac},\\
\Omega_{u,abc}&=\Omega_{u,a}\cap\Omega_{u,b}\cap\Omega_{u,c}, & \Omega_{v,abc}&=\Omega_{v,a}\cap\Omega_{v,b}\cap\Omega_{v,c}, & \Omega_{abc}&=\Omega_{u,abc}\sqcup\Omega_{v,abc}.\end{aligned}$$
Since by Remark \[rem:novalence1\] we assume that neither of graphs ${\Gamma}_H$, ${\Gamma}_K$, ${\Gamma}_{H\cap K}$ can have a vertex of valence $1$, we observe that each vertex in $\Omega_{u,a}$ is also the origin of either another $b$–edge, or $c$–edge, or both. Thus, $$\Omega_{u,a}=\Omega_{u,ab}\bigvee_{\Omega_{u,abc}}\Omega_{u,ac},$$ where $K=L\bigvee_NM$ means that $K=L\cup M$ and $L\cap M=N$. Similarly, each vertex in $\Omega_{v,a}$ is also the terminus of either another $b$–edge, or $c$–edge, or both. Thus, $$\Omega_{v,a}=\Omega_{v,ab}\bigvee_{\Omega_{v,abc}}\Omega_{v,ac},$$ Of course, a completely similar statement holds for all other graphs in place of $\Omega_{u,a}$, $\Omega_{v,a}$.
Clearly, $\Omega_{ab}\cap\Omega_{bc}=\Omega_{bc}\cap\Omega_{ab}=\Omega_{ac}\cap\Omega_{ab}=\Omega_{abc}$. Thus we can denote $$A=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{ac},\qquad B=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{bc},\qquad
C=\Omega_{ac}\bigvee_{\Omega_{abc}}\Omega_{bc}.$$
Notice that, by construction, all graphs $A$, $B$, $C$ are subgraphs of $\Omega=\Omega_u\sqcup\Omega_v$.
In what follows we will depict connected components of $\Omega$ against the following ‘trefoil’ Venn diagram, illustrating the relation $$\Omega_{ab}\cap\Omega_{bc}=\Omega_{ab}\cap\Omega_{ac}=\Omega_{bc}\cap\Omega_{ac}=\Omega_{abc},$$ see Figure \[fig:venn\].
(0,3.464) arc (60:-60:2) arc (240:60:2) arc (360:180:2) arc (120:60:2); (0,2.35) node [$\Omega_{abc}$]{}; (0,0.85) node [$\Omega_{bc}$]{}; (-1.25,3) node [$\Omega_{ab}$]{}; (1.25,3) node [$\Omega_{ac}$]{};
(-2,4) – (2,4); (0,4.65) node [$A$]{};
(-0.6,-0.25) – (-2.5,3); (-2.15,1) node [$B$]{};
(2.5,3) – (0.6,-0.25); (2.15,1) node [$C$]{};
Now, for $A$, we have: $$\label{eq:A}
\begin{multlined}
A=\Omega_{ab}\cup\Omega_{ac}=(\Omega_{u,ab}\sqcup\Omega_{v,ab})\cup(\Omega_{u,ac}\sqcup\Omega_{v,ac})=\\[1ex]
(\Omega_{u,ab}\cup\Omega_{u,ac})\sqcup(\Omega_{v,ab}\cup\Omega_{v,ac})=\Omega_{u,a}\sqcup\Omega_{v,a}\cong \Omega_a\sqcup\Omega_a,
\end{multlined}$$ and similarly for $B$, $C$.
Thus we established the following duality, discovered by Dicks [@Di]:
\[prop:dicksdual\] Each of the graphs $A,B,C$ defined above consists of an even number of connected components, which are isomorphic in pairs. If $\{Z,Z'\}$ is such a pair of components of $A$, then one of $Z,Z'$ is a component of $\Omega_{u,a}$ and the other is a component of $\Omega_{v,a}$, and the isomorphism between them preserves the bipartite structure. Similar statements are true for $B$ and $C$ in place of $A$.
Modeling the topological pushout on the Dicks graphs {#ssec:modeling}
----------------------------------------------------
Notice that the Dicks graphs $\Omega_u$, $\Omega_v$, $\Omega_a$, $\Omega_b$, $\Omega_c$ express the equivalence relation $\sim$ used to define the topological pushout in subsection \[ssec:tp\]: two vertices $z,z'$ are connected by an edge in the graph $\Omega_x$ (where $x$ stands for any of $u,v,a,b,c$) if and only if they are the two projections under $\Pi_H$, $\Pi_K$ of the same element of ${\Gamma}_{H\cap K}$. Thus, the connected components of $\Omega_u\sqcup\Omega_v$ are exactly the vertices of the topological pushout, and the connected components of $\Omega_a\sqcup\Omega_b\sqcup\Omega_c$ are the edges of the topological pushout, with inclusions $\tilde o$, $\tilde t$ defined above being the origin and the terminus maps. In particular, we see from Figure \[fig:dicks-uvabc\] that the topological pushout ${\mathcal T}$ from Example \[ex:main\] (see Figure \[fig:mainpic\]) has exactly two vertices corresponding to the connected graphs $\Omega_u$ and $\Omega_v$ and four directed edges: one $a$–edge for the connected graph $\Omega_a$, one $b$–edge for the connected graph $\Omega_b$ and two $c$–edges for the two connected components of $\Omega_c$.
It will be useful for us to recast the topological pushout in terms of graphs $A,B,C$ defined above:
\[prop:toppush\] The topological pushout ${\mathcal T}$ admits the following description:
**Vertices:** connected components of $\Omega=\Omega_u\sqcup\Omega_v=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{bc}\bigvee_{\Omega_{abc}}\Omega_{ac} = A\cup B\cup C$.
**Edges:** pairs of connected components of $A$, $B$, $C$ from the pairing in Proposition \[prop:dicksdual\].
If $e=\{Z,Z'\}$ is such a pair, viewed as a directed edge, it inherits its type from the corresponding graph it belongs to: if $e\subset A$, then $e$ is an $a$–edge, if $e\subset B$, then $e$ is a $b$–edge, and if $e\subset C$ then $e$ is a $c$–edge. The origin and terminus maps are defined as follows. If $e\subset A$, then one of $Z,Z'$ is a connected component of $\Omega_{u,a}$, and the other, of $\Omega_{v,a}$. Let $Z\subset\Omega_{u,a}$, $Z'\subset\Omega_{v,a}$, say. Then the origin of $e$ is the connected component of $(A\cup B\cup C)\cap \Omega_{u}$ in which $Z$ lies, and the terminus of $e$ is the connected component of $(A\cup B\cup C)\cap \Omega_{v}$ in which $Z'$ lies. Similar definitions apply to $B$ and $C$ in place of $A$.
By the definition of ${\mathcal T}$, its vertices are the connected components of $\Omega=\Omega_u\sqcup\Omega_v$. Since every vertex of ${\Gamma}_H$, ${\Gamma}_K$ is incident to either an $a$–edge, a $b$–edge, or a $c$–edge, the same is true for ${\mathcal T}$. Hence $\Omega_u=\Omega_{u,a}\cup\Omega_{u,b}\cup\Omega_{u,c}$ and $\Omega_v=\Omega_{v,a}\cup\Omega_{v,b}\cup\Omega_{v,c}$. From we see that $\Omega_{u,a}\sqcup\Omega_{v,a}=A$, $\Omega_{u,b}\sqcup\Omega_{v,b}=B$, and $\Omega_{u,c}\sqcup\Omega_{v,c}=C$. It follows that $\Omega_u\sqcup\Omega_v=A\cup B\cup C$.
The edges of ${\mathcal T}$, by definition, are the connected components of $\Omega_a\sqcup\Omega_b\sqcup\Omega_c$, with the attaching maps $\tilde o$, $\tilde t$, defined above. If $Z_0$ is a connected component of $\Omega_a$, say, it defines two isomorphic connected subgraphs $Z=\tilde o(Z_0)\subset\Omega_{u,a}$ and $Z'=\tilde t(Z_0)\subset\Omega_{v,a}$. Since $\Omega_{u,a}$ and $\Omega_{v,a}$ are, by definition, isomorphic copies of $\Omega_a$, the subgraphs $Z,Z'$ are the whole connected components of $\Omega_{u,a}$, $\Omega_{v,a}$, respectively. By , $A=\Omega_{u,a}\sqcup\Omega_{v,a}$, therefore the pair $\{Z,Z'\}$ is a pair of connected components of $A$ determined by $Z_0$. The attaching maps for $\{Z,Z'\}$ described in the Proposition are induced by $\tilde o$ and $\tilde t$ applied to $Z_0$.
(0,3.464) arc (60:-60:2) arc (240:60:2) arc (360:180:2) arc (120:60:2);
(0,3.464) arc (60:-60:2) arc (240:60:2) arc (360:240:2);
(0,3.464) arc (120:240:2) arc (-60:120:2) arc (180:300:2);
(0,0) arc (0:120:2) arc (180:360:2) arc (60:180:2);
(2,1) node \[rotate=-30\] [$\supset$]{}; (-2,1) node \[rotate=30\] [$\subset$]{}; (0,4.7) node \[rotate=90\] [$\supset$]{};
Figure \[fig:toppush\] shows the topological pushout for the subgroups of Example \[ex:main\], modeled by subsets $\Omega_{ab}$, $\Omega_{bc}$, $\Omega_{ac}$, $\Omega_{abc}$, in accordance with Proposition \[prop:toppush\]. Notice that the connected components of $\Omega_a,\Omega_b,\Omega_c$ establish bijections between parts $A,B,C$ of $\Omega_u$ (the left ‘trefoil’) and the corresponding parts of $\Omega_v$ (the right ‘trefoil’). In particular, the connected graph $\Omega_a$ from Figure \[fig:dicks-uvabc\] acts as the $a$–edge of ${\mathcal T}$ and establishes a bijection between $\tilde o(\Omega_a)=\{1{-}7{-}3\}$ with $\tilde t(\Omega_a)=\{2{-}6{-}4\}$. Also, the connected graph $\Omega_b$ from Figure \[fig:dicks-uvabc\] is the $b$–edge of ${\mathcal T}$ and it establishes a bijection between $\tilde o(\Omega_b)=\{5{-}3{-}7\}$ and $\tilde t(\Omega_b)=\{8{-}2{-}6\}$. Finally, each connected component of $\Omega_c$ from Figure \[fig:dicks-uvabc\] serves as a $c$–edge of ${\mathcal T}$ and they establish bijections of the two connected components $1{-}7$, $3{-}5$ of $\Omega_{u,c}$ with $2{-}8$, $4{-}6$ of $\Omega_{v,c}$. Notice also that this bijection does not preserve subsets $\Omega_{ac}$ and $\Omega_{bc}$ individually but leaves invariant their union $C=\Omega_{ac}\bigvee_{\Omega_{abc}}\Omega_{bc}$: the subgraph $1{-}7$ of $\Omega_{u,ac}$ gets paired with the subgraph $2{-}8$ of $\Omega_{v,bc}$, and also $3{-}5$ of $\Omega_{u,bc}$ gets paired with $4{-}6$ of $\Omega_{v,ac}$. The curved edges $3{-}7$ and $2{-}6$ reflect the fact that vertices $3$ and $7$ (and also $2$ and $6$) are adjacent in $\Omega_{ab}$, but not in $\Omega_{abc}$.
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Now we are ready to relate ranks of $H$, $K$, $H\cap K$ and ${\mathcal T}$ with the structure of the Dicks graph $\Omega=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{bc}\bigvee_{\Omega_{abc}}\Omega_{ac}$. We are going to prove the following Theorem \[thm:ranks\] assuming the validity of Proposition \[prop:cn\], which will be proved in Section \[sec:cn\]. Recall that a *cycle* in an undirected graph is a path $x_0e_0x_1e_1\dots x_ke_kx_0$ such that all vertices $x_i$ are pairwise different. We denote by $\#W$ the cardinality of a finite set $W$ and, as before, by ${\operatorname{rr}}(H)$ the reduced rank of group $H$, i.e. the quantity $\max(0,{\operatorname{rk}}(H)-1)$.
\[thm:ranks\] Let $F$ be a free group and $H,K\le F$ be finitely generated subgroups. Let ${\Gamma}_H$, ${\Gamma}_K$ and ${\Gamma}_{H\cap K}$ be the core graphs of the corresponding subgroups, and let ${\mathcal T}$ be their topological pushout. Let also $\Omega$ and $\Omega_{abc}$ be the Dicks graphs defined above. Then
1. $\Omega_{abc}$ is a bipartite graph with $2{\operatorname{rr}}(H)$ vertices in one part and $2{\operatorname{rr}}(K)$ vertices in the other;
2. $\Omega_{abc}$ has $2{\operatorname{rr}}(H\cap K)$ edges;
3. (\# connected components of $\Omega_{abc}$) $\ge 2{\operatorname{rr}}({\mathcal T})$, with the equality taking place if and only if every cycle of $\Omega$ lies entirely in one of the subgraphs $\Omega_{ab}$, $\Omega_{bc}$, $\Omega_{ac}$ (with different cycles possibly lying in different subgraphs).
The graph $\Omega_{abc}$ has as its vertices all vertices of valence $3$ of ${\Gamma}_H$ and ${\Gamma}_K$. Since graphs ${\Gamma}_H$, ${\Gamma}_K$, ${\Gamma}_{H\cap K}$ are normalized as in Remark \[rem:novalence1\], all their vertices have valence either $2$ or $3$. Computing Euler characteristic of ${\Gamma}$ (where ${\Gamma}$ stands for any of ${\Gamma}_H$, ${\Gamma}_K$, ${\Gamma}_{H\cap K}$) gives: $$1-{\operatorname{rk}}({\Gamma}) = \#V({\Gamma})-\#\big(E^+({\Gamma})\big)= \sum_{v\in V({\Gamma})}\left(1-\frac{{\operatorname{val}}(v)}2\right),$$ which is equivalent to $$2{\operatorname{rr}}({\Gamma})=\sum_{v\in V({\Gamma})}\big(\!{\operatorname{val}}(v)-2\big)=\text{\ \big(\# vertices of valence $3$ in ${\Gamma}$\big)},$$ the last equality being true since vertices of valence $2$ contribute $0$ to the sum. This proves part (1).
Edges of $\Omega_{abc}$ are exactly the vertices of valence $3$ in ${\Gamma}_{H\cap K}$, hence the computation above establishes part (2) as well.
For part (3), exactly as above, we have $$\label{eq:val}
2{\operatorname{rr}}({\mathcal T})=\sum_{v \in V({\mathcal T})} \big(\!{\operatorname{val}}(v)-2\big).$$ According to Proposition \[prop:toppush\], the valence of a vertex $v$ of ${\mathcal T}$ (represented by some connected component $D$ of $A\cup B\cup C$) is the sum of numbers of connected components of $A\cap D$, $B\cap D$ and $C\cap D$. Our goal is to understand the relationship between components of $\Omega_{abc}$, components of $A\cup B\cup C$, and components of $A$, $B$ and $C$, taken separately. The difficulty lies in an observation that two components $P$, $Q$ of $\Omega_{abc}$ may be connected by a path outside $\Omega_{abc}$, i.e. by a path all edges of which lie in one of the graphs $\Omega_{ab}\setminus\Omega_{abc}$, $\Omega_{bc}\setminus\Omega_{abc}$, $\Omega_{ac}\setminus\Omega_{abc}$. If this is the case, then $P$ and $Q$ actually correspond to the same connected component $D$ of $A\cup B\cup C$, i.e. to the same vertex of ${\mathcal T}$, and their contribution to the valence of ${\mathcal T}$ may be different from the value $2\cdot 3$ expected otherwise. A careful treatment of this situation is given in Section \[sec:cn\], where we take an abstract approach and study a certain class ${\mathcal C_n}$ of graphs ${\Gamma}$ with a function $\Sigma$ associated to them, which encode the connectedness of components of $\Omega_{abc}$ to each other through the three graphs $\Omega_{ab}$, $\Omega_{bc}$, $\Omega_{ac}$, and their joint contribution to the right-hand side of . To get the input for the main result of Section \[sec:cn\], Proposition \[prop:cn\], we form the following undirected graph ${\Gamma}$, which we will call the *component connectivity graph* (CCG) of $\Omega_{abc}$.
Vertices of ${\Gamma}$ are connected components of $\Omega_{abc}$.
If $p$, $q\in V({\Gamma})$, they are some components $P$, $Q$ of $\Omega_{abc}$. If there exists a path between $P$ and $Q$ which does not contain any vertices of $\Omega_{abc}$ (except the first vertex of the path and the last one), and all edges of which lie in $\Omega_{ab}\setminus\Omega_{abc}$, we connect vertices $p,q$ in $E({\Gamma})$ with an undirected edge and assign the color *magenta* to it. Similarly, if a $\Omega_{abc}$-avoidant path between $P$ and $Q$ lies in $\Omega_{ac}\setminus\Omega_{abc}$, we add an edge to $E({\Gamma})$ connecting $p$ and $q$ and assign the color *yellow* to it. Lastly, if such path lies in $\Omega_{bc}\setminus\Omega_{abc}$, we add an edge to $E({\Gamma})$ connecting $p$ and $q$ and assign the color *cyan* to it. Thus every two vertices of ${\Gamma}$ may be connected by up to three undirected edges, each having a different color. (The choice of names for the colors is suggested by mixing the basic colors red, blue and green, which we used to depict $a$–edges, $b$–edges and $c$–edges, respectively. Hence, edges from $\Omega_{ab}\setminus\Omega_{abc}$ get color red-blue, i.e. magenta, edges from $\Omega_{ac}\setminus\Omega_{abc}$ get color red-green, i.e. yellow, and edges from $\Omega_{bc}\setminus\Omega_{abc}$ get color blue-green, i.e. cyan.)
Thus, the graph ${\Gamma}$ encodes the connectedness information (within $A\cup B\cup C$) between different connected components of $\Omega_{abc}$. The contribution of vertices of ${\mathcal T}$ to the sum in (i.e. the right-hand side of ) is equal to the function $\Sigma({\Gamma})$, defined in equation of Section \[sec:cn\].
Having formed the input for Proposition \[prop:cn\], we can use its conclusion, which reads: $\Sigma({\Gamma})\le n$. Here $n$ is the number of vertices of ${\Gamma}$, i.e. the number of connected components of $\Omega_{abc}$, and $\Sigma({\Gamma})$ is the right-hand side of . This proves the inequality in part (3) of the Theorem. Proposition \[prop:cn\] also specifies when we have the equality in $\Sigma({\Gamma})\le n$: this happens if and only if all cycles of ${\Gamma}$ are *monochromatic* in the terminology of Section \[sec:cn\]. In terms of the Dicks graphs, this means that every cycle of $\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{bc}\bigvee_{\Omega_{abc}}\Omega_{ac}$ lies entirely in either $\Omega_{ab}$, or $\Omega_{bc}$, or $\Omega_{ac}$. This finishes the proof of part (3) of the Theorem.
Situations when the inequality in part (3) of the Theorem \[thm:ranks\] is strict appear quite frequently, see for example Figure \[fig:k23\] in Section \[sec:guzman4\].
Class Cn {#sec:cn}
========
The goal of this technical section is to prove Proposition \[prop:cn\] needed for the proof of part (3) of Theorem \[thm:ranks\].
Consider a class ${\mathcal C_n}$ of pairs $({\Gamma},c)$ where ${\Gamma}$ is an undirected graph with multiple edges allowed (but not loops), and $c\colon E({\Gamma})\to \{\text{{\it magenta, yellow, cyan}}\}$ is an edge-coloring map, with the following properties:
1. Each graph ${\Gamma}$ from ${\mathcal C_n}$ has exactly $n$ vertices.
2. Let $E({\Gamma},p,q)=E({\Gamma},q,p)$ denote the set of all undirected edges between two different vertices $p,q\in V({\Gamma})$. Then the edge-coloring map $c$ is injective on each set $E({\Gamma},p,q)$.
In other words, any two different vertices $p,q$ of ${\Gamma}$ may be joined by up to three undirected edges of different colors from the set [*{magenta, yellow, cyan}*]{}.
For any ${\Gamma}\in{\mathcal C_n}$ we define three subgraphs ${\Gamma}_{my}$, ${\Gamma}_{yc}$, ${\Gamma}_{mc}$ of ${\Gamma}$ as follows:
- $V({\Gamma}_{my})=V({\Gamma}_{yc})=V({\Gamma}_{mc})=V({\Gamma})$;
- $E({\Gamma}_{my})=\{\text{all {\it magenta} and {\it yellow} edges of ${\Gamma}$}\}$;
- $E({\Gamma}_{yc})=\{\text{all {\it yellow} and {\it cyan} edges of ${\Gamma}$}\}$;
- $E({\Gamma}_{mc})=\{\text{all {\it magenta} and {\it cyan} edges of ${\Gamma}$}\}$;
(0,3.464) arc (60:-60:2) arc (240:60:2) arc (360:180:2) arc (120:60:2);
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(2.5,3) – (0.6,-0.25); (2.25,1) node [${\Gamma}_{yc}$]{};
(-1.25,3) node \[align=center,magenta,font=\] [magenta\
edges]{}; (1.25,3) node \[align=center,orange,font=\] [yellow\
edges]{}; (0,0.75) node \[align=center,cyan,font=\] [cyan\
edges]{}; (0,3) circle (2pt); (-0.5,2) circle (2pt); (0.5,2) circle (2pt); (0,2.35) node [$V({\Gamma})$]{};
We define a function $\Sigma\colon{\mathcal C_n}\to{\mathbb Z}_{\ge0}$ as follows: $$\label{eq:sigma}
\Sigma({\Gamma})=\sum_{C\in CC({\Gamma})} \big({{\operatorname{val}}_{my}}(C)+{{\operatorname{val}}_{yc}}(C)+{{\operatorname{val}}_{mc}}(C)-2\big),$$ where $CC({\Gamma})$ denotes the set of all connected components of ${\Gamma}$, and $${\operatorname{val}}_{colors}(C)=\text{ \# connected components of $(C\cap {\Gamma}_{colors})$}$$ for $\mathit{colors}\in\{my,yc,mc\}$.
Recall that a [*cycle*]{} in an undirected graph ${\Gamma}$ is a sequence $x_0e_0x_1e_1\dots x_ke_kx_0$ of pairwise different vertices $x_i$ and edges $e_i\in E({\Gamma},x_i,x_{i+1})$, $e_k\in E({\Gamma},x_{k},x_0)$. A cycle is called [*monochromatic*]{} if all its edges are of the same color (from the set {[*magenta, yellow, cyan*]{}}).
\[prop:cn\] For any ${\Gamma}\in{\mathcal C_n}$, we have: $\Sigma({\Gamma})\le n$, with the equality taking place if and only if all cycles of ${\Gamma}$ are monochromatic (with different cycles possibly having different colors).
Our first observation is:
Indeed, in that case ${\Gamma}$ has $n$ connected components which are singleton vertices and ${\Gamma}_{my}={\Gamma}_{yc}={\Gamma}_{mc}={\Gamma}$, so $\Sigma({\Gamma})=n\cdot(1+1+1-2)=n$.
We are going to prove that adding an edge to an arbitrary graph ${\Gamma}\in{\mathcal C_n}$ may only decrease $\Sigma$, and we identify all cases when the decrease does not happen.
Choose two vertices $p,q\in V({\Gamma})$ such that $|E({\Gamma},p,q)|<3$ and consider $${\Gamma}'={\Gamma}\cup e,$$ where $e$ is a new edge between $p$ and $q$ of a color that is not present in $E({\Gamma},p,q)$. Without loss of generality, we may assume that the color of edge $e$ is [*magenta*]{}.
Let $[p],[q]$ denote the connected components of ${\Gamma}$ containing $p,q$, respectively, and let $[p]_{colors}$, $[q]_{colors}$ be the connected components of ${\Gamma}_{colors}$ containing $p,q$, respectively, for $colors\in\{my,yc,mc\}$.
Also, let $[p]',[q]'$ denote the connected components of ${\Gamma}'$ containing $p,q$, respectively, and let $[p]'_{colors}$, $[q]'_{colors}$ be the connected components of ${\Gamma}'_{colors}$ containing $p,q$, respectively, for $colors\in\{my,yc,mc\}$.
Let also $${\operatorname{val}}'_{colors}(C')=\text{ \# connected components of $(C'\cap {\Gamma}'_{colors})$}$$ for $C'$ a connected component of ${\Gamma}'$ and $\mathit{colors}\in\{my,yc,mc\}$.
We look at several cases.
. Then $[p]_{colors}\ne[q]_{colors}$ for any $colors\in \{my,yc,mc\}$, and adding [*magenta*]{} edge $e$ to $p,q$ makes $[p]'=[q]'$, $[p]'_{my}=[q]'_{my}$ and $[p]'_{mc}=[q]'_{mc}$ while $[p]'_{yc}\ne[q]'_{yc}$ (since ${\Gamma}_{yc}$ by definition has only yellow and cyan edges, so if $[p]'_{yc}=[q]'_{yc}$ then $[p]_{yc}=[q]_{yc}$). Thus $$\Sigma({\Gamma})=\big({{\operatorname{val}}_{my}}[p]+{{\operatorname{val}}_{yc}}[p]+{{\operatorname{val}}_{mc}}[p] - 2\big) + \big({{\operatorname{val}}_{my}}[q]+{{\operatorname{val}}_{yc}}[q]+{{\operatorname{val}}_{mc}}[q] - 2\big) + \sum_{\substack{C\ne[p],[q]\\ C\in CC({\Gamma})}}(\dots),$$ and $$\Sigma({\Gamma}')=\big({{\operatorname{val}}_{my}}'[p]'+{{\operatorname{val}}_{yc}}'[p]'+{{\operatorname{val}}_{mc}}'[p]' - 2\big) + \sum_{\substack{C'\ne[p]'\\ C'\in CC({\Gamma}')}}(\dots),$$
Now, $$\begin{gathered}
{{\operatorname{val}}_{my}}'[p]'=\big(\text{\# components of $[p]'\cap {\Gamma}'_{my}$}\big) = \\
\big(\text{\# components of $[p]\cap {\Gamma}_{my}$}\big) + \big(\text{\# components of $[q]\cap {\Gamma}_{my}$}\big) - 1 = \\
{{\operatorname{val}}_{my}}[p]+{{\operatorname{val}}_{my}}[q]-1.\end{gathered}$$
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Similarly, $$\begin{gathered}
{{\operatorname{val}}_{mc}}'[p]'=\big(\text{\# components of $[p]'\cap {\Gamma}'_{mc}$}\big) = \\
\big(\text{\# components of $[p]\cap {\Gamma}_{mc}$}\big) + \big(\text{\# components of $[q]\cap {\Gamma}_{mc}$}\big) - 1 = \\
{{\operatorname{val}}_{mc}}[p]+{{\operatorname{val}}_{mc}}[q]-1.\end{gathered}$$
And $$\begin{gathered}
{{\operatorname{val}}_{yc}}'[p]'=\big(\text{\# components of $[p]'\cap {\Gamma}'_{yc}$}\big) = \big(\text{\# components of $[p]'\cap {\Gamma}_{yc}$}\big)=\\
\big(\text{\# components of $([p]\cup[q])\cap {\Gamma}_{yc}$}\big) = \big(\text{\# components of $([p]\cap {\Gamma}_{yc})\cup([q]\cap {\Gamma}_{yc})$}\big) = \\
{{\operatorname{val}}_{yc}}[p]+{{\operatorname{val}}_{yc}}[q].\end{gathered}$$
Comparing the contributions of the left- and right-hand sides to the function $\Sigma$, and noticing that the components $C\in CC({\Gamma})\setminus\{[p],[q]\}$ and $C'\in CC({\Gamma}')\setminus\{[p]'\}$ pairwise coincide, we conclude that the two sums are equal: $\Sigma({\Gamma}')=\Sigma({\Gamma})$ in Case I. This proves, in particular, that
Indeed, every forest can be obtained from an edgeless graph by adding edges which connect disjoint components.
. Thus $[p]'=[q]'$ and the four subcases are possible:
- $[p]_{my}\ne[q]_{my}$, $[p]_{mc}\ne[q]_{mc}$;
- $[p]_{my}=[q]_{my}$, $[p]_{mc}\ne[q]_{mc}$;
- $[p]_{my}\ne[q]_{my}$, $[p]_{mc}=[q]_{mc}$;
- $[p]_{my}=[q]_{my}$, $[p]_{mc}=[q]_{mc}$.
$[p]=[q]$, $[p]_{my}\ne[q]_{my}$, $[p]_{mc}\ne[q]_{mc}$.
These conditions mean that there exists a path $p{-}q$ in ${\Gamma}$, but not in ${\Gamma}_{my}$ or ${\Gamma}_{mc}$. This means that some edge of the path $p{-}q$ must be [*cyan*]{}, and some other [*yellow*]{}. Thus adding a [*magenta*]{} edge $e$ between $p$ and $q$ does create a non-monochromatic cycle.
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Computing the contribution of $[p]=[q]$ to $\Sigma({\Gamma})$ and of $[p]'=[q]'$ to $\Sigma({\Gamma}')$ we see that: $$\begin{aligned}
{{\operatorname{val}}_{my}}'[p]'=&{{\operatorname{val}}_{my}}[p]-1,\\
{{\operatorname{val}}_{mc}}'[p]'=&{{\operatorname{val}}_{mc}}[p]-1,\\
{{\operatorname{val}}_{yc}}'[p]'=&{{\operatorname{val}}_{yc}}[p],\end{aligned}$$ and $$\begin{aligned}
\Sigma({\Gamma})=&\sum_{[p]\in CC({\Gamma})}\big({{\operatorname{val}}_{my}}[p]+{{\operatorname{val}}_{yc}}[p]+{{\operatorname{val}}_{mc}}[p]-2\big)\\
\Sigma({\Gamma}')=&\sum_{[p]'\in CC({\Gamma}')}\big({{\operatorname{val}}_{my}}'[p]'+{{\operatorname{val}}_{yc}}'[p]'+{{\operatorname{val}}_{mc}}'[p]'-2\big).\end{aligned}$$ Hence, $\Sigma({\Gamma}')=\Sigma({\Gamma})-2$.
Subcases (2) and ($2'$) are symmetric, so we consider only subcase (2).
$[p]=[q]$, $[p]_{my}=[q]_{my}$, $[p]_{mc}\ne[q]_{mc}$.
This means that $p$ and $q$ are connected by a path $p{-}q$ in ${\Gamma}_{my}$, but not in ${\Gamma}_{mc}$. In particular, every such path $p{-}q$ must contain a [*yellow*]{} edge (otherwise all edges of $p{-}q$ would be [*magenta*]{} and and $p{-}q$ would lie in ${\Gamma}_{mc}$). Thus adding a new [*magenta*]{} edge $e$ between $p$ and $q$ does create a non-monochromatic cycle.
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Computing the contribution of $[p]=[q]$ to $\Sigma({\Gamma})$ and of $[p]'=[q]'$ to $\Sigma({\Gamma}')$ in this case, we observe that: $$\begin{aligned}
{{\operatorname{val}}_{my}}'[p]'=&{{\operatorname{val}}_{my}}[p],\\
{{\operatorname{val}}_{mc}}'[p]'=&{{\operatorname{val}}_{mc}}[p]-1,\\
{{\operatorname{val}}_{yc}}'[p]'=&{{\operatorname{val}}_{yc}}[p],\end{aligned}$$ hence $\Sigma({\Gamma}')=\Sigma({\Gamma})-1$.
$[p]=[q]$, $[p]_{my}=[q]_{my}$, $[p]_{mc}=[q]_{mc}$.
Every path $p{-}q$ consists entirely of [*magenta*]{} edges. In this case, adding a new [*magenta*]{} edge does not create a non-monochromatic cycle. And in this case, $$\begin{aligned}
{{\operatorname{val}}_{my}}'[p]'=&{{\operatorname{val}}_{my}}[p],\\
{{\operatorname{val}}_{mc}}'[p]'=&{{\operatorname{val}}_{mc}}[p],\\
{{\operatorname{val}}_{yc}}'[p]'=&{{\operatorname{val}}_{yc}}[p],\end{aligned}$$ so that $\Sigma({\Gamma}')=\Sigma({\Gamma})$.
There exists a path $p{-}q$ having a [*non-magenta*]{} edge, let it be [*yellow*]{}. Then there is a path $pPq$ in ${\Gamma}_{my}$ with at least one edge [*yellow*]{} and a path $pQq$ in ${\Gamma}_{mc}$. Let $p{-}p'$ be the maximal by inclusion common initial subpath of $pPq$ and $pQq$ consisting entirely of [*magenta*]{} edges. Similarly, let $q'{-}q$ be the maximal by inclusion common terminal subpath of $pPq$ and $pQq$ consisting entirely of [*magenta*]{} edges. Thus, we can denote $pPq=p{-}p'P'q'{-}q$ and $pQq=p{-}p'Q'q'{-}q$ for some subpaths $P',Q'$. We observe at once that the union of the paths $p'P'q'$ and $p'Q'q'$ is a non-monochromatic cycle already existing in ${\Gamma}$.
(-3,0)–(-1.5,0); (1.5,0)–(3,0); (-1.5,0) to \[out=45, in=180\] (0,1); (1.5,0) to \[out=135, in=0\] (0,1); (-1.5,0) to \[out=-45, in=180\] (0,-1); (0,-1) to \[out=0, in=-135\] (1.5,0);
(-3,0) circle (1.5pt); (-3,-0.35) node [$p$]{}; (-1.5,0) circle (1.5pt); (-1.6,-0.35) node [$p'$]{}; (1.5,0) circle (1.5pt); (1.6,-0.35) node [$q'$]{}; (3,0) circle (1.5pt); (3,-0.35) node [$q$]{}; (0,1) circle (1.5pt); (0.5,1.25) node [$P'$]{}; (0,-1) circle (1.5pt); (-0.5,-1.25) node [$Q'$]{};
We notice that, in this situation, adding a [*magenta*]{} edge $e$ to $p,q$ does not change $\Sigma({\Gamma})$ since, as before, $$\begin{aligned}
{{\operatorname{val}}_{my}}'[p]'=&{{\operatorname{val}}_{my}}[p],\\
{{\operatorname{val}}_{mc}}'[p]'=&{{\operatorname{val}}_{mc}}[p],\\
{{\operatorname{val}}_{yc}}'[p]'=&{{\operatorname{val}}_{yc}}[p],\end{aligned}$$ so that $\Sigma({\Gamma}')=\Sigma({\Gamma})$.
However, we can show by an inductive reasoning on the number of edges that in the situation (3b) we have $\Sigma({\Gamma})$ already less than $n$.
Indeed, we can construct ${\Gamma}$ from the edgeless graph on $n$ vertices ${\Gamma}_0$ by adding edges one at a time. We get a sequence of graphs: $${\Gamma}_0,{\Gamma}_1,{\Gamma}_2,\dots, {\Gamma}_i,\dots, {\Gamma}_m={\Gamma},$$ all belonging to ${\mathcal C_n}$ and such that for each $i$, ${\Gamma}_{i+1}={\Gamma}_{i}\cup e_{i+1}$ for a new edge $e_{i+1}$.
If ${\Gamma}_i$ is a forest, we showed above that $\Sigma({\Gamma}_i)=\Sigma({\Gamma}_0)=n$.
Suppose that we have already proved by induction on the number $k$ of edges that if ${\Gamma}_i$ has all cycles monochromatic and $|E({\Gamma}_i)|\le k$, then $\Sigma({\Gamma}_i)=n$. Consider a new edge $e_{i+1}$ such that ${\Gamma}_{i+1}={\Gamma}_i\cup e_{i+1}$.
If $e_{i+1}$ does not create a cycle, then $e_{i+1}$ joins two components of ${\Gamma}_i$, and ${\Gamma}_{i+1}$ is a forest. Hence, $\Sigma({\Gamma}_{i+1})=\Sigma({\Gamma}_i)$.
If $e_{i+1}$ creates a non-monochromatic cycle then we are in the subcase (1), (2) or ($2'$) above, and we see that in this case $\Sigma({\Gamma}_{i+1})<\Sigma({\Gamma}_i)$.
If $e_{i+1}$ creates a monochromatic cycle, i.e. $e_{i+1}$ joins vertices $p,q\in V({\Gamma})$, and there exists a path $p{-}q$ in ${\Gamma}_i$ of the same color as $e_{i+1}$, then we are in the situation (3a), and $\Sigma({\Gamma}_{i+1})=\Sigma({\Gamma}_i)$.
The last two cases are mutually exclusive, since, by the inductive hypothesis, ${\Gamma}_i$ does not have a non-monochromatic cycle.
This inductive reasoning, together with the consideration of the cases and all the subcases above, shows the following:
- adding an edge to a graph can only decrease the value of $\Sigma$;
- graphs ${\Gamma}$ with all cycles monochromatic have $\Sigma({\Gamma})=n$;
- creating a non-monochromatic cycle (when there were none) decreases the value of $\Sigma$ by 1 or 2.
This proves the Proposition.
Proof of Theorems \[thm:h2\] and \[thm:guzman4\].\[sec:guzman4\]
================================================================
We will prove Theorem \[thm:guzman4\] first and Theorem \[thm:h2\] at the end of the section. We start with translating Theorem \[thm:guzman4\] into a statement about the Dicks graphs.
\[prop:recast\] Let $F$ be a free group, and suppose there exist subgroups $H,K\le F$ such that ${\operatorname{rk}}(H)$, ${\operatorname{rk}}(K)\ge 2$, ${\operatorname{rk}}(H\vee K)={\operatorname{rk}}(H)+{\operatorname{rk}}(K)-i$, and ${\operatorname{rk}}(H\cap K)=\frac{i(i-1)}{2}+1$, for some $i\ge 3$. Then the corresponding Dicks graphs have the following properties:
1. One component of $\Omega_{abc}$ is isomorphic to the complete bipartite graph $K_{i,i-1}$, while all others are singleton vertices.
2. Every cycle of $\Omega=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{bc}\bigvee_{\Omega_{abc}}\Omega_{ac}$ lies entirely in one of the subgraphs $\Omega_{ab}$, $\Omega_{bc}$, or $\Omega_{ac}$ (with different cycles possibly lying in different subgraphs).
Denote $h={\operatorname{rr}}(H)$, $k={\operatorname{rr}}(K)$. Then ${\operatorname{rr}}(H\cap K)=\frac{i(i-1)}{2}$ and from Theorem \[thm:ranks\] we conclude that $\Omega_{abc}$ is a bipartite graph with $2h$ vertices in one part and $2k$ vertices in the other, and that $\Omega_{abc}$ has exactly $i(i-1)$ edges. The condition ${\operatorname{rk}}(H\vee K)={\operatorname{rk}}(H)+{\operatorname{rk}}(K)-i$ can be written in terms of the reduced ranks as ${\operatorname{rr}}(H\vee K)=h+k-(i-1)$. Also, from the discussion in subsection \[ssec:tp\] we know that ${\operatorname{rr}}({\mathcal T})\ge{\operatorname{rr}}(H\vee K)$. Thus, part (3) of Theorem \[thm:ranks\] gives us: $$\label{eq:ccO}
\text{\big(\# connected components of $\Omega_{abc}$\big)} \ge 2{\operatorname{rr}}({\mathcal T})\ge 2h+2k-2(i-1).$$ Let $\Omega_{abc}$ have $p$ single-vertex components in the part corresponding to vertices from $V({\Gamma}_H)$, $q$ single-vertex components in the part corresponding to vertices from $V({\Gamma}_K)$, and $\ell$ components $C_j$, ($j=1,\dots,\ell$) each of which has at least one edge. Then inequality implies: $$p+q+\ell\ge 2h+2k-2(i-1),$$ or $$\label{eq:8}
(2h-p)+(2k-q)-\ell \le 2(i-1).$$ Let $s_j$, $t_j$ denote the number of vertices of $C_j$ in each of the two parts of the bipartite graph $\Omega_{abc}$. Then we have the following equalities: $\sum_{j=1}^\ell s_j = 2h-p$, $\sum_{j=1}^\ell t_j = 2k-q$, and , together with the condition $2{\operatorname{rr}}(H\cap K)=i(i-1)$, becomes: $$\begin{aligned}
\sum_{j=1}^\ell (s_j + t_j - 1)&\le 2(i-1),\label{eq:9}\\
\sum_{j=1}^\ell \text{(\# edges of $C_j$)}&=i(i-1).\label{eq:10}\end{aligned}$$ Notice that the quantity $s_j+t_j-1$ is the number of edges in a spanning tree of the component $C_j$. The following Lemma shows that the total number of edges in the left-hand side of attains its maximum $i(i-1)$ under the constraint if and only if $\ell=1$ and $C_1=K_{i,i-1}$.
Let’s call a component of a bipartite graph ${\Gamma}$ having at least one edge, *nontrivial*.
\[lem:forest\] Let ${\Gamma}$ be a bipartite graph having $\ell$ nontrivial connected components $C_1,\dots,C_\ell$. Under the constraint: $$\sum_{j=1}^\ell (\text{\# edges of a spanning tree of $C_j$})\le 2m,$$ for some $m\ge 2$, the maximal possible number of edges for ${\Gamma}$ is achieved when $\ell=1$ and $C_1=K_{m,m+1}$.
(This is essentially the reasoning of Ivanov [@Iva (4.2)] with few more details provided.) Let’s show first that for any two components $C_1$, $C_2$ of ${\Gamma}$ with the number of edges in their spanning trees $m_1$, $m_2$ respectively, a single component $C_0$ with a spanning tree having $m_1+m_2$ edges can have a bigger total number of edges than $C_1$ and $C_2$ together. Indeed, let $s_1,t_1$ be numbers of vertices in the two parts of $C_1$ and $s_2,t_2$ be numbers of vertices in the two parts of $C_2$ (we refer to the bipartite structure of ${\Gamma}$ here). Then the maximal total number of edges in $C_1$ is $s_1t_1$ which is achieved when $C_1=K_{s_1,t_1}$ and, similarly, the maximal total number of edges in $C_2$ is $s_2t_2$. Now let’s consider a join $C_0$ of $C_1$ and $C_2$ at a pair of vertices either in one part or in the other. In the first case we will get a bipartite graph $C_0$ on $s_1+s_2-1$ vertices in one part and $t_1+t_2$ vertices in the other, and in the second case $C_0$ will have $s_1+s_2$ vertices in one part and $t_1+t_2-1$ ones in the other. In both cases, a spanning tree for $C_0$ will have $m_1+m_2$ edges, so that the total number of edges in all spanning trees remains invariant. Notice that the number of edges of the complete bipartite graph $K_{s_1+s_2-1,t_1+t_2}$ is $$(s_1+s_2-1)(t_1+t_2)=(s_1t_1+s_2t_2)+(s_1-1)t_2+(s_2-1)t_1 > s_1t_1+s_2t_2, \text{ if either $s_1\ge 2$ or $s_2\ge 2$.}$$ Similarly, the number of edges of the complete bipartite graph $K_{s_1+s_2, t_1+t_2-1}$ is $$(s_1+s_2)(t_1+t_2-1)=(s_1t_1+s_2t_2)+(t_1-1)s_2+(t_2-1)s_1 > s_1t_1+s_2t_2, \text{ if either $t_1\ge 2$ or $t_2\ge 2$.}$$ We conclude that joining two components $C_1$ and $C_2$ allows us to have a bigger total number of edges in $C_0$ than the sum of edges in $C_1$ and $C_2$, unless $s_1=s_2=t_1=t_2=1$, when joining two $K_{1,1}$’s produces a $K_{1,2}$ with the same number of edges. But since $2m>2$, we are going to deal with components having more than two edges, and we can proceed by induction, joining components together, and each time (after possibly joining two $K_{1,1}$’s the very first time) we increase the maximal possible number of edges while preserving the total number of edges of all spanning trees. Hence we prove by induction that the maximal number of edges is achieved when there is only one nontrivial component, and it should be a complete bipartite graph $K_{s,t}$ with $s+t-1=2m$. Clearly, the number $st$ of edges of $K_{s,t}$ is maximized under the constraint $s+t-1=2m$ if and only if $s$ and $t$ are closest to being equal. Since $s$ and $t$ have opposite parity, we conclude that $s=m$ and $t=m+1$, or vice versa.
Hence, part (1) is established. Notice also that the only solution to and implies the equality in , and this is equivalent to having two equalities in . Hence we may apply the last clause of part (3) of Theorem \[thm:ranks\], which proves part (2).
We will need the following graph theoretic construction.
Let $\Omega=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{bc}\bigvee_{\Omega_{abc}}\Omega_{ac}=\Omega_u\sqcup\Omega_v$, $\Omega_a$, $\Omega_b$, $\Omega_c$ be the the Dicks graphs defined for given core graphs ${\Gamma}_H$, ${\Gamma}_K$, ${\Gamma}_{H\cap K}$, and let $\tilde o$, $\tilde t$ be the embeddings of $\Omega_x\hookrightarrow \Omega_y$ ($x\in\{a,b,c\}$, $y\in\{u,v\}$) defined in Section \[sec:dicks\]. Let, as before, $A=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{ac}$, $B=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{bc}$, $C=\Omega_{ac}\bigvee_{\Omega_{abc}}\Omega_{bc}$. For any finite connected bipartite undirected graph $\Delta$ (with a fixed bipartite structure) define the following directed graph, which we will call the *subgraph isomorphism graph* for $\Delta$, and denote it ${\operatorname{SIG}}(\Delta)$:
Vertices of ${\operatorname{SIG}}(\Delta)$: $$V\big(\!{\operatorname{SIG}}(\Delta)\big)=\{\text{subgraphs } {\Gamma}\subset \Omega, \text{ such that } {\Gamma}\subset A, \text{ or } {\Gamma}\subset B, \text{ or } {\Gamma}\subset C, \text{ and } {\Gamma}\cong\Delta\},$$ where $\cong$ is the isomorphism of bipartite graphs, i.e. it is required to send parts of bipartite structure of ${\Gamma}$ (induced by that of $\Omega$) into the corresponding parts of $\Delta$.
Since $\Omega=\Omega_u\sqcup\Omega_v$, we may define the set of directed edges of ${\operatorname{SIG}}(\Delta)$ by specifying the stars of the ‘source’ vertices ${\Gamma}\subset\Omega_u$. Let ${\Gamma}$ be a vertex of ${\operatorname{SIG}}(\Delta)$ such that ${\Gamma}$, viewed as a subgraph of $\Omega$, lies in $A\cap\Omega_u$. Then ${\Gamma}$ lies in the image under $\tilde o$ of some connected component $Q\subset \Omega_a$. Hence ${\Gamma}'=\tilde t\circ (\tilde o|_Q)^{-1}({\Gamma})$ is another vertex of ${\operatorname{SIG}}(\Delta)$, with ${\Gamma}'\subset A\cap\Omega_v$. We connect ${\Gamma}$ and ${\Gamma}'$ in ${\operatorname{SIG}}(\Delta)$ with a directed edge labeled $a$ with the origin ${\Gamma}$ and the terminus ${\Gamma}'$. Similarly, if ${\Gamma}\subset B\cap\Omega_u$, the star of ${\Gamma}$ in ${\operatorname{SIG}}(\Delta)$ will have an outgoing $b$–edge, and if ${\Gamma}\subset C\cap\Omega_u$, the star of ${\Gamma}$ will have an outgoing $c$–edge, with their termini defined correspondingly. Thus, a vertex ${\Gamma}$ of ${\operatorname{SIG}}(\Delta)$ may have valence $1$, $2$, or $3$, if ${\Gamma}$ lies in only one of the subsets $A$, $B$, $C$, or in only two of them, or in all three, respectively. Clearly, the same is true for vertices ${\Gamma}'\subset\Omega_v$.
In other words, edges of ${\operatorname{SIG}}(\Delta)$ are in $1{-}1$ correspondence with the subgraphs of $\Omega_a$, $\Omega_b$, $\Omega_c$ which are isomorphic to $\Delta$, with the restrictions of $\tilde o$, $\tilde t$ as the origin and the terminus maps.
Notice also that ${\operatorname{SIG}}(\Delta)$ admits a natural immersion into the topological pushout ${\mathcal T}$, since the vertices and edges of ${\operatorname{SIG}}(\Delta)$ are naturally mapped into the vertices and edges of ${\mathcal T}$, and this mapping is injective on stars.
Figure \[fig:sig\] shows the graph ${\operatorname{SIG}}(K_{1,1})$ for the Dicks graphs in Figure \[fig:toppush\].
(0,0) circle (2.5pt); (1,0) circle (2.5pt); (0,0)–(1,0); (0,-0.3) node [$8$]{}; (1,-0.3) node [$2$]{};
(0,0) circle (2.5pt); (1,0) circle (2.5pt); (0,0)–(1,0); (0,-0.3) node [$7$]{}; (1,-0.3) node [$3$]{};
(0,0) circle (2.5pt); (1,0) circle (2.5pt); (0,0)–(1,0); (0,-0.3) node [$7$]{}; (1,-0.3) node [$1$]{};
(0,0) circle (2.5pt); (1,0) circle (2.5pt); (0,0)–(1,0); (0,-0.3) node [$6$]{}; (1,-0.3) node [$2$]{};
(0,0) circle (2.5pt); (1,0) circle (2.5pt); (0,0)–(1,0); (0,-0.3) node [$6$]{}; (1,-0.3) node [$4$]{};
(0,0) circle (2.5pt); (1,0) circle (2.5pt); (0,0)–(1,0); (0,-0.3) node [$5$]{}; (1,-0.3) node [$3$]{};
(-2.75,2.75) to (-4,0.75); (-2.75,-3) to (-4,-0.75); (5,-0.75) to (3.75,-3); (5,0.75) to (3.75,2.75); (-0.85,3.5) to (1.85,3.5); (-0.85,-3.5) to (1.85,-3.5);
(0,0) circle (6pt); (2,0) circle (6pt); (1,3) circle (6pt); (0,0)–(1,3); (2,0)–(1,3); (-0.4,-0.9) node [(1,2)]{}; (2.4,-0.9) node [(3,4)]{}; (1,3.9) node [(7,6)]{};
(0,0) circle (6pt); (2,0) circle (6pt); (1,3) circle (6pt); (0,0)–(1,3); (2,0)–(1,3); (-0.4,-0.9) node [(1,2)]{}; (2.4,-0.9) node [(3,4)]{}; (1,3.9) node [(7,6)]{};
(1,0) circle (6pt); (0,3) circle (6pt); (2,3) circle (6pt); (1,0)–(0,3); (2,3)–(1,0); (1,-0.7) node [(3,2)]{}; (-0.4,3.8) node [(5,8)]{}; (2.4,3.8) node [(7,6)]{};
(1,0) circle (6pt); (0,3) circle (6pt); (2,3) circle (6pt); (1,0)–(0,3); (2,3)–(1,0); (1,-0.7) node [(3,2)]{}; (-0.4,3.8) node [(5,8)]{}; (2.4,3.8) node [(7,6)]{};
(0,0) circle (6pt); (0,3) circle (6pt); (0,0)–(0,3); (0,-0.7) node [(1,2)]{}; (0,3.8) node [(7,8)]{};
(2,0) circle (6pt); (2,3) circle (6pt); (2,0)–(2,3); (2,-0.7) node [(3,4)]{}; (2,3.8) node [(5,6)]{};
An undirected graph ${\Gamma}$ is called *$k$–connected*, for $k\in{\mathbb N}$, if $\#V({\Gamma})>k$ and ${\Gamma}\setminus Y$ is connected for every $Y\subset V({\Gamma})$ with $\#Y<k$. We will make use of the following global version of Menger’s theorem, see [@Dies Th.3.3.6(i)].
A graph is $k$–connected if and only if it contains $k$ independent paths between any two vertices.
Recall that a *path* in an undirected graph is a sequence $x_1e_1x_2e_2\dots x_ke_kx_{k+1}$ of pairwise distinct vertices $x_i$ and undirected edges $e_i$ such that for all $i$, vertices $x_i,x_{i+1}$ are incident to edge $e_i$. Two paths from $x$ to $y$ are *independent* if they share no other vertices except $x$ and $y$.
Finally, we are ready to prove Theorem \[thm:guzman4\].
Construct all the Dicks graphs $\Omega_\bullet$ for the core graphs ${\Gamma}_H$, ${\Gamma}_K$, ${\Gamma}_{H\cap K}$. Then graphs $\Omega$ and $\Omega_{abc}$ satisfy conditions (1) and (2) of Proposition \[prop:recast\]. In particular, one connected component of $\Omega_{abc}$ is isomorphic to the complete bipartite graph $K_{i,i-1}$, and all other components are singleton vertices. Clearly, graph $K_{i,i-1}$ is $2$–connected, if $i\ge 3$. For the rest of the proof, let $\Delta$ denote $K_{i,i-1}$. (The proof will be valid for an arbitrary $2$–connected graph $\Delta$.)
Consider the graph ${\operatorname{SIG}}(\Delta)$ built for the Dicks graphs constructed above. It has a single vertex of valence $3$, since, by part (1) of Proposition \[prop:recast\], only one subgraph isomorphic to $\Delta$ exists in $\Omega_{abc}=A\cap B\cap C$. We claim that all other vertices of ${\operatorname{SIG}}(\Delta)$ have valence $2$.
Suppose the contrary, that there exists a vertex ${\Gamma}$ of ${\operatorname{SIG}}(\Delta)$ which has valence $1$. This means that subgraph ${\Gamma}\subset\Omega$ lies in only one of subgraphs $A$, $B$, $C$ of $\Omega$, let’s say ${\Gamma}\subset A=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{ac}$. Since ${\Gamma}\not\subset B$ and ${\Gamma}\not\subset C$, we conclude that ${\Gamma}\cap (\Omega_{ab}\setminus\Omega_{abc})\ne\varnothing$ and ${\Gamma}\cap (\Omega_{ac}\setminus\Omega_{abc})\ne\varnothing$.
If there exist vertices $p$, $q$ such that $p\in V({\Gamma})\cap (\Omega_{ab}\setminus\Omega_{abc})$ and $q\in V({\Gamma})\cap (\Omega_{ac}\setminus\Omega_{abc})$, then by Menger’s theorem above, there exist two independent paths from $p$ to $q$, and their union is a cycle which does not lie entirely in either of the subgraphs $\Omega_{ab}$, $\Omega_{ac}$, $\Omega_{bc}$, thus contradicting condition (2) of Proposition \[prop:recast\].
Assume now that $V({\Gamma})\cap (\Omega_{ab}\setminus\Omega_{abc})=\varnothing$ but there exists an edge $e\in E({\Gamma})\cap (\Omega_{ab}\setminus\Omega_{abc})$. Let $s,t$ denote the vertices incident to the edge $e$. The last two conditions imply that $s,t\in V(\Omega_{abc})$. Form a new graph ${\Gamma}'$ by subdividing the edge $e$ into a sequence $e_1pe_2$ of two edges $e_1, e_2$ and a new vertex $p$ such that the vertices $p$ and $s$ are incident to $e_1$ and the vertices $p$ and $t$ are incident to $e_2$. We claim that the graph ${\Gamma}'$ obtained this way is also $2$–connected. First, observe that $s\ne t$ since $\Omega$ has no loops (being bipartite). If we remove the vertex $p$ from ${\Gamma}'$ the result is the same as if we remove the edge $e$ from ${\Gamma}$. Since ${\Gamma}$ is $2$–connected, Menger’s theorem guarantees the existence of another path $s{-}t$ in ${\Gamma}$ which doesn’t contain the edge $e$. Hence, ${\Gamma}\setminus e$ is still connected, and so is ${\Gamma}'\setminus p$. Also, the removal of any other vertex $p'\ne p$ from ${\Gamma}'$ doesn’t make the resulting graph disconnected. Indeed, for any two vertices $p_1$, $p_2$ of ${\Gamma}$ there exist at least two independent paths between them, by Menger’s theorem, and only one of them may contain $p'$. This means that $p_1$, $p_2$ are still connected via the other path in ${\Gamma}\setminus p'$ and hence in ${\Gamma}'\setminus p'$. Also, the vertex $p$ is connected to any other vertex of ${\Gamma}'\setminus p'$ since $s\ne t$.
If $V({\Gamma})\cap (\Omega_{ac}\setminus\Omega_{abc})\ne\varnothing$, pick a vertex $q$ in that subset. Otherwise, as before, there is some edge $e'\in E({\Gamma})\cap (\Omega_{ac}\setminus\Omega_{abc})$, with the endpoints $s'\ne t'\in V(\Omega_{abc})$, and we perform the above operation of subdivision of edge again, applied to $e'$, thus obtaining another $2$–connected graph with $e'$ changed into $e'_1qe'_2$. (For simplicity, we will still denote this graph by ${\Gamma}'$.)
Applying Menger’s theorem again, we see that there exist two independent paths from $p$ to $q$ in ${\Gamma}'$. Their union is a cycle in ${\Gamma}'$, which has subpaths $se_1pe_2t$ and $s'e'_1qe'_2t'$, see Figure \[fig:cycle\]. Going back to the original graph ${\Gamma}$ and replacing these subpaths with the subpaths $s\,e\,t$ and $s'e't'$, respectively, (the latter only if we performed the subdivision of edges twice), we get a cycle in the original graph ${\Gamma}$ which does not lie entirely in either of the subgraphs $\Omega_{ab}$, $\Omega_{ac}$, $\Omega_{bc}$, thus contradicting condition (2) of Proposition \[prop:recast\].
(0,0) arc (0:120:2) arc (180:360:2) arc (60:180:2); (-0.25,1.25) .. controls (-1,2) and (-2,1) .. (-0.75,0.25); (-1.25,1.3) circle (1.5pt); (-0.75,1.65) node [$e_1$]{}; (-1.3,0.5) node [$e_2$]{}; (-1.25,1.3) circle (1.5pt); (-1.1,1.2) node [$p$]{};
(0.25,1.25) .. controls (1,2) and (2,1) .. (0.75,0.25); (1.25,1.3) circle (1.5pt); (0.75,1.7) node [$e'_1$]{}; (1.3,0.5) node [$e'_2$]{}; (1.25,1.3) circle (1.5pt); (1.1,1.2) node [$q$]{};
(-0.25,1.25) circle (1.5pt); (-0.3,1.1) node [$s$]{}; (-0.75,0.25) circle (1.5pt); (0.3,1.1) node [$s'$]{}; (0.25,1.25) circle (1.5pt); (-0.7,0.45) node [$t$]{}; (0.75,0.25) circle (1.5pt); (0.7,0.45) node [$t'$]{};
(-0.25,1.25) to \[out=-30, in=210\] (0.25,1.25); (-0.75,0.25) to \[out=-30, in=210\] (0.75,0.25);
(-1.7,1.5) node [$\Omega_{ab}$]{}; (1.7,1.5) node [$\Omega_{ac}$]{}; (0,0.65) node [$\Omega_{abc}$]{};
Therefore, all vertices of the graph ${\operatorname{SIG}}(\Delta)$ have valence $2$, except for a single vertex of valence $3$. But this is impossible, since in any graph the number of vertices of odd valence must be even (otherwise the count for the number of edges, $\#E^+({\Gamma})=\frac12\sum_{v\in V({\Gamma})}{\operatorname{val}}(v)$, would be a half-integer).
The obtained contradiction proves that the values of the ranks of $H$, $K$, $H\vee K$, and $H\cap K$ in Theorem \[thm:guzman4\] are non-realizable.
\[ex:k23\] Interestingly, condition (1) alone in Proposition \[prop:recast\] does not make the Dicks graphs non-realizable, as the following example shows. Let subgroups $H,K\le \theta(F)\le F(a,b,c)$ be given by $H=\theta\big(\langle b,a^3,ab^{-1}a,ab^2a^{-1}\rangle\big)=\langle cb^{-1}, (ca^{-1})^3, ca^{-1}ba^{-1},ca^{-1}(cb^{-1})^2ac^{-1}\rangle$, $K=\theta\big(\langle a^{-1}b,b^{-2}ab^2\rangle\big)=
\langle ab^{-1},bc^{-1}ba^{-1}cb^{-1}cb^{-1}\rangle$. Then $\Omega_{abc}=K_{2,3}\cup\{\text{three vertices}\}$, see Figure \[fig:k23\]. The graph SIG($K_{2,3}$) has one vertex of valence $3$ and three vertices of valence $1$. Note also that $\text{(\# connected components of $\Omega_{abc}$)}=4>2=2{\operatorname{rr}}({\mathcal T})$, cf. part (3) of Theorem \[thm:ranks\].
(1,-0.5) to \[out=90, in=90,looseness=0.75\] (2,-0.5); (1,-0.5) to \[out=-90, in=-90,looseness=0.75\] (2,-0.5); (1,-0.5) to \[out=90, in=90,looseness=0.6\] (4,-0.5); (3,-0.5) to \[out=90, in=90,looseness=0.6\] (6,-0.5); (3,-0.5) to \[out=180, in=0,looseness=1\] (2,-0.5); (3,-0.5) to \[out=0, in=180,looseness=1\] (4,-0.5); (5,-0.5) to \[out=180, in=0,looseness=1\] (4,-0.5); (5,-0.5) to \[out=90, in=90,looseness=0.75\] (6,-0.5); (5,-0.5) to \[out=-90, in=-90,looseness=0.75\] (6,-0.5); (1,-0.9) node [$1$]{}; (2,-0.9) node [$2$]{}; (3,-0.9) node [$3$]{}; (4,-0.9) node [$4$]{}; (5,-0.9) node [$5$]{}; (6,-0.9) node [$6$]{};
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in [1,2,3,4,5,6]{}
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in [1,2,3,4,5]{} [ (,) circle (1.5pt); ]{}
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(0,0) circle (3pt); (1,0) circle (3pt); (2,0) circle (3pt); (0,2) circle (3pt); (1,2) circle (3pt); (2,2) circle (3pt); (0,0)–(0,2)–(1,0)–(1,2)–(0,0); (0,0)–(2,2)–(2,0)–(0,2); (1,0)–(2,2); (1,2)–(2,0); (-1,1) node [$\Omega_u$:]{}; (0,-0.5) node [7]{}; (1,-0.5) node [9]{}; (2,-0.5) node [11]{}; (0,2.5) node [1]{}; (1,2.5) node [3]{}; (2,2.5) node [5]{};
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(0,3.464) arc (60:-60:2) arc (240:60:2) arc (360:180:2) arc (120:60:2);
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(-3,2) node [${\mathcal T}:$]{};
The case $i=3$ of Theorem \[thm:guzman4\] resolves the remaining open case $m=4$ of Guzman’s “Group-Theoretic Conjecture” in the affirmative:
Let $F$ be a free group. If two subgroups $H,K\le F$ both have ranks equal to $4$, and ${{\operatorname{rk}}(H\cap K)\ge 4}$, then ${\operatorname{rk}}(H\vee K)\le 4$.
Indeed, looking at the locus of known realizable values and the region of proved non-realizable values for ${\operatorname{rk}}(H)={\operatorname{rk}}(K)=4$, see Figure \[fig:guzman4\], we conclude that the GTC for $m=4$ holds true if and only if the tuple $\big(\!{\operatorname{rk}}(H\vee K), {\operatorname{rk}}(H\cap K)\big)=(5,4)$ is not realizable. But this is exactly what Theorem \[thm:guzman4\] says for ${\operatorname{rk}}(H)={\operatorname{rk}}(K)=4$ and $i=3$.
Invoking the implication theorem from [@Gu], we obtain a proof of the “Geometric Conjecture” for $k=6$:
Let $M$ be a closed, orientable, hyperbolic $3$–manifold. If $\pi_1(M)$ is $6$–free then there exists a point $P$ in $M$ such that the set of all elements of $\pi_1(M,P)$ represented by loops of length less than $\log(11)$ is contained in a free subgroup of $\pi_1(M)$ of rank at most $3$.
[(3.0cm,0.75cm) node [${\operatorname{rk}}(H\cap K)$]{}; (-0.3cm,-1.6cm) node \[rotate=90\] [${\operatorname{rk}}H\vee K$]{}; ]{}
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(13pt,-91pt) – ++(143pt,0) – ++ (0,91pt) – ++(-143pt,0) – ++(0,-91pt); (65pt,-91pt) – ++(0,52pt) – ++(91pt,0);
Now we are going to prove Theorem \[thm:h2\].
Let $k$, $v$ and $c$ be as in the statement of Theorem \[thm:h2\]. The existence of subgroups $H$, $K\le F$ with ${\operatorname{rk}}(H)=2$, ${\operatorname{rk}}(K)=k$, ${\operatorname{rk}}(H\vee K)=v$, and ${\operatorname{rk}}(H\cap K)=c$, such that $c\le k+2-v$, follows from Theorem \[thm:real\]. Let’s prove that if $c>k+2-v$, such subgroups do not exist.
Suppose the contrary, that subgroups $H$, $K$ with ${\operatorname{rk}}(H)=2$, ${\operatorname{rk}}(K)=k$, ${\operatorname{rk}}(H\vee K)=v$, ${\operatorname{rk}}(H\cap K)=c$, and $c>k+2-v$ do exist. Denote $i=k+2-v$ and let $d\ge1$ be such that $c=i+d$. From Theorem \[thm:ranks\] we see that $\Omega_{abc}$ is a bipartite graph with $2$ vertices in the $V({\Gamma}_H)$–part of $\Omega_{abc}$ and $2(k-1)$ vertices in the $V({\Gamma}_K)$–part (note that now $k$ denotes ${\operatorname{rk}}(K)$ so that ${\operatorname{rr}}(K)=k-1$), and that $\Omega_{abc}$ has $2(c-1)=2(i-1)+2d$ edges. Denote nontrivial components of $\Omega_{abc}$ as $C_1$, $C_2$, …, $C_\ell$ and call the edges of a spanning tree of $C_j$ the *spanning edges* of $C_j$. Arguing as in the proof of Proposition \[prop:recast\], we get: $$\label{eq:11}
\sum_{j=1}^\ell \big(\text{\# spanning edges of $C_j$}\big)\le 2(i-1)$$ and $$\label{eq:12}
\sum_{j=1}^\ell \big(\text{\# edges of $C_j$}\big)=2(i-1)+2d.$$
Denote the two vertices in the $V({\Gamma}_H)$–part of $\Omega_{abc}$ as $z$ and $w$. We claim that $z$ and $w$ belong to the same nontrivial component $C_j$. Indeed, if $z$ and $w$ belong to different components, then these components are trees, and, in particular, the count in is equal to that of , which contradicts the condition $d\ge 1$. Hence, the component $C_j$ containing $z$ and $w$ is the only nontrivial component of $\Omega_{abc}$, i.e. $j=\ell=1$.
Denote $$\begin{aligned}
s&=\big(\text{\# spanning edges of $C_1$}\big),\label{eq:s}\\
m&=\big(\text{\# vertices of valence $2$ in the $V({\Gamma}_K)$--part of $\Omega_{abc}$}\big),\label{eq:m}\\
q&=\big(\text{\# singleton vertices in the $V({\Gamma}_K)$--part of $\Omega_{abc}$}\big),\label{eq:q}\end{aligned}$$ see Figure \[fig:Omega2\].
(0,0) circle (2pt); (0.5,0) circle (2pt); (1,0) circle (2pt); (1.5,0) circle (2pt); (2,0) circle (2pt); (2.5,0) circle (2pt); (3,0) circle (2pt); (3.5,0) circle (2pt); (4,0) circle (2pt); (4.5,0) circle (2pt); (5,0) circle (2pt); (5.5,0) circle (2pt); (0.5,1) circle (2pt) node \[above=5pt\] [$z$]{}; (2,1) circle (2pt) node \[above=5pt\] [$w$]{}; (0,0)–(0.5,1)–(0.5,0)–(2,1)–(1,0)–(0.5,1)–(1.5,0)–(2,1)–(2,0)–(0.5,1); (2.5,0)–(2,1)–(3,0); (2.1,-0.15) – (0.4,-0.15); (1.25,-0.65) node [$m$]{}; (5.6,-0.15) – (3.4,-0.15); (4.5,-0.65) node [$q$]{}; (-1,0.5) node [$\Omega_{abc}:$]{};
In particular, we see that $m\ge 2d+1$ (indeed, $m-1$ is the rank of $C_1$, i.e. the number of edges of $C_1$ minus the number of spanning edges of $C_1$, hence is at least $2d$), and that $\Omega_{abc}$ contains a subgraph $\Delta$ isomorphic to the complete bipartite graph $K_{2,m}$.
Consider the graph ${\operatorname{SIG}}(K_{2,m})$, as defined after the proof of Proposition \[prop:recast\]. Arguing as in the proof of Theorem \[thm:guzman4\], we observe that ${\operatorname{SIG}}(K_{2,m})$ has a unique vertex of valence $3$, and hence must have another vertex of odd valence, that is of valence $1$. Let’s call the subgraph of $\Omega$, corresponding to this valence $1$ vertex, $\Delta'$. Without loss of generality we can assume that $\Delta'$ lies in $A=\Omega_{ab}\bigvee_{\Omega_{abc}}\Omega_{ac}$, and hence that $\Delta'\cap \big(\Omega_{ab}\setminus\Omega_{abc}\big)\ne\varnothing$ and $\Delta'\cap \big(\Omega_{ac}\setminus\Omega_{abc}\big)\ne\varnothing$. We conclude that $\Delta'\cap\Omega_{abc}\ne\varnothing$, and, since the isomorphisms involved in the definition of ${\operatorname{SIG}}$ preserve the bipartite structure, the image of $m$ vertices of $K_{2,m}$ in $\Delta'$ (call this subset $M$) is a subset of the $q$ singleton vertices of the $V({\Gamma}_K)$–part of $\Omega_{abc}$. In particular, $m\le q$, see Figure \[fig:val1\].
(0,0) arc (0:120:2) arc (180:360:2) arc (60:180:2);
(-0.75,0.25) circle (2pt); (-0.25,0.25) circle (2pt); (0.25,0.25) circle (2pt); (0.75,0.25) circle (2pt); (-1,1.3) circle (2pt); (1,1.3) circle (2pt); (-1,1.3)–(-0.75,0.25)–(1,1.3); (-1,1.3)–(-0.25,0.25)–(1,1.3); (-1,1.3)–(0.25,0.25)–(1,1.3); (-1,1.3)–(0.75,0.25)–(1,1.3);
(-1.6,1.3) node [$\scriptstyle\Omega_{ab}$]{}; (1.6,1.3) node [$\scriptstyle\Omega_{ac}$]{}; (0,1.35) node [$\scriptstyle\Omega_{abc}$]{}; (0.8,0.3) – (-0.8,0.3); (0,-0.05) node [$m\le q$]{};
To estimate ${\operatorname{rr}}({\mathcal T})$ we recall equation from Section \[sec:dicks\]: $$2{\operatorname{rr}}({\mathcal T})=\sum_{v \in V({\mathcal T})} \big(\!{\operatorname{val}}(v)-2\big).\tag{5}$$ From Proposition \[prop:toppush\] we know that the vertices of ${\mathcal T}$ are connected components of $\Omega$. Vertices of ${\mathcal T}$ of valence $\ge 3$ correspond to certain subgraphs of $\Omega_{abc}$, with the exact relation between components of $\Omega_{abc}$ and the valence of the corresponding vertex of ${\mathcal T}$ given by the component connectivity graph ${\Gamma}$, as constructed in the proof of Theorem \[thm:ranks\]. The vertices of ${\Gamma}$ are connected components of $\Omega_{abc}$. Two vertices $p$, $q$ of ${\Gamma}$ may be connected by up to three undirected edges (colored magenta, yellow, and cyan) in ${\Gamma}$, if there exists an $\Omega_{abc}$-avoidant path connecting components $p$ and $q$ which lies entirely in $\Omega_{ab}$, $\Omega_{ac}$, or $\Omega_{bc}$, respectively. The right-hand sum of equation equals the value of the function $\Sigma$ on ${\Gamma}$, as defined in of Section \[sec:cn\]. The main conclusion of Proposition \[prop:cn\] is that this value, and hence, the value $2{\operatorname{rr}}({\mathcal T})$, is bounded above by the number of vertices of ${\Gamma}$.
The component connectivity graph ${\Gamma}$ in the situation we are considering will have one vertex for the only nontrivial component $C_1$ of $\Omega_{abc}$ and $q$ vertices for the remaining singleton components. However, if we identify the $m$ vertices of $M$ in $\Delta'$ into a single vertex, thus forming a new graph ${\Gamma}'$, we observe that the value of function $\Sigma$ on ${\Gamma}$ and ${\Gamma}'$ is the same. Indeed, these $m$ vertices belong to the same component in each of the subgraphs $A$, $B$, $C$ of $\Omega$, and hence their contribution to the number of $a$–edges, $b$–edges, and $c$–edges of ${\mathcal T}$ is the same as if they were a single vertex of $\Omega_{abc}$. Hence, we can use the graph ${\Gamma}'$ for computing the quantity $2{\operatorname{rr}}({\mathcal T})$, and we conclude that the latter is bounded above by $n=\#V({\Gamma}')$. We now estimate $n$.
We have: $n=1+1+(q-m)$, where the ones correspond to the component $C_1$ and the subset $M$, which is one vertex of ${\Gamma}'$. Since a spanning tree of $C_1$ contains only one vertex of valence $2$ in the $V({\Gamma}_K)$–part (for otherwise there would be a cycle in it), all other vertices of the spanning tree that lie in the $V({\Gamma}_K)$–part have valence $1$. Now from and above we deduce that: $$q=2(k-1)-(s-1).$$ Also from and it follows that: $$m=\big(\text{\# edges of $C_1$}\big)-\big(\text{\# spanning edges of $C_1$}\big) + 1=2(i-1)+2d-s+1.$$ Hence, we get: $$n=2+q-m=2+2(k-1)-(s-1)-2(i-1)-2d+s-1=2(k-i-d+1),$$ and we deduce from formula above (and the discussion following it) that $${\operatorname{rr}}({\mathcal T})\le k-i-d+1.$$ Now recall that $v={\operatorname{rk}}(H\vee K)=k+2-i$ and that ${\operatorname{rk}}(H\vee K)\le {\operatorname{rk}}({\mathcal T})={\operatorname{rr}}({\mathcal T})+1$. We get: $$v={\operatorname{rk}}(H\vee K)\le {\operatorname{rk}}({\mathcal T})=(k+2-i)-d = v-d,$$ which yields a contradiction with the condition that $d\ge1$. Since $d$ was defined as $d=c-(k+2-v)$, this proves that $c+v\le k+2$.
[99]{} Ian Agol, Marc Culler, Peter B. Shalen, Singular surfaces, mod $2$ homology, and hyperbolic volume. I. *Trans. Amer. Math. Soc.* 362 (2010), no. 7, 3463–3498. Frédérique Bassino, Cyril Nicaud, Pascal Weil, Random generation of finitely generated subgroups of a free group. *Internat. J. Algebra Comput.* 18 (2008), no. 2, 375–405. Gilbert Baumslag, Alexei G. Myasnikov, Vladimir Shpilrain, Open problems in combinatorial and geometric group theory. <http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html> Oleg Bogopolski, *Introduction to group theory.* (EMS Textbooks in Mathematics.) European Mathematical Society (EMS), Zürich, 2008. x+177 pp. Marc Culler, Peter B. Shalen, $4$–free groups and hyperbolic geometry. *J. Topol.* 5 (2012), no. 1, 81–136. Warren Dicks, Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture. *Invent. Math.* 117 (1994), 373–389. Warren Dicks, Joel Friedman’s proof of the strengthened Hanna Neumann conjecture, pp. 91–101 in [@Fr]. Warren Dicks, Simplified Mineyev. Preprint at <http://mat.uab.cat/~dicks/SimplifiedMineyev.pdf> Reinhard Diestel, *Graph Theory*, Fifth edition. Graduate Texts in Mathematics, 173. Springer, Berlin, 2017. xviii+428 pp. Matthew Fayers, The [genyoungtabtikz]{} package, version 1.14, 2016-10-05. <http://www.maths.qmul.ac.uk/~mf/genyoungtabtikz.html>. Joel Friedman, Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks. *Mem. Amer. Math. Soc.* 233 (2015), no. 1100, xii+106 pp. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.0, 2018, <https://www.gap-system.org>. Rosemary K. Guzman, Hyperbolic $3$–manifolds with $k$–free fundamental group. *Topology Appl.* 173 (2014), 142–156. Rosemary K. Guzman, Peter B. Shalen, The geometry of $k$–free hyperbolic $3$–manifolds. *J. Topol. Anal.* <https://doi.org/10.1142/S1793525320500016>. Joshua E. Hunt, The Hanna Neumann Conjecture and the rank of the join. [arXiv:1509.04449](https://arxiv.org/abs/1509.04449) Wilfried Imrich, Thomas Müller, On Howson’s theorem. *Arch. Math. (Basel)* 62 (1994), no. 3, 193–198. Sergei V. Ivanov, On a conjecture of Imrich and Müller. *J. Group Theory* 20 (2017), no. 4, 823–828. Sergei V. Ivanov, On joins and intersections of subgroups in free groups. *J. Comb. Algebra* 2 (2018), 1–18. Andrei Jaikin-Zapirain, Approximation by subgroups of finite index and the Hanna Neumann conjecture. *Duke Math. J.* 166 (2017), 1955–1987. Ilya Kapovich, Alexei Myasnikov, Stallings foldings and subgroups of free groups. *J. Algebra* 248 (2002), no. 2, 608–668. Richard P. Kent IV, Achievable ranks of intersections of finitely generated free groups. *Internat. J. Algebra Comput.* 15 (2005), no. 2, 339–341. Richard P. Kent IV, Intersections and joins of free groups. *Algebr. Geom. Topol.* 9 (2009), no. 1, 305–325. Larsen Louder, D.B. McReynolds, Graphs of subgroups of free groups. *Algebr. Geom. Topol.* 9 (1) (2009), pp. 327–355. Igor Mineyev, Groups, graphs, and the Hanna Neumann conjecture. *J. Topol. Anal.* 4 (2012), no. 1, 1–12. Christian Sievers, FGA (Free Group Algorithms) – a GAP package, Version 1.4.0, 2018, <http://www.icm.tu-bs.de/ag_algebra/software/FGA>. John R. Stallings, Topology of finite graphs. *Invent. Math.* 71 (1983), no. 3, 551–565. Till Tantau, The TikZ and PGF Packages, manual for version 3.1, <http://sourceforge.net/projects/pgf>, 2019-01-05.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present [98]{} spectroscopic binary orbits resulting from our ongoing radial-velocity survey of the old (7 Gyr) open cluster NGC 188. All but [13]{} are high-probability cluster members based on both radial-velocity and proper-motion membership analyses. [15]{} of these member binaries are double lined. Our stellar sample spans a magnitude range of [10.8$\leq$V$\leq$16.5]{} (1.14-0.92 [M$_{\odot}$]{}) and extends spatially to 17 pc ($\sim$13 core radii). All of our binary orbits have periods ranging from a few days to on the order of 10$^3$ days, and thus are hard binaries that dynamically power the cluster. For each binary, we present the orbital solutions and place constraints on the component masses. Additionally, we discuss a few binaries of note from our sample, identifying a likely blue straggler - blue straggler binary system (7782), a double-lined binary with a secondary star which is under-luminous for its mass (5080), two potential eclipsing binaries (4705 and 5762), and two binaries which are likely members of a quadruple system (5015a and 5015b).'
author:
- 'Aaron M. Geller[^1] and Robert D. Mathieu$^1$,'
- 'Hugh C. Harris'
- 'Robert D. McClure'
title: 'WIYN Open Cluster Study. XXXVI. Spectroscopic Binary Orbits in NGC 188'
---
Introduction
============
Within an open cluster, dynamical interactions with hard binaries[^2] provide energy to the cluster, and can foster a complex interplay of stellar evolution, stellar dynamical exchanges, mass transfer, and even stellar collisions. Such interactions have the potential to result in the formation of “anomalous” stars that defy standard stellar evolutionary theory, such as blue stragglers (BSs). Recent $N$-body simulations [e.g., @hur05] are beginning to illuminate the likely formation mechanisms of such anomalous stars within open clusters, and it has become clear that the binary population plays a significant role. Detailed studies of open cluster binary populations are critical to constrain such models so that we can study cluster evolution as well as the formation mechanisms of anomalous stars. Furthermore, accurate and comprehensive surveys of binary populations are essential for our understanding of the onset of mass transfer, tidal interactions, initial and present-day mass functions, stellar dynamics, and even star formation processes.
Radial-velocity (RV) surveys offer an efficient way to identify single[^3] and binary open cluster members as well as to solve for binary orbital solutions. Open clusters are ideally suited for such surveys as they offer a coeval sample of stars that are generally easily accessible through ground-based observations using even modest-sized telescopes. Spectroscopic binary surveys have been carried out for a few well known clusters (e.g., Hyades @deb00; Praesepe, @mer99, @abt99, @deb00; Pleiades, @mer92; and M67; @mat90). Today, the advent of multi-object spectrographs permits surveys of larger stellar samples in more distant open clusters, allowing us to explore binary populations as a function of age, stellar density, metallicity and stellar mass.
We present [98]{} binary orbits in the old (7 Gyr) open cluster NGC 188, derived from our ongoing RV survey of the cluster, covering a magnitude range of [10.8$\leq$V$\leq$16.5]{} (1.14-0.92 [M$_{\odot}$]{}), a 1 diameter region on the sky (roughly 13 core radii[^4]) and, for some binaries, a timespan of up to thirty years. This survey of NGC 188 is part of the WIYN Open Cluster Study [WOCS; @mat00]. Our detectable binaries all have periods ranging from a few days to on the order of 10$^3$ days. Given an internal velocity dispersion of 0.64 $\pm$ 0.04 [km s$^{-1}$]{} [@gel08 hereafter Paper 1], these binaries constitute much of the hard-binary population that dynamically powers the cluster. In Paper 1, we describe our observations, data reduction and the precision of our measurements. We also provide RV membership probabilities ([P$_{RV}$]{}) for stars observed $\geq$3 times and identify RV variable stars. In this second paper in the series, we present our complete current RV database on the cluster (Section \[data\]). In Section \[orbits\], we provide the [70]{} single-lined (SB1) and [15]{} double-lined (SB2) binary cluster-member orbital solutions derived from this survey. For each binary, we provide the plotted orbital solution, tabulated orbital parameters, and constraints on the component masses. In Section \[anom\] we discuss a few binaries of note, including a likely blue straggler - blue straggler binary system (7782), a SB2 binary with a secondary star which is under-luminous for its mass (5080), two potential eclipsing binaries (4705 and 5762), and two binaries which are likely members of a quadruple system (5015a and 5015b). Finally, in the Appendix we provide the orbital solutions and parameters for the [13]{} field binaries that we have serendipitously discovered over the course of our survey. The third paper in this series will study the binary frequency of the cluster and analyze the binary distributions in period, eccentricity and secondary mass. With the data analyzed in this series of papers, we will gain a detailed understanding of the cluster dynamics, the properties of the hard-binary population and their influence on the formation of anomalous stars like BSs, and thereby provide valuable constraints for future $N$-body models of NGC 188.
Data
====
Our NGC 188 stellar sample spans a magnitude range of [10.8$\leq$V$\leq$16.5]{} and a 1 diameter region on the sky. Our magnitude limits include solar-mass main-sequence stars, subgiants, giants, and BSs, and our spatial coverage extends radially to $\sim$13 core radii. The IDs and coordinates for our stellar sample are taken from the @pla03 proper-motion (PM) study. As explained in Paper 1, our full RV database is composed of two data sets, one from WIYN[^5] and one from the Dominion Astrophysical Observatory (DAO). The DAO dataset is composed of RVs measured at the DAO 1.2m and the Palomar 5m telescopes both converted to the DAO Radial-Velocity Spectrometer (RVS) system. Here we present our complete current RV database for each of our observed stars in the field of NGC 188 in Table \[RVtable\]. We include in this table both cluster members and nonmembers as well as stars without sufficient observations to derive membership information. We refer the reader to Paper 1 for thorough descriptions of our stellar sample and its completeness, and where we provide our findings on cluster membership and velocity variability.
We show data for two stars, one SB2 binary and one single star, in Table \[RVtable\], and provide the full table electronically. For individual RV measurements, we list the reduced Heliocentric Julian Date (HJD-2400000 d), the observatory at which the observations were taken, using “W” for WIYN, “D” for DAO and “P” for Palomar, the measured RV, and the cross-correlation peak height for WIYN measurements as a guide to the quality of measurement (with a maximum value of 1; see Paper 1 for a detailed description of the precision of our data as a function of the peak height). For the binaries with orbital solutions, we also provide the residual (O-C), derived as the observed minus the expected RV from the orbital solution, and the phase. For SB2 binaries with orbital solutions, we provide RVs and cross-correlation peak heights (where available) for both stars and their respective residuals.
Observations taken at the WIYN 3.5m range in date from October 1995 through August 2008. Observations made at the DAO 1.2m range in date from February 1980 through November 1996. All observations prior to 1980 were taken at the Palomar 5m, with the earliest observations taken in December 1973. We have found no zero-point offset between the WIYN and DAO data sets (Paper 1), and have thus integrated both sets of measurements without modification into the single RV data set presented here. The precision of the WIYN data is 0.4 [km s$^{-1}$]{} and of the DAO data is 1.0 [km s$^{-1}$]{} (Paper 1).
Spectroscopic Binary Orbits {#orbits}
===========================
In the following section, we present our [85]{} orbital solutions of the binary members of NGC 188. We first discuss our [70]{} SB1 binaries and then our [15]{} SB2 binaries. For both sets, we provide the tabulated orbital parameters, plotted orbit curves and component mass estimates.
Single-Lined Orbital Solutions {#SB1}
------------------------------
For each SB1 binary, we solve for the orbital solution using the data given in Table \[RVtable\]. We provide the plotted orbital solutions in Figure 1; for each binary we plot the orbit in the top panel and the RV residuals in the bottom panel. In Table \[SB1tab\] we provide the orbital elements for each binary in two rows, where the first row includes the binary ID, the orbital period ($P$), the number of orbital cycles observed, the center-of-mass RV ($\gamma$), the orbital amplitude ($K$), the eccentricity ($e$), the longitude of periastron ($\omega$), a Julian Date of periastron passage ($T_\circ$), the projected semi-major axis ($a \sin i$), the mass function ($f(m)$), the rms residual velocity from the orbital solution ($\sigma$), and the number of RV measurements ($N$). Where applicable, the second row contains the respective errors on each of these values. In Table \[SB1masstab\], we present physical properties for each SB1, including the WOCS ID, the $V$ magnitude and the $(\bv)$ color [both from @ste04], the radial distance from the cluster center (in arcminutes), the RV membership probability (*[P$_{RV}$]{}*; from Paper 1), the PM membership probability [*[P$_{PM}$]{}*; from @pla03], a photometric estimate for the mass of the primary ($M_1$), a lower limit for the mass of the secondary ($M_2$ min), and finally a photometric estimate for the mass of the secondary ($M_2$).
The photometric estimates for the primary and secondary masses are derived simultaneously across the available $UBVRI$ photometry for each binary using a photometric deconvolution technique. We use the observed $(U\!-\!V)$, $(\bv)$, $(V\!-\!R)$, and $(V\!-\!I)$ colors, where available, and $V$ magnitudes (as compiled by @ste04) along with a 7 Gyr, solar-metallicity Padova isochrone[^6] [@gir02] to produce a set of synthetic binaries. This set of binaries contains primary stars within a range of masses whose magnitudes extend from the observed $V$ magnitude to this magnitude plus 0.75 (as this would be the contribution from an equal mass companion) and, for each primary star, a set of secondary stars of equal or lesser mass. The component masses of the synthetic binary that has a composite $V$ magnitude and colors in the available photometric bands that most closely match the observed $V$ magnitude and colors, in both color-magnitude and color-color space, are taken as the photometric primary and secondary mass estimates.
We only attempt to quote masses for the main-sequence, sub-giant and giant binaries. We caution the reader that, for binaries with mass ratios $\lesssim$0.5, the photometric masses are less certain, as solar-type binaries with these low mass ratios fall very near to the isochrone [e.g., @hur98]. Also the morphology of the isochrone near the turnoff, makes the masses for binaries in this region more sensitive to selection of the distance modulus. In certain cases (e.g., when the observed binary lies directly on the isochrone to within the photometric errors, or the binary is found blueward of the main-sequence or redward of the giant branch), we cannot derive reliable mass estimates in the manner described above. For such cases, we use the observed $V$ magnitude to estimate an upper limit on the mass of the primary star. We have found that the secondary must be at least 2.5 magnitudes fainter than the primary at a central wavelength of 5250 Å (the central wavelength of the WIYN spectra) for the binary to be observed as single lined. Thus, in these cases, we use this resulting upper limit on the $V$ magnitude for the secondary to derive the upper limit on its mass (and note this in the table). Finally, for all SB1 binaries we use the primary mass estimate along with the orbital mass function to derive a lower limit on the secondary mass.
For two binaries, 4965 and 4688, we notice a clear trend with time in the residuals of the orbital solutions fit to the observed RVs. We assume that this trend is due to the presence of an additional long-period companion (or companions). Therefore, for each of these two binaries, we fit a polynomial function (of first and second order, respectively) to the residuals, subtract this fit from the observed RVs, and refit the orbit to these corrected RVs. There is no trend in the resulting residuals from the corrected orbital solutions for either of these binaries. We note that all of the orbital parameters derived from the corrected orbital solutions agree with those of the uncorrected orbital solutions to within the errors, except for two parameters in 4688; the orbital amplitude, $K$, increased from 6.7 $\pm$ 0.5 in the uncorrected orbit to 9.6 $\pm$ 1.7 in the corrected orbit, and the orbital eccentricity, $e$, increased from 0.57 $\pm$ 0.05 in the uncorrected orbit to 0.70 $\pm$ 0.04 in the corrected orbit. We show the corrected orbital solution plots in Figure 1 and parameters in Table \[SB1tab\]. In our RV data table, Table \[RVtable\], we include the observed RVs and the residuals to the corrected orbital solutions.
Curiously, this SB1 photometric deconvolution technique has yielded three cases where we would expect to see the secondary. Binaries 4524 and 4843 lie well blueward of the giant branch, and binary 4390 lies well redward of the main sequence. We also note that some spectra of 4710 reveal an additionally component for which we have no current explanation. This binary is located near the main-sequence turnoff. The rest of the mass estimates yield luminosity ratios in which we indeed would not expect to observe the secondary star, given our observing setup.
We use a Monte Carlo technique to estimate the mean uncertainty on our mass estimates, assuming this uncertainty to be derived from two main sources: the uncertainties on the photometry and on the isochrone fit. For binaries in which we can estimate masses from the photometric deconvolution technique, we find a mean uncertainty for the primary mass of 0.09 [M$_{\odot}$]{} and on the secondary of 0.14 [M$_{\odot}$]{}. The standard deviations about these means are 0.15 [M$_{\odot}$]{} and 0.20 [M$_{\odot}$]{}, respectively. Uncertainties on the minimum secondary masses are found in a similar manner, using the derived primary mass uncertainty along with the error on the mass function resulting from the orbital solution, and result in a mean uncertainty of 0.04 [M$_{\odot}$]{} with a standard deviation about the mean of 0.10 [M$_{\odot}$]{}. Finally, for binaries in which we can only give limits on the primary and secondary masses, we note that the mean uncertainty on the $V$ magnitudes for all binaries is 0.011 magnitudes. For solar-type stars, a shift of this amount to the observed magnitude of a main-sequence star results in a shift in mass of 0.003 [M$_{\odot}$]{}.
Double-Lined Orbital Solutions {#SB2}
------------------------------
The RV measurements for the primary and secondary stars of a given SB2 binary are found using a TwO Dimensional CORelation (TODCOR) technique formulated by @zuk94. TODCOR uses two template spectra to derive the two RVs of an SB2 binary simultaneously, greatly increasing our ability to recover reliable RVs even for those observations that appear highly blended in a one-dimensional cross-correlation function. As all of our detected SB2 binaries have mass ratios $\gtrsim$0.7, we choose to use the same solar template that we use to derive RVs for all single stars and SB1 binaries as both template spectra in TODCOR. Our procedure in deriving the orbital solutions is to first solve for the orbit of the primary in the manner discussed in Section \[SB1\] and then use the derived orbital elements to solve for the full SB2 orbit (including the RVs of the secondary star). We provide the plotted orbital solutions in Figure 2; the plots are of the same format as for the SB1 binaries, except here, the primary RVs are plotted using filled circles while secondary RVs are plotted with open circles. Additionally, we present the tabulated orbital elements in Table \[SB2tab\], in similar format to Table \[SB1tab\], except here, in place of the mass function, we provide the quantity $m$ $\sin^3$ $i$ and the mass ratio ($q$).
We also include Table \[SB2masstab\] that contains similar information on the SB2 binaries as we provide in Table \[SB1masstab\] for the SB1 binaries. Here we do not quote a lower limit on the secondary mass as the mass ratio can be calculated directly from the orbital solution. We use the same photometric deconvolution procedure as for the SB1 binaries to derive the photometric mass estimates, except, here, we keep the mass ratio fixed. For the red-giant binary 3118, we cannot use this technique, as the system is observed to lie redward of the giant branch. Therefore, we use the Padova isochrone to formulate a mass-luminosity relation of $L \propto M^{11}$, valid for this region on the NGC 188 giant branch, to derive the appropriate correction to the observed $V$ magnitude, from which we can estimate the primary mass. (Specifically, we observe a mass ratio for 3118 of $q$ = 0.795, which implies a correction to the observed $V$ magnitude of $V_1 = V + 0.08$, and we use this $V_1$ to estimate the mass of the primary.) Given this primary mass estimate and the mass ratio, we can easily derive the secondary mass.
Again, we utilize a Monte Carlo technique to estimate the uncertainties on our mass estimates in a similar manner to Section \[SB1\]. The mean uncertainty on the primary mass estimates is similar to that of the SB1 binaries. We can then use the mass ratio, primary mass and their respective uncertainties to derive a mean uncertainty on the secondary-mass estimates of 0.09 [M$_{\odot}$]{}, with a standard deviation about this mean of 0.02 [M$_{\odot}$]{}. Additionally, we utilize our SB2 binaries to check the accuracy of this photometric deconvolution technique by first estimating masses with the mass ratio fixed and then estimating masses for the same binaries without fixing the mass ratio (essentially, treating the systems as SB1 binaries and using the technique described in Section \[SB1\]). For the primary mass, we find a mean difference between these two techniques of 0.01 [M$_{\odot}$]{}, and for the secondary mass estimates, we find a mean difference of 0.03 [M$_{\odot}$]{}. The standard deviations about these means are 0.02 [M$_{\odot}$]{} and 0.06 [M$_{\odot}$]{}, respectively. These values lie within our estimated uncertainties, and demonstrate the robustness of the mass estimates for both SB1 and SB2 binaries derived using our photometric deconvolution technique.
Binaries of Note {#anom}
================
In the following section, we discuss the properties of various intriguing binaries that we have discovered in NGC 188. We first discuss three binaries that contain potential encounter products. We then include our photometric variables and X-ray sources, and present evidence that 5015 is in fact a quadruple system composed of two SB1 binary cluster members.
Binaries Containing Potential Encounter Products
------------------------------------------------
#### 5078:
5078 has a period of 4.78303 $\pm$ 0.00012 days, well below the circularization period of 14.5 days in NGC 188 [@mei05]. However this binary has a significantly higher than circular eccentricity, at 0.121 $\pm$ 0.006. 5078 is a particularly intriguing binary as it is a BS with an SB2 orbital solution. This relatively high eccentricity may be a sign of a recent dynamical interaction or an additional companion [@maz90]. Triple systems are not uncommon within binary populations, with observational evidence ranging from 5-50% [@may87; @duq91; @pou04; @tok06]. Furthermore, @tok06 showed that for solar-type binaries, the frequency of additional companions increases towards shorter inner-binary periods, finding a frequency of tertiary companions for binaries with periods $\sim$5 days of $\sim$65%.
#### 5080 :
5080 is a SB2 binary found right above the main-sequence turnoff at $V$ = 14.624 and $(\bv)$ = 0.668. The system is located at 0.7 core radii from the cluster center, and has a [P$_{RV}$]{} = 96% and a [P$_{PM}$]{} = 98%. From our orbital solution, we find a mass ratio of 1.01 $\pm$ 0.07, and we estimate that both stars have masses of $\sim$1.02 [M$_{\odot}$]{}. However, from inspection of the cross-correlation functions, it is clear that the two stars have different luminosities. We checked for a potential template mismatch using a set of solar-metallicity synthetic spectral templates ranging from a 0.5 [M$_{\odot}$]{} main-sequence star to a 1.14 [M$_{\odot}$]{} star at the tip of the giant branch. For all spectra of 5080 in which we detect the secondary, a combination of two solar templates returns the highest two-dimensional correlation peak height and therefore the best fit to the data. Hence we proceed to use our standard solar spectrum as the template for both the primary and secondary stars in order to derive the luminosity ratio. The majority of the correlation functions are highly blended. Consequently we ran TODCOR on the four observations that show the largest RV separations and derive a luminosity ratio ($L_2/L_1$) of 0.32, with a standard deviation of 0.04. Thus the secondary star appears to be under-luminous for its mass. We note that, if we take the lowest value for the mass ratio allowed by the error, of 0.94, then we could be observing a binary containing a primary star that has evolved just past the turnoff with a main-sequence secondary star. If we take the mass of the primary star to be 1.02 [M$_{\odot}$]{}, as derived in Section \[SB2\], then the secondary star could have a mass as low as 0.96 [M$_{\odot}$]{}. Using these values with the Padova isochrone, we derive a luminosity ratio of 0.65, which is certainly much larger than what we observe.
#### 7782 :
7782 is a BS SB2 binary located at 9.7 core radii with a [P$_{RV}$]{} = 95% and a [P$_{PM}$]{} = 11%. 7782 is the second bluest of our detected BSs in NGC 188 with a $(\bv)$ = 0.494. Interestingly, we find the system to have a mass ratio of 1.005 $\pm$ 0.013, meaning that both stars in the system are likely more massive than the main-sequence turnoff mass. Utilizing TODCOR, we select the 11 observations with well separated peaks to find a luminosity ratio of 0.739 with a standard deviation of 0.026. We suggest that 7782 may be a BS - BS binary system.
Photometric Variables and X-ray Sources
---------------------------------------
#### 4289 :
4289 is a SB1 binary found at the base of the giant branch at a radius of 2.5 core radii. The binary is a secure cluster member with both [P$_{RV}$]{} and [P$_{PM}$]{} = 98 %. We derive an orbital solution with a period of 11.4877 $\pm$ 0.0009 days and an eccentricity consistent with circular of 0.012 $\pm$ 0.010. We estimate that the primary star is likely a red giant with a mass of $<$ 1.12 [M$_{\odot}$]{}, and the secondary star is on the main sequence with a mass of $<$ 0.78 [M$_{\odot}$]{}. This binary was observed to be one of the brightest X-ray sources, GX28, in the @gon05 survey. They point out that one would not expect a giant star in NGC 188 to show rapid rotation or surface activity unless the star is a member of a tight binary system in which rapid rotation has been maintained by synchronization. We do not see any evidence for line broadening due to rotation in our spectra, which corresponds to an upper limit of $\sim$10 [km s$^{-1}$]{} (derived from similar analysis to that of @rho01). With a period of $\sim$11.5 days and assuming an appropriate radius for the primary star of $\sim$2.3 [R$_{\odot}$]{}, we would expect a maximum rotational velocity of $\sim$10 [km s$^{-1}$]{} resulting from tidal synchronization. According to @gon05a, even this relatively slow rotation may be sufficient to increase the surface coverage of magnetic-loop structures in giants like 4289 enough to produce the observed X-ray emission.
#### 4705 :
This SB2 binary is found at 1.3 core radii, and lies near the giant branch with $V$ = 13.933 and $(\bv)$ = 0.938. The binary is a high-probability cluster member with both [P$_{RV}$]{} and [P$_{PM}$]{} = 98 %. We derive a kinematic orbital solution with a period of 35.178 $\pm$ 0.005 days and an eccentricity of 0.487 $\pm$ 0.005. This star was observed as a photometric variable, V11, by @kal90, who noted a dimming of almost 0.4 magnitudes over the course of the night of December 13, 1986. Kaluzny et al. conjecture that this variability and the location of 4705 on the CMD can be explained if 4705 is an eclipsing binary with a relatively unevolved red-giant primary and an upper-main-sequence secondary star. The observed photometric dimming occurred at a phase of $\sim$0.02 in our derived orbit (when the RVs of both the primary and secondary stars were very near the $\gamma$-velocity of the system). We used the program NIGHTFALL[^7] to determine the phase at which one would expect to observe an eclipse in this system, and find that we would indeed expect an eclipse to occur at a phase of $\sim$0.02. Thus 4705 may be an eclipsing binary system in NGC 188. Furthermore, we estimate the primary mass to be 1.14 [M$_{\odot}$]{} and find a mass ratio of 0.956 $\pm$ 0.013. This would allow for an upper main-sequence secondary star as predicted. Additionally, 4705 was found to be an X-ray variable, GX18, by @gon05, who observed low-amplitude brightness variations on the time scale of weeks. They suggest that these variations are due to slow rotation, as rotating giants can produce high X-ray luminosities, possibly related to the existence of magnetic fields induced by turbulent motion in their deepening convective zones. It has also been suggested by @zha02 and @gon05 that 4705 may be an RS CVn system.
#### 5379 :
This SB1 BS binary lies at 1.6 core radii from the cluster center and is a secure cluster member with both [P$_{PM}$]{} and [P$_{RV}$]{} = 98 %. This binary is a BS, with a $V$ magnitude of 15.373 and a $(\bv)$ color of 0.542. We derive a period of 120.21 $\pm$ 0.04 days with an eccentricity of 0.24 $\pm$ 0.03. Additionally, @kaf03 found this binary to be a photometric variable (WV3) with a period of 0.18148 days. We cannot derive a kinematic orbital solution with this short period. We do observe signs of above average rotation in the 5379 spectra, and we have used the procedure of @rho01 to derive a $v \sin i$ of 15.4 $\pm$ 0.5 [km s$^{-1}$]{}. If this photometric variability is due to chromospheric activity or star spots at this short period, we would expect a rotational velocity for the star of $>$250 [km s$^{-1}$]{}, which can be ruled out for all inclination angles greater than $\sim$3.5. @kaf03 suggested that 5379 may be a member of the short-period end of the NGC 188 W UMa population. This now seems less likely given our lack of observed rapid rotation. We note that the photometric period, amplitude of the oscillations, and the observed $v \sin i$ lie within the observed range of $\delta$ Sct variable stars [@rod00]. However 5379 does not lie near the instability strip.
#### 5762 :
5762 is a SB2 binary found at the main-sequence turnoff at 3.4 core radii from the cluster center. The binary has a [P$_{PM}$]{} = 97% and [P$_{RV}$]{} = 66%. We derive a circular orbit with a period of 6.50430 $\pm$ 0.00004 days, a mass ratio near unity of 0.977 $\pm$ 0.008, and a minimum separation between the primary and secondary of 18.95 $\pm$ 0.08 [R$_{\odot}$]{}. Zhang et al. (2002, 2004) identified this system as an eclipsing binary (V12). The observed photometric eclipse in @zha02 occurred at a phase of 0.88 in our orbital solution, when both stars in the system were moving near the $\gamma$-velocity. This provides further evidence for the eclipsing nature of the system. @mei09 discuss this eclipsing binary in detail. We simply point out that even if we are viewing this system at a low inclination angle, the true separation between the two stars will likely be very favorable to mass transfer as both stars evolve up the giant branch. As such, 5762 may be a pre-mass-transfer system which could represent a BS precursor.
A Possible Quadruple System : 5015
----------------------------------
5015 is a 90% PM member, and upon preliminary inspection of the observed spectra and the resulting cross-correlation functions, we presumed that 5015 was a typical SB2 binary. There are two clear peaks in most of the 1D correlation functions, and both RVs are easily recovered using TODCOR for all but one observation. We followed the usual procedure of fitting an orbital solution to the primary, then using the derived orbital parameters to fit the full orbital solution, including the secondary velocities. However, we were unable to derive an SB2 orbit using the parameters from the fit to the primary. We then proceeded to fit a separate orbital solution to the secondary RVs, and found that the two solutions had entirely different parameters. We show the individual orbits in Figure \[5015aborbs\] and give the respective orbital parameters in Table \[5015ab\]. Individually, each of the derived $\gamma$-velocities results in a [P$_{RV}$]{} = 0%. Interestingly, though, if we take the average of the two $\gamma$-velocities, we get -41.9 $\pm$ 0.3 [km s$^{-1}$]{}, which is very close to the cluster mean RV of [-42.36 $\pm$ 0.04 [km s$^{-1}$]{}]{} (Paper 1). Thus we have two options: either the two observed binaries are a chance superposition of two field binaries, or we are observing a quadruple system that is a likely member of NGC 188.
[l r@c@l r@c@l]{} P (days) & 312.5 & $\pm$ & 0.9 & 8.3291 & $\pm$ & 0.0004\
$\gamma$ ([km s$^{-1}$]{}) & -46.50 & $\pm$ & 0.24 & -37.2 & $\pm$ & 0.6\
K ([km s$^{-1}$]{}) & 11.4 & $\pm$ & 0.4 & 45.6 & $\pm$ & 0.7\
e & 0.10 & $\pm$ & 0.03 & 0.008 & $\pm$ & 0.016\
$\omega$ (deg) & 74 & $\pm$ & 21 & 70 & $\pm$ & 150\
T$_\circ$ (HJD-2400000 d) & 51599 & $\pm$ & 19 & 51875 & $\pm$ & 4\
a$\sin$ i (10$^6$ km) & 48.6 & $\pm$ & 1.5 & 5.22 & $\pm$ & 0.08\
f(m) ([M$_{\odot}$]{}) & 4.7e-2 & $\pm$ & 0.4e-2 & 8.2e-2 & $\pm$ & 0.4e-2\
$\sigma$ ([km s$^{-1}$]{}) & &\
N & &\
\[5015aborbs\]
If we assume that the two binaries are not cluster members, then we can ask what is the likelihood that we are observing a superposition of two binaries in the field. To answer this question, we utilized the theoretical Besançon model of the Milky Way [@rob03] to derive the expected number of field stars within one square degree, covering our observed magnitude range, towards the direction of NGC 188. We then assume that the locations of these field stars are described by a Poisson distribution and proceed to calculate the conditional probability that we would observe two field stars within a three arcsecond diameter fiber, given that we observe at least one, and find a 0.04% probability. Furthermore, since 5015 contains two binaries within a three arcsecond diameter region, we then multiply this value twice by the field binary fraction of 51%, as observed by @duq91. Finally, we must account for the RVs of the two binaries. To do so, we again use the Besançon model to calculate the percentage of field stars with RVs within five [km s$^{-1}$]{} from the mean RV for NGC 188 (i.e., only including field stars with -47 [km s$^{-1}$]{} $\leq$ RV $\leq$ -37 [km s$^{-1}$]{}), and find these stars to populate 20% of the field towards NGC 188. Including these constraints, the probability of observing two field binaries in the direction of NGC 188 within a three arcsecond diameter fiber that have RVs within five [km s$^{-1}$]{} from the mean RV for NGC 188 is decidedly small, at 0.002%. To date, we have observed a total of 1116 stars in the direction of NGC 188. Though this is a relatively large number of stars, it is certainly not enough for us to expect to observe such a chance superposition of two field binaries. Therefore, this scenario seems unlikely.
Conversely, we can assume that these two binaries are members of a quadruple system in which the two binaries orbit each other about the system’s center of mass. Observations of field solar-type binary populations find the frequency of triples and higher-order systems to be 5-50% [e.g. @may87; @duq91; @tok97]. Additionally, there is observational evidence for the presence of multiple-star systems in a few well studied open clusters (e.g., M67, @mat90; Praesepe, @mer94; Pleiades, @bou97; Hyades, @pat98). Recent $N$-body simulations by @hur05 suggest that in an old open cluster, we might expect up to $\sim$7% of the sources to reside in dynamically-formed triple or higher-order systems. Thus we should not be surprised to find a few such star systems in NGC 188.
Using TODCOR, we derive a luminosity ratio of 0.36 $\pm$ 0.02. From the Padova isochrone, we find a luminosity ratio of $L \propto M^{4.5}$, valid for this region of the NGC 188 main-sequence, which results in a mass ratio of 0.80 $\pm$ 0.04. Therefore, the true center-of-mass RV of the quadruple system would be -42.4 $\pm$ 0.3 [km s$^{-1}$]{}, which would result in a [P$_{RV}$]{} = 98%. This along with the @pla03 [P$_{PM}$]{} = 90% provides strong evidence for cluster membership.
Summary
=======
In this paper, we present [98]{} binary orbits resulting from our ongoing RV survey of the old open cluster NGC 188. This is the second paper in a series aimed at characterizing the solar-type single- and binary-star populations within the cluster. These data will enable us to investigate the formation mechanisms and evolution of anomalous stars, like BSs, as they are influenced by the binary population, through comparison with detailed theoretical models of the cluster.
We provide our complete current RV database for NGC 188 in Table \[RVtable\], including the measured RVs for all stars observed in the direction of NGC 188 over the course of our RV survey of the cluster. We use these data to derive the [70]{} SB1 (Section \[SB1\]) and [15]{} SB2 (Section \[SB2\]) orbital solutions for the NGC 188 cluster member binaries presented in this paper, and provide the results both graphically and as tabulated orbital elements. For the main-sequence, sub-giant and giant binaries we use a photometric deconvolution technique to estimate the masses of the primary and secondary stars relative to a 7 Gyr solar-metallicity isochrone, and we provide the SB1 results in Table \[SB1masstab\] and the SB2 results in Table \[SB2masstab\]. For SB1 systems, we also provide a lower limit on the secondary mass, derived using the orbital mass function.
In Section \[anom\] we identify a few binaries of note, including a likely quadruple system, 5015. Notably, 4705 and 5762 are both SB2 systems that may also be eclipsing binaries (5762 is studied in detail by @mei09). We also observe the BS 7782 as an SB2 system with a mass-ratio near unity, which suggests that the system may contain two BS stars. We use TODCOR to investigate the luminosity ratio for the equal mass SB2 binary 5080 and find that the secondary star appears to be under-luminous for its mass. Finally we discuss the additional photometric variables and X-ray sources that are in binaries in NGC 188. The binaries of note discussed in Section \[anom\] are ripe for further study.
The WIYN Open Cluster Study will continue its survey of NGC 188 in order to provide orbital solutions for all binaries in the cluster out to periods of 1000 days as well as a fraction of longer period binaries. In future papers, we will analyze the binary distribution in period, eccentricity and secondary mass, and constrain the cluster binary fraction. These data will form critical constraints on future detailed $N$-body models of NGC 188 as well as other open clusters, allowing us to study the complex interplay of stellar evolution and dynamics amongst the single- and binary-cluster members as they interact in the open cluster environment.
The authors would like to express their gratitude to the staff of the WIYN Observatory without whom we would not have been able to acquire these thousands of superb stellar spectra. We also thank the many undergraduate and graduate students who have helped to obtain these spectra over the years at WIYN for this project. We would like to acknowledge R. F. Griffin and J. E. Gunn for contributing their NGC 188 RVs to our project, who, in turn, wish to express their thanks to the Palomar Observatory for the use of the 5m telescope. Thanks to Murray Fletcher for his expertise in developing the DAO RVS instrument, and to Jim Hesser who acquired a portion of the DAO NGC 188 data. Finally, we wish to thank to anonymous referee for the helpful suggestions in improving this paper. This work was funded by the National Science Foundation grant AST-0406615 and the Wisconsin Space Grant Consortium.
Facilities: , ,
APPENDIX
========
Field Binaries
--------------
In our survey to find binary cluster members, we have serendipitously derived orbital solutions for [13]{} field binaries, all with either [P$_{RV}$]{} or [P$_{PM}$]{} = 0%. We note that some of these binaries appear to be kinematic members of NGC 188 from either PM or RV evidence, but none are cluster members in all three dimensions. In the interest of studies of the field binary population, we present these orbital plots (Figures 4 and 5) and parameters (Tables \[fieldSB1tab\] and \[fieldSB2tab\]) here.
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[^1]: Visiting Astronomer, Kitt Peak National Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.
[^2]: A hard binary is defined as having an internal energy that is much greater than the energy of the relative motion of a single star moving within the cluster [@heg74]. For solar mass stars in a cluster with a one-dimensional velocity dispersion equal to 1 [km s$^{-1}$]{}, all hard binaries have periods less than $\sim$10$^5$ days.
[^3]: In the following, as in @gel08 (Paper 1) we use the term “single” to identify stars with no significant RV variation; a star is termed single if the standard deviation of its RV measurements is less than four times our precision. Certainly, many of these stars are also binaries, although generally with longer periods and/or lower total mass than the binaries identified in this study.
[^4]: We adopt a core radius of 1.3 pc [@bon05] at a distance of 1.9 kpc, which corresponds to 2.35 arcminutes on the sky.
[^5]: The WIYN Observatory is a joint facility of the University of Wisconsin - Madison, Indiana University, Yale University, and the National Optical Astronomy Observatories.
[^6]: For the isochrone, we set $E(\bv) =$ 0.025 and $(m-M)_V =$ 11.23 [@for07].
[^7]: NIGHTFALL is copyright (c) 1998-2002 Rainer Wichmann, (c) 2001-2002 Markus Kuster, (c) 2001- 2002 Patrick Risse and can be downloaded from http://www.hs.uni-hamburg.de/DE/Ins/Per/Wichmann/Nightfall.html.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report high-resolution measurements of voltage ($V$) noise in the mixed state of micrometre-sized thin films of amorphous Nb$_{0.7}$Ge$_{0.3}$, which is a good representative of weak-pinning superconductors. There is a remarkable difference between the noise below and above the irreversibility field $B_{irr}$. Below $B_{irr}$, in the presence of measurable pinning, the noise at small applied currents resembles shot noise, and in the regime of flux flow at larger currents decreases with increasing voltage due to a progressive ordering of the vortex motion. At magnetic fields $B$ between $B_{irr}$ and the upper critical field $B_{c2}$ flux flow is present already at vanishingly small currents. In this regime the noise scales with $(1-B/B_{c2})^2 V^2$ and has a frequency ($f$) spectrum of $1/f$ type. We interpret this noise in terms of the properties of strongly driven depinned vortex systems at high vortex density.'
author:
- 'D. Babić'
- 'T. Nussbaumer'
- 'C. Strunk'
- 'C. Schönenberger'
- 'C. Sürgers'
title: 'Vortex Motion Noise in Micrometre-Sized Thin Films of the Amorphous Nb$_{0.7}$Ge$_{0.3}$ Weak-Pinning Superconductor'
---
\[intro\] Introduction
======================
When set in motion by a current $I$, vortices in superconductors generate a voltage $V$. The resulting $V(I)$ curve may be either non-linear, implying depinning phenomena, or linear, indicating flux flow (FF). Such $V(I)$ characteristics do not provide complete information on the nature of vortex motion, especially if the pinning is weak. This is a point where information available from the voltage noise becomes a powerful indicator of the underlying physics. The finding [@gurp] that vortices moving as bundles composed of $N$ magnetic-flux quanta $\phi_0$ may produce shot noise attracted considerable attention, and resulted in extensive subsequent work which was eventually extended beyond a simple shot-noise approach.[@clem] Samples used in these studies were mainly polycrystalline conventional superconductors with appreciable pinning and non-linear $V(I)$ characteristics up to very close to $B_{c2}$. Noise experiments have also been carried out on high-$T_c$ superconductors,[@woltgens; @danna; @kim] which are in a “liquid” state of negligible pinning over a large portion of the magnetic field vs. temperature plane, displaying linear $V(I)$ curves. However, the intricate anisotropic character of vortex matter in these compounds is a serious obstacle to the understanding of the mechanisms that contribute to voltage noise related to motion of vortices against a weak pinning potential.
Thus, a number of phenomena in the weak-pinning regime have remained largely unexplored from the point of view of vortex motion noise. The same holds for the noise properties in the depinned state, i.e. for $B >
B_{irr}$. For instance, the interplay of bulk pinning and surface barriers,[@bl; @geom] which are both obstacles for vortex motion (and can be of similar strengths when pinning is weak), has been studied mostly by analysing the $V(I)$ curves and the magnetoresistance $R(B,T)$.[@richard] Similarly, dynamic ordering of vortex motion has also been explored by measuring the average transport properties.[@aarts] Noise measurements can reveal effects which are beyond reach of measurements of the average voltage. For example, if pinning is absent the $V(I)$ is linear, but one could ask does this mean that vortices really move completely “silently” or are there some dynamic effects which introduce fluctuations in their velocity? Moreover, it is known that shot noise probes the properties of “granular magnetic-flux charge”, $N \phi_0$, but the details of this process are still subject to discussion - especially if $N$ is small (characteristic of weak pinning).
In this paper we present high-resolution noise measurements which address the above topics. We have chosen a system particularly suitable for such research, namely Nb$_{0.7}$Ge$_{0.3}$ amorphous thin films of thickness $d$ comparable to the coherence length $\xi$. These films are conventional, isotropic, weak-coupling $s$-wave BCS superconductors in the dirty limit, and for $\xi \sim d$ they exhibit an extended ($B$,$T$) range of easily-movable vortices.[@theun] In contrast to the complicated situation in high-$T_c$ compounds, here vortices can be considered as undeformed “cylinders” of a volume $\xi^2 d$ and the Ginzburg-Landau (GL) parameters can be found straightforwardly. We also note that our shaping the samples in the form of [*narrow wires*]{} turned out to be crucial for observing the overall properties of the noise, i.e. for both $B < B_{irr}$ and $B > B_{irr}$.
In the regime where $V(I)$ and $R(B,T)$ still indicate the presence of pinning we find a noise similar to that in Ref., i.e. which for small applied currents resembles shot noise, being linear in $V$ and frequency independent at low frequencies, and decreases for more ordered vortex motion at high currents. Closer to $B_{c2}$, over a rather extended range, we find no evidence for pinning in neither $V(I)$ or $R(B,T)$. The lower boundary of this region is therefore taken as the irreversibility field $B_{irr}$. The noise for $B_{irr} < B < B_{c2}$ is qualitatively different from that in the presence of pinning. It exhibits a $1/f$ frequency spectrum and is quadratic in $V$. Moreover, it scales with $(1-B/B_{c2})^2 V^2$. The monotonic increase with increasing $V$, and in particular the scaling which involves no pinning dependent parameters, motivates us to propose that the noise of this kind is a peculiar property of strongly driven vortices at high vortex density.
\[exp\] Experiment
==================
Our samples (20 nm thick) were produced by magnetron sputtering of Nb and Ge on to oxidised silicon wafers through masks prepared by electron-beam lithography, using a double-layer resist (PMMA/PMMA-MA). The measurements were carried out in a $^4$He cryostat, above the $\lambda$-point of liquid helium. Voltage noise, $V(I)$ and $R(B,T)$ were measured extensively on a $W = 5$ $\mu$m wide and $L=50$ $\mu$m long wire connected to two wide contact pads (sample S5). In order to investigate size effects in the noise we performed a less comprehensive set of measurements on a $W=1$ $\mu$m and $L=10$ $\mu$m sample (sample S1). By analysing the low-current (10 nA; 10 Acm$^{-2}$) $R(B,T)$ measurements within the framework of a model appropriate for dirty weak-coupling superconductors [@kes] we characterised sample S5 in detail. The transition temperature $T_c = 2.91$ K is determined as the midpoint of the 10 % - 90 % (0.1 K) zero-field transition curve. The transition curve is smooth and free of “kinks” that would indicate the presence of inhomogeneities, and we ascribe the rather wide transition (in units of $T/T_c$) to a pronounced two-dimensional character of the sample. A similar conclusion was drawn in Ref. for an YBa$_2$Cu$_3$O$_{7- \delta}$ single crystal investigated systematically with respect to different $\delta$-values and consequently different anisotropies. Very weak temperature dependence of the normal state resistivity $\rho_N$ above $T_c$ permits the estimation of $\rho_N (T=0) = (2.3 \pm 0.2)$ $\mu \Omega$m. Using this value and $ - (dB_{c2} / dT)_{T=T_c} \approx 2.05$ TK$^{-1}$, determined from the $R( B={\rm const.},
T)$ measurements (not shown), we calculate[@kes] the GL parameters: $\xi (0) = 7.4$ nm, $\kappa = 77$ and $\lambda (0) =
1.63 \kappa \xi(0) = 930$ nm. The parameters of sample S5 are in good agreement with published work.[@theun] Sample S1 had a slightly lower $T_c$ (2.55 K) and larger $\rho_N$, but otherwise showed fairly the same properties as sample S5. The method of noise measurements is described in detail in Ref.. In short, the signal from a sample is processed through two low-noise amplifiers the outputs of which are cross-correlated in a spectrum analyser. The noise setup is calibrated against the equilibrium Nyquist noise $4 k_B T R_N$ in the normal state ($R_N$ is the normal state resistance). By this approach we have obtained a resolution of $\alt 10^{-20}$ V$^2$s, necessary for measurements of small noise signals appearing in the case of weak pinning. For both samples the frequency window for the noise measurements was 106.5 - 114 kHz, except for the measurements of the frequency dependence of the noise power spectrum $S_V$, performed at several frequencies between 20 kHz and 250 kHz.
All the noise measurements were carried out at fixed temperatures, $T=2.4$ K ($T/T_c = 0.82$) for sample S5 and $T=2.25$ K ($T/T_c = 0.88$) for sample S1. Since sample S1 had lower $T_c$, we had to choose a larger value of $T/T_c$ in order to avoid temperature instabilities that appear in the vicinity of the $\lambda$-point.
\[average\] Magnetoresistance and current-voltage characteristics
=================================================================
First we analyse the $R(B,T)$ and $V(I)$ results. In the lower inset to Fig.1 we show $R(B , 2.4 \; {\rm K})$ for sample S5. Above $\sim 0.65$ T we found good agreement with the FF theory of Larkin and Ovchinnikov (LO).[@lo] The LO FF conductivity is given by $$%
\sigma_{FF} = \frac{1}{\rho_N}
\left[ 1+\frac{1}{(1-T/T_c)^{1/2}}
\left( \frac{B_{c2}}{ B} \right)
%g \left( \frac{B}{B_{c2}} \right) \right] \; ,
g \left( B/B_{c2} \right) \right] ,
%
\label{LOFF}$$ where (for $z > 0.315$) $g(z) = (1 - z)^{3/2} [0.43+ 0.69 (1-z)]$. The solid line, representing the LO FF resistance $R_{FF} = R_N /
\sigma_{FF} \rho_N$, is drawn by taking $T_c = 2.91$ K, $B_{c2}=
1.18$ T and $R [B_{c2}(2.4 \; {\rm K})] = R_N = 1375$ $\Omega$ ($\rho_N = 2.75$ $\mu \Omega m$). The mentioned uncertainty in $\rho_N$ implies a certain range of the $B_{c2}$ values that do not deteriorate the fit. This range is $\sim 1.14 - 1.22$ T and agrees fairly well with $B_{c2} \sim 1.09 - 1.12$ T obtained by the extrapolation method of Ref.. Henceforth we use $B_{c2} = 1.18$ T. Taking different values of $T_c$ (within the transition width) has little effect on the quality of the fit. We conclude that for the fields above $\sim
0.65$ T the vortices flow freely even at very small applied currents, and thus $B_{irr} (2.4 \; {\rm K}) \sim 0.65$ T, which is, as we show below, in agreement with the $V(I)$ results.
{width="75mm"}
For magnetic fields below $\sim 0.65$ T the LO theory does not explain the magnetoresistance data, and $R$ is smaller than $R_{FF}$. This indicates that the vortices are slowed down by experiencing a pinning potential. However, $R$ is finite even at magnetic fields as low as $\sim 0.05 B_{c2}$, which implies a very weak pinning. In the upper inset to Fig.1 we show a log-log plot of a typical $V(I)$ in this region, for 0.27 T. Over two decades in $I$ the $V(I)$ is ohmic ($R = 25$ $\Omega$) before it turns upwards. This suggests a hopping vortex motion (HVM), most probably thermally activated. In the model of thermally activated HVM, vortex velocity is given by $v_\phi = l (\nu_+ - \nu_-)$, where $l$ is the hop length and $\nu_{\pm} \propto \exp[-(U \mp U_F)/k_B T]$ the hopping rates over a potential $U$ in the direction (+) and opposite (-) to the driving force ${\bf F} = - \nabla U_F$. Since $F \propto IB$ and $V=BLv_\phi$, for $I \rightarrow 0$ the $V(I)$ is linear. From our measurements of $R (B= {\rm const.}, T)$ we can estimate the values of $U/k_B$, which are remarkably small. At 2.4 K, $U/k_B$ is lower than 10 K and is a decreasing function of $B$. At higher currents the $V(I)$ gradually changes to a $V (I) \propto(I - I_c)$ dependence, as we show for 0.27 T in Fig.1 by open circles. This suggests a force-induced transition to flux flow, i.e. an ordering of the vortex motion with increasing driving force. This assumption will be supported further by the noise results presented in Section \[lownoise\]. Finally, at even higher currents $V$ jumps to a value of the order of $V_N = R_N I$ (Fig.1, upper inset) due to the appearance of non-linear FF described in the LO theory [@lo] and observed experimentally for similar films.[@aarts; @lefloch] Above 0.65 T, where FF takes place even at vanishingly small currents, the $V(I)$ is simple: linear starting from $I \rightarrow 0$ and all the way up to the appearance of non-linear effects in FF, as shown for 0.67 T in Fig.1 by full squares.
\[noise\] Noise results
=======================
In the rest of the paper we present and discuss the results of our noise measurements, which if not specified otherwise refer to sample S5. We introduce $\Sigma_V$ to denote the excess noise, which is the difference between the total measured noise $S_V$ and the thermal (Nyquist) noise $4 k_B T (dV/dI)$. The currents used in the noise measurements were always kept below those corresponding to the appearance of the high-current non-linearities mentioned in Section \[average\], since we are interested in situations where the average transport properties are still unaffected by the high-current dynamical processes described in the LO theory. [@lo] In Section \[lownoise\] we analyse the noise in the regime of non-linear $V(I)$ curves, i.e. for $B < B_{irr}$, and in Section \[highnoise\] we turn to the noise for $B_{irr} < B < B_{c2}$, where the $V(I)$ is linear and $R(B, 2.4 \; {\rm K})$ agrees well with the LO FF theory.
\[lownoise\] Noise in the regime of non-linear $V(I)$
-----------------------------------------------------
In Fig.2 we show a typical $\Sigma_V (V)$ curve in the regime of non-linear $V(I)$, i.e. for 0.27 T (corresponding to the $V(I)$ curve in Fig.1). The maximum background Nyquist noise is $\sim
2.5 \times 10^{-20}$ V$^2$s. $\Sigma_V (V)$ first increases linearly up to $V\simeq 0.2$ mV which is close to the upper limit of HVM in $V(I)$. At higher voltages, where $V(I)$ becomes proportional to $(I - I_c)$, $\Sigma_V$ gradually decreases with increasing $V$. From this decrease of $\Sigma_V (V)$ we infer that the vortex motion becomes more and more ordered when the driving force progressively dominates over the pinning potential. At large driving force the pinning potential causes not only a finite offset $I_c$ in $V(I)$ but also random fluctuations of the vortex velocity, which is most probably the origin of the small residual noise above $V \sim 0.5$ mV. This residual noise is expected to vanish together with $I_c$ at $B_{irr}$, which is indeed observed in our experiment. It is worthwile to note that the onset of collective vortex motion has stronger effect on $\Sigma_V$ than on $V(I)$. In $\Sigma_V$ the depinning threshold $I_c$ is indicated by a pronounced maximum above which an ordering of the vortex motion occurs. On the other hand, $V(I)$ shows no sharp feature at $I_c$, implying that $I_c$ has to be determined by extrapolation of the linear part of $V(I)$ down to $V=0$. Since the linear regime extends only over a small current range between the HVM regime and the high-current non-linearities, the determination of $I_c$ is more ambiguous than in $\Sigma_V$. The non-monotonic character of $\Sigma_V (V)$ supports our interpretation more strongly, and also supplements research on dynamic vortex ordering studied[@aarts] by analysing the average transport properties.
{width="75mm"}
Similarly to shot noise (in current) of electrons, which is a linear function of $I$, shot noise (in voltage) of vortices is a linear function of $V$. [@gurp] To check whether the linear increase of $\Sigma_V (V)$ in the low-voltage regime can be interpreted as shot noise we investigated the frequency and sample-width dependences of $\Sigma_V$. The studies[@gurp; @habjoin; @knoedler] of vortex-motion shot noise offer different models for the slopes $\Gamma$ of linear $\Sigma_V (V)$ plots, as we discuss later, but they agree in predicting a frequency-independent $\Sigma_V(f)$ up to a frequency $f_c \sim v_{\phi} /W = V/B L W$. [*Because the wire width W is small*]{}, in the present case the (calculated) $f_c$ is large, more than 500 kHz for all the measured points, except for a few ones very close to $V = 0$. We measured $\Sigma_V(f)$ at a characteristic point ($V = 65$ $\mu$V) of a linear $\Sigma_V (V)$ curve, and found that $\Sigma_V (f)$ is essentially flat between 20 kHz and 250 kHz, as shown in the lower inset to Fig.2. This result meets the above-mentioned expectation for shot noise.
In the first work[@gurp] on vortex shot noise the factor $\Gamma$ was related to the “charge” of a vortex bundle, i.e. $\Gamma = 2 N \phi_0$. The low level of noise found in this study for Corbino disc geometry implied that the noise in the samples of bar geometry was produced essentially at their edges. This finding can be understood in terms of the surface barriers (of Bean-Livingston[@bl] or geometrical[@geom] type) for a vortex entering and leaving a sample. In short, a depinned vortex bundle “shoots” accross a bar-geometry sample, interacts only weakly with the pinning centres and the rest of (pinned or slowly moving) vortices, and the noise is created by the bundle overcoming the barriers at the entry and exit from the sample. In this case $\Gamma$ does not depend on sample width. However, in later studies[@habjoin; @knoedler] it was found that if vortex bundles travel a distance $x \ll W$ before their motion is interrupted by the pinning centres, $\Gamma$ should be inversely proportional to $W$. The reason for this can be inferred from the Josephson relation $V = \phi_0 (d \varphi / dt) / 2 \pi$, where $\varphi$ is the phase of the superconducting order parameter. A moving vortex causes the phase shift of $2 \pi$ only if it moves over the whole distance $W$. If the actual distance $x$ is shorter than $W$, the phase change associated with one voltage pulse is a factor $x/W$ less than $2 \pi$, and the consequence is $\Gamma = 2 \phi_0 N x / W$.[@knoedler] Note that in this case the noise is produced in the bulk, i.e. at the pinning centres. The reduction factor $x/W$ explains the result of Ref. that the noise produced in the bulk (by slow vortices moving over small distances, or by local bundle-velocity fluctuations) was much smaller than that due to the “shooting” bundles overcoming the surface barriers. A more complicated expression for $\Gamma$ was derived in Ref., where it was found that if there is a distribution of the strengths and positions of pinning centres the above expression becomes $\Gamma = 2 \phi_0 \langle N^2
\rangle \langle x^2 \rangle / \langle N \rangle \langle x \rangle
W$, where the brackets denote averages over the distribution function.
In the upper inset to Fig.2 we plot $\Gamma(B)$ for sample S5 and $\Gamma (B) / 8$ for sample S1. Over the whole field range where we found well defined linear $\Sigma_V (V)$ curves the slopes $\Gamma(B)$ for both samples decrease with increasing field in the same manner, and $\Gamma$ for sample S1 is approximately eight times larger. If we take into account slightly different experimental conditions for the two samples this is in fair agreement with $\Gamma \propto 1/W$. At magnetic fields lower than $\sim 0.20$ T the resistances of the samples, the measured voltage and the corresponding voltage noise are small, which leads to a larger error in $\Gamma$.
We now address the question of whether the noise is produced by the pinning or by the surface barriers. The surface barriers are important at applied magnetic fields of the order of, or lower than, the thermodynamic critical field $H_c = B_{c2} / \mu_0 \kappa
\sqrt{2}$. In our case, $\mu_0 H_c \sim 11$ mT is much lower than the fields at which we found the noise of a measurable magnitude. In addition, the approximate scaling of $\Gamma$ with sample width suggests that the bulk pinning, and not the sample edges, dominates the noise. In turn, measurements of the width dependence of $\Gamma$ may be an alternative to other experiments[@richard] for determining whether the surface barriers influence the measured transport properties.
The fact that our measurements allow us to exclude the surface barriers as the main origin of the noise in our samples also sheds more light on the nature of the $B_{irr}$ and the meaning of the potential $U$ of HVM. It is known that for some samples (e.g. single crystals of the Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ high-$T_c$ superconductor) surface barriers may have considerable effect on both the irreversibility field[@zeldov] and the thermally activated transport.[@richard] This is not the case in the present situation, the $B_{irr}$ can be attributed to a transition to a depinned vortex state and the $U$ is related to bulk pinning, as we have anticipated in Section \[average\].
We attribute the decrease of $\Gamma$ with increasing $B$ to the weakening of pinning as $B$ approaches $B_{irr}$, since for $B > B_{irr}$ we found no linear $\Sigma_V
(V)$ curves and, moreover, the overall noise magnitude decreases as $B$ increases towards $B_{irr}$. The decrease of $\Gamma
(B)$ [*for*]{} $B$ [*well below*]{} $B_{irr}$ could be explained within the framework of the models of Refs., if the unknown parameters $x$, $N$, and, respectively, $\langle x \rangle, \langle
x^2 \rangle, \langle N \rangle$ and $\langle N^2 \rangle$ depend on magnetic field in the right way. Since $\Gamma$ comprises these parameters as products and ratios (see above), they cannot be extracted independently from our data. However, both models break down in the limit $B \rightarrow
B_{irr}$. This can be understood as follows. The effects of pinning are (1) formation of vortex bundles in order to increase the driving force and thus facilitate their motion against the pinning potential, (2) reduction of the hopping distance below the sample width $W$. The pinning force vanishes at $B_{irr}$, implying $\langle N^2 \rangle ^{1/2}
\rightarrow \langle N \rangle \rightarrow 1$ and $\langle x^2
\rangle ^{1/2} \rightarrow \langle x \rangle \rightarrow W$, i.e. $\Gamma \rightarrow 2 \phi_0$, which is in contrast to the experimental observation shown in Fig.2.
A reason for this breakdown of the classical models can possibly be inferred from the comparison of the transports of (normal) electrons and vortices close to the limit of perfect transmission. Our experimental realisation - where vortices are created at the entry into a sample and vanish at the exit, is equivalent to a two-terminal mesoscopic conductor - where electrons have their source and drain in the reservoirs. Whenever the transmission coefficient $\Theta$ for electron transport through such a mesoscopic conductor is close to unity, shot noise is suppressed by a factor $(1 - \Theta)$.[@buetles] In the ballistic limit ($\Theta = 1$) there is no noise associated with electron transport. If vortices are not slowed down by bulk pinning and/or surface barriers, their motion is determined by the viscous drag only. This situation represents perfect vortex motion, conceptually similar to ballistic transport of electrons. Therefore, if there are no dynamic effects present (see Section \[highnoise\]), in the limit of perfect vortex transmission accross a sample the noise should vanish. A more quantitative treatment of vortex motion shot noise at high transmittance requires further research.
\[highnoise\] Noise in the regime of linear $V(I)$
--------------------------------------------------
Above $B_{irr} \sim 0.65$ T, where the vortex density is large and $V \approx R_{FF} I$ for all our noise measurements, no noise described in Section \[lownoise\] was found. Instead, as we show in Fig.3a, $\Sigma_V$ is a monotonic function of $V$, increasing as $V^2$, and as a function of magnetic field it decreases as $B$ approaches $B_{c2}$. Moreover, as shown in Fig.3b, there is a scaling $\Sigma_V \propto (1 - B/B_{c2})^2 V^2$ which holds over $B_{irr} < B < B_{c2}$ and is insensitive to variations of $B_{c2}$ in the range $1.14 - 1.22$ T. The frequency dependence of $\Sigma_V$ in this regime is of $1/f$ type, more precisely $1/f^\alpha$ with $\alpha = 1.5 \pm 0.1$ (Fig.3b, upper inset). In the normal state, above $B_{c2}$, $\Sigma_V = 0$ and $S_V$ is simply the voltage-independent Nyquist noise.
{width="75mm"}
The existence of any noise in the regime where the vortices are most likely to be completely depinned, as seen from the $R(B,T)$ and $V(I)$, is rather surprising, since in the pinned state the magnitude of the noise described in Section \[lownoise\] is becoming progressively smaller as $B \rightarrow B_{irr}$. Furthermore, if the background pinning would still influence the noise significantly one would not expect an increase of $\Sigma_V$ with increasing $V$, because at larger driving force the role of pinning is less important. Therefore, the origin of the noise shown in Fig.3 has to be sought in dynamic properties of depinned vortices far from equilibrium, with a guideline along the LO theory of non-equilibrium phenomena in flux-flow dissipation.[@lo] In addition, a possible partial or complete melting of the vortex lattice, which could occur at $B_{irr}$,[@dbjrc] should also be taken into account.
There is experimental evidence in support of our assumption that the peculiar noise observed is not related to depinning processes. In Fig.4 we show $\Sigma_V(V)$ for 0.61 T, i.e. just at the crossover from HVM to LO FF in $R(B,T)$. For low voltages, $\Sigma_V (V) \propto V$ (as indicated by the dashed line), suggesting that the vortices undergo the HVM. At $V \sim 0.8$ mV the noise starts to deviate from the linear dependence, showing in a small voltage range a tendency to decrease, typically for the vortex motion becoming more ordered with increasing driving force. However, at higher $V$ the decrease of $\Sigma_V (V)$ does not continue but instead $\Sigma_V$ approaches the same $\Sigma_V (V) \propto V^2$ behaviour as for the higher fields (the solid line in Fig.4 indicates the scaling in Fig.3b). Although the vortex motion is becoming more and more uniform the noise increases, which can hardly be explained in terms of vortex interaction with a pinning potential.
Quadratic voltage dependence and $1/f$ power spectrum are generally known to be the properties of resistance fluctuations.[@weissman] Hence, a possibility that our finding represents resistance fluctuations, i.e. vortex velocity fluctuations, requires attention. At a fixed $(B,T,I)$ point, two parameters influence vortex velocity and consequently FF resistance: $\rho_N$ and vortex core area $A_c$. Thus, if there are fluctuations in either $\rho_N$ or $A_c$, the FF resistance fluctuates as well. The fact that the measured noise above $B_{c2}$ is just the Nyquist noise rules out fluctuations of $\rho_N$, leaving us with a possibility that $A_c$ fluctuates. We argue below that such fluctuations may occur if the vortex velocity is large and the vortex density high.
{width="65mm"}
The non-equilibrium properties of vortex cores and the related influence on flux-flow dissipation were studied theoretically by LO,[@lo] and in Ref. . If the electric field generated in moving vortex cores is sufficiently strong, quasiparticles in the cores can gain enough energy to overcome the potential barriers at vortex edges and to escape into the surrounding superfluid. This leads to a reduction of the core size, the vortex viscosity decreases[@comvisc] and the vortex velocity increases, resulting in the non-linearities in $V(I)$ at large currents and finally the jump shown in the upper inset to Fig.1. At low vortex density the electron-phonon relaxation processes are sufficiently efficient to cool the hot quasiparticles to the bath temperature, as the heating occurs in the cores only and the cooling over the whole volume.
However, the situation changes at [*large vortex density*]{}. With increasing vortex density the cooling efficiency decreases and the quasiparticles are heated-up to an elevated temperature.[@lo; @bez] This may cause an increase of thermal fluctuations of the quasiparticle density. As a consequence, the quasiparticle pressure on the vortex “walls” may fluctuate, which would then result in the fluctuations of $A_c$. The related fluctuations of $v_\phi$ are measured as voltage fluctuations.
Since the average transport properties can be for $B > B_{irr}$ consistently described by the LO theory, it is tempting to check whether the LO expression for $\sigma_{FF}$ (see Section \[average\]) allows to relate the possible core-size fluctuations and the measured fluctuations in voltage. Because the observed noise occurs where $V(I)$ is linear, the $\Sigma_V \propto V^2$ dependence can be explained by assuming fluctuations of the [*conductivity*]{}, i.e. $\Sigma_V\Delta
f=(\delta V)^2 = (\delta \sigma_{FF})^2 / \sigma_{FF}^2\;V^2$. $\Delta f$ is the frequency interval over which the noise spectrum is averaged. To relate the fluctuations $\delta
\sigma_{FF}$ and $\delta A_c$ we can rewrite $\sigma_{FF}$ in terms of the vortex core area $A_c \sim \xi^2 \sim \phi_0 /
B_{c2}$ and the intervortex distance $l_B \sim \sqrt{\phi_0 /B}$, so that $z = B/B_{c2} = A_c / l_B^2$. Then we calculate $\delta
\sigma_{FF} = (1/l_B^2) (\partial \sigma_{FF} /
\partial z) \delta A_c$ from Eq. \[LOFF\] and obtain $$%
\frac{(\delta \sigma_{FF})^2}{\sigma_{FF}^2} = G(B/B_{c2}, T/T_c)\;
\frac{(\delta A_c)^2}{A_c^2}\;\;,
%
\label{LOFF_noise}$$ where $G(z, T/T_c) =[dg(z)/dz - g(z)/z]^2/ [(1- T/T_c)^{1/2} +
g(z)/z]^2$.
The form of $(\delta A_c)^2/A_c^2$ is not known [*a priori*]{}. However, it can be deduced by combining Eq.\[LOFF\_noise\] and the experimentally observed behaviour $(\delta \sigma_{FF})^2 / \sigma_{FF}^2 = (\delta V)^2/V^2 = \gamma (1 - B/B_{c2})^2 \Delta f$ (see Fig.3b). This results in $$%
\frac{(\delta A_c)^2}{A_c^2} =
\gamma \frac{(1 - B/B_{c2})^2\;\Delta f}{G(B/B_{c2},
T/T_c)} \;\; .
%
\label{deltaAc}$$ In Fig.5 we plot this expression against $B/B_{c2}$ in order to check whether there is any approximation that would lead to a simple picture of the fluctuations. It is seen that $(\delta A_c)^2/A_c^2$ can be well approximated for $B/B_{c2} \lesssim 0.92$ by a power law, i.e., $(\delta A_c)^2/A_c^2\propto (B/B_{c2})^{-n}$ with $n \approx 2$. The simulations for other values of $T/T_c$ show that the power-law approximation holds well for essentially any value of $T/T_c$. The power $n$ weakly depends on $T/T_c$ but is reasonably close to 2 in the region $0.7 < T/T_c <0.95$.
The apparent $(B/B_{c2})^{-2}$-decrease of $(\delta A_c)^2/A_c^2$ has a simple visualisation: such a functional dependence corresponds to a plausible assumption that the fluctuations $\delta A_c$ of the vortex area are proportional to the space $\sim l_B^2$ available, so that $(\delta A_c)/A_c \propto
(l_B^2/\xi^2)\propto(B/B_{c2})^{-2}$. The above modelling based on the LO conductivity hence shows that the assumption of core-size fluctuations may reproduce the measured voltage and magnetic field dependences of the voltage noise.
![Solid line: log-log plot of $(\delta A_c)^2/A_c^2$ given by Eq.\[deltaAc\], for $f=110$ kHz, $\Delta f = 7.5$ kHz, $T/T_c = 0.82$, $\gamma = 2.1 \times 10^{-13}$ s (corresponding to the data shown in Fig.3b). Crosses indicate the $(B/B_{c2})^{-2}$ approximation of $(\delta A_c)^2/A_c^2$, discussed in text.](babicfig5){width="65mm"}
With the experimentally determined value of the prefactor $\gamma = 2.1 \times 10^{-13}$ s for $f = 110$ kHz and $\Delta f = 7.5$ kHz we obtain the relative fluctuation amplitude $\delta A_c / A_c$ of the order of $10^{-5}$. However, as we discuss below, the $1/f$ spectrum implies that the fluctuations are distributed over a range of relaxation times. As a consequence, the small value of $\delta A_c / A_c$ only represents the contribution of those core-size fluctuations which occur in this frequency window around the given frequency.
The observed $1/f$ spectrum cannot be explained if all vortex cores fluctuate in exactly the same manner. The fluctuation of the size of a vortex core is assumed to be a random process with a characteristic time $\tau$. If $\tau$ would be the same for all cores, this would result in a Debye-Lorentzian spectrum of the fluctuations, white up to the cutoff frequency $1 / \tau$. On the other hand, a distribution of $\tau$ and a superposition of Debye-Lorentzian spectra may result in a $1/f$ spectrum.[@weissman] Properties of the distribution then also determine how much the fluctuations with a given $\tau$ contribute to $\delta A_c / A_c$ measured at $(f,\Delta f)$. Such a distribution may arise, for example, as a consequence of [*different local correlations*]{}.
That vortex motion can strongly depend on local conditions was demonstrated in Ref., where it was found that, in the presence of pinning, vortices move in a form of intermittent “rivers” between the pinned islands. In our case one can hardly discuss a motion around the pinned islands, since any important pinned fraction would affect the average transport properties significantly, which is not observed (see the discussion of Fig.1). This however does not necessarily imply that there are no “floating islands”, i.e. vortex lattice domains moving together with the “liquid” phase. The average flux-flow dissipation in such a (depinned) system would be still well described by the LO theory, since the ratio $B/B_{c2}$ influences the magnetoresistance much more strongly than the exact geometry of a system of moving vortices.[@lo] However, [*the local vortex correlations could be different*]{} for vortices deeply in the islands, in the “liquid”, close to the island boundaries, etc., which could lead to different relaxation times for the core fluctuations. These different relaxation times would then give a $1/f$ noise spectrum.
We are aware that our arguments offer only a qualitative picture, and that further clarification of the above ideas is required. However, at the moment we do not know of any quantitative theoretical model which would account for the observed peculiarities of vortex motion noise above $B_{irr}$, nor are we aware of any related systematic experimental work dealing with a range $B_{irr} < B < B_{c2}$ as large as $\sim 50$ % of $B_{c2}$. Thus, we believe that the results and discussion of this Section could be used as a possible starting point for further experimental and theoretical work.
\[summary\] Summary and conclusions
===================================
We have measured voltage noise in the mixed state of micrometre-sized wires of amorphous Nb$_{0.7}$Ge$_{0.3}$ thin films. The samples are well described by conventional theories for dirty weak-coupling superconductors, have weak pinning, relatively low irreversibility field $B_{irr}$, and the vortex structure is much simpler than in high-$T_c$ superconductors. These properties make the samples suitable for exploring the vortex motion noise in the weak-pinning regime.
At low magnetic fields, i.e. for $B < B_{irr}$, and small applied currents the voltage-current curves exhibit properties characteristic of thermally activated hopping of vortices. The related noise is a linear function of voltage, with the slopes $\Gamma$ of noise vs. voltage curves inversely proportional to the sample width, and is basically frequency independent up to 250 kHz. This behaviour is in agreement with the shot noise model and the assumption that the noise is generated by bulk pinning and not by surface barriers. $\Gamma$ decreases with increasing $B$ over the whole magnetic field range of the shot-noise-like behaviour, which does not contradict the presently available models of vortex motion shot noise. These models however fail to explain the disappearance of the shot noise as $B \rightarrow B_{irr}$. For $B < B_{irr}$ but at larger currents the vortex motion becomes more uniform and the noise decreases. The decrease and the low level of the noise is ascribed to the ordering of vortex motion with increasing driving force.
In a narrow range of $B$ slightly below $B_{irr}$, at low $V$ one still observes the above-mentioned two types of noise but at large $V$ the noise becomes quadratic in $V$. This signifies the appearance of the dynamic effects inherent to large vortex density, a behaviour fully developed for $B > B_{irr}$. For $B > B_{irr}$ the $V(I)$ curves are linear over the whole range of our measurements and the magnetoresistance agrees well with the flux-flow theory of Larkin and Ovchinnikov. The noise in this regime is completely different from that for $B < B_{irr}$. Over the whole voltage range it increases quadratically with increasing voltage, its frequency spectrum is of $1/f$ type, and it scales with $(1-B/B_{c2})^2 V^2$. The origin of this noise is not entirely clear. We present a qualitative explanation in terms of the non-equilibrium properties of moving vortex cores which are subjected to fluctuations of their radius.
We are greateful to J. Aarts for contributing to this work in its initial stage. Valuable discussions with K. E. Nagaev, V. B. Geshkenbein, G. B. Lesovik, G. Blatter, H. v. Löhneysen, F. Nori, E. H. Brandt and J. R. Cooper are greatfully acknowledged. This work was supported by the Swiss National Science Foundation.
[99]{} D. J. van Ooijen, and G. J. van Gurp, Phys. Lett. [**17**]{}, 239 (1965); G. J. van Gurp, Phys. Rev. [**166**]{}, 436 (1968). For an overview see: J. R. Clem, Phys. Rep. [**75**]{}, 1 (1981). This paper also contains details of theoretical approach to vortex motion noise. P. J. M. Wöltgens, C. Dekker, S. W. A. Gielkens and H. W. de Wijn, Physica C [**247**]{}, 67 (1995). G. D’Anna, P. L. Gammel, H. Safar, G. B. Alers and D. J. Bishop, J. Giapintzakis and D. M. Ginsberg, Phys. Rev. Lett. [**75**]{}, 3521 (1995). D. H. Kim, K. E. Gray, N. Jukam, D. J. Miller, Y. H. Kim, J. M. Lee, J. H. Park and T. S. Hahn, Phys. Rev. B [**60**]{}, 3551 (1999). C. P. Bean and J. D. Livingston, Phys. Rev. Lett. [**12**]{}, 14 (1964). E. Zeldov, A. I. Larkin, V. B. Geshkenbein, M. Konczykowski, D. Majer, B. Khaykovich, V. M. Vinokur and H. Shtrikman, Phys. Rev. Lett. [**73**]{}, 1428 (1994). D. T. Fuchs, R. A. Doyle, E. Zeldov, S. F. W. R. Rycroft, T. Tamegai, S. Ooi, M. L. Rappaport and Y. Myasoedov, Phys. Rev. Lett. [**81**]{}, 3944 (1998); S. F. W. R. Rycroft, R. A. Doyle, D. T. Fuchs, E. Zeldov, R. J. Drost, P. H. Kes, T. Tameagi, S. Ooi and D. T. Foord, Phys. Rev. B [**60**]{}, R757 (1999). J. M. E. Geers, C. Attanasio, M. B. S. Hesselberth, J. Aarts and P. H. Kes, Phys. Rev. B [**63**]{}, 094511 (2001). M. H. Theunissen and P. H. Kes, Phys. Rev. B [**55**]{}, 15183 (1997); For a comprehensive study see: M. H. Theunissen, PhD. Thesis, University of Leiden, 1997. F. Habbal and W. C. H. Joiner, J. Low Temp. Phys. [**28**]{}, 83 (1977); J. D. Thompson and W. C. H. Joiner, Phys. Rev. B [**20**]{}, 91 (1979). P. H. Kes and C. C. Tsuei, Phys. Rev. B [**28**]{}, 5126 (1983). D. Babić, J. R. Cooper, J. W. Hodby and Chen Changkang, Phys. Rev. B [**60**]{}, 698 (1999). M. Henny, S. Oberholzer, C. Strunk and C. Schönenberger, Phys. Rev. B [**59**]{}, 2871 (1999). A. I. Larkin and Yu. N. Ovchinnikov, in [*Nonequilibrium Superconductivity*]{}, edited by D. N. Lengenberg and A. I. Larkin (North Holland, Amsterdam, 1986). F. Lefloch, C. Hoffmann and O. Demolliens, Physica C [**319**]{}, 258 (1999). C. M. Knoedler and R. F. Voss, Phys. Rev. B [**26**]{}, 449 (1982). E. Zeldov, D. Majer, M. Konczikowski, A. I. Larkin, V. M. Vinokur, V. B. Geshkenbein, N. Chikumoto and H. Shtrikman, Europhys. Lett. [**30**]{}, 367 (1995). G. B. Lesovik, Pisma Zh. Eksp. Teor. Fiz. [**49**]{}, 515 (1989) \[JETP Lett. [**49**]{}, 594 (1989)\]; M. Büttiker, Phys. Rev. Lett. [**65**]{}, 2901 (1990). For an overview of the various aspects of $1/f$ noise and resistance fluctuations see e.g. M. B. Weissman, Rev. Mod. Phys. [**60**]{}, 537 (1988). A. I. Bezuglyj and V. A. Shklovskij, Physica C [**202**]{}, 234 (1992). From the LO theory it follows that out of equilibrium the vortex viscosity is velocity dependent and given by $\eta(v_\phi) = \eta_0 [1 + (v_\phi/ v_\phi^*)^2]^{-1}$, where $\eta_0$ is the viscosity for small vortex velocities and $v_\phi^*$ a characteristic vortex velocity. Also, $A_c (v_\phi) = A_c(0) [1 + (v_\phi/ v_\phi^*)^2]^{-1}$, which leads to $\eta(v_\phi) = \eta_0 A_c (v_\phi) / A_c(0)$. Therefore, one should not confuse the vortex-size dependences of $\eta_0$ and $\eta(v_\phi)$: while $\eta_0$ is larger for smaller $A_c(0)$, $\eta(v_\phi)$ is smaller for smaller $A_c(v_\phi)$. T. Matsuda, K. Harada, H. Kasai, O. Kanimura and A. Tonomura, Science [**271**]{}, 1393 (1996).
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'D. Maurin'
- 'R. Taillet'
bibliography:
- 'ms3378.bib'
date: 'Received 6 December 2002 / Accepted 27 March 2003'
title: 'Spatial Origin of Galactic Cosmic Rays in Diffusion Models: II- Exotic Primary Cosmic Rays'
---
Introduction
============
A great amount of work has been done these last twenty years on the astrophysical signatures that could unravel new physics. In the eighties, there were great hopes that the antiproton signal, which showed an excess at an energy of a few hundreds of MeV in the first balloon experiments, could be such a signature. However, this hope was swept away by the progress in measurements – see e.g. [bess]{} [@Orito00; @Maeno01] or [heat]{} [@Beach01] and [caprice]{} [@Boezio01] at higher energy – and a better determination of the cosmic ray propagation parameters (see e.g. @Maurin02b). It was shown that the measured antiproton flux was indeed compatible with the sole secondary standard spallative production [@Bergstrom99b; @Donato01] (see the first paper for a comprehensive historical discussion and a panel of references dealing with exotic antiproton production).
@Donato00 showed that the antideuteron ($\bar{d}\,$) signal could lead to a clearer signature of SUSY. However, as discussed in many other studies on SUSY antiprotons [@Rudaz88; @Stecker89; @Jungman94; @Bottino95; @Bottino98; @Wells99; @Bergstrom99b], the indeterminacy in the dark matter distribution, as well as its possible clumpiness [@Bergstrom99], might severely change the conclusions. In contrast, the Hawking evaporation of Primordial Black Holes (PBH) could also yield a new source of cosmic rays [@Maki96], but the precise shape of the dark matter in this case is not crucial [@Barrau02; @Barrau03]. Nevertheless, in the latter case, it was shown that even considering only the propagation parameters giving a good fit to B/C data, the remaining degeneracy for example in the diffusive halo height has sizeable effects on the primary flux [@Barrau02].
Hence, at least two different phenomena can affect the conclusions reached in papers dealing with exotic flux calculations. The first one, related to the spatial distribution of SUSY sources, is usually thoroughly discussed [@Bergstrom99b], but the second point - namely the influence of various propagation parameters - is generally skipped, due to the simplicity of the propagation models used. The aim of the paper is [*not*]{} to compare the predicted $\bar{p}$, $\bar{D}$ fluxes with observations for different series of models, but rather to point out which characteristics of the models actually play a role, in order to give some physical insights and milestones for studies specifically devoted to exotic flux evaluations.
We apply the method described in @Taillet03 to determine the volumes from which a fraction $f$ of cosmic rays reaching the Solar neighborhood were emitted, or equivalently the volumes that contribute to the fraction $f$ of the total flux detected in the Solar neighborhood. These volumes will be referred to as the $f$-volumes throughout the paper.
We find that depending on the diffusion parameters (evaluated from a systematic study of standard CR, @Maurin02b) as well as on the source spatial distribution, the spatial origin of cosmic rays may be quite local, the particles detected in the Solar neighborhood having mostly been created a few kpc away from the Solar neighborhood in some cases, or a few kpc away from the Galactic center in others.
Evaluation of the $f$-volumes {#general_formula}
=============================
In a companion paper [@Taillet03], we presented a method to compute the region from which a cosmic ray detected in the Solar neighborhood has a given probability of originating. This method was applied to standard sources located in the disk, and we now use it for (exotic) sources in the halo. A schematic view of our model is presented in Fig. \[galaxy\] where the isothermal dark matter profile has been superimposed on the Galaxy to compare their typical scales (the reader is referred to @Taillet03 for all the details concerning the model, such as the functional form of the galactic wind and the geometry of the box). The probability that a particle detected in the Solar neighborhood was emitted from any finite volume ${\cal V}$ can be computed as $${\cal P} \left\{ {\cal V} | \vec{r}_o \right\}=
\frac{\int_{\cal V} w(\vec{r}_s) N_{\rm r_s}(\vec{r}_o) d^3\vec{r}_s}
{\int_{\cal V_{\rm tot}} w(\vec{r}_s) N_{\rm r_s}(\vec{r}_o) d^3\vec{r}_s}\;\;,
\label{proba_integree}$$ where the source distribution $w(\vec{r}_s)$ has been introduced and $N_{\rm r_s}(\vec{r}_o)$ is the density in $\vec{r}_o$ resulting from a point source located in $\vec{r}_s$. In this paper, we are interested in determining $f$-volumes, i.e. volumes ${\cal V}(f)$ from which a given fraction $f$ of cosmic rays detected in the Solar neighborhood were emitted. They are defined by $${\cal P} \left\{ {\cal V}(f) | \vec{r}_o \right\}= f\;.
\label{def_f_vol}$$ Actually, even for a given value of $f$, there are many different volumes, delimited by different closed surfaces, fulfilling this condition. We focus on the smallest of these volumes, precisely delimited by an isodensity surface. Monte Carlo integration is then particularly well adapted to evaluate the integrals in Eq. \[proba\_integree\]. In a typical run, $\sim 10^6$ points are required to reach a $\lesssim 0.5$% convergence and the integral is performed inside all isodensity surfaces at once, so that the $f$-volume defined by Eq. (\[def\_f\_vol\]) are simple to recover.
Influence of the propagation parameters {#PL-sources}
---------------------------------------
The quantity $N_{\rm r_s}(\vec{r}_o)$ appearing in Eq. (\[proba\_integree\]) is evaluated by solving the diffusion equation with a point-like source, in the geometry depicted in Fig. \[galaxy\]. Propagation is affected, at different levels, by three effects: escape, galactic wind and spallations. First, escape happens when a particle reaches one of the boundaries of the diffusive volume. As discussed in the companion paper, this limits the range from which cosmic rays can travel to the Solar neighborhood. It was also shown that the side boundary plays only a minor role, and one can assume that the box has an infinite radial extension. Second, a convective wind $V_c$ directed out from the Galactic plane blows the charged nuclei away, so that it is more difficult to reach the plane from high $z$ sources. Finally, spallations may happen when a nucleus crossing the thin disk interacts with the interstellar matter. The nuclei are then destructed at a rate $\Gamma_{\rm inel}=2hn_{\rm ISM}.v.\sigma_{\rm inel}$. A particle emitted from a remote source is more affected by spallations as it is likely to have crossed the disk many times before reaching the Solar neighborhood. In the companion paper, this effect was shown to be important for heavy species created in the disk. Here, we focus on very light species, having smaller cross-sections, which are mostly created in the halo. They are affected by the wind in the whole halo, i.e. from the moment of their creation, whereas they are only affected by spallations when they cross the disk, which is less likely for halo sources than for disk sources. As a result, spallations play only a minor role in the present study (this effect is nevertheless included in our treatment).
When these three effects are taken into account, the density in $O$ due to a Dirac source $\delta(\vec{r}-\vec{r}_s)$ can be computed. Because of the cylindrical symmetry present for an infinite disk, it is equivalent to consider a source term $\delta(z-z_s)
\delta(r-r_s)/2\pi r_s$, which leads to $$N_{(r_s,z_s)}(0)=e^{-z_s/r_{\rm w}}
\sum_{i=1}^{\infty}
\frac{J_0\left(\zeta_i r_s/R \right)}{\pi J_1^2(\zeta_i)R^2A_i}\times
\frac{\sinh\left[S_i(L-z_s)/2\right]}{\sinh (S_iL/2)}
\label{sol_vent}$$ with $$S_i = \sqrt{\frac{4}{r_{\rm w}}+\frac{4\zeta_i^2}{R^2}}
\;\; \mbox{and} \;\;
A_i = K\left( \frac{2}{r_{\rm sp}} +\frac{2}{r_{\rm w}} +
S_i \coth \left[\frac{S_iL}{2} \right]\right)
\label{ai_reduit}$$ and where the parameters $$\left\{
\begin{array}{l}
\displaystyle
r_{\rm w} \equiv \frac{2K}{V_c} \approx
5.87 {\; \mbox{kpc}} \times \frac{K(E)}{0.03
{\; \mbox{kpc}}^2{\; \mbox{Myr}}^{-1}}
\; \frac{ 10 {\; \mbox{km}} {\; \mbox{s}}^{-1}}{V_c},
\label{f_rw}\vspace{0.2cm}\\\displaystyle
r_{\rm sp}\equiv \frac{2K}{2h \Gamma_{\rm inel}} \approx
\frac{3.17 {\; \mbox{kpc}}}{\beta} \times\frac{K(E)}{0.03
{\; \mbox{kpc}}^2{\; \mbox{Myr}}^{-1}}
\; \frac{ 100 {\; \mbox{mb}}}{\sigma},
\label{f_rspal}
\end{array}
\right.$$ give the order of magnitude of the typical distance over which the associated process affects propagation.
![Isodensity surfaces in the $(z,r)$ plane for $L=5$ kpc and $L=10$ kpc (side boundary $R=20$ kpc). Inner contours correspond to $d{\cal P}(r_s,z_s|O)/d^3\vec{r}_s=0.01 \,
\mbox{kpc}^{-3}$ and the contours are spaced by a factor $1/4$. From top to bottom, $r_{\rm sp}=3.17$ kpc (no wind), no wind and no spallations, $r_{\rm w}=2.935$ kpc (no spallations). These numbers correspond, respectively, for a reasonable choice of $K(E)$ at 1 GeV, to $\sigma_{\rm sp}\approx100$ mb and $V_c\approx20$ kpc Myr$^{-1}$. The additional thick line in each panel delimitates contours ${\cal P}({\cal V}(f)|O)=99\%$.[]{data-label="fig:prim_wind_spal"}](ms3378_f1.eps){width="\columnwidth"}
In practice, large values of $R$ have been used in Eq. (\[sol\_vent\]) so that the hypothesis $R\rightarrow \infty$ is actually recovered. The effect of escape, wind and spallations are compared in Fig. \[fig:prim\_wind\_spal\], which shows the shape of the isodensity surfaces for two values $L=5$kpc (left panels) and $L=10$ kpc (right panels), and for typical values of $r_{\rm w}$ and $r_{\rm sp}$. The value $r_{\rm sp}=3.17$ kpc has been retained because it corresponds to the antideuteron destruction cross section for a typical value of the diffusion coefficient $K=0.03$ kpc$^2$ Myr$^{-1}$. In the upper panels, one can see the shrinking of the contours in the vicinity of the disk, due to the effect of spallations. The effect of the wind is rather to flatten the contours, as can be seen in the lower panels. The probability density also decreases more rapidly when convection or spallations are included than when diffusion alone is considered. As a result, the 99%-volumes are reduced, as indicated by the thick lines. It is thus of importance to use realistic values for $K(E)$, $L$ and $V_c$ in order to give confident $f$-volumes for real situations ($\sigma_{\rm inel}$ is not a free parameter, it solely depends on the species we consider).
To summarize the previous results about the origin of exotic primaries in diffusion/convection/spallation models: i) the pure diffusive regime provides an upper limit that is strongly dependent on the halo size; ii) the Galactic wind lessens the $f$-volumes: either propagation is convection-dominated – in this case, the origin depends only on the value of $L$ and $r_{\rm w}$, i.e. $V_c$ and $K(E)$ – or it is escape-dominated and the geometrical upper limit (sole dependence on $L$, not $K$) is recovered; iii) spallations also systematically lessen the $f$-volumes: the heavier the nucleus, the larger its destruction rate, the closer it comes from. However, as a particle created in the halo is less likely to cross the disk, this effect is negligible compared to the wind for $r_{\rm w}\gtrsim r_{\rm sp}$. We show below that all these effects are more pronounced for annihilating SUSY than for evaporating PBH because the density profile $h_{\rm DM}(r,z)$ appears with a square.
Dark matter distribution
------------------------
The dark matter distribution in our Galaxy is poorly known, and several dark matter profiles can be used. The first constraint is that the observed rotation curve of our Galaxy is almost flat beyond a few kpc from the center. For a spherical halo, it follows that the density decreases as $1/r^2$ outside the central regions. In the inner regions, the situation is far from clear. Numerical simulations indicate that the central distribution of dark matter is cuspy, with a $r^{-\gamma}$ dependence with $\gamma \sim 0.5 -
1.5$ [@Ghez98], but this seems to be in contradiction with observations [@Binney01].
In the absence of a clear answer to this problem, we use several profiles for the Dark Matter distribution, with the generic form $$h_{\rm DM}(r,z) = \left(
\frac{R_\odot}{\sqrt{r^2+z^2}}
\right)^\gamma
\left( \frac{R_c^\alpha + R_\odot^\alpha}{R_c^\alpha +
(\sqrt{r^2+z^2})^\alpha}
\right)^\epsilon
\label{dep_spatiale}$$ where spherical symmetry has been assumed. Numerical simulations point toward singular profiles with $\gamma = 1.5$, $\alpha = 1.5$, $\epsilon = 1$ and $R_c=33.2
\, \mbox{kpc}$ [@Moore99] or $\gamma = 1$, $\alpha = 1$, $\epsilon = 2$ and $R_c=27.7
\, \mbox{kpc}$ ([@NFW], hereafter NFW). We also considered an isothermal profile with $\gamma = 0$, $\alpha =
2$, $\epsilon = 1$ and $R_c=3$ kpc (the modified isothermal profile would give very similar results).
As already said, exotic SUSY particles (resp. PBH) are supposed to fill (resp. follow) the dark matter halo profile $h_{\rm DM}(r,z)$. However, the nature of the cosmic ray creation process is different in these two cases, leading to very different effective source terms, i.e. different weight $w(r,z)$ in Eq. (\[proba\_integree\]). For evaporating Primordial Black Holes, the particle production is proportional to the density of the objects $w_{\rm PBH}(r,z) \propto
h_{\rm DM}(r,z)$. In contrast, the production term for supersymmetric particles is proportional to the square of the density because two dark matter particles must be present for annihilation to occur. In this case $w_{\rm SUSY}(r,z) \propto h_{\rm DM}(r,z)^2$.
They are displayed in Fig. \[profils\] both for SUSY and PBH weight (see above).
![Effective source weight (PBH or SUSY) for several profiles (see text).[]{data-label="profils"}](ms3378_f2.eps){width="\columnwidth"}
![Schematic view of our Galaxy: diffusive and convective propagation plus spallations in the thin disk. Effective primary exotic sources follow either the dark matter profile or its square (isothermal profile is depicted).[]{data-label="galaxy"}](ms3378_f3.eps){width="\columnwidth"}
The Moore and NFW profiles are singular at the Galactic center, so that the source term is much stronger there. The probability that a cosmic ray detected in the Solar neighborhood was emitted from this region is enhanced for these profiles. A crude estimate of this effect is obtained by a mere count of the effective (PBH or SUSY) source numbers in this critical region. For example, in the range \[$0-2$\] kpc, a Moore profile leads to an enhancement $\times 2.7$ for PBHs and $\times 90$ for SUSY annihilations, compared to the isothermal case. Stretching this interval decreases the enhancement factor, and for \[$0-4$\] kpc, it is respectively $\times 1.5$ and $\times 25$, and finally for \[$0-8$\] kpc, the numbers are $\times 1.1$ and $\times 20$. The enhancement is far smaller for PBH than for SUSY particles. Notice that the upper limits on the PBH density derived from antiproton flux measurements in @Barrau02 were of the same order of magnitude for an isothermal halo and for cuspy halos. This result is definitively not transposable to the SUSY case.
This is not the final word. The center of our Galaxy contains a supermassive black hole (SBH) of a few $10^6$ M$_\odot$. During its formation, it probably accreted the surrounding dark matter, leading to a local enhancement of the density. Gondolo & Silk (1999) (hereafter GS) found that if the SBH grows adiabatically in the center of the Galaxy, the cuspy profile ($\rho(r)\propto r^{-\gamma}$ with $0<\gamma<2$) becomes spiky and $\rho(r)\propto
r^{-A}$ with $2.25<A<2.5$ in a region of a few parsecs around the black hole. The presence of the spike would have dramatic consequences for several predictions of the signal from annihilating dark matter particles, e.g. $\gamma$ and neutrinos [@Gondolo_Silk99] or synchrotron emission of $e^+e^-$ pairs [@Gondolo00; @Bertone01]. The signal coming from the direction of the Galactic Center is obtained by integrating along the line of sight, and the contribution of the central region is very different with or without a spike. In the case of the isothermal profile, the central region (around the SBH) contributes at the level of $\sim 10^{-9}$ whereas this contribution is greater than $\sim 10^5$ for a Moore profile [@Gondolo_Silk99]. However, these results are expected to be overoptimistic, and it is doubtful that such a spike exists in our Galaxy, as indicated by a more careful dynamical modelling of the SBH growth [@Ullio01]. These authors review several effects (adiabatic growth versus instantaneous growth, models with off-centered black holes) and recover some results that were known before the Gondolo & Silk paper: only the peculiar case in which the SBH forms adiabatically at the exact center of the dark matter profile can lead to an enhancement such as described in GS. Finally, in a recent study, @Merritt02 have observed that, taking into account the quite large probability that the Milky Way experienced a major merger in its history, the ensuing dark matter profile and resulting annihilation fluxes could be several order of magnitudes smaller than obtained with dark matter profile not disturbed by a SBH.
The points discussed above are mostly relevant for particles travelling in straight lines. For charged particles, due to the diffusive nature of propagation, the probability to come from a sphere ${\cal S}$ of radius $r=10$ pc around on the Galactic center ($\sim 8$ kpc away) is $\int_{\cal S}(d{\cal P}/d^3\vec{r}_s)d^3\vec{r}_s$, which is $\lesssim 10^{-10}$ ($d{\cal P}/d^3\vec{r}_s$ is given for example in Fig. \[fig:prim\_wind\_spal\]). Due to the very narrow scale where the SBH may affect the distribution, even enhancement such as obtained in @Gondolo_Silk99 – and which is not very realistic – cannot yield a significant contribution for charged particles. Eventually, the dark matter profile remains of importance (isothermal or cuspy). In the following, most results will be presented for the isothermal case, the influence of the cusp being discussed at the end.
$f$-volumes for SUSY and PBH weights and different values of $L$ {#susy_et_les_jaunes}
----------------------------------------------------------------
We now have all the elements to compute the $f$-volumes, inserting the source distributions described above in Eq. (\[proba\_integree\]). The function entering the integral does not possess cylindrical symmetry, so that the full three-dimensional integral must be computed. We first neglect spallations and galactic wind to consider only the effect of $L$. This parameter is expected to play an important role, as the charged particles created outside of the magnetic halo of our Galaxy do not penetrate inside it and are not detected [@Barrau02; @Barrau03].
![Contours ${\cal P}({\cal V_{\rm SUSY,~PBH}}|R_\odot)=99\%$ origin for $L=2$, $L=5$ and $L=10$ kpc. $f$-volumes have been respectively evaluated with weight $w_{\rm PBH}(r,z)\propto h_{\rm DM}(r,z)$ (solid lines) and $w_{\rm SUSY}(r,z)\propto h^2_{\rm DM}(r,z)$ (dashed lines) for $R\gg L$, but the result remain mostly unchanged using $R=20$ kpc (but in case of a halo size $L=10$ kpc that requires $R=30$ kpc). Upper panel: $V_{\rm SUSY}(99\%)$ and $V_{\rm PBH}(99\%)$ in the $x_s-y_s$ plane ($z_s=0$). Lower panel: same quantities but in the $x_s-z_s$ plane ($y_s=0$). In both panels, the dot marks the Galactic center, and $\odot$ denotes the Sun location (it is set to $R_\odot=8$ kpc).[]{data-label="proba_vol_weighted"}](ms3378_f4.eps){width="0.5\columnwidth"}
Fig. \[proba\_vol\_weighted\] shows the $99$%-volumes in the Galactic plane $z_s=0$ (upper panel), for the PBH and SUSY case. Their shape reflects the fact that the probability density has a maximum at $r_s=R_\odot$, while the effective source distribution peaks at the Galactic center $O$. Because of the quadratic dependence on $h_{\rm DM}(r,z)$, ${\cal V}_{\rm SUSY}(99$%) is smaller than ${\cal V}_{\rm PBH}(99$%). Three halo sizes are displayed ($L=2$, $5$ and $10$ kpc): for larger halos, the surfaces are more deformed towards the Galactic center (the contribution of this region to the flux is larger), whereas they remain grossly unaffected in the anti-center direction. This effect is less pronounced in the case of a PBH-like source distribution. The same contours are also plotted for $y_s=0$ in the lower panel. The shapes are almost maximally distorted towards rectangular contours. This is less and less pronounced, as either $L$ is enhanced, or larger powers of $h_{\rm DM}(r,z)$ are chosen.
The figures above show clearly that we are only sensitive to a well-defined region of the source distribution: first to the region which is embedded in the diffusive halo, and then, even within this region, to a sub-region between the Galactic center and the Solar neighborhood. These sub-regions represent a fraction of the total number of sources given by $$f^{\rm tot}(L) = \frac{\int_{{\cal V}(99\%)} w(r,z)\;
d^3\vec{r}}{\int w(r,z)\; d^3\vec{r}}\;,$$ where $\int w(r,z)\; d^3\vec{r}$ is the total number of sources. It is also of interest to compare the number of sources located in the same sub-regions to the number of sources in the diffusive halo $$f^{\rm cyl}(L) = \frac{\int_{{\cal V}(99\%)} w(r,z)\;
d^3\vec{r}}{\int_{\cal V_{\rm cyl}} w(r,z)\; d^3\vec{r}}\;;$$ where ${\cal V}_{\rm cyl}$ is the volume of the diffusive halo. The corresponding numbers are given in Table \[tab1\] for various halo sizes.
------------ -------------------------- ------------------
$f^{\rm cyl}(L)$ $f^{\rm tot}(L)$
PBH / SUSY PBH / SUSY
$L=10$ kpc $\sim 1.$ $\sim 1.$ 0.023 0.76
$L=5$ kpc 0.70 0.85 0.010 0.54
$L=2$ kpc 0.31 0.60 0.002 0.21
------------ -------------------------- ------------------
: Fraction of the number of exotic primaries emitted in $V_{\rm PBH}(99\%)$ and $V_{\rm SUSY}(99\%)$ for various $L$, compared to the total number of exotic primaries emitted either in the bounded geometry (halo size $L$ and radial extension $R=20$ kpc) – denoted $f^{\rm cyl}(L)$ –, or in the whole dark halo – denoted $f^{\rm tot}(L)$.[]{data-label="tab1"}
The fraction $f^{\rm cyl}(L)$ decreases with $L$, much faster for PBH than for SUSY. This can be understood as the number of contributors outside of the dark halo core radius rapidly vanishes for SUSY particles (see Fig. \[galaxy\]). As regards the results for $f^{\rm tot}(L)$, we remark that this number is particularly small for PBH, i.e. only a very small fraction of primordial halos distributed in the Galaxy contribute to the charged primary cosmic rays detected in the Solar neighborhood.
Finally, it is also interesting to give the fraction of primaries that escape before reaching the Solar neighborhood. It is defined as $$f^{\rm esc}({\cal V})\equiv 1 -
f^{\rm detect}({\cal V})
= 1- \frac{\int_{{\cal V}} w(r,z)
N_{\rm cyl}(\vec{r}|R_\odot)\;d^3\vec{r}}{
\int_{{\cal V}} w(r,z)
N_{\infty}(\vec{r}|R_\odot)\;d^3\vec{r}} \;\;,
\label{rir_SN}$$ where $N_{\rm cyl}(\vec{r}|R_\odot)$ and $N_{\infty}(\vec{r}|R_\odot)$ are respectively related to the flux of particles detected at $R_\odot$, in the cylindrical geometry and in an unbounded space, from the same sources emitting from inside the volume ${\cal V}$. Estimations for $L=10$ kpc and $L=2$ kpc are compiled in Tab. \[tab2\].
$L=10$ kpc $L=2$ kpc
----------------------------------------------- ------------ -----------
$f^{\rm esc}_{\rm PBH}:~{\cal V}$(50-90-99%) 40-55-64% 45-75-88%
$f^{\rm esc}_{\rm SUSY}:~{\cal V}$(50-90-99%) 49-52-55% 59-92-95%
: Fraction of primaries $f^{\rm esc}$ emitted from the (50-90-99)%-volumes of the cylindrical geometry (see above) that escape through upper and lower boundary located at $L=10$ kpc or $L=2$ kpc (for PBH and SUSY effective source distribution), before they can reach the Solar neighborhood.[]{data-label="tab2"}
The trends are conform to intuition. Forming greater fractions of the detected flux requires more distant sources, the latter more easily escape through boundaries. For large diffusive halo $L$, the fraction that escape increases more quickly for PBH sources than for SUSY sources, whereas the converse is true for small halos. This is related to the fact that one has to compare the shape and typical extension of the source distribution to the parameter $L$. The fraction of primaries which are emitted inside ${\cal V}$ but which never reach the solar neighborhood is actually greater than $f^{\rm esc}$, as even in the case of diffusion in unbounded space, there are many trajectories which start in ${\cal V}$ and never reach the solar neighborhood (diffusion in three dimensions is a transient process).
Realistic propagation parameters {#BC_induced}
================================
The previous section considered simplified diffusion situations with a typical value $K\sim0.03$ kpc$^2$ Myr$^{-1}$. Actually, $K(E)$ is energy dependent, and more precisely, $$K(E)=K_0\beta {\cal R}^\delta.$$ Here, $\delta$ is the diffusion slope and $K_0$ the normalization of the diffusion coefficient. In a previous study (see Paper Ia, Ib), we show that various combinations of parameters $K_0$, $\delta$, diffusive halo height $L$ and Galactic wind magnitude $V_c$ are equivalent, in the sense that they give a B/C spectrum that is consistent with the observations. In this section, we use these combinations to provide a realistic range of values for $r_{\rm w}$ and $r_{\rm sp}$ and to explore the consequences on the origin of exotic primary antiprotons and antideuterons. The heavier antinuclei will not be considered here, as it was shown by Chardonnet, Orloff & Salati (1997) that their formation is suppressed because of the low probability of coalescence of many antinucleons.
Parameter range allowed
-----------------------
To compute the parameters introduced in Eqs. (\[f\_rspal\]), the spallation cross sections of antiprotons and antideuterons are taken from the Particle Data Group[^1]. In this work, we only consider spallation on pure hydrogen. It would be straightforward to take into account the spallations on the Helium component of the interstellar medium, but the effect is too small to be worth the complication. The four parameters $K_0$, $\delta$, $L$ and $V_c$ are taken from our comprehensive study of standard secondary to primary B/C ratio [@Maurin02b]. Three values (two extremes and a medium value) have been retained for both the diffusion slopes ($\delta=0.35$, 0.60 and 0.85) and the halo sizes ($L=2$ kpc, $L=6$ kpc and $L=10$ kpc). We emphasize that the values of all these parameters come from the study of standard sources of cosmic rays and do not depend on the exotic sources, which do not produce B nor C. We do not take reacceleration and energy losses into account in this work. These effects, though necessary to study the spectra of cosmic rays, are not so crucial here as they only amount to a redistribution of the cosmic rays at different energies. A particle detected at an energy of 1 GeV/nuc was just created at a slightly different energy and its origin is not drastically different.
The values of $r_{\rm sp}$ and $r_{\rm w}$ are plotted in Fig. \[fig:rw\_rs\] for antiprotons and antideuterons. The left panel shows that propagation is convection-dominated ($r_{\rm w}\ll 1$) at low energy when large $\delta$ values are considered and escape-dominated at all energies for small $\delta$. Notice that although at a given $\delta$, the quantity $r_{\rm w}/L$ is fairly independent of $L$, the origin is definitely not the same for $L=2$ kpc as for $L=10$ kpc.
![Left panel: evolution of $r_{\rm w}$ as a function of kinetic energy per nucleus for primary $\bar{p}$ and $\bar{d}$; from top to bottom, $\delta=0.35$, $\delta=0.60$ and $\delta=0.85$. The parameter $r_{\rm w}/L\equiv\chi_{\rm w}$, as well as $r_{\rm sp}/L\equiv\chi_{\rm sp}$, are not very sensitive to the halo size $L$ (for $\bar{d}$, only $L=6$ kpc is displayed) but $r_{\rm w}$ and $r_{\rm sp}$ do. Right panel: $r_{\rm sp}/L$ as a function of $E_k$/nuc for the same $\delta$ values and for the halo size $L=6$ kpc. The values of $r_{\rm w}$ are different between $\bar{p}$ and $\bar{d}$ because they depend on rigidity (through $K$), i.e. on $Z/A$ (it is $1$ for $\bar{p}$ and $1/2$ for $\bar{d}$). For $r_{\rm sp}$, there is an additional strong dependence on the species because of the destruction cross sections.[]{data-label="fig:rw_rs"}](ms3378_f5.eps){width=".85\columnwidth"}
The right panel shows that spallation is not the dominant effect for the light nuclei considered here. Only for large diffusion slopes $\delta$ and more particularly for antideuterons this effect becomes sizeable and comparable to the diffusive escape. The comparison of the two panels shows that spallations are always less efficient than convective wind or boundary escape. Finally, whatever the value of $\delta$, propagation is escape-dominated above a few tens of GeV/nuc and the origin of primary cosmic rays is solely dependent on the halo size.
Antiprotons and antideuterons {#pbar_dbar_realiste}
-----------------------------
We are now able to draw the $f$-volumes for the realistic propagation parameters being considered. We focus on the antideuteron signal as it seems to be the most promising species to look for in cosmic rays. An interstellar energy of 1 GeV/nuc is chosen; the nuclei that reach the detector are solar modulated so that they are detected with a final energy of $400-800$ MeV/nuc, where the signal is the more interesting. Table \[tab3\] summarizes the values of $r_{\rm w}$ and $r_{\rm sp}$ at this energy for antideuterons.
(kpc) $\delta=0.35$ $\delta=0.60$ $\delta=0.85$
------------ --------------- --------------- --------------- ---------------
$L=10$ kpc $r_{\rm w}=$ $\infty$ 8. 2.9
$r_{\rm sp}=$ 21. 7.6 3.5
$L=6$ kpc $r_{\rm w}=$ $\infty$ 5.5 2.1
$r_{\rm sp}=$ 15.5 5.5 2.6
$L=2$ kpc $r_{\rm w}=$ $\infty$ 2.1 0.85
$r_{\rm sp}=$ 6. 2.2 1.05
: $r_{\rm w}$ and $r_{\rm sp}$ for three halo sizes $L$ and three diffusion slopes $\delta$: these numbers are for 1 GeV/nuc (interstellar energy) antideuterons.[]{data-label="tab3"}
The situation is very different for small or large $\delta$. For small values (corresponding roughly to a Kolmogorov power spectrum $\delta = 1/3$), only spallations affect the propagation ($V_c=0$, $r_{\rm w}=\infty$) and this effect was shown to be weak; for large $\delta$ – the value $\delta=0.85$ is the one preferred in our B/C analysis (Paper Ib) –, models are convection-dominated though $r_{\rm sp}$ and $r_{\rm w}$ have about the same strength.
Fig. \[fig:final1\] displays ${\cal P}_{\rm cyl}({\cal V}(f)|O)=99$% for the values reported in Tab. \[tab3\]. For $\delta=0.35$ (external contours), the [*geometrical*]{} (upper limit) contours are recovered. However, for larger $\delta$ (internal contours), these contours shrink. All comments made in Fig. \[proba\_vol\_weighted\] as regards halo size, or SUSY and PBH behavior, remain valid. Actually, the diffusion coefficient slope $\delta$, as $L$ for the geometrical limit, is a key parameter to trace back the CR origin, because of the values of $r_{\rm w}$ it implies, through $V_c$ and $K_0$.
![99%-volumes for exotic primaries (no side boundaries). Upper panels: cut in the $z_s=0$ kpc plane; lower panels: cut in the $y_s=0$ kpc plane. Left panels correspond to $L=2$ kpc, middle panels to $L=6$ kpc and right panels to $L=10$ kpc. In each panel, we plot either the PBH case (solid lines) or the SUSY case (dotted lines). From external lines to internal lines correspond the values of the diffusion coefficient slope $\delta=0.35$, $\delta=0.60$, $\delta=0.85$.[]{data-label="fig:final1"}](ms3378_f6.eps){width="0.85\columnwidth"}
It is also of interest to have a closer look at the first % that contribute to the flux. As the $f$-volumes with $f\lesssim50$% correspond to isodensity contours that are quite insensitive to the boundaries (or to other effects) they present the axial symmetry around the $x_s$ axis, so that a single cut through, e.g. the $x_s-z_s$ plane, delivers all the information about their shape. Fig. \[fig:final3\] displays the $f$-volumes $f=$10-25-50-75% for $L=10$ kpc. The difference observed in Fig. \[fig:final3\] between small (lower panels) and large $\delta$ (upper panels) is readily explained: a large value of $\delta$ also corresponds to a large value of $K_0$ (see @Maurin03a for details), so that a greater wind is needed in order to prevent from too many spallations occurring at low energy. The net result is that the wind blow the particles away, reducing the effective zone from where they come from. This is not the case for small $\delta$ where the [*geometrical*]{} limit (pure diffusion) is almost reached.
![${\cal P}_{\rm cyl}({\cal V}(f)|O)=10\%-25\%-50\%-75\%$ for $L=10$ kpc in the $z_s=0$ plane (except for the $75$%-volumes, other $f$-volumes with $f\lesssim50$% are not deformed by the boundaries so that they present symmetry around the $x_s$ axis). Upper panels correspond to $\delta=0.85$, and lower panels to $\delta=0.35$. Both the PBH case (solid lines) and the SUSY case (dotted lines) are plotted.[]{data-label="fig:final3"}](ms3378_f7.eps){width="1.\columnwidth"}
The consequences for indirect dark matter searches are important. In the case of an isothermal profile (left panels), the particles created in the Galactic center have a small probability to reach the detector for large $\delta$, whereas the converse is true for small $\delta$. In the latter case, the predictions and the limits that can be put on a supersymmetric signal depend heavily on the central shape of the dark matter halo, which is precisely the part we know the least about. These contours for smaller diffusive halo sizes $L$ have not been presented; they have a smaller extent, meaning that we are less sensitive to the distribution of dark matter far from the Solar neighborhood. As a result, the question of the dark matter density profile cusp is less crucial for small $L$.
Similar contours for the NWF profile are drawn in the right panels of Fig. \[fig:final3\]. Combining information from the above surfaces to the relative enhancement of sources going from the isothermal case to the cuspy case allows several complementary remarks: for small $\delta$, about half the SUSY Cosmic rays come from the range \[$0-3$\] kpc. Thus, the $\sim 50$ enhancement factor on the production provided by the cusp translates directly into a factor $\times 50$ in the detected flux. For PBH case, the origin is less localized and the enhancement factor is smaller, so that the net gain is more probably about $10-20$%. For large $\delta$, contours look like boxes encompassing both the Solar position and the Galactic center. In the SUSY case, the addition of a cusp strongly deforms the box towards the Galactic center, but it is not straightforward to estimate the enhancement without considering specific values for the diffusion parameters. For PBH, the contours, and hence the flux, are not expected to be very sensitive to the parameters. This discussion is of less importance for small halo sizes.
From the above discussion, it appears that the most important parameters are $L$ and $V_c/2K$. The value $\delta = 1/3$ (Kolmogorov spectrum) corresponding to $V_c = 0$, has been preferred these last years (see e.g. @Strong98). However, our previous studies [@Maurin01; @Maurin02b; @Maurin03a] show that large values of $\delta$, and non-null values of $V_c $, are preferred. This trend is confirmed the most recent results of @Moskalenko02 who now tend to prefer $\delta=0.42-0.52$. To conclude, if the value of $\delta$ happens to be large or more precisely if a strong Galactic wind is preferred, the discussions about the dark matter profile, including about the existence of a spike, are not so crucial. If conversely $\delta$ is small (no wind), the dark matter cusp as well as the exact location of the Solar system should be accurately known before exploring the SUSY parameter space.
Finally, all the remarks made for antideuterons in the previous sections apply as well for antiprotons. According to Fig. \[fig:rw\_rs\], for a given $\delta$ at a given energy, the corresponding $r_{\rm w}$ is about twice its antideuteron value. The resulting $f$-volumes are larger than those for antideuterons, but the conclusions remain the same.
Electrons and positrons {#subsec:electrons_positrons}
-----------------------
Exotic sources in the halo also emit electrons and positrons. Positrons are more promising to study supersymmetric signals as the background of standard positrons is much lower than electron’s ($e^+/(e^++e^-)<0.1$), being predominantly secondary. Recently, the [heat]{} experiment [@Coutu99] reported an excess at about 7 GeV (see also the [mass]{}-91 experiment, @Grimani02).
These particles are lighter than nuclei, so that they are subject to much stronger energy losses, due to synchrotron radiation and inverse Compton. This results in an effective lifetime given by [@Aharonian95; @Atoyan95; @Baltz99] $$\tau_{\rm loss} \sim 300 {\; \mbox{Myr}} \times \frac{1 {\; \mbox{GeV}}}{E} \,.$$ @Aharonian95 and @Atoyan95 showed that in that case, all boundaries have negligible effects on positrons and electrons above a few GeV, so that the characteristic distance travelled by these species is $r_{\rm loss} \sim \sqrt{K \tau_{\rm loss}}$ (random walk through the tangled magnetic fields), or $$r_{\rm loss} \sim 1 {\; \mbox{kpc}} \times \sqrt{\frac{1 {\; \mbox{GeV}}}{E}}
\sqrt{\frac{K}{0.03 {\; \mbox{kpc}}^2{\; \mbox{Myr}}^{-1}}}\;.$$ The result is an exponential cutoff that depends on the energy, i.e. the probability density reads $d{\cal P}_{\rm rad}/d^3\vec{r}_s\propto
\exp(-r_s/r_{\rm loss})/r_s$ (see also Sec.4.3 in @Taillet03). In the case considered here of sources in the whole diffusive volume, the normalized probability density is given by $$d{\cal P}_{\rm rad}=
\frac{\exp(-r_s/r_{\rm loss})}{4\pi r_s \,.\,r_{\rm loss}^2} d^3\vec{r}_s\;.$$ It is quite different from the case of a source distribution located in the disk only (see Eq. (12) in @Taillet03). The resulting $f$-volumes (spheres) are given by $${\cal P}_{\rm rad}(r<r_{\rm lim}|O)=1-\left(1+\frac{r_{\rm lim}}{r_{\rm loss}}\right)
\exp\left(-\frac{r_{\rm lim}}{r_{\rm loss}}\right)\;.$$ It means that sources that contribute to the fraction $f=(50-90-99)$% of the detected positrons emitted in the halo are located inside the sphere of radius $r_{\rm lim}\approx(1.7-4.8-6.6)\times r_{\rm loss}$. For the realistic values of $K(E)$ used above (see also Paper Ib), we compile in Tab. \[tab\_positrons\] the range covered by $r_{\rm loss}$ at $E=7$ GeV.
$\delta=0.35$ $\delta=0.60$ $\delta=0.85$
------------ --------------- --------------- ---------------
$L=10$ kpc 1. .65 .48
$L=6$ kpc .85 .55 .41
$L=2$ kpc .53 .35 .26
: The quantity $r_{\rm loss}$ (kpc) is given for three halo sizes $L$ and three diffusion slopes $\delta$ at the total energy $E=7$ GeV.[]{data-label="tab_positrons"}
Because of the very small scale involved along with the exponential decrease, $f$-volumes for positrons are expected to be only slightly deformed by the dark matter distribution, except for small $\delta$ and large $L$ whose 99%-volumes extend up to $\sim 7$ kpc.
It is possible now to make a few quantitative comments on the HEAT results and on the conclusion of @Baltz02 about this signal. They argued that, defining a boost factor related to the clumpiness of dark matter, one can accommodate with $e^+$ data without enhancing too much the antiproton signal. The point is that antiprotons come from further than positrons, so that if a clump exists close to us, its contribution of antiprotons is averaged over a larger zone than positrons. A comparison of Figs. \[fig:final1\] and numbers presented above gives a relative distance $$r^{\rm origin}_{e^+,~e^-}/
r^{\rm origin}_{\bar{p},~\bar{d}}\sim 0.1$$ for all reasonable stationary propagation models. However, considering large or small $\delta$, the effect of the clumpiness factor is expected to be different in different propagation models. Hence, the enhancement factor for the antiproton signal used in @Baltz02 should also depend on the diffusion efficiency, i.e. combination of diffusion plus convection (that is not considered in the above reference). To summarize, the relation between SUSY positron and antiproton signals is not straightforward, if the dark matter halo is clumpy. Thus it seems a hard task to combine constraints from these two different signals, unless they are obtained with the same analysis. Depending on $\delta$ and $L$, their origin is more or less local, and the size of the clumps as well as the typical distance between the clumps may be of importance.
Summary and conclusions
=======================
This paper analyzes the spatial origin of exotic particles created from a dark matter profile. We presented the $f$-volumes inside which a given fraction of the cosmic rays detected in the Solar neighborhood were emitted. At high energy ($E \gg 1$ GeV/nuc), the shape of the isodensity surfaces is set by the geometry of the diffusive halo, in particular on its height $L$, the influence of the side boundary at $r=R$ being small. We then showed that the $f$-volumes defined are smaller when spallations and convection are taken into account, but in a very different way: for particles in the diffusive halo, the wind exponentially decreases the probability of reaching the Galactic plane, whereas spallations have about a null effect on the latter. The parameters $L$ and $2V_c/K$ indicate whether the propagation is convection or escape-dominated. In Table \[tab5\] we summarize the parameters that act as a cut-off in various situations.
--------- ---------------------- -------------------------------- --------------------------------
Cut-off Escape-dominated Convection-dominated Losses-dominated
($\chi_{\rm w}\gg1$) ($\chi_{\rm w}\ll1$) ($e^-e^+\gtrsim$ GeV)
Halo $L$ $L^*\approx 3K/V=3r_{\rm w}/2$ $
\approx 5r_{\rm loss}$
Radial $\min(R,3L)$ $\min(R,3L^*)$ $\approx 5
r_{\rm loss}$
--------- ---------------------- -------------------------------- --------------------------------
Two source distribution for the isothermal dark matter profile were considered: production related to the density of the source (e.g. PBH evaporation), or production related to the square of the density of the sources (e.g. SUSY annihilation). The 99%-volumes are strongly stretched toward the Galactic center, corresponding to the maximum of the source distribution. This follows from the competition between the effective source which is maximum at the Galactic center, and the probability density which steadily decreases from our position $R_{\odot}$ to reach $\sim 10^{-4}-10^{-5}$ kpc$^{-3}$ for purely diffusive regime (or even less when convection is included) at the Galactic center. The fluxes in the Solar neighborhood are found to be far more sensitive to the dark matter profile in the SUSY case than in the PBH case. In both cases, the side boundary of the diffusive volume is observed to play a negligible role as long as $R\gtrsim 20-30$ kpc.
As a last step, realistic propagation parameters were implemented, and the key parameters were found to be the halo size $L$ and the diffusion slope $\delta$ (actually $V_c/K_0$). For the species considered here (antiprotons and antideuterons), spallations always play a negligible role in the origin. It was found that this origin is far more local in case of large $\delta$ and small $L$ than in case of small $\delta$ and large $L$. Moreover, the shape of the dark matter distribution near the Galactic center does not matter so much for the PBH case, whereas it may be crucial for SUSY annihilating particles. We emphasized that in any discussion of the annihilation signal in charged particles, the propagation parameter $\delta$ or more precisely, the presence of a Galactic wind, should be considered, with the same importance of the parameter $L$ or the choice of the dark matter profile.
Two last points are worth noting. First, even though the work presented here does not allow a quantitative estimation of the effect of possible clumpiness of the dark matter halo (for SUSY annihilations), we observed that the comparison between the electron and antiproton SUSY signals should involve a careful inspection of the corresponding boost factors. Second, whereas the use of B/C-induced propagation parameters is justified for standard antiprotons (corresponding $f$-surfaces can be seen in @Taillet03), there is no guarantee that these parameters are valid in the $f$-volumes depicted here.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has benefited from the support of PICS 1076, CNRS and of the PNC (Programme National de Cosmologie).
Numerical evaluation of the point source solution in Bessel basis {#numerique}
=================================================================
One needs to evaluate numerically point source solutions such as $$N^{\rm cyl}_{\delta}(r,z)=\frac{1}{\pi KR} \;
\sum_{i=1}^{n_{\rm tronc}}
\frac{J_0\left(\zeta_i r/R\right)}{\zeta_i J_1^2(\zeta_i)} \frac{\sinh
\left[\zeta_i(L-z)/R\right]}{\cosh
\left(\zeta_i L/R\right)}
\label{cyl_pure_diff}$$ In the above expression, $(r,z)$ is the position of the $\delta$ source in polar coordinate and $R$ is the radial extension of the Galaxy. $N^{\rm cyl}_i(z)$ can be evaluated for each $i$ and need to be summed till the $n_{\rm tronc}$-th order, which should formally tend to infinity. For evident reasons, $n_{\rm tronc}$ is chosen to be the smallest possible with the constraint that the rebuilt series $N^{\rm cyl}_\delta(r,z)$ has reached a good convergence. In the case of $\delta(\vec{r})$ point source, profiles are singular near the source and convergence of the series appear to be very slow. The ansatz depicted in @Taillet03 is useless as soon as sources are outside the disk. We present below two alternatives to evaluate this sum.
Average value of the oscillating series with $r$
------------------------------------------------
In analogy with classical Fourier analysis, resummation of coefficients provide oscillating behavior around the [*true*]{} value. This can be understood if we recall that at the $n$-th order, the function added is $\propto J_0(x\equiv\zeta_n\rho)$: $\rho$ lying in $[0-1]$, the argument of $J_0$ takes values $x=\{ \zeta_1,\zeta_2,\dots\zeta_n\}$, i.e. at the $n$-th order, the corrective function has $n$ roots. Thus convergence will be more quickly reached if for a given order $n_{\rm cutoff}$, instead of evaluating $N^{\rm cyl}_{\delta}(r,z)$, one averages $$N^{\rm cyl}_{\delta}(r_n,z)=\frac{N^{\rm cyl}_{\delta}(r_{n-1},z)+
N^{\rm cyl}_{\delta}(r_{n+1},z)}{2}\;;$$ where $r_{n-1}$, $r_n$ and $r_{n+1}$ are ordered realizations of $r$. The sole condition is that the $\{r_n\}^{n=1,\dots n_{\rm cutoff}}$ belong to the grid $r=\{0,\; R/(2 n_{\rm cutoff}),\;
2R/(2 n_{\rm cutoff}),\dots, \;R\}$, i.e. $2 n_{\rm cutoff}$ linear steps between 0 and 1. To summarize, around the oscillating value, if the appropriate step is chosen, it ensures that the averaged two points are not both above or below the true value, and furthermore, that two opposite extrema of the oscillating function are averaged.
Step-like source: $\theta$ function
-----------------------------------
An alternative way is to consider solution from a step-like source, e.g. $\theta(a-r)$, in order to smooth the problematic behavior observed near the origin for the $\delta$ source. With the suitable normalization in the source term, i.e. $$q_{\theta}(r,z)=\frac{\theta(a-r)}{\pi a^2}\delta(z)\;\;,$$ and using the property $\int \rho J_0(\rho)d\rho=\rho J_1(\rho)$, it leads to a solution which is equivalent to the delta solution $N^{\rm cyl}_{\delta}(r,z)$ – Eq. (\[cyl\_pure\_diff\]) –, as long as the distance $r_o$ of the observer $X_o$ from the source satisfies the relation $r_o\gg a$. The Bessel coefficients of $\delta$ and $\theta$ solutions are related through $$N_i^{\theta}(z_o)=2\times\frac{J_1(\zeta_i a/R)}{(\zeta_i a/R)}
\times N_i^{\delta}(z_o)\;.
\label{theta_delta}$$ The acceleration of convergence can be understood as, in Eq. (\[theta\_delta\]), the additional term behaves at least as $1/i$ ($J_1$ is bounded and $\zeta_i\approx i\pi$). Here $a$ should be taken such as to verify $a/R\ll 1$ (with $R=20$ kpc for the Galaxy, one can safely take $a\sim10$ pc).
Thus, a $\theta$-like source slightly underestimates the result close to $\vec{r}=\vec{r_s}$, but this zone corresponds to very small volumes that add a negligible contribution when one evaluates integrated probabilities. For practical purposes, both methods (average or $\theta$ source) give the desired results with about the same number of Bessel functions, i.e. $n_{\rm cutoff}\sim 100$.
[^1]: http://pdg.lbl.gov/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove necessary conditions on pairs of measures $(\mu,\nu)$ for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p<\infty$, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Stein [@MR1232192], and the measure $\mu$ satisfies a weak doubling condition related to the non-degeneracy of the kernel. We also show similar results for pairs of measures $(\mu,\sigma)$ for the operator $T_\sigma f = T(f\,d\sigma)$, which has come to play an important role in the study of weighted norm inequalities. Our major tool is a careful analysis of the strong type inequalities for averaging operators; these results are of interest in their own right. Finally, as an application of our techniques, we show that in general a singular operator does not satisfy the endpoint strong type inequality $T : L^1(\nu) \rightarrow L^1(\mu)$. Our results unify and extend a number of known results.'
address:
- |
David Cruz-Uribe, OFS\
Department of Mathematics\
University of Alabama\
Tuscaloosa, AL 35487, USA
- |
John-Oliver MacLellan, Department of Mathematics, University of Alabama\
Tuscaloosa, AL 35487, USA
author:
- 'David Cruz-Uribe, OFS'
- 'John-Oliver MacLellan'
bibliography:
- 'necessity.bib'
date: '04/18/2020'
title: Neccessary conditions for two weight inequalities for singular integral operators
---
Introduction
============
The goal of this paper is to establish necessary conditions for two weight, weak type inequalities for Calderón-Zygmund operators. This problem has a long history. In the one weight case it is well known that if each of the Riesz transforms is of weak type $(p,p)$ with respect to a weight $w$, then $w \in A_p$. See for example [@MR807149 Theorem 3.7, p. 417]. Stein [@MR1232192 p. 210], showed that if any convolution type singular integral operator whose kernel satisfies a weak non-degeneracy condition is bounded on $L^p(w)$, then $w \in A_p$. The necessity of two weight $A_p$ for the weak $(p,p)$ inequality for the Hilbert transform was established by Muckenhoupt and Wheeden [@MR0417671].
In this paper we consider two versions of this problem. First suppose $(\mu, \nu)$ is a pair of positive regular Borel measures on $\mathbb{R}^n$, where $\mu$ satisfies a weak doubling condition (see Definition \[defi:directionaldoubling\]) and $T$ is a Calderón-Zygmund operator whose kernel satisfies a weak non-degeneracy condition (see Definition \[defi:nondegeneracy\]). We first prove the following result on the necessity of the two-weight $A_p$ condition (see Definition \[def:A\_p\]).
\[thm:necessity measures\] Let $T$ be a Calderón-Zygmund operator with a non-degenerate kernel in the direction $u_0$. Suppose that for some $ 1 \leq p < \infty$, and a pair of positive regular Borel measures $(\mu,\nu)$, with $\mu$ directionally doubling in the direction $u_0$, $$\|Tf\|_{L^{p,\infty}(\mu)} \leq C \|f\|_{L^p(\nu)}.$$ Then:
1. $d\nu=d\nu_s+vdx$ where $v \in L^1_{\text{loc}}$ and $\nu_s$ is singular;
2. $\mu \ll \nu $, and $\mu \ll dx$, so $d\mu=u dx$ where $u\in L^1_{\text{loc}}$;
3. $(u,v) \in A_p$ and $u(x) \leq Cv(x)$ a.e.
The second version of the problem is to let $( \mu,\sigma)$ be a pair of positive regular Borel measures on $\mathbb{R}^n$, and consider the singular integral operator $T_\sigma$ defined by $$T_\sigma f(x)= T( f\,d\sigma)(x) = \int_{\mathbb{R}^n} K(x,y) f(y) \; d\sigma(y).$$ This approach to weighted norm inequalities first appeared implicitly in [@MR676801] for the maximal operator. We establish necessary conditions on $(\mu,\sigma)$ for $T_\sigma$ to satisfy the weak type inequality $$\mu(\{ x: |T_\sigma f(x)| > \lambda \}) \leq \frac{C}{\lambda^p} \int |f|^p d\sigma$$ for $1<p<\infty$. (See Definition \[def:truncated weak (p,p)\] for a more careful definition of this operator and the meaning of this inequality.) This problem was considered in [@MR2957550], where they proved the necessity of the $A_p$ condition for measures (see Definition \[def: Ap condition for measures\]) assuming a strong ellipticity condition. More precisely, they assumed that there exists a family of kernels $\{K_j\}_{j=1}^N$ such that given any unit direction vector $u$, there exists $j$ such that $K_j$ satisfies a non-degeneracy condition in the direction $u$ (see Definition \[defi:nondegeneracy\]). In [@MR3470665], they were able to prove the stronger $PA_p$ condition (see Definition \[def:Poisson Ap\]) is necessary with a similar hypothesis assuming a strong $(p,p)$ inequality. Our results are similar but we only assume that we have a single operator $T_{\sigma}$ whose kernel is non-degenerate in one direction. We obtain the necessity of $A_p$ under the additional hypothesis that $\mu$ satisfies a weak doubling condition related to the non-degeneracy condition (see Defintion \[defi:directionaldoubling\]). We prove this result for completeness since it is a simple application of the techniques used to prove Theorem \[thm:necessity measures\].
\[thm:necessity Tsigma\] Let $(\mu,\sigma)$ be a pair of positive regular Borel measures with $\mu$ directionally doubling in the direction $u_0$. Suppose the operator $T_\sigma$ has a non-degenerate kernel in the direction $u_0$, and that for some $ 1 < p < \infty$, $$\label{weak(p,p) Tsigma}
\|T_\sigma f\|_{L^{p,\infty}(\mu)} \leq C \|f\|_{L^p(\sigma)}.$$ Then $(\mu, \sigma) \in A_p$.
More importantly, we also establish the necessity of the $PA_p$ condition with the additional assumption that $\sigma$ is doubling.
\[thm:necessity Tailed Ap\] Let $(\mu, \sigma)$ be a pair of positive regular Borel measures with $\mu$ directionaly doubling in the direction $u_0$ and $\sigma$ doubling. Suppose $T_\sigma$ has a non-degenerate kernel in the direction $u_0$ and for some $1<p<\infty$, $$\|T_\sigma f\|_{L^{p,\infty}(\mu)} \leq C \|f\|_{L^p(\sigma)}.$$ Then $(\mu, \sigma) \in PA_p$.
Finally as an application of our techniques we consider the question of whether a Calderón-Zygmund operator can be bounded from $L^1(\mu)$ to $L^1(\nu)$ for a pair of positive Borel measures $(\mu, \nu)$. If the operator under consideration is translation invariant and the measures $(\mu,\nu)$ are regular, it is known that this is impossible. See [@MR807149 p. 468]. In Muckenhoupt and Wheeden derived a necessary condition for the Hilbert transform to be bounded from $L^1(v)$ to $L^1(u)$, where $u$ and $v$ are weights. We obtain an analogous estimate (see in the proof of Theorem \[thm:strong 1-1\]), but give a more complete characterization in terms of measures.
\[thm:strong 1-1\] Let $T$ be a Calderón Zygmund operator with a non-degenerate kernel in the direction $u_0$, and let $(\mu,\nu)$ be positive Borel measures on $\mathbb{R}^n$.
1. If $\nu$ is singular with respect to Lebesgue measure and $T:L^1(\nu) \rightarrow L^1(\mu)$, then $\mu= 0$.
2. If $\mu$ is a regular measure with $d\mu=d\mu_s + udx$ where $\mu_s$ is singular with respect to Lebesgue measure, $u \not\equiv 0$, $d\nu= d\nu_s + vdx$ where $\nu_s$ is singular with respect to Lebesgue measure, and $v$ is a non-negative measurable function such that $v(x)< \infty$ a.e, then $T$ is not bounded from $L^1(\nu)$ to $L^1(\mu)$.
3. If $\mu$ is a regular measure that is singular with respect to Lebesgue measure, and directionaly doubling in the direction $u_0$, and $\nu$ is a positive regular Borel measure, then $T$ is not bounded from $L^1(\nu)$ to $L^1(\mu)$.
The following example shows that the hypothesis in $(2)$ that $v(x)< \infty$ a.e. is needed. Let $d\mu=\chi_{[-1,1]} \; dx$, $d\nu= \chi_{\mathbb{R} \setminus [-2,2]} \; dx+ \infty \cdot
\chi_{[-2,2]}\; dx$, and let $Tf(x)=Hf(x)$. Then $(\mu,\nu)$ satisfies the key estimate $\eqref{eq:strong 1-1 3}$ below, and for $f$ with $\text{supp}(f) \subset \{x:|x|>2\}$ we have that $$\begin{gathered}
\int_{\mathbb{R}} |Hf(x)| \; d\mu(x) = \int_{-1}^1 |Hf(x)| \; dx
\leq \int_{-1}^1 \int_{|y|>2} \frac{|f(y)|}{|x-y|} \; dy \; dx \\
= \int_{|y|>2} \int_{-1}^1 \frac{1}{|x-y|} \; dx \; |f(y)| \; dy
\leq 2 \int_{|y|>2} |f(y)| \; dy = 2 \int_{\mathbb{R}} |f(y)| \; d\nu(y) . \end{gathered}$$
The following example shows that the hypothesis in $(2)$ that $\mu$ is not totally singular with respect to Lebesgue measure is needed. Let $\mu=\delta(0)$, $\nu= \frac{1}{x} dx$ and let $Tf(x)=Hf(x)$. Then for any $f \in L^1(\nu)$, $$\int_{\mathbb{R}} |Hf(x)| d\mu(x)= |Hf(0)|= \left| \int_{\mathbb{R}} \frac{f(y)}{y} \; dy \right | \leq \int_\mathbb{R} |f(y)| d\nu(y).$$
The main idea in our proofs is to reduce the problem of obtaining necessary conditions for the $L^p$ boundedness of singular integrals to that of averaging operators (see Definitions \[def: averaging operator\] and \[def:averaging operator Asigma\]). For singular integrals $T$, we work with the averaging operator $A_Q$. When $\mu=udx$, $\nu= vdx$ it is well known that the $A_p$ condition characterizes the strong type inequality for $A_Q$; see Jawerth [@MR833361]. For completeness we prove this result (see Theorem \[thm:Ap neccessity averaging operators\]). Furthermore, we obtain a characterization of the strong type inequality for $A_Q$ when $(\mu, \nu)$ are positive regular Borel measures (see Theorem \[thm: necessity averaging operators measures\]). This result is new and is interesting in its own right.
To study the singular integrals $T_\sigma$ we introduce the analogous averaging operator $A_{Q,\sigma}f = A_Q(f\,d\sigma)$. We show that the $A_p$ condition for measures is also necessary and sufficient (see Theorem \[thm: Neccessity Asigma\]) for these operators to be bounded, $1<p<\infty$.
The rest of the paper is organized as follows. In Section 2 we give preliminary definitions and notation used in this paper. In Section 3 we prove Theorem \[thm:necessity measures\]. In Section 4 we prove Theorem \[thm:necessity Tsigma\] and Theorem \[thm:necessity Tailed Ap\]. Finally, in Section 5 we prove Theorem \[thm:strong 1-1\].
Preliminaries
=============
Throughout this paper will use the following notation. The symbol $n$ will denote the dimension of the Euclidean space $\mathbb{R}^n$. $Q(x,r)$ denotes the cube with center $x \in \mathbb{R}^n$ and sidelength $2r$, while $B(x,r)$ denotes the ball with center $x \in \mathbb{R}^n$ and radius $r$. For a cube $Q$, $rQ$ is the cube with the same center as $Q$ and with side length $r$ times the length of $Q$. Positive constants $C,c$ may change value at each appearance. Sometimes we will indicate the dependence on certain parameters by writing for instance, $C(n,p)$ etc. We will work extensively with average integrals and use the notation, $$\avgint_Q u \; dx = \ \frac{1}{|Q|} \int_Q u \; dx.$$
We now define the singular integral operators we are interested in. For further, details, see [@MR1800316].
We say that an operator $T$ defined on measurable functions is a Calderón-Zygmund operator if $T$ is bounded on $L^2(\mathbb{R}^n)$ and for any $ f\in L^2_c(\mathbb{R}^n)$ we have the representation $$Tf(x)= \int_{\mathbb{R}^n} K(x,y)f(y) \; dy, \quad x \notin \text{supp}(f).$$ Here $K(x,y)$ is a kernel defined for all $ x \neq y$ in $ \mathbb{R}^n \times \mathbb{R}^n$, that satisfies the standard estimates $$\label{eq:size}
|K(x,y)| \leq \frac{C_0}{|x-y|^n}$$ and $$\label{eq:smoothness}
|K(x+h,y)- K(x,y)| + |K(x,y+h) - K(x,y)| \leq C_0\frac{|h|^ \delta }{|x-y|^{n+\delta}}$$ for all $|h|< \frac{1}{2} |x-y|$ and some fixed $\delta >0$.
We want to define the operators $T_\sigma$ more carefully; to do so we follow the treatment given in [@MR3688149]. Let $(\mu,\sigma)$ be a pair of regular Borel measures. Fix a Calderón-Zygmund operator $T$ with kernel $K$. Let $\{\eta_{\epsilon, R}\}_{0< \epsilon< R <\infty}$ be a family of non-negative truncation functions with supports in the annuli $\epsilon<|x|<R$, and such that $\eta_{\epsilon,R}(x)=1$ if $2\epsilon<|x|<\frac{R}{2}$. For example, we can take $\eta_{\epsilon, R}= \chi_{\{\epsilon<|x|<R\}}$, but other choices are possible. Define the family of truncated kernels ${K_{\epsilon,R} (x,y) = \eta_{\epsilon,R}(x-y)K(x,y)}$. These are bounded with compact support for a fixed $x$ or $y$. Thus, the truncated operators defined by $$T^{\epsilon, R}_{\sigma} f(x)= \int_{\mathbb{R}^n} K_{\epsilon,R}(x,y) f(y) \; d\sigma(y), \quad x\in \mathbb{R}^n,$$ are pointwise well defined for $f\in L^1_{\text{loc}}$. Hereafter, we will assume that each of the truncated kernels $\{K_{\epsilon,R}\}_{0< \epsilon< R <\infty}$ satisfies the standard kernel estimates and with uniform constants.
\[def:truncated weak (p,p)\] Given a Calderón-Zygmund operator $T$ with kernel $K$, we say that $T_\sigma$ satisfies the weak $(p,p)$ inequality, $1<p<\infty$, provided that there exists a family of truncations $\{\eta_{\epsilon, R}\}_{0<
\epsilon< R <\infty}$ such that for all $f\in L^p(\sigma)$, $$\label{eq:truncated weak (p,p)}
\|T_\sigma^{\epsilon,R} f \|_{L^{p,\infty}(\mu)} \leq C\|f\|_{L^p(\sigma)}$$ with constant independent of $\epsilon$ and $R$. In this case we write $$\|T_\sigma f\|_{L^{p,\infty}(\mu)} \leq C\|f\|_{L^p(\sigma)}.$$
Given this definition, in our proofs below we will need to fix particular values of $\epsilon$ and $R$ and apply inequality . We will, however, generally write $T_\sigma$ instead of $T^{\epsilon, R}_{\sigma}$ when there is no possibility of confusion.
While we need to fix a family of truncations to define $T_\sigma$, the choice is less important than it might seem at first. In [@MR3688149], they showed that if the pair $(\mu,\sigma)$ satisfies the $A_p$ condition for measures, below, then the corresponding strong $(2,2)$ inequality for $T_\sigma$ holds independent of the choice of truncations used.
\[defi:nondegeneracy\] Given a Calderón-Zygmund operator $T$ with kernel $K(x,y)$, we say $T$ has a non-degenerate kernel if there exists $a>0$, and a unit vector $u_0$ such that for $x,\,y \in \mathbb{R}^n$, $x-y = t u_0$, $t\in\mathbb{R}$, $$\label{eq:nondegeneracy}
|K(x,y)| \geq \frac{a}{|x-y|^n}.$$
For example, holds for the Hilbert transform as well as of any of the Riesz transforms in the direction $e_j$. However, not all singular integrals satisfy this property. See for example [@MR1376747 Lemma 1.4] where they construct a “one-sided” Calderón-Zygmund kernel with support in $(0,
\infty)$; they establish that a sufficient condition for this operator to be bounded is a “one-sided” $A_p$ condition that is strictly weaker than the conditions we consider.
\[def:A\_p\] Let $u,\,v$ be non-negative, measurable functions. We say the pair $(u,v) \in A_p$, $1<p< \infty$, if $$\label{eq:A_p}
[u,v]_{A_p}= \sup_{Q} \left( \avgint_Q u \; dx \right) \left( \avgint_Q v^{1-p'} \; dx \right)^{p-1} < \infty,$$ and in $A_1$ if $$\label{eq:A1}
\avgint_Q u \; dx \leq [u,v]_{A_1} \operatorname*{ess\,inf}_{x \in Q} v(x).$$
\[def: Ap condition for measures\] If $(\mu, \sigma)$ are positive Borel measures, we say that $(\mu,
\sigma) \in A_p$, $ 1 < p<\infty$, if $$\label{eq:Ap condition for measures}
[\mu, \sigma]_{A_p}= \sup_Q \frac{\mu(Q)}{|Q|} \left(\frac{\sigma(Q)}{|Q|} \right)^{p-1} < \infty.$$
If $d\mu= u \; dx$, $d\sigma=v^{1-p'}\; dx$, then is equivalent to .
\[remark: Ap balls cubes\] It is straightforward to see that if , , or hold for any cube $Q \subset \mathbb{R}^n$, then they also hold for any ball $ B \subset \mathbb{R}^n$. We will use this fact below.
Inequality implies that $\mu, \sigma$ do not share a common point mass: if there exists a point $a$ such that $\sigma \{a\} \mu \{a\}>0$, then the expression in blows up as $Q$ shrinks to $\{a\}$.
\[def:Poisson Ap\] We say the pair $(\mu,\sigma)$ is in $PA_p$, $1<p< \infty$ if for any cube $Q(y_0,r)$, $$\label{eq:Poisson Ap}
\left(\frac{\mu(Q(y_0,r))}{|Q(y_0,r)|}\right) \left(\int_{\mathbb{R}^n} \left( \frac{r^{p'-1}}{(|x-y_0|+r)^{p'}} \right)^n d\sigma(x) \right)^{p-1} \leq C.$$
This condition first appeared in [@MR0417671] in one dimension where they proved that it was necessary for the strong type inequality for the Hilbert transform to hold. The $n$-dimensional version first appeared in [@MR1175693] in the context of the fractional integral operator. When $p=2$, this condition is sometimes called “Poisson $A_p$". This is because the second term on the left-hand side of is approximately the Poisson extension of $\sigma$ evaluated at a point in the upper half plane given by $y_0$ and $r$. It is straightforward to see that the $PA_p$ condition implies the $A_p$ condition.
A positive measure $\mu$ is said to be doubling if there exists a constant $ C>0$ such that for any cube $ Q$, $\mu(2Q) \leq C \mu(Q)$.
Equivalently $\mu$ is doubling if $\mu(P) \leq C \mu(Q)$, whenever $P,Q$ are adjacent cubes with $|Q|=|P|$. (We say the cubes $P,Q$ are adjacent if the boundaries of $P$ and $Q$ share a point in common.)
For our results we do not need to assume the full doubling condition, but rather a “directional” doubling condition.
\[defi:directionaldoubling\] Let $\mu$ be a positive Borel measure, and fix a unit vector $u_0$. We say $\mu$ is directionally doubling in direction $u_0$ if there exists a constant $C_\mu>0$ such that given adjacent cubes $P(x_0,r),\,Q(y_0,r)$ whose centres satisfy $x_0-y_0=tu_0$, $t \in \mathbb{R}$, $$\label{eq:directionaldoubling}
\mu(P(x_0,r)) \leq C_\mu \mu(Q(y_0,r)).$$
Definition \[defi:directionaldoubling\] is weaker than the doubling condition. For example, for $ E \subset \mathbb{R}^2$, define $\mu(E)=\iint_E e^{-|x|} \; dx dy$. Then it is straightforward to show that $\mu$ is directionally doubling in the direction $e_2$ but is not doubling.
Proof of Theorem \[thm:necessity measures\]
===========================================
In order to proceed with the proofs of our first main result we will need to prove some preliminary results about averaging operators.
\[def: averaging operator\] Given a cube $Q$, define the averaging operator $A_Q$ on a function $ f \in L^1_{\text{loc}}$ by $$A_Q f(x)= \avgint_Q f(y) \; dy \;\chi_Q(x).$$
The following result first appeared in [@MR833361] but to the best of our knowledge a proof does not appear in the literature. For completeness we sketch the proof.
\[thm:Ap neccessity averaging operators\] Given a cube $Q$, $1 \leq p < \infty$, and $(u,v) \in A_p$, for all $f \in L^p(v)$ $$\|A_Q f \|_{L^p(u)} \leq [u,v]_{A_p}^{1/p} \|f\|_{L^p(v)}.$$ Conversely, given $1\leq p<\infty$, if $(u,v)$ are a pair of weights such that for every cube $Q$, $$\label{eq:Ap neccessity averaging operators 1 }
\|A_Q f \|_{L^p(u)} \leq K \|f\|_{L^p(v)},$$ then $(u,v) \in A_p$. Moreover, $[u,v]_{A_p} \leq K^p $.
Let $Q \subset \mathbb{R}^n$. We first prove the sufficiency of the $A_1$ condition when $p=1$. Indeed, $$\|A_Q f\|_{L^1(u)}= \int_{\mathbb{R}^n} \left\lvert \avgint_Q f \; dy \chi_Q(x) \right\rvert u \; dx
\leq \int_Q \frac{u(Q)}{|Q|} |f| \; dy \leq [u,v]_{A_1} \int_{\mathbb{R}^n} |f|v \; dy.$$
If $p>1$, by Hölder’s inequality, $$\begin{aligned}
\|A_Q\|_{L^p(u)}^p &= \int_{\mathbb{R}^n} \left\lvert \avgint_Q f \; dy \; \chi_Q(x) \right\rvert ^p u\; dx\\
& \leq \left( \avgint_Q |f|v^\frac{1}{p} v^\frac{-1}{p} \; dy \right)^p u(Q) \\
&\leq \left( \int_Q |f|^p v \; dy \right) \left(\avgint_Q u \; dy \right) \left(\avgint_Q v^{1-p'} \; dy \right)^{p-1}\\
& \leq [u,v]_{A_p} \int_{\mathbb{R}^n} |f|^p v \; dy.\end{aligned}$$ To prove necessity, let $S \subset Q$ be measurable and set $ f= \chi_S$. Then becomes $$\label{eq:Ap neccessity averaging operators 2}
u(Q) \left(\frac{|S|}{|Q|} \right)^p \leq Kv(S).$$ In [@MR807149 p. 388] they show that if holds, then $(u,v) \in A_p$, and that $[u,v]_{A_p} \leq K^p$. This completes the proof.
We now prove an analogue of Theorem \[thm:Ap neccessity averaging operators\] for measures.
\[thm: necessity averaging operators measures\] Let $(\mu,\nu)$ be a pair of positive regular Borel measures. Given $1 \leq p < \infty$, suppose that there exists a constant $C$ such that for every cube $Q$ $$\label{eq:necessity averaging operators measures 1}
\|A_Qf \|_{L^p(\mu)} \leq C \|f\|_{L^p(\nu)}.$$
Then:
1. $d\nu=d\nu_s+vdx$ where $v \in L^1_{\text{loc}}$ and $\nu_s$ is singular;
2. $\mu \ll \nu $, and $\mu \ll dx$, so $d\mu=u \,dx$ where $u\in L^1_{\text{loc}}$;
3. $(u,v) \in A_p$ and $u(x) \leq Cv(x)$ a.e.
Fix a cube $Q \subset \mathbb{R}^n$ and let $ S \subset Q$ be measurable. Let $f=\chi_S$ in ; then arguing as before we obtain $$\label{eq:necessity averaging operators measures 2}
\left(\frac{|S|}{|Q|} \right)^p \mu(Q) \leq C \nu(S).$$
Suppose $\nu$ were singular with respect to Lebesgue measure and $|S|>0$. Then there exists a set $A \subset \mathbb{R}^n$ such that $|A|=0$ and $\nu(S)= \nu(S \cap A)$. If we replace $S$ with $ S \setminus A$ in , we find $ \mu(Q)=0$ for any cube $Q \supset S$. This implies that $\mu=0$. Hence, $ d\nu= d\nu_s + v \; dx$ where $v \in L^1_{\text{loc}}$ $v \neq 0$, and $\nu_s$ is singular.
Now fix any set $S \subset Q$ with $ \nu(S)=0$. Since $\nu$ is regular, for any $\epsilon>0$ there exists an open set $E \supset S$ such that $\nu(E) < \epsilon$. Since $E$ is open, $ E = \cup_j Q_j$ where $ \{Q_j\}$ is a disjoint collection of dyadic cubes. If we let $Q=S=Q_j$ in , we have $$\mu(S) \leq \mu(E)= \sum_j \mu(Q_j) \leq C \sum_j \nu(Q_j)= C \nu(E) < C \epsilon.$$ Since $\epsilon>0$ was arbitrary we have $\mu(S)=0$, and so $ \mu \ll \nu$.
We can now write as $$\mu(Q) \left\lvert \avgint_Q f \; dx \right\rvert^p \leq C \left( \int_{\mathbb{R}^n} |f|^p \; d\nu_s + \int_{\mathbb{R}^n} |f|^p v \; dx \right).$$ Let $A= {\mathrm{supp}}( \nu_s)$. Since $|A|=0$, if we set $ f = \chi_{S \setminus A}$, we have\
$$\label{eq:necessity averaging operators measures 3}
\left( \frac{|S|}{|Q|} \right)^p \mu(Q) \leq C v(S).$$ Using the same argument that showed $\mu \ll \nu$, replacing $\nu$ with $v$, we can see that $ \mu \ll vdx \ll dx$. Hence, $d\mu=u dx$ for some $u\in L^1_{\text{loc}}$. Let $S=Q$ in , then by the Lebesgue differentiation theorem we have that $u(x) \leq C v(x)$ a.e. Moreover, by we have that $$\left(\frac{|S|}{|Q|} \right)^p u(Q) \leq C v(S).$$ Hence, the fact that $(u,v) \in A_p$ follows as in the proof of Theorem \[thm:Ap neccessity averaging operators\].
We will show that given any cube $Q$ the averaging operator satisfies $A_{Q}: L^p(\nu) \rightarrow L^p(\mu)$. The desired conclusion then follows from Theorem \[thm: necessity averaging operators measures\]. Choose a constant $t \geq 4$ such that $2C_0(1+2^{n+ \delta})t^{-\delta} \leq a$. Here, $a$ is the constant in , and $\delta,C_0$ are as in . We further require $t= \frac{N C_2}{\sqrt{n}}$, where $ \frac{1}{\sqrt{n}} \leq C_2 \leq 1$ and $N$ is an integer. The exact choice of the constant $C_2$ will be made clear below. Let $x_0,\,y_0$ be two points satisfying $x_0-y_0=tr\sqrt{n} u_0$, $r>0$, and consider the cubes $Q(x_0,r)$, $Q(y_0,r)$. Given any point $ x \in Q(x_0,r)$ we can write $x= x_0 + h$, where $|h| < r \sqrt{n}$. Similarly, given $y \in Q(y_0,r)$, $ y= y_0+k$ where $|k| < r \sqrt{n}$. We claim that for such $x$ and $y$, $$\label{eq:necessity measures 1}
| K(x,y)-K(x_0,y_0)| \leq \frac{1}{2} | K(x_0,y_0)|.$$ To prove this we will apply which is possible since $|h|< r\sqrt{n} \leq \frac{1}{2}|x_0-y_0|$, and $$|x_0+h-y_0| \geq |x_0-y_0| -|h| \geq tr\sqrt{n} - r \sqrt{n} \geq \frac{t}{2} r \sqrt{n} \geq 2 |k|.$$ Thus, we can estimate as follows: $$\begin{aligned}
\label{eq:necessity measures 2}
|K(x,y)&- K(x_0,y_0)|\\
& \leq |K(x_0+h, y_0+k)-K(x_0+h,y_0)|+|K(x_0+h,y_0)-K(x_0,y_0)|\nonumber \\
& \leq \frac{C_0|k|^\delta}{|x_0+h-y_0|^{n+\delta}} + \frac{C_0|h|^\delta}{|x_0-y_0|^{n+\delta}}\nonumber \\
&= I_1 + I_2 \nonumber.\end{aligned}$$ We can bound $I_2$ immediately: $$I_2 \leq \frac{C_0(r \sqrt{n})^\delta}{(tr\sqrt{n})^\delta |x_0-y_0|^n} = C_0 \frac{ t^{-\delta}}{|x_0-y_0|^n}.$$ To estimate $I_1$, note that $$|x_0+h-y_0| \geq \frac{t}{2} r\sqrt{n}= \frac{1}{2}|x_0-y_0|.$$ Hence, $$I_1 \leq \frac{ C_0 2^{n+\delta}( r\sqrt{n})^\delta}{(tr\sqrt{n})^\delta|x_0-y_0|^n}= C_0\frac{ 2^{n+\delta}t^{-\delta}}{|x_0-y_0|^n }.$$ If we combine these estimates, by our choice of $t$ and we have $$I_1+I_2 \leq \frac{a}{2} \frac{1}{|x_0-y_0|^n} \leq \frac{1}{2}|K(x_0,y_0)|,$$ which proves .
It now follows that for any $ x\in Q(x_0,r)$, $ y\in Q(y_0,r)$, the kernel $K(x,y)$ always has the same sign. Therefore, if we fix a non-negative function $f$ with ${\mathrm{supp}}(f)\subset Q(x_0,r) $, then
$$\begin{aligned}
\label{eq:necessity measures 4}
|Tf(y)|&= \left\lvert \int_{Q(x_0,r)} K(x,y) f(x) \; dx \right\rvert \\
&= \int_{Q(x_0,r)} |K(x,y)| f(x) \; dx \nonumber\\
& \geq \int_{Q(x_0,r)} |K(x_0,y_0)|f(x) \; dx - \int_{Q(x_0,r)} |K(x,y) -K(x_0,y_0)|f(x) \; dx; \nonumber\\
\intertext{again by \eqref{eq:necessity measures 1} and \eqref{eq:nondegeneracy},\nonumber}
& \geq \frac{1}{2}|K(x_0,y_0)| \int_{Q(x_0,r)} f(x) \; dx \nonumber\\
& \geq \frac{a}{2|x_0-y_0|^n} \int_{Q(x_0,r)} f(x) \; dx \nonumber \\
& \geq \frac{a}{2(tr \sqrt{n})^n} \int_{Q(x_0,r)} f(x) \; dx \nonumber\\
&= c(a,t,n) \avgint_{Q(x_0,r)} f(x) \; dx.\nonumber\end{aligned}$$
Given this inequality and the assumption that $T$ satisfies a weak $(p,p)$ inequality we have for any $0<\lambda< c(a,t,n) \avgint_{Q(x_0,r)} f \; dx$: $$\mu(Q(y_0,r)) \leq \mu( \{x:|Tf(y)| > \lambda \} \leq \frac{C}{\lambda^p} \int_{Q(x_0,r)} |f|^p \; d\nu.$$ If we take the supremum over all such $\lambda$, we get $$\label{eq:necessity measures 3}
\mu(Q(y_0,r)) \left(\avgint_{Q(x_0,r)} f \; dx \right)^p \leq c(a,t,n,p) \int_{Q(x_0,r)} |f|^p \; d\nu.$$
Now fix a value of $C_2$, depending only on $u_0$, so that starting from $Q(y_0,r)$ we can form a chain of adjacent cubes $Q(x_j,r)$, $j=1, \dots N$ in the direction $-u_0$ such that $x_1=y_0$ and $x_N=x_0$. Each $Q(x_j,r)$ satisfies $ \mu(Q(x_{j+1},r)) \leq C_\mu \mu(Q(x_j ,r))$, where where $C_\mu$ is the directional doubling constant from . The number of cubes $N$, lying between $Q(y_0, r)$ and $Q(x_0,r)$ depends only on $t$ and $n$. Thus, there exists constant $ C=C(C_\mu,t,n)$ such that $ \mu(Q(x_0,r)) \leq C \mu(Q(y_0,r))$. Hence, $$\begin{aligned}
\mu(Q(x_0,r))\left(\avgint_{Q(x_0,r)} f \; dx \right)^p \leq C \mu(Q(y_0,r)) \left( \avgint_{Q(x_0,r)} f \; dx \right)^p \leq C \int_{Q(x_0,r)} |f|^p \; d\nu.\end{aligned}$$ Since the resulting constant depends only on $C_1,\,p,\,t,\,n,\,a$ and not on $Q(x_0,r)$ we have shown that the averaging operators $A_Q : L^p( \nu) \rightarrow L^{p}(\mu)$ uniformly for all $Q$. Therefore, by Theorem \[thm: necessity averaging operators measures\] we get the desired conclusion.
Proofs of Theorems \[thm:necessity Tsigma\] and \[thm:necessity Tailed Ap\]
===========================================================================
Before proceeding with the proof of Theorem \[thm:necessity Tsigma\] we first define the related averaging operator.
\[def:averaging operator Asigma\] Given a non-negative measure $\sigma$ and a cube $Q$, define the averaging operator $A_{Q,\sigma}$ acting on a function $f\in L^1_{\text{loc}}(\sigma)$ by $$\label{eq:Asigma}
A_{Q,\sigma} f(x) = \frac{1}{|Q|} \int_Q f(y) \; d\sigma(y) \chi_Q(x).$$
The following result characterizes the $L^p$ boundedness of $A_{Q,\sigma}$.
\[thm: Neccessity Asigma\] Given a cube $Q$, $1 \leq p < \infty$, and a pair of positive regular Borel measures $(\mu, \sigma)$, suppose that $(\mu,\sigma)$ satisfy the $A_p$ condition . Then for all $f \in L^p(\sigma)$, $$\|A_{Q,\sigma} f \|_{L^p(\mu)} \leq [\mu,\sigma]_{A_p}^{1/p} \|f\|_{L^p(\sigma)}.$$ Conversely given $1\leq p<\infty$, if $(\mu,\sigma)$ is a pair of positive regular Borel measures such that for every cube $Q$, $$\label{eq:Neccessity Asigma 1}
\|A_{Q,\sigma} f \|_{L^p(\mu)} \leq K \|f\|_{L^p(\sigma)},$$ then $(\mu,\sigma) \in A_p$. Moreover $[\mu,\sigma]_{A_p} \leq K^p .$
We first prove necessity. Fix a cube $Q$ and let $1 \leq p < \infty$. If $\sigma(Q)=0$, then is immediate. If $\sigma(Q)>0$ let $f=\chi_Q$ in . Then we obtain $$\mu(Q) \left( \frac{\sigma(Q)}{|Q|} \right)^p \leq K^p \sigma(Q).$$ Dividing by $\sigma(Q)$ and taking the supremum over all cubes $Q$ we have $(\mu,\sigma) \in A_p$ and $[\mu,\sigma]_{A_p} \leq K^p$. The proof of sufficiency is similar to Theorem \[thm:Ap neccessity averaging operators\] so we omit the details.
We can now prove Theorems \[thm:necessity Tsigma\] and \[thm:necessity Tailed Ap\].
The proof is a straightforward modification of the proof of Theorem \[thm:necessity measures\]. Fix the cubes $Q(x_0,r)$ and $Q(y_0,r)$ as before. Then with the same notation as before, we have that if $x\in Q(x_0,r)$ and $y\in Q(y_0,r)$, $$|x-y|= |x_0-y_0+h-k| < tr\sqrt{n} +2r\sqrt{n} < 2tr\sqrt{n}.$$ Similarly, we have $|x-y|> \tfrac{1}{2}tr\sqrt{n}$. Therefore, if we choose $0<\epsilon < \tfrac{1}{4}tr\sqrt{n}$ and $R>4tr\sqrt{n}$, we have that the kernel $K_{\epsilon,R}(x,y)=K(x,y)$ and so satisfies the non-degeneracy condition with a uniform constant. We also have that it satisfies the standard estimates and .
For simplicity, we now write $T_\sigma$ instead of $T_{\sigma}^{\epsilon,R}$ and $K$ for $K_{\epsilon,R}$. If we repeat the previous argument, we have that $K$ satisfies the estimate . We can then repeat the proof of , using the fact that $T_\sigma$ satisfies the weak $(p,p)$ inequality with uniform constant, to get $$|T_\sigma f(y) | \geq \frac{c(a,t,n)}{|Q(x_0,r)|} \int_{Q(x_0,r)} f(x) \; d\sigma(x).$$ Given this inequality we continue to argue as we did in the proof of Theorem \[thm:necessity measures\] to get that the averaging operator $A_{Q,\sigma}=A_{Q(x_0,r),\sigma}$ satisfies $A_{Q,\sigma}: L^p(\sigma) \rightarrow L^p(\mu)$. This estimate holds for every cube $Q(x_0,r)$ with constants independent of $\epsilon$ and $R$, and so $(\mu,\sigma) \in A_p$ by Theorem \[thm: Neccessity Asigma\].
We adapt the proof of Theorem \[thm:necessity measures\], exchanging the roles of $x_0$ and $y_0$. Fix a cube $Q(y_0,r)$. Choose $t \geq 4$ as in the proof of Theorem \[thm:necessity measures\]. Rather than considering the cube $Q(x_0,r)$ we replace it with a ball. Fix $S>r$ and for each $r\leq
s \leq S$ let $B_s= B(x_s, s \sqrt {n})$, where $x_s=y_0+ts\sqrt{n}u_0$. If we now argue as we did in the proof of Theorem \[thm:necessity Tsigma\], if we fix $R>4tS\sqrt{n}$ and $\epsilon<\tfrac{1}{4}tr\sqrt{n}$, then for $y\in Q(y_0,r)$ and $x\in
B_s$, $K_{\epsilon,R}$ satisfies the non-degeneracy condition with a uniform constant. We also have that it satisfies the standard estimates and . Again, we will write $T_\sigma$ for $T_{\sigma}^{\epsilon,R}$ and $K$ for $K_{\epsilon,R}$.
We can now argue as follows: for all $y \in Q(y_0,r)$, $y=y_0 + k$ where $|k| \leq r \sqrt{n}$ and for $x\in B_s$, $x= x_s +h$ where $|h| \leq s\sqrt{n}$. As in the proof of Theorem \[thm:necessity measures\] we have $|h| \leq s\sqrt{n} \leq \frac{1}{2}|x_s-y_0|$, and $$|x_s+h-y_0| \geq |x_s -y_0| - |h| \geq ts \sqrt{n}- s\sqrt{n} \geq \frac{t}{2} s \sqrt{n} \geq \frac{t}{2} r \sqrt{n} \geq 2|k|.$$ We can now apply as in estimate to get that for $y \in Q_r$ and $ x \in B_s$, $$|K(x,y)-K(x_s,y_0)| \leq \frac{1}{2} |K(x_s,y_0)|.$$ This implies that for any $y \in Q(y_0,r)$ and $x\in B_s$ $K(x,y)$ always has the same sign. Moreover, we have $$|K(x,y)| \geq \frac{1}{2} |K(x_s, y_0)|.$$ Therefore, $$\begin{gathered}
|K(x,y)| \geq \frac{1}{2}|K(x_s,y_0)| \geq \frac{a}{2} \frac{1}{|x_s-y_0|^n}
\geq \frac{a}{2} \frac{1}{(|x-x_s|+|x-y_0|)^n} \\
\geq c(a,n) \frac{1}{(|x-y_0|)^n}
\geq c(a,n) \frac{1}{(|x-y_0|+r)^n};\end{gathered}$$ the second to last inequality follows since $|x-x_s| \leq s \sqrt{n} \leq ts \sqrt{n}= |x-y_0|$.
Define the truncated cone $$C_r = \bigcup_{s \geq r} B_s .$$ Notice $C_r$ has a central axis of $y_0 + su_0$, for $s \geq r$. For $S>0$ let $$f_{r,S}(x)= \left(\frac{1}{|x-y_0| + r)} \right)^{n(p'-1)} \chi_{C_r \cap B(y_0,S)}.$$
Then for all $ y \in Q(x_0,r)$ we have $$\begin{gathered}
\label{eq: necessity Tailed Ap 1}
|T_\sigma f_{r,S}(y)| = \int_{C_r \cap B(y_0,S)} |K(x,y)|
\left(\frac{1}{(|x-y_0| + r )}\right)^{n(p'-1)} \; d\sigma(x) \\
\geq c(a,n) \int_{C_r \cap B(y_0,S)}
\left( \frac{1}{(|x-y_0|+ r)^{p'} } \right)^n d \sigma(x).\end{gathered}$$
We have that $T_\sigma$ satisfies the weak $(p,p)$ inequality with uniform constant, so we can argue as we did to derive in the proof of Theorem \[thm:necessity measures\] to get $$\begin{gathered}
\mu(Q(y_0,r)) \left( \int_{C_r \cap B(y_0,S)}
\left( \; \frac{1}{( |x-y_0|+ r)^{p'} } \right)^{n} d\sigma(x)
\right)^p \\
\leq C \int_{C_r\cap B(y_0,S)}
\left(\frac{1}{|x-y_0| +r } \right)^{np(p'-1)} d\sigma(x).\end{gathered}$$ Since $p(p'-1)=p'$ we have $$\frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left( \int_{C_r \cap B(y_0,S)} \left(\frac{r^{p'-1}}{(|x-y_0|+r)^{p'}} \right)^n d\sigma(x) \right)^{p-1} \leq C.$$ Since the constant $C$ is independent of $\epsilon$ and $R$, and so of $S$, we can take the limit as $S\rightarrow \infty$, and by the monotone convergence theorem we get $$\label{eq: necessity Tailed Ap 2}
\frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left( \int_{C_r} \left(\frac{r^{p'-1}}{(|x-y_0|+r)^{p'}} \right)^n d\sigma(x) \right)^{p-1} \leq C.$$
We will now extend inequality to all of $\mathbb{R}^n$. Let $$A_k= B(y_0, 2^{k+1} tr \sqrt{n}) \setminus B(y_0, 2^k tr
\sqrt{n}).$$ Consider the ball $B(x_k, 2^{k+2}tr \sqrt{n})$, where $$x_k= y_0 + (\frac{2^{k+1} +2^{k}}{2})tr \sqrt{n} u_0= y_0+ \tfrac{3}{8}(2^{k+2})tr \sqrt{n} u_0.$$ This is the ball of radius $2^{k+2}tr \sqrt{n}$ centered at the midpoint of the portion of the central axis of $C_r$ that lies inside $A_k$. We claim $A_k \subset B(x_k,2^{k+2} tr \sqrt{n})$. To see this, fix $x\in A_k$; then $$|x-x_k| \leq |x-y_0|+|x_k -y_0| \leq 2^{k+1}tr\sqrt{n}+ \tfrac{3}{8} (2^{k+2})tr\sqrt{n} \leq 2^{k+2} tr\sqrt{n}.$$ Since the ball $B(x_k, \frac{3}{8} (2^{k+2}) r \sqrt{n})$ is one of the balls $B_s$ that defines $C_r$, it is immediate that $$\bigcup^\infty_{k=0} B(x_k, \tfrac{3}{8} (2^{k+2}) r\sqrt{n}) \subset C_r.$$ Since $\sigma$ is doubling there exists a constant $C=C(t,n, \sigma)$ such that $$\sigma(B(x_k, 2^{k+2} tr \sqrt{n})) \leq C \sigma \left(B(x_k, \tfrac{3}{8} (2^{k+2})r \sqrt{n}) \right).$$
Hence, we can estimate as follows: $$\begin{aligned}
&\frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left( \int_{\mathbb{R}^n \setminus B(y_0 ,tr\sqrt{n})} \left( \frac{r^{p'-1}}{(|x-y_0| + r)^{p'}} \right)^n d\sigma(x) \right)^{p-1} \\
& \quad= \frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left ( \sum^\infty_{k=0}\int_{A_k} \left( \frac{r^{p'-1}}{(|x-y_0| + r)^{p'}} \right)^n d\sigma(x) \right)^{p-1} \\
& \quad\leq \frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left(\sum^\infty_{k=0} \left( \frac{r^{p'-1}}{(2^ktr\sqrt{n} + r)^{p'}} \right)^n \sigma(A_k) \right)^{p-1} \\
& \quad\leq \frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left( \sum_{k=0}^\infty\left(\frac{r^{p'-1}}{(2^ktr\sqrt{n} + r)^{p'}} \right)^n \sigma(B(x_k, 2^{k+2} tr\sqrt{n} )) \right)^{p-1} \\
& \quad \leq C \frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left( \sum_{k=0}^\infty\left(\frac{r^{p'-1}}{(2^ktr\sqrt{n} + r)^{p'}} \right)^n \sigma(B(x_k, \tfrac{3}{8}(2^{k+2}) r\sqrt{n} )) \right)^{p-1} \\
& \quad\leq C \frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left( \sum_{k=0}^\infty\int_{B(x_k, \frac{3}{8}(2^{k+2})r\sqrt{n})}\left(\frac{r^{p'-1}}{(|x-y_0| + r)^{p'}} \right)^n d\sigma(x) \right)^{p-1} \\
& \quad \leq C \frac{\mu(Q(y_0,r))}{|Q(y_0,r)|} \left( \int_{C_r} \left(\frac{r^{p'-1}}{(|x-y_0|+r)^{p'}} \right)^n d\sigma(x) \right)^{p-1} \\
& \quad \leq C.\end{aligned}$$ The third to last inequality holds since $2^{k}tr\sqrt{n} \geq
\frac{1}{2} |x-y_0|$ for any $x \in B(x_k,
\tfrac{3}{8}(2^{k+2})r\sqrt{n})$.
By Remark \[remark: Ap balls cubes\], we can apply the result of Theorem \[thm:necessity Tsigma\] to the ball $B(y_0, tr\sqrt{n})$ to get $$\begin{aligned}
& \left(\frac{\mu(Q(y_0,r))}{|Q(y_0,r)|}\right)
\left(\int_{B(y_0, tr\sqrt{n})}
\left( \frac{r^{p'-1}}{(|x-y_0|+r)^{p'}}
\right)^n d\sigma(x) \right)^{p-1}\\
& \qquad \qquad \qquad \leq \left(\frac{\mu(Q(y_0,r))}{|Q(y_0,r)|}\right) \left(
\frac{\sigma(B(y_0, tr\sqrt{n}))}{|B(y_0,tr
\sqrt{n})|}\right)^{p-1} \\
& \qquad \qquad \qquad \leq \left(\frac{ \mu(B(y_0,
tr\sqrt{n}))}{|B(y_0,tr\sqrt{n})|}
\right) \left( \frac{\sigma(B(y_0,
tr\sqrt{n}))}{|B(y_0,tr
\sqrt{n})|}\right)^{p-1} \\
& \qquad \qquad \qquad \leq C.\end{aligned}$$ If we combine this inequality with the previous estimate, we get $$\left(\frac{\mu(Q(y_0,r))}{|Q(y_0,r)|}\right)
\left(\int_{\mathbb{R}^n} \left( \frac{r^{p'-1}}{(|x-y_0|+r)^{p'}}
\right)^n d\sigma(x) \right)^{p-1} \leq C.$$ Since this holds for every cube $Q(y_0,r)$, it follows that $(\mu,\sigma)$ satisfy the $PA_p$ condition.
Strong $(1,1)$ Inequalities
===========================
For the proof of Theorem \[thm:strong 1-1\] we first give some preliminary lemmas.
\[lem:essential infimum\] Let $v$ be a measurable function. Then for a.e. $x \in \mathbb{R}^n$, $$\label{eq:essential infimum 1}
\lim_{r \rightarrow 0^+} [\operatorname*{ess\,inf}_{ y \in Q(x,r)} v(y)] \leq v(x).$$
The proof of Lemma \[lem:essential infimum\] is implicit in [@MR0417671 Theorem 4] in one dimension; the proof is the same in higher dimensions.
A family $\{E_r\}_{r>0}$ of Borel subsets of $\mathbb{R}^n$ is said to shrink nicely to $x \in \mathbb{R}^n$ if $$E_r \subset B(x,r) \quad \text{for each}\; r,$$ and there exists a constant $\alpha$ independent of $r$ such that $$|E_r|> \alpha | B(x,r)|.$$
\[lem:differentiation theorem\] Let $\mu$ be a regular Borel measure on $\mathbb{R}^n$, and let $d\mu=d\mu_s +u\; dx$ be its Lebesgue Radon-Nikodym decomposition. Then for a.e. $x \in \mathbb{R}^n$, $$\lim_{r \rightarrow 0} \frac{ \mu(E_r)}{|E_r|} = u(x).$$
The proof of Lemma \[lem:differentiation theorem\] can be found in [@MR1681462 Theorem 3.22, p.99].
First suppose that the measure $\nu$ is singular with respect to Lebesgue measure. As in the proof of Theorem \[thm:necessity Tailed Ap\] fix a cube $Q(y_0,r)$ and define the truncated cone $C_r$. Let $f$ be a non negative function with ${\mathrm{supp}}(f) \subset Q(y_0,r)$. Then, if we estimate as in the proof of Theorem \[thm:necessity Tailed Ap\] to get , we have for all $x \in C_r$, $$|Tf(x)| \geq c(a,n) \int_{Q(y_0,r)} \frac{f(y)}{(r+|x-y_0|)^n} \; dy.$$ By assumption $T: L^1(\nu) \rightarrow L^1(\mu)$, so we have that $$\begin{aligned}
\label{eq:strong 1-1 1}
\int_{Q(y_0,r)} f(x)\; d\nu(x) &\geq c \int_{\mathbb{R}^n} |Tf(x)| \; d\mu(x) \\ \nonumber
& \geq c \int_{C_r} \int_{Q(y_0,r)} \frac{f(y)}{(r+|x-y_0|)^n} dy \; d\mu(x) \\ \nonumber
& = c \int_{Q(y_0,r)} f(y) \; dy \int_{C_r} \frac{1}{(r+|x-y_0|)^n} \; d\mu(x).\end{aligned}$$
If $\mu \neq 0$, since $\mu$ is a Borel measure, there exists a ball $B$ such that $\mu(B) >0$. Fix a point $y_0$ and $r>0$ such that $B \subset C_r$. Let $f= \chi_{Q(y_0, r) \setminus {\mathrm{supp}}(\nu)}$ in inequality ; then the left-hand side equals $0$. Since $\nu$ is singular with respect to Lebesgue measure, $|{\mathrm{supp}}(\nu)|=0$, so the first term on the right-hand side is positive. Since the integrand in the second term on the right-hand side is bounded away from $0$, the second term is positive unless $\mu(C_r)=0$, a contradiction. Hence, $\mu =0$.
Now let $\mu$ be a regular measure with Lebesgue decomposition $d\mu= d\mu_s + u \; dx$, where $u \not\equiv 0$, and suppose $d\nu=d\nu_s + vdx$, where $v$ is a non-negative function such that $v(x)< \infty$ a.e. Fix a point $y_0$ such that $ 0< u(y_0) < \infty$. We can further assume that $y_0$ is a Lebesgue point for $\mu$ in the sense of Lemma \[lem:differentiation theorem\], and that the conclusion of Lemma \[lem:essential infimum\] holds for the function $v$ at $y_0$. Let $a= \operatorname*{ess\,inf}_{ x \in Q(y_0,r) } v(x)$. Given $\epsilon>0$, let $E= \{ x \in Q(y_0,r): v(x) < a + \epsilon \}$, $A=E \setminus {\mathrm{supp}}(\nu_s)$, and set $ f= |A|^{-1} \chi_{A}$ in inequality . By the definition of the essential infimum, $|E|>0$, and since $|{\mathrm{supp}}(\nu_s)|=0$, we have that $|A|>0$. Thus, $$\int_{C_r} \frac{1}{(r+|x-x_0|)^n} \; d\mu(x)
\leq C\frac{\nu(A)}{|A|} \leq C\frac{v(A)}{|A|}
\leq C(a+ \epsilon)
= C [ \operatorname*{ess\,inf}_{x \in Q(y_0,r)} v(x) + \epsilon].$$ Since $\epsilon>0$ was arbitrary, this inequality holds with $\epsilon=0$. As $r\rightarrow 0$, $C_r$ converges to the cone $C_0$ with central axis $y_0+ ts\sqrt{n} u_0, s\geq 0$. Therefore, by the monotone convergence theorem and Lemma \[lem:essential infimum\] we have $$\label{eq:strong 1-1 3}
\int_{C_0} \frac{1}{|x-y_0|^n} \; d\mu(x) \leq C v(y_0).$$ Let $B_j=B(y_0, 2^{-j}), A_j=(C_0\cap B_j) \setminus B_{j+1}$. Since $C_0$ has constant aperture, there exists $0<\alpha<1$ such that $|A_j|=\alpha|B_j|$, so the collection $\{A_j\}$ shrinks nicely to $y_0$. Then we have that $$\label{eq:strong 1-1 4}
\lim_{j\rightarrow \infty} \frac{\mu(A_j)}{|B_j|} = \alpha u(y_0).$$ Fix $j_0$ such that for all $j \geq j_0$ we have $\frac{\mu(A_j)}{|B_j|} \geq \frac{\alpha}{2} u(y_0)$. Hence, $$\begin{gathered}
v(y_0) \geq c \sum_{j \geq j_0} \int_{A_j} \frac{1}{|x-y_0|^n} d\mu(x)\\
\geq c\sum_{j \geq j_0} 2^{nj} \mu(A_j)
\geq c\sum_{j \geq j_0} \frac{\mu(A_j)}{|B_j|}
\geq c \sum_{j \geq j_0} u(y_0)= \infty. \end{gathered}$$ Let $E= \{ x: 0<u(x)< \infty \}$ which has positive measure since $ u\not\equiv 0$. Then we have $v(x)= \infty$ for a.e $x\in E$. This contradicts the fact that $ v(x)< \infty$ a.e.
Finally, suppose $\mu$ is a regular measure that is singular with respect to Lebesgue measure, and is directionaly doubling in the direction $u_0$, and $\nu$ is a positive regular Borel measure. If $T: L^1(\nu) \rightarrow L^1(\mu)$, then it satisfies a weak $(1,1)$ inequality, and so by Theorem \[thm:necessity measures\] $\mu$ is absolutely continuous; a contradiction.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Study of methods of resolved top quarks kinematic reconstruction in the ${\ensuremath{{t\bar{t}}}}\rightarrow \ell+$jets channel is presented at the particle level as well as the fast-simulation detector level. Previous and current pseudo-top quark reconstruction algorithms are compared with suggestions presented on how to improve the reconstructed top-quark mass line shape, including the check of performance on physics observables in terms of correlations between detector, particle and parton levels, and in unfolding, with implications for current high energy physics experiments.'
address: 'Regional Centre of Advanced Technologies and Materials, Joint Laboratory of Optics of Palacký University and Institute of Physics AS CR, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic'
author:
- 'J. Kvita'
bibliography:
- 'main.bib'
title: 'Study of methods of resolved top quark reconstruction in semileptonic ${\ensuremath{{t\bar{t}}}}$ decay'
---
HEP,pseudo-top quark ,kinematic reconstruction ,unfolding
Introduction
============
Top quark is the heaviest fermion in the Standard Model, its large mass [@1674-1137-40-10-100001] leading to a corresponding mean life time below the typical hadronization time, although the decay dynamics is governed by the weak interaction. The top quark decays within the third generation of quarks to a $W$ boson and a $b$ quark in almost 100% cases.
In hadron collisions, top quarks are produced either singly with the participation of the weak interaction, or in pairs via the strong interaction, although interference between these two leading-order pictures is present in higher orders of the perturbation theory. Production of multiple top quark final states is a subject of experimental searches.
When produced at low transverse momentum (${\ensuremath{{p_{\rm T}}}}$) w.r.t. the beam axis ($p^{t}_\mathrm{T} \lesssim m_t / 2$), top quark decay products can be identified via angularly resolved objects in a detector. With increasing transverse momentum, however, top quark decay products become collimated and merged into “boosted” objects requiring dedicated experimental techniques.
While the high-momentum top quarks are interesting in accessing the physics of a heavy quark at high momentum transfers and possibly probing new physics in the TeV regime, the resolved topology still constitutes the bulk of the statistics delivered in proton-proton ($pp$) collisions by the LHC accelerator and serves as a useful tool in high energy physics (HEP). Improved methods of top quark identification and reconstruction can thus lead to a better understanding of not only the physics of the top quark, but also of phenomena where top quark events form a background to more exotic or beyond-the-standard model (BSM) processes.
Kinematics of resolved top quarks can be reconstructed using the so-called pseudo-top algorithm [@Aad:2015eia] which is a frequent and useful tool in extracting full kinematic information in the ${\ensuremath{{t\bar{t}}}}$ environment in $pp$ collisions. Objects with a high correspondence to the kinematics of the original top quarks at the parton level are constructed from stable particles or detector-level objects using the same algorithm. Measured and fully corrected (for detector effects) spectra of these objects are used to tune and validate Monte Carlo (MC) generator tunes as well as challenge perturbative quantum chromodynamics (pQCD) calculations at various precision, search for new physics and constrain spectra shapes in the ${\ensuremath{{t\bar{t}}}}$ sample which is an important background for searches for e.g. the ${\ensuremath{{t\bar{t}}}}{}+$Higgs boson production.
Measurements unfolded to the particle level in well-defined fiducial phase-space volumes close to the detector level are useful for parameters tuning and validation of fixed-order MC generators at various precision and of different models of processes like hadronization, initial and final state radiation or underlying event [@ATL-PHYS-PUB-2016-020].
A solid definition of particle-level objects with a good correspondence to top-quarks kinematics is important in order not to dilute the information at both detector and particle levels. Using parton-level top quarks as the reference level to which measured spectra are corrected involves large corrections to the full phase-space as well as theoretical ambiguities of defining top quarks as partons. The definition of variables at the particle level with a good correlation to the four-momenta of parton top quarks is preferred as it provides a weaker model dependence of the measured cross sections compared to the definition at the parton level, yielding more robust results in time as a heritage of current high-energy physics experiments.
The goal of the presented study is to compare various modifications of the pseudo-top algorithm and their performance in terms of the resolution of the reconstructed top quark and ${\ensuremath{{t\bar{t}}}}{}$ mass as well as in terms of the degree of correlation between parton, particle and detector levels. The physics objects and event selection are described in Section \[sec:select\]. Events where ${\ensuremath{{t\bar{t}}}}{}$ pairs are produced in $pp$ collisions at the central-mass-energy of [[13$\,\textrm{TeV}$]{}]{} were generated at particle level, with the subsequent detector level simulated using simple yet realistic tools as described in Section \[sec:samples\] with the focus on the approximate ATLAS experiment geometry and resolutions. Only events in the semileptonic ${\ensuremath{{t\bar{t}}}}$ decay channel are generated as this channel provides optimal signal-to-background ratio and large statistics in current experimental data, as well as reasonably constraint kinematics. Results are presented in Section \[sec:results\] while \[app1\] summarizes the analytic solutions to various conditions used to reconstruct the missing kinematic information carried away by the neutrino.
Objects Definition and Selection {#sec:select}
================================
This study focuses on cases where the ${\ensuremath{{t\bar{t}}}}{}$ pair decays semileptonically, i.e. one $W$ boson from either top quark decays hadronically while the other decays leptonically into a pair of a lepton and a neutrino. Decays to a $\tau$ lepton are considered when the $\tau$ lepton decays to an electron or a muon (and the corresponding neutrino), which can then pass the selection criteria.
The [<span style="font-variant:small-caps;">Rivet</span>]{} [@Buckley:2010ar] version 2.5.4 and the [<span style="font-variant:small-caps;">Rivet</span>]{} analysis `ATLAS_2015_I1404878` [@Aad:2015mbv] of the 8 TeV measurement of differential spectra in $pp \rightarrow {\ensuremath{{t\bar{t}}}}{}$ events by the ATLAS experiment have been used as the baseline of the objects selection and the pseudo-top algorithm definition, which was then modified (see Section \[sec:results\]).
Collimated hadronic final states dubbed “jets” reconstructed from stable particles except neutrinos by the Anti-$k_t$ algorithm [@Cacciari:2008gp] with the distance parameter of 0.4 are required to be within pseudorapidity [^1] $|\eta| < 2.5$ and to pass the requirement on their transverse momentum (w.r.t. the beam, i.e. the $z$, axis) of [[${\ensuremath{{p_{\rm T}}}}> 25$$\,\textrm{GeV}$]{}]{}. Jets are further labelled (tagged) as $b$-jets if a $b$-hadron with ${\ensuremath{{p_{\rm T}}}}> 5\,$GeV is found within $\Delta R < 0.4$ around the jet axis. The presence of two $b$-jets is an important event signature and is part of most event selection in HEP analyses concerning top quarks.
Leptons (electron or muons) are selected within the same kinematic limits, but are first “dressed” in terms of adding four-momenta of photons within 0.1 in a cone of radius defined as $\Delta R = \sqrt{\Delta\eta^2 + \Delta\phi^2}$ around the lepton, to account for final-state photon radiation which typically is included in the lepton final states in a detector. Particle jets overlapping with the selected lepton within $\Delta R < 0.2$ are removed.
In summary, at least four jets are expected in the event, two of which are required to be $b$-tagged, and a high-${\ensuremath{{p_{\rm T}}}}$ lepton and a large transverse energy imbalance in the event due to the escaping neutrino. In practice, the requirement of two $b$-jets often yields sufficiently pure ${\ensuremath{{t\bar{t}}}}{}$ sample that additional selection criteria on the missing transverse energy are not needed. While events with one $b$-tagged jets are often used e.g. for measuring the inclusive cross-section, they are not considered in this study as the requirement of two $b$-tagged jets removes combinatorial ambiguities in the jet assignment to top quark decay products.
Samples {#sec:samples}
=======
All events were generated for the case of $pp$ collisions at the centre-of-mass energy of [[13$\,\textrm{TeV}$]{}]{} using the [<span style="font-variant:small-caps;">MadGraph</span>]{} version [2.5.5]{} simulation toolkit [@Alwall:2014hca] which was chosen for its versatility and ability to generate all processes considered in this analysis. This generator has also been used for data comparison by the CMS collaboration and gradually also by the ATLAS collaboration. In total, $2\,$M events were generated for parton-shower-to-matrix-element matched processes $pp \rightarrow {\ensuremath{{t\bar{t}}}}{}+$jet at the leading (LO) order in pQCD and $pp \rightarrow {\ensuremath{{t\bar{t}}}}{}$ at the next-to-leading (NLO) order using the Standard Model matrix elements. For the purpose of studying the unfolding performance, an alternative sample of 2 M ${\ensuremath{{t\bar{t}}}}{}$ events was generated at the LO only, to provide a sample with slightly different spectra. Finally, 1 M events were generated for the process of a hypothetical additional neutral heavy vector boson $Z'$ decaying as $pp \rightarrow Z' \rightarrow {\ensuremath{{t\bar{t}}}}{}$ (using the model [@FeynModelZprime; @Christensen:2008py; @Wells:2008xg]). Parton shower and hadronization were simulated using the integrated [<span style="font-variant:small-caps;">Pythia</span>]{}8 [@Sjostrand:2007gs; @Pythia8] generator and the top-quark mass of $172\,$GeV ([<span style="font-variant:small-caps;">MadGraph</span>]{} default) was used for all simulated samples. The detector-level simulation is described in Sect. \[sec:delphes\].
Pseudotop algorithm studies {#sec:results}
===========================
Hadronic pseudo-$W$
-------------------
The particle-level candidate for the hadronically decaying $W$ boson is composed from non-$b$-tagged jets by either using such two highest-[${p_{\rm T}}$]{} light jets or by finding the pair of light-flavour jets with an invariant mass closest to the $W$ boson mass $m_W = 80.4\,$GeV. The two scenarios, as defined and used in ATLAS 7 TeV [@Aad:2015eia]; and ATLAS 8 TeV [@Aad:2015mbv] and 13 TeV [@Aaboud:2017fha] analyses, respectively, are compared at the particle level in Fig. \[pst:mttlep\_nupz\_study1\] using the privately simulated samples as detailed in Sec. \[sec:samples\]. The plots show that the original definition (denoted “old $W^\mathrm{had}$” in plot legends) using the pair of highest-[${p_{\rm T}}$]{} non-$b$-tagged jets was improved (in what is now the standard option) by using the pair of jets with invariant mass closest to $m_W$. Improvement is seen terms of the line shapes of both the hadronic pseudo-$W$ and hadronic pseudo-top masses ($m^{W,\mathrm{had}}$ and $m^{t,\mathrm{had}}$), namely providing a less-pronounced tail towards larger masses. A change in slope of the transverse momentum spectra ($p_{\mathrm{T}}^{W,\mathrm{had}}$ and $p_{\mathrm{T}}^{t,\mathrm{had}}$) is also seen, although if reproduced at both particle and detector levels this is not a priory a problem in using either definition e.g. for MC tuning studies. Still, any improvement in the mass line shape is of course a preferred option, as it possibly improves also the correlation to the parton level.
![The particle-level hadronic pseudo-$W$ mass (top left) and [${p_{\rm T}}$]{} (top right), and the hadronic pseudo-top quark mass (bottom left) and [${p_{\rm T}}$]{} (bottom right) for different choices of the light jets to form the hadronic pseudo-$W$ in the event: as the pair of jets with invariant mass closest to $m_W$ (dashed), or as the pair of highest-[${p_{\rm T}}$]{} non-$b$-tagged jets (dotted). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator.[]{data-label="pst:mttlep_nupz_study1"}](figure_1.pdf "fig:"){width="50.00000%"} ![The particle-level hadronic pseudo-$W$ mass (top left) and [${p_{\rm T}}$]{} (top right), and the hadronic pseudo-top quark mass (bottom left) and [${p_{\rm T}}$]{} (bottom right) for different choices of the light jets to form the hadronic pseudo-$W$ in the event: as the pair of jets with invariant mass closest to $m_W$ (dashed), or as the pair of highest-[${p_{\rm T}}$]{} non-$b$-tagged jets (dotted). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator.[]{data-label="pst:mttlep_nupz_study1"}](figure_2.pdf "fig:"){width="50.00000%"}\
![The particle-level hadronic pseudo-$W$ mass (top left) and [${p_{\rm T}}$]{} (top right), and the hadronic pseudo-top quark mass (bottom left) and [${p_{\rm T}}$]{} (bottom right) for different choices of the light jets to form the hadronic pseudo-$W$ in the event: as the pair of jets with invariant mass closest to $m_W$ (dashed), or as the pair of highest-[${p_{\rm T}}$]{} non-$b$-tagged jets (dotted). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator.[]{data-label="pst:mttlep_nupz_study1"}](figure_3.pdf "fig:"){width="50.00000%"} ![The particle-level hadronic pseudo-$W$ mass (top left) and [${p_{\rm T}}$]{} (top right), and the hadronic pseudo-top quark mass (bottom left) and [${p_{\rm T}}$]{} (bottom right) for different choices of the light jets to form the hadronic pseudo-$W$ in the event: as the pair of jets with invariant mass closest to $m_W$ (dashed), or as the pair of highest-[${p_{\rm T}}$]{} non-$b$-tagged jets (dotted). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator.[]{data-label="pst:mttlep_nupz_study1"}](figure_4.pdf "fig:"){width="50.00000%"}
Pseudo-top quarks
-----------------
The four-momentum of the leptonically decaying pseudo-top quark is defined by adding the four-momenta of the $b$-jet closest to the lepton and of the reconstructed leptonically decaying pseudo-$W$ candidate detailed later. Finally, the four-momenta of the hadronically decaying pseudo-top quark is defined as the sum of the four-momenta of the remaining highest-[${p_{\rm T}}$]{} $b$-jet and of the hadronic pseudo-$W$ candidate.
Optimization of the $p_z^\nu$ choice
------------------------------------
As the undetected neutrino from the leptonic $W$ decay carries away kinematic information, its momentum has to be reconstructed. The transverse component of its momentum can be easily estimated using the vector of the reconstructed missing transverse energy, defined as the negative sum of the neutrinos transverse momenta at the particle level or as the negative sum of calorimeter transverse energy deposits at the detector level. Neutrino’s longitudinal momentum ($p_z^\nu$) has to be computed from an additional reasonable physics constrain. The following choices are tried for the computation of $p_z^\nu$ and compared for distributions of rapidities of the leptonic $W$ and leptonic top quark ($y^{W,\mathrm{lep}}$ and $y^{t,\mathrm{lep}}$) and checking also their hadronic counterparts ($y^{W,\mathrm{had}}$ and $y^{t,\mathrm{had}}$).
1. The usual (denoted as “standard” in plot legends) definition of the leptonic pseudo-$W$ and leptonic pseudo-top relies on the solution of $p^\nu_z$ from a quadratic equation stemming from the $m_{\ell\nu}= m_W$ condition. If a complex solution is found, the imaginary part is dropped, when two real solutions exist, the one with smaller $|p_z^\nu|$ is taken. This choice has some physics motivation, e.g. in the fact that top quark pairs are produced in $gg$, i.e. same-parton species, collisions, and on average a large imbalance in the $p_z$ of the $gg$ system is not expected. However, this neutrino solution leads to visibly different spectra of rapidities of leptonic pseudo-$W$ and pseudo-top quark candidates (see Fig. \[pst:Wrap\_cmp2\]), compared to those of their hadronic counterparts, namely being significantly more central by construction.
2. As a test and a check, the more forward $p_z$ solution is also tried, denoted as “more forward” in plot legends.
3. As a modification, a new condition (denoted as “closest $m_t$”) based on the minimal difference $|m_{t,\mathrm{had}} - m_{t,\mathrm{lep}}|$ is used to choose the best $p_z^\nu$ solution. This simple reconsideration leads to a rapidity spectrum of the leptonic pseudo-$W$ as well as of the leptonic pseudo-top be closer in shape to those of their hadronic counterparts (see Fig. \[pst:Wrap\_cmp2\]–\[pst:pseudotop\_nupz\_mt\_study3\]), though slightly broader. However, as seen in Figure \[pst:pseudotop\_nupz\_mt\_study3\], the leptonic pseudo-top mass spectrum is improved in the low-mass tail and especially in the peak of the distribution.
4. Next, a novel solution (denoted as “same $m_t$” in plot legends) to the $p^\nu_z$ problem is defined as a solution to the $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ condition, taking again a more central solution in case of a positive quadratic equation discriminant (see again Fig. \[pst:pseudotop\_nupz\_mt\_study3\]). Although this algorithm further diminishes the low-mass tail for the leptonic pseudo-top, it largely increases the large-mass tail and decreases magnitude in the peak region and leads to large tails in the mass distribution of the leptonic pseudo-$W$ (not shown).
5. Returning to the $p_z^\nu$ solution from the $m_{\ell\nu}= m_W$ condition, a swap in the $b$-jets assignment is also newly allowed, and both neutrino solutions are also tried similarly as in the “closest $m_t$” solution, so in total the best choice out of four is selected in terms of minimal $|m_{t,\mathrm{had}} - m_{t,\mathrm{lep}}|$; this algorithm is denoted “best $m_t$” in plot legends.
Other methods, like trying the “same $m_t$” solution first when in the case of a negative discriminant the standard solution is tried next, were also tested, but these approaches did not lead to significant improvements in performance.
![The particle-level leptonic (top left) and hadronic (top right) pseudo-$W$ rapidity and leptonic (bottom left) and hadronic (bottom right) pseudo-top rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed), more forward (dotted) and “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:Wrap_cmp2"}](figure_5.pdf "fig:"){width="50.00000%"} ![The particle-level leptonic (top left) and hadronic (top right) pseudo-$W$ rapidity and leptonic (bottom left) and hadronic (bottom right) pseudo-top rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed), more forward (dotted) and “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:Wrap_cmp2"}](figure_6.pdf "fig:"){width="50.00000%"}\
![The particle-level leptonic (top left) and hadronic (top right) pseudo-$W$ rapidity and leptonic (bottom left) and hadronic (bottom right) pseudo-top rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed), more forward (dotted) and “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:Wrap_cmp2"}](figure_7.pdf "fig:"){width="50.00000%"} ![The particle-level leptonic (top left) and hadronic (top right) pseudo-$W$ rapidity and leptonic (bottom left) and hadronic (bottom right) pseudo-top rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed), more forward (dotted) and “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:Wrap_cmp2"}](figure_8.pdf "fig:"){width="50.00000%"}
![The particle-level leptonic (top left) and hadronic (top right) pseudo-top mass and rapidity of the leptonic (bottom left) and hadronic (bottom right) pseudo-top for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid); and “same $m_t$” (dot-dashed). The hadronic pseudo-top spectra are unaffected by the choices on the leptonic side of the event, showing however the similarity of the $y_{t,\mathrm{had}}$ rapidity spectrum to the leptonic one from the “closest $m_t$” solution. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:pseudotop_nupz_mt_study3"}](figure_9.pdf "fig:"){width="50.00000%"} ![The particle-level leptonic (top left) and hadronic (top right) pseudo-top mass and rapidity of the leptonic (bottom left) and hadronic (bottom right) pseudo-top for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid); and “same $m_t$” (dot-dashed). The hadronic pseudo-top spectra are unaffected by the choices on the leptonic side of the event, showing however the similarity of the $y_{t,\mathrm{had}}$ rapidity spectrum to the leptonic one from the “closest $m_t$” solution. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:pseudotop_nupz_mt_study3"}](figure_10.pdf "fig:"){width="50.00000%"}\
![The particle-level leptonic (top left) and hadronic (top right) pseudo-top mass and rapidity of the leptonic (bottom left) and hadronic (bottom right) pseudo-top for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid); and “same $m_t$” (dot-dashed). The hadronic pseudo-top spectra are unaffected by the choices on the leptonic side of the event, showing however the similarity of the $y_{t,\mathrm{had}}$ rapidity spectrum to the leptonic one from the “closest $m_t$” solution. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:pseudotop_nupz_mt_study3"}](figure_11.pdf "fig:"){width="50.00000%"} ![The particle-level leptonic (top left) and hadronic (top right) pseudo-top mass and rapidity of the leptonic (bottom left) and hadronic (bottom right) pseudo-top for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid); and “same $m_t$” (dot-dashed). The hadronic pseudo-top spectra are unaffected by the choices on the leptonic side of the event, showing however the similarity of the $y_{t,\mathrm{had}}$ rapidity spectrum to the leptonic one from the “closest $m_t$” solution. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:pseudotop_nupz_mt_study3"}](figure_12.pdf "fig:"){width="50.00000%"}
Performance on the line shape of a hypothetical $Z'$ particle
-------------------------------------------------------------
Performance of one of the new choice of the neutrino $p_z$ solution w.r.t. the standard one was checked on the shape of the reconstructed mass peak of a hypothetical particle $Z'$ particle decaying to a ${\ensuremath{{t\bar{t}}}}$ pair. Its mass of $m_{Z'} = 700\,$GeV was selected such that the resolved topology of top quark decay products is still dominant over the boosted one. The results are presented in Fig. \[pst:mtt\_nupz\_study4\], showing a sharper peak of the pseudo-[${t\bar{t}}$]{} mass ($m^{{\ensuremath{{t\bar{t}}}}{}}$) distribution for the novel proposed method (“closest $m_t$”).
![The particle-level pseudo-[${t\bar{t}}$]{} invariant mass distribution for the [${t\bar{t}}$]{} sample (left) and for the hypothetical $Z'$ boson of mass of $700\,$GeV and decaying to a ${\ensuremath{{t\bar{t}}}}$ pair for the different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed) and “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mtt_nupz_study4"}](figure_13.pdf "fig:"){width="50.00000%"} ![The particle-level pseudo-[${t\bar{t}}$]{} invariant mass distribution for the [${t\bar{t}}$]{} sample (left) and for the hypothetical $Z'$ boson of mass of $700\,$GeV and decaying to a ${\ensuremath{{t\bar{t}}}}$ pair for the different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed) and “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mtt_nupz_study4"}](figure_14.pdf "fig:"){width="50.00000%"}
Of course, the physical binning is driven by the experimental resolution and cannot be this fine, however, a $10$% improvement in the peak region is possible, which is at the level of the typical experimental uncertainties and resolution.
Kinematic variables {#sec:vars}
-------------------
By construction, the changes in the $p_z^\nu$ choice do not affect ${\ensuremath{{p_{\rm T}}}}$-related quantities of the leptonic top quark nor the [${t\bar{t}}$]{} system, nor the out-of-plane variable $p_{\rm out}$ [@Aad:2015mbv] used in initial and final state radiation tuning [@ATL-PHYS-PUB-2016-020]. However, improvement may be searched for in the line shape of the mass and rapidity of the leptonic top quark ($m^{t,\mathrm{lep}}$, $y^{t,\mathrm{lep}}$) and of the [${t\bar{t}}$]{} system ($y^{{\ensuremath{{t\bar{t}}}}{}}$), and the mass ($m^{{\ensuremath{{t\bar{t}}}}{}}$) of the [${t\bar{t}}$]{} system, or other variables composed from the two top quarks which also use the longitudinal momentum, like the $\cos\theta^*$ (angle between a top quark and the $z$ axis in a frame where the ${\ensuremath{{t\bar{t}}}}{}$ system has zero momentum along the $z$ axis) and the laboratory opening angle between the two top quarks ($\delta_{{\ensuremath{{t\bar{t}}}}{}}$). Further variables studied later are the transverse momentum of the ${\ensuremath{{t\bar{t}}}}$ system ($p_\mathrm{T}^{{\ensuremath{{t\bar{t}}}}{}}$) and the out-of-plane momentum $p_{\mathrm{out}}$ which has two entries per event due to the possible rôle swap of a top quark to define a plane together with the $z$ axis direction, to which the momentum of the other top quark is projected; and the ${y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}{}}}$ and ${\chi^{t\bar{t}}}$ variables, defined as $${p_{\mathrm{out}}}\equiv \vec{p}^{\,t, \mathrm{had}} \cdot \frac{\vec{p}^{\,t,\mathrm{lep}} \times \hat{z}}{|\vec{p}^{\,t,\mathrm{lep}}\times \hat{z}|} \,, \quad \mathrm{and\,\, had \leftrightarrow lep}$$ $${y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}{}}}\equiv \frac12 \left| y^{t,\mathrm{had}} + y^{t,\mathrm{lep}} \right|$$ $${\chi^{t\bar{t}}}\equiv \exp \left| y^{t,\mathrm{had}} - y^{t,\mathrm{lep}} \right| \,.$$ These are sensitive to final state radiation, the boost of the ${\ensuremath{{t\bar{t}}}}{}$ system and thus also to PDFs; and to new physics via their sensitivity to the production angle in central mass system. Their shapes also differ for the “same $m_t$” and “best $m_t$” options.
Performance on the [<span style="font-variant:small-caps;">Delphes</span>]{} detector level {#sec:delphes}
-------------------------------------------------------------------------------------------
In order to check a possible improvement in the correspondence between particle and detector levels, the [<span style="font-variant:small-caps;">Delphes</span>]{} simulation package [@deFavereau:2013fsa] was used with a modified ATLAS card (to allow storage of partons, $b$-hadrons and photons needed for dressing of leptons) to simulate the passage of particles through a realistic particle detector. The ATLAS card was validated by [<span style="font-variant:small-caps;">Delphes</span>]{} authors as described in Section 5 of [@deFavereau:2013fsa]. A cross-check of using similarly modified CMS card in this analysis was also performed, finding very similar results. For these studies, $2\,$M ${\ensuremath{{t\bar{t}}}}{}$ events were generated by [<span style="font-variant:small-caps;">MadGraph</span>]{} to provide a larger sample also at the [<span style="font-variant:small-caps;">Delphes</span>]{} detector level due to finite detector efficiency to select the objects within the phase-space defined in Section \[sec:select\]. The efficiency was found to be about 7%, similar as in real experiments and analyses.
Independent implementations of the aforementioned pseudo-top algorithms were used both at the particle (using [<span style="font-variant:small-caps;">Pythia</span>]{}8 stable particles) and [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels.
At the particle level, selected leptons (electrons or muons) were dressed by photons with $\Delta R < 0.1$ w.r.t. the lepton. The $b$-tagging at the particle level was performed by matching a particle jet to an open-beauty $b$-hadron (meson or a baryon) with ${\ensuremath{{p_{\rm T}}}}> 5\,$GeV based on the PDG-ID codes [@1674-1137-40-10-100001]. If a match was found within $\Delta R < 0.4$, the particle jet was considered as $b$-tagged.
First the performance on the line shape of the leptonic pseudo-top mass is checked in Fig. \[pst:mt\_nupz\_study5\], showing a very similar behaviour compared to the pure [<span style="font-variant:small-caps;">Rivet</span>]{} study in the preceding Section at the particle level, and a slightly modified performance at the [<span style="font-variant:small-caps;">Delphes</span>]{} detector level where the new approach (“closest $m_t$”) still yields smaller low-mass tail while the “same $m_t$” yields a slightly sharper peak, although producing a more pronounced tail to higher masses. The “best $m_t$” choice yields even smaller low-mass tail, but returns even more pronounced tail towards larger masses.
The performance on the hypothetical $Z'$ particle (using the sample of $1\,$M events) at the [<span style="font-variant:small-caps;">Delphes</span>]{} detector level is compared in Fig. \[pst:zp\_study8\] showing unfortunately a completely washed-out peak compared to a more pronounced peak of the “closest $m_t$” at the particle level, similar to what was found using [<span style="font-variant:small-caps;">Rivet</span>]{} in the previous Section.
![Leptonic pseudo-top mass for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closet $m_t$” (solid), “same $m_t$” (dot-dashed), and the one giving the best top quark masses allowing also the $b$-jets swap (“best $m_t$”, dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. Top (bottom) plots are in the logarithmic (linear) scale.[]{data-label="pst:mt_nupz_study5"}](figure_15.pdf "fig:"){width="99.00000%"}\
![Leptonic pseudo-top mass for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closet $m_t$” (solid), “same $m_t$” (dot-dashed), and the one giving the best top quark masses allowing also the $b$-jets swap (“best $m_t$”, dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. Top (bottom) plots are in the logarithmic (linear) scale.[]{data-label="pst:mt_nupz_study5"}](figure_16.pdf "fig:"){width="99.00000%"}
![Pseudo-[${t\bar{t}}$]{} invariant mass distribution for the hypothetical $Z'$ boson generated at mass of $700\,$GeV and decaying to a ${\ensuremath{{t\bar{t}}}}$ pair at the particle level (left) and [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS [<span style="font-variant:small-caps;">Delphes</span>]{} card (right) for the different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed) and the “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. Top (bottom) plots are in the logarithmic (linear) scale. []{data-label="pst:zp_study8"}](figure_17.pdf "fig:"){width="99.00000%"} ![Pseudo-[${t\bar{t}}$]{} invariant mass distribution for the hypothetical $Z'$ boson generated at mass of $700\,$GeV and decaying to a ${\ensuremath{{t\bar{t}}}}$ pair at the particle level (left) and [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS [<span style="font-variant:small-caps;">Delphes</span>]{} card (right) for the different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (dashed) and the “closest $m_t$” (solid). Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. Top (bottom) plots are in the logarithmic (linear) scale. []{data-label="pst:zp_study8"}](figure_18.pdf "fig:"){width="99.00000%"}
Correlations between levels {#sec:migra}
---------------------------
Migration matrices between the particle and the detector (provided by [<span style="font-variant:small-caps;">Delphes</span>]{}) levels were obtained and normalized so that each element of the matrix $\mathcal{M}_{ij}$ stands for the fraction of events migrating from a given particle-level bin $i$ to various detector-level bins labelled $j$. As rapidities of the leptonic pseudo-top quark and of the ${\ensuremath{{t\bar{t}}}}$ system depend on the choice of the neutrino $p_z$ solution, migration matrices for these variables were studied. Compared to the standard choice, worse performance in terms of the correlation between the particle and detector levels was found for the “same $m_t$” method (not shown) while similar (though slightly lower) for the “closest $m_t$” method, as displayed in Fig. \[pst:migra\_study6\_ptcl\_det\]. Correlations between the particle and detector levels for more kinematic variables and all the studied algorithms are summarized in Tab. \[tab:corrs:particle\_detector\].
Matching between the particle and detector levels {#sec:match}
-------------------------------------------------
In order to further improve the correlation between the detector and particle levels, current HEP experiments also restrict the analysis phase-space to events where corresponding objects forming the pseudo-top quarks (i.e. the lepton, light jets and $b$-tagged jets) are well angularly matched between the particle and detector levels, using usually a $\Delta R$ cut of 0.02 for leptons and 0.35 for jets. This leads to much more diagonal migration matrices, as can be seen in Fig. \[pst:migra\_study6\_ptcl\_det\_match\]. The price for this is an additional matching efficiency of the order of 0.5–0.7 which needs to be compensated for using a dedicated bin-by-bin correction, while the advantage is that the migration matrix then accounts only for resolution and not for combinatorial effects. In particular, for the “best $m_t$” case, the matching condition between the two $b$-jets had to be relaxed in order to allow for the swap of the $b$-jets, as the strict assignment was otherwise only about 20% efficient. The performance of the algorithms on the line shape of the leptonic pseudo-top mass as shown in Fig. \[pst:mt\_nupz\_study5\_match\] is similar to the case without the matching requirement (Fig. \[pst:mt\_nupz\_study5\]). Correlations between the particle and detector levels for the case of matched events are summarized in Tab. \[tab:corrs:particle\_detector\_match\], with the highlighted best performing algorithm.
![Migration matrices between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). No angular matching between the particle and detector level objects forming the pseudo-tops was performed. The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns. []{data-label="pst:migra_study6_ptcl_det"}](figure_19.pdf "fig:"){width="100.00000%"}\
![Migration matrices between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). No angular matching between the particle and detector level objects forming the pseudo-tops was performed. The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns. []{data-label="pst:migra_study6_ptcl_det"}](figure_20.pdf "fig:"){width="100.00000%"}
![Migration matrices between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for matched events for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns. []{data-label="pst:migra_study6_ptcl_det_match"}](figure_21.pdf "fig:"){width="100.00000%"}\
![Migration matrices between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for matched events for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns. []{data-label="pst:migra_study6_ptcl_det_match"}](figure_22.pdf "fig:"){width="100.00000%"}
![Leptonic pseudo-top mass for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the one giving the “best $m_t$”. Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. Top (bottom) plots are in the logarithmic (linear) scale. []{data-label="pst:mt_nupz_study5_match"}](figure_23.pdf "fig:"){width="100.00000%"}\
![Leptonic pseudo-top mass for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the one giving the “best $m_t$”. Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. Top (bottom) plots are in the logarithmic (linear) scale. []{data-label="pst:mt_nupz_study5_match"}](figure_24.pdf "fig:"){width="100.00000%"}
observable standard closest $m_t$ same $m_t$ best $m_t$
------------------------------------------------------ ---------- --------------- ------------ ------------
$m^{{\ensuremath{{t\bar{t}}}}{}}$ **0.77** 0.72 0.68 0.69
$y^{{\ensuremath{{t\bar{t}}}}{}}$ **0.93** **0.93** 0.89 0.91
$\delta^{{\ensuremath{{t\bar{t}}}}{}}$ **0.69** **0.69** 0.64 0.58
$|\cos\theta^*|$ **0.65** 0.63 0.61 0.48
$y^{t,\,\mathrm{lep}}$ **0.96** 0.92 0.85 0.86
${y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}{}}}$ **0.84** 0.82 0.78 0.78
${\chi^{t\bar{t}}}$ **0.73** 0.67 0.68 0.59
: Correlation coefficients of the migration matrices between the particle and [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for different observables (with largest values, within 1%, highlighted in bold) and various ways to reconstruct the pseudo-${\ensuremath{{t\bar{t}}}}$ related observables.[]{data-label="tab:corrs:particle_detector"}
observable standard closest $m_t$ same $m_t$ best $m_t$
------------------------------------------------------ ---------- --------------- ------------ ------------
$m^{{\ensuremath{{t\bar{t}}}}{}}$ **0.95** 0.93 **0.95** 0.91
$y^{{\ensuremath{{t\bar{t}}}}{}}$ **0.99** **0.99** **0.99** 0.97
$\delta^{{\ensuremath{{t\bar{t}}}}{}}$ **0.97** **0.96** **0.96** 0.85
$|\cos\theta^*|$ **0.92** 0.90 **0.93** 0.77
$y^{t,\,\mathrm{lep}}$ **0.99** 0.97 **0.98** 0.92
${y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}{}}}$ **0.98** 0.96 **0.97** 0.91
${\chi^{t\bar{t}}}$ **0.96** 0.92 **0.96** 0.84
: Correlation coefficients of the migration matrices between the particle and [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for matched events for different observables (with largest values, within 1%, highlighted in bold) and various ways to reconstruct the pseudo-${\ensuremath{{t\bar{t}}}}$ related observables.[]{data-label="tab:corrs:particle_detector_match"}
In addition, a comparison to parton-level top quarks was performed, taking the last top quarks in the [<span style="font-variant:small-caps;">Pythia</span>]{}8 parton chain, corresponding to top quarks after the final state radiation. For simplicity, the leptonic top quark at the parton level is taken as the one angularly closer to the particle or detector level leptonic pseudo-top. Migration matrices between the parton and particle, and parton and detector levels were studied with the following observations.
The correlation between the parton and particle levels is shown in Fig. \[pst:migra\_study6\_parton\_ptcl\] where only a slight decorrelation is observed for the novel “closest $m_t$” method. The resulting correlation coefficients for all the studied spectra between the parton and particle or detector levels for more variables are summarized in Tab. \[tab:corrs:parton\_particle\] or Tab. \[tab:corrs:parton\_detector\], respectively. It can be observed that the correlation between the parton and [<span style="font-variant:small-caps;">Delphes</span>]{} detector level is worse for the pseudo-[${t\bar{t}}$]{} mass using the “same $m_t$” method compared to the standard one, but all correlations are very similar for the standard and the “closest $m_t$” methods. Still, the improved and more careful treatment of the rapidity of the neutrino in the “closest $m_t$” method leads to the removal of the “tilt” in migration matrices of the rapidities of the pseudo-${\ensuremath{{t\bar{t}}}}$ as well as the leptonic pseudo-top compared to the standard method, and the parton-to-detector level correspondence is thus more linear (Fig. \[pst:migra\_study6\_parton\_det\]).
No tilt observed in migration between the particle and detector levels means the rapidities are similarly biased for these two levels compared to the parton level, as can also be checked in bottom plots of Fig. \[pst:migra\_study6\_parton\_ptcl\]. As the rapidities are used in fits of parton distribution functions (PDF), the “compression” of the rapidities of the top quark and the ${\ensuremath{{t\bar{t}}}}{}$ system using the standard reconstruction method possibly dilutes the information and diminishes the potential to constrain the PDF functions, while it could be partially recovered using the proposed “closest $m_t$” method.
observable standard closest $m_t$ same $m_t$ best $m_t$
------------------------------------------------------ ---------- --------------- ------------ ------------
$m^{{\ensuremath{{t\bar{t}}}}{}}$ **0.74** 0.72 0.63 0.70
$y^{{\ensuremath{{t\bar{t}}}}{}}$ **0.93** **0.92** 0.89 0.91
$\delta^{{\ensuremath{{t\bar{t}}}}{}}$ **0.66** **0.66** 0.56 0.57
$|\cos\theta^*|$ 0.46 **0.48** 0.36 0.44
$y^{t,\,\mathrm{lep}}$ **0.90** **0.89** 0.78 0.87
${y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}{}}}$ **0.81** 0.78 0.76 0.78
${\chi^{t\bar{t}}}$ **0.57** **0.56** 0.47 0.50
: Correlation coefficients for the migration matrices between the parton and particle levels for different observables (with largest values, within 1%, highlighted in bold) and various ways to reconstruct the pseudo-${\ensuremath{{t\bar{t}}}}$ related observables.[]{data-label="tab:corrs:parton_particle"}
observable standard closest $m_t$ same $m_t$ best $m_t$
------------------------------------------------------ ---------- --------------- ------------ ------------
$m^{{\ensuremath{{t\bar{t}}}}{}}$ **0.64** 0.62 0.54 0.59
$y^{{\ensuremath{{t\bar{t}}}}{}}$ **0.90** **0.90** 0.83 **0.89**
$\delta^{{\ensuremath{{t\bar{t}}}}{}}$ **0.65** **0.66** 0.56 0.59
$|\cos\theta^*|$ **0.42** **0.43** 0.40 **0.42**
$y^{t,\,\mathrm{lep}}$ **0.88** **0.87** 0.74 0.85
${y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}{}}}$ **0.77** 0.75 0.68 0.73
${\chi^{t\bar{t}}}$ **0.56** 0.53 0.51 0.50
: Correlation coefficients for the migration matrices between the parton and [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for different observables (with largest values, within 1%, highlighted in bold) and various ways to reconstruct the pseudo-${\ensuremath{{t\bar{t}}}}$ related observables.[]{data-label="tab:corrs:parton_detector"}
![Migration matrices between the parton and particle levels for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns.[]{data-label="pst:migra_study6_parton_ptcl"}](figure_25.pdf "fig:"){width="100.00000%"}\
![Migration matrices between the parton and particle levels for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns.[]{data-label="pst:migra_study6_parton_ptcl"}](figure_26.pdf "fig:"){width="100.00000%"}
![Migration matrices between the parton and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns.[]{data-label="pst:migra_study6_parton_det"}](figure_27.pdf "fig:"){width="100.00000%"}\
![Migration matrices between the parton and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the pseudo-[${t\bar{t}}$]{} rapidity (top) and the leptonic pseudo-top rapidity (bottom) obtained using the ATLAS card; for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). The solid bold line is the diagonal, $\rho$ stands for the correlation coefficient evaluated before the normalization of columns.[]{data-label="pst:migra_study6_parton_det"}](figure_28.pdf "fig:"){width="100.00000%"}
Unfolding performance {#sec:unf}
---------------------
In order to check the performance of correcting the detector-level spectra for resolution effects (unfolding), a Python implementation [@fbu_py] of the Fully Bayesian Unfolding technique [@FBU] was used to unfold the rapidity spectra of the pseudo-${\ensuremath{{t\bar{t}}}}$ system to the parton level. In detail, the [<span style="font-variant:small-caps;">Delphes</span>]{} detector level spectrum from the projection of the response matrix was used as input pseudo-data and comparison was made after unfolding to the original parton-level spectrum from the projection of the response matrix on the other axis. It was checked that the unfolded posterior distributions are very well Gaussian and the posterior mean was taken as the unfolded result in each bin. Results in Fig. \[pst:unf\_study\_parton\_closure\] show, besides the largely more central detector-level spectrum for the standard neutrino $p_z$ choice (empty triangles), that a perfect closure (full points) is reached for both standard and “closest $m_t$” choice in terms of the $\chi^2/\mathrm{ndf} \leq 0.01$, by comparing the unfolded histogram divided by the parton-level spectrum to unity. Thus the two options are equivalent in unfolding performance in terms of a closure test within the same sample.
In reality, however, more stringent unfolding tests are needed as the spectrum in data is not the same as in simulation. Different simulation samples lead to different migration matrices and efficiency corrections, which are thus model-dependent. Larger difference between spectra at the detector and parton level can lead to unfolding non-closure which needs to be treated as a systematics.
The following tests are motivated by one of the dominant systematics uncertainties in real measurements which is often due to the choice of the ${\ensuremath{{t\bar{t}}}}$ generator to derive the corrections. A more realistic closure test was thus performed using the LO ${\ensuremath{{t\bar{t}}}}$ sample and unfolding it using the migration matrix derived from the NLO ${\ensuremath{{t\bar{t}}}}$ sample. The difference between the spectra at the LO and NLO is depicted in Fig. \[pst:study\_parton\_det\_LO\_NLO\_diff\]. The unfolding closure test without scaling to the full partonic phase space is shown in Fig. \[pst:unf\_study\_parton\_det\_gensyst\] while the full closure test, i.e. including the efficiency correction to the full partonic phase-space, is shown in Fig. \[pst:unf\_study\_parton\_det\_gensyst\_eff\]. Due to the fact that the efficiency derived using the NLO sample is about 15% higher than that of the LO sample because of kinematics, the closure test was performed between normalized distributions and the number of degrees of freedom (ndf) was lowered by one. In both cases, a more stable and slightly better performance in terms of the $\chi^2$ test can be observed for the case of the “closest $m_t$” option.
![Unfolding closure test (ratio of the unfolded [<span style="font-variant:small-caps;">Delphes</span>]{} detector level to the parton level) for the pseudo-[${t\bar{t}}$]{} rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). []{data-label="pst:unf_study_parton_closure"}](figure_29.pdf "fig:"){width="45.00000%"} ![Unfolding closure test (ratio of the unfolded [<span style="font-variant:small-caps;">Delphes</span>]{} detector level to the parton level) for the pseudo-[${t\bar{t}}$]{} rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). []{data-label="pst:unf_study_parton_closure"}](figure_30.pdf "fig:"){width="45.00000%"}\
![Comparison of the LO (dashed) and NLO (solid) shapes of the pseudo-[${t\bar{t}}$]{} rapidity distribution at the parton (left) and [<span style="font-variant:small-caps;">Delphes</span>]{} detector (right) level obtained using the ATLAS card and for the “closest $m_t$” option at the detector level.[]{data-label="pst:study_parton_det_LO_NLO_diff"}](figure_31.pdf){width="95.00000%"}
![Unfolding the LO ${\ensuremath{{t\bar{t}}}}$ sample using the migration matrix from the NLO ${\ensuremath{{t\bar{t}}}}$ sample. Closure test (ratio of the unfolded [<span style="font-variant:small-caps;">Delphes</span>]{} detector level to the parton level) for the pseudo-[${t\bar{t}}$]{} rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). []{data-label="pst:unf_study_parton_det_gensyst"}](figure_32.pdf "fig:"){width="45.00000%"} ![Unfolding the LO ${\ensuremath{{t\bar{t}}}}$ sample using the migration matrix from the NLO ${\ensuremath{{t\bar{t}}}}$ sample. Closure test (ratio of the unfolded [<span style="font-variant:small-caps;">Delphes</span>]{} detector level to the parton level) for the pseudo-[${t\bar{t}}$]{} rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). []{data-label="pst:unf_study_parton_det_gensyst"}](figure_33.pdf "fig:"){width="45.00000%"}
![Unfolding the LO ${\ensuremath{{t\bar{t}}}}$ sample to the full partonic phase-space using the migration matrix and the efficiency correction from the NLO ${\ensuremath{{t\bar{t}}}}$ sample. Normalized closure test (ratio of the normalized unfolded [<span style="font-variant:small-caps;">Delphes</span>]{} detector level to the normalized parton level) for the pseudo-[${t\bar{t}}$]{} rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). []{data-label="pst:unf_study_parton_det_gensyst_eff"}](figure_34.pdf "fig:"){width="45.00000%"} ![Unfolding the LO ${\ensuremath{{t\bar{t}}}}$ sample to the full partonic phase-space using the migration matrix and the efficiency correction from the NLO ${\ensuremath{{t\bar{t}}}}$ sample. Normalized closure test (ratio of the normalized unfolded [<span style="font-variant:small-caps;">Delphes</span>]{} detector level to the normalized parton level) for the pseudo-[${t\bar{t}}$]{} rapidity for different choices of the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition: the standard choice (left) and the “closest $m_t$” (right). []{data-label="pst:unf_study_parton_det_gensyst_eff"}](figure_35.pdf "fig:"){width="45.00000%"}
Spectra comparison
------------------
Additional information is provided by the comparison of shapes of several physics observables used in applications like tuning; these are shown in Figs. \[pst:mt\_nupz\_study6\_match\]–\[pst:mt\_nupz\_study9\_match\] which show the spectra at particle and detector levels with the angular matching required between objects forming the pseudo-tops at the two levels (see Sec. \[sec:match\]). For spectra of transverse momenta of leptonic and hadronic pseudo-tops (Fig. \[pst:mt\_nupz\_study6\_match\]) and the ${\ensuremath{{t\bar{t}}}}$ system (Fig. \[pst:mt\_nupz\_study7\_match\]), and of the out-of-plane momentum $p_\mathrm{out}$ (Fig. \[pst:mt\_nupz\_study8\_match\]) all solutions are equivalent except for the “best $m_t$” case where large slope changes are observed, disfavouring this option, however well-motivated it had seemed in allowing also the $b$-jets swap (thus affecting also the hadronic-top and ${\ensuremath{{p_{\rm T}}}}$-related quantities). The standard and “same $m_t$” choices lead to unnaturally more central rapidities of the leptonic pseudo-top and of the ${\ensuremath{{t\bar{t}}}}{}$ system (Figs. \[pst:mt\_nupz\_study7\_match\]–\[pst:mt\_nupz\_study8\_match\]). Interestingly, large slope differences are also observed for higher values of the ${y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}{}}}{}$ and ${\chi^{t\bar{t}}}{}$ variables (Fig. \[pst:mt\_nupz\_study9\_match\]) which are of interest for new physics searches using top quarks, and a proper choice of the pseudotop algorithm could be done based on the performance of these variables for particular models. But this task is beyond the scope of this study.
Conclusions
===========
A detailed study of the past, current as well as further modified pseudo-top algorithms and their details used in recent HEP measurements was presented at both the particle and detector levels using ${\ensuremath{{t\bar{t}}}}{}$ events generated by [<span style="font-variant:small-caps;">MadGraph</span>]{} and detector response simulated by [<span style="font-variant:small-caps;">Delphes</span>]{}, with particle level analyzed also within the standard [<span style="font-variant:small-caps;">Rivet</span>]{} framework. Correlations and unfolding to the parton level were also studied.
Differences are highlighted between the different pseudo-top algorithms in their behaviour especially for the rapidity of objects based on the choice of the neutrino longitudinal momentum from the generally two solutions of the quadratic equation based on the $m_{\ell\nu}= m_W$ or $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ condition.
An improvement in the pseudo-top algorithm is possible for rapidities of the leptonic pseudo-top and the pseudo-${\ensuremath{{t\bar{t}}}}$ system and also seen in the peak the reconstructed leptonic pseudo-top mass when the neutrino $p_z$ choice is done upon the smallest difference of the reconstructed pseudo-top quark masses (the “closest $m_t$” case).
Improvement is also checked in terms of the invariant mass of the pseudo-top quark pair for a hypothetical $Z'$ particle of mass of $700\,$GeV and decaying to a ${\ensuremath{{t\bar{t}}}}$ pair, where a sharper line is observed at the particle level, indicating better resolution reached in this variable, important for searches for new physics, although the performance at the detector level is smeared due to detector resolution effects.
A summary of pro’s and con’s of the presented methods is presented in Table \[tab:pros\_cons\]. In particular, the suggested novel “closest $m_t$” approach keeps almost the same correlations between detector and particle or parton levels as the standard choice of the neutrino $p_z$, while it has been shown that it provides more realistic spectra (especially less centrally biased rapidities) and outperforms the standard choice in a realistic unfolding test to the parton level, including the efficiency correction. While the “same $m_t$” or “best $m_t$” methods were motivated in further constraining the leptonic pseudo-top mass (and actually performing better around the peak for the leptonic pseudo-top mass distribution) or allowing the swap of $b$-jets, respectively, they result in undesired tails in the leptonic pseudo-top mass distribution and large slopes in spectra of physics interest.
In conclusion, the current pseudo-top algorithm used at LHC seems to be sufficient and robust enough for current observables. Still, improvements in terms of correlations between parton, particle and detector levels could be reached using the “closest $m_t$” method, namely by performing more linearly for the rapidity of the leptonic pseudo-top and the pseudo-[${t\bar{t}}$]{} system, and showing better unfolding closure via smaller sensitivity to spectra of models used to define the efficiency and migration matrix. These variables in particular are useful and used in PDF fitting efforts [@Czakon:2016olj].
Last, \[app1\] details explicit forms of solutions to quadratic equations for the $p_z^\nu$ problems.
standard closest $m_t$ same $m_t$ best $m_t$
------- ------------------ --------------------------------------- ------------------------------ ----------------------------------
Pro’s already used, better linearity sharper $m_t^{\mathrm{lep}}$ smaller low $m_t^{\mathrm{lep}}$
good general for rapidity spectra, peak tail
performance less central $y_t^{\mathrm{lep}}$,
smaller low $m_t^{\mathrm{lep}}$ tail
better unfold. closure
Con’s not optimized slight decorrelation too hard modified spectra
for new energies for some variables high $m_t^{\mathrm{lep}}$ higher $m_t^{\mathrm{lep}}$
tail tail
: A summary table of pro’s and con’s of the studied pseudo-top algorithms.[]{data-label="tab:pros_cons"}
![Distributions of the leptonic (top) and hadronic (bottom) pseudo-top quark transverse momentum for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study6_match"}](figure_36.pdf "fig:"){width="100.00000%"}\
![Distributions of the leptonic (top) and hadronic (bottom) pseudo-top quark transverse momentum for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study6_match"}](figure_37.pdf "fig:"){width="100.00000%"}
![Distributions the pseudo-${\ensuremath{{t\bar{t}}}}$ transverse momentum (top) and mass (bottom) for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study7_match"}](figure_38.pdf "fig:"){width="100.00000%"}\
![Distributions the pseudo-${\ensuremath{{t\bar{t}}}}$ transverse momentum (top) and mass (bottom) for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study7_match"}](figure_39.pdf "fig:"){width="100.00000%"}
![Distributions of the pseudo-${\ensuremath{{t\bar{t}}}}$ rapidity (top) and the ${p_{\mathrm{out}}}$ (bottom) variable for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study8_match"}](figure_40.pdf "fig:"){width="100.00000%"}\
![Distributions of the pseudo-${\ensuremath{{t\bar{t}}}}$ rapidity (top) and the ${p_{\mathrm{out}}}$ (bottom) variable for matched events for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study8_match"}](figure_41.pdf "fig:"){width="100.00000%"}
![Distributions of the $y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}}$ (top) and $\chi^{{\ensuremath{{t\bar{t}}}}}$ (bottom) variables for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study9_match"}](figure_42.pdf "fig:"){width="100.00000%"}\
![Distributions of the $y_{\mathrm{boost}}^{{\ensuremath{{t\bar{t}}}}}$ (top) and $\chi^{{\ensuremath{{t\bar{t}}}}}$ (bottom) variables for different choices of the neutrino $p_z$ solution: the standard choice (dashed), “closest $m_t$” (solid), “same $m_t$” (dot-dashed), and the “best $m_t$” (dotted). Left: particle level, right: [<span style="font-variant:small-caps;">Delphes</span>]{} detector level obtained using the ATLAS card. Ratios to the standard option are provided in lower panels, the yellow band indicating the statistical uncertainty in the denominator. []{data-label="pst:mt_nupz_study9_match"}](figure_43.pdf "fig:"){width="100.00000%"}
Acknowledgements
================
The author gratefully acknowledges the support by the project LO1305 of the Ministry of Education, Youth and Sports of the Czech Republic.
Analytic solutions to the neutrino $p_z$ {#app1}
========================================
Solution to the $m_{\ell\nu}= m_W$ condition
--------------------------------------------
The condition $m_{\ell\nu}= m_W$ leads to a quadratic equation for the longitudinal neutrino momentum $p_z^\nu$ with coefficients in standard notation given by $$a = ( E^\ell )^2 - ( p_z^{\ell} )^2$$ $$b = -2 \, k^2 \, p_z^{\ell}$$ $$c = ( E^\ell \, \slashed{E}_T )^2 - k^4$$ where $$k^2 = \frac12 \left[ ( m_W^2 - ( m^{\ell} )^2 ) \right] + (p_x^{\ell}\, \slashed{E}_x + p_y^{\ell}\, \slashed{E}_y)$$ and the the nature of the solution (complex, one real or two real) is governed by the sign of the usual discriminant $D \equiv b^2 - 4ac$ (here of dimension GeV${}^{6}$).
Solution to the $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ condition
-------------------------------------------------------------------
The condition $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ leads to a quadratic equation for the longitudinal neutrino momentum $p_z^\nu$ with coefficients in standard notation given by $$a = 4 \, \left[ (p_z^{\mathrm{sum}})^2 - \Sigma_E^2 \right]$$ $$b = 4 \, \Delta m^2 \, p_z^{\mathrm{sum}}$$ $$c = \Delta m^2 - 4 \, \slashed{E}_T^2 \, \Sigma_E^2$$ where $$\Sigma_E^2 \equiv (E^\ell + E^{b_\ell})^2$$ $$\Sigma^2 \equiv (m^{b_\ell})^2 + (m^\ell)^2 - 2 \, \Delta p^2 + 2 \, E^\ell \, E^{b_\ell} - 2 \, \left[ (p^\ell_x + p^{b_\ell}_x ) \, \slashed{E}_x + ( p^\ell_y + p^{b_\ell}_y ) \, \slashed{E}_y \right]$$ $$\Delta p^2 = \vec{p}_\ell \cdot \vec{p}_{b_\ell}$$ $$p_z^{\mathrm{sum}} \equiv p_z^\ell + p_z^{b_\ell}$$ $$\Delta m^2 \equiv m^2_{t,\mathrm{had}} - \Sigma^2$$ $$\slashed{E}_T^2 \equiv (\slashed{E}_x)^2 + (\slashed{E}_y)^2 \,.$$
Migration matrices for the discriminants
----------------------------------------
The migration matrices between the particle and detector levels (without the matching requirement) for the signed discriminant of the above quadratic equations (to the power of $1/6$ to keep the unit of GeV) are shown in Fig. \[pst:pseudotop\_Disc\_study1\] and for particle-to-detector matched events (in terms of objects forming the pseudo-top quarks, as described in Section \[sec:migra\]) in Fig. \[pst:pseudotop\_Disc\_study\_match\]. More detailed studies do not show large differences in the correlation between observables at the particle and detector levels when split into categories where the signs of the discriminants are the same or opposite at the two levels. Thus, the requirement of the diagonality of discriminants cannot substitute the performance of the matching correction.
![The migration matrix between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the signed discriminant (to the power of $1/6$ to keep the GeV unit) for the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition (left) and based on the $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ condition (right). []{data-label="pst:pseudotop_Disc_study1"}](figure_44.pdf "fig:"){width="45.00000%"} ![The migration matrix between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the signed discriminant (to the power of $1/6$ to keep the GeV unit) for the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition (left) and based on the $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ condition (right). []{data-label="pst:pseudotop_Disc_study1"}](figure_45.pdf "fig:"){width="45.00000%"}
![The migration matrix between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the signed signed discriminant (to the power of $1/6$ to keep the GeV unit) for matched events for the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition (left) and based on the $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ condition (right). []{data-label="pst:pseudotop_Disc_study_match"}](figure_46.pdf "fig:"){width="45.00000%"} ![The migration matrix between the particle and the [<span style="font-variant:small-caps;">Delphes</span>]{} detector levels for the signed signed discriminant (to the power of $1/6$ to keep the GeV unit) for matched events for the neutrino $p_z$ solution based on the $m_{\ell\nu}= m_W$ condition (left) and based on the $m_{t,\mathrm{had}} = m_{t,\mathrm{lep}}$ condition (right). []{data-label="pst:pseudotop_Disc_study_match"}](figure_47.pdf "fig:"){width="45.00000%"}
[^1]: The pseudorapidity $\eta$ is defined using the polar angle $\theta$ from the positive $z$ axis coinciding with one of the colliding proton beam as $\eta \equiv -\ln\tan\frac{\theta}{2}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using the concept of finite-size scaling, Monte Carlo calculations of various models have become a very useful tool for the study of critical phenomena, with the system linear dimension as a variable. As an example, several recent studies of Ising models are discussed, as well as the extension to models of polymer mixtures and solutions. It is shown that using appropriate cluster algorithms, even the scaling functions describing the crossover from the Ising universality class to the mean-field behavior with increasing interaction range can be described. Additionally, the issue of finite-size scaling in Ising models above the marginal dimension ($d^*=4$) is discussed.'
address:
- 'Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, D-55099 Mainz, Germany.'
- 'Department of Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands.'
author:
- Kurt Binder
- Erik Luijten
- Marcus Müller
- 'Nigel B. Wilding'
- 'Henk W.J. Blöte'
title: 'Monte Carlo investigations of phase transitions: status and perspectives'
---
critical phenomena, Ising model, crossover scaling, polymers, finite-size scaling
Introduction
============
It is a common belief that at the present time, about 30 years after the renormalization-group theory of critical phenomena was invented [@1], static critical behavior of systems in thermal equilibrium is rather well understood. In particular, this is expected to be true for the most intensively studied case, the Ising universality class [@2], to which systems such as uniaxial ferromagnets, binary alloys, simple fluids, fluid mixtures, polymer solutions and polymer blends belong [@3]. However, in the present work, we shall draw attention to some aspects of critical behavior in Ising-like spin systems which are, even today, still incompletely understood. The first of these concerns the problem of crossover between the Ising universality class and mean-field critical behavior. This crossover occurs, for instance, when the interaction range (and hence the “Ginzburg number” $G$ entering the Ginzburg criterion [@4]) is varied [@5; @6; @7; @8; @9; @10; @11]. A closely-related crossover is found for symmetrical polymer mixtures when the chain length $N$ of the polymers is varied [@3; @12; @13; @14; @15; @16; @17; @18; @19; @20; @21]. A part of this crossover (though typically not the full extent of the crossover scaling function) can be probed experimentally near the critical point of fluids and fluid binary mixtures [@22; @23; @24]. While the Ginzburg criteria [@4; @13; @14] provide a qualitative understanding of this crossover, the quantitatively accurate theoretical prediction of the crossover scaling function is a challenging problem [@25; @26; @27; @28; @29; @30; @31; @32; @33], and hence Monte Carlo studies [@5; @6; @7; @8; @9; @10; @11; @15; @16; @17] are of great potential benefit. In particular, the question as to what extent (if at all) such crossover scaling functions are universal is an intriguing one [@9; @10; @11; @22; @23; @33].
Another very interesting crossover which can also be studied is that which occurs near the critical point of unmixing for polymer solutions in a bad solvent [@12; @34; @35; @36; @37; @38; @39; @40; @41; @42; @43]. For chain length $N\to\infty$ the critical temperature $T_{\rm c}(N)$ moves towards the $\Theta$-temperature, where a single coil undergoes a transition from a swollen coil to a collapsed globule. This limit corresponds to a tricritical point [@12].
Monte Carlo analyses of critical phenomena typically apply finite-size scaling concepts [@44; @45; @46; @47; @48; @49]. However, care is necessary in the proper application of these methods in the mean-field limit. In fact, the standard formulation of finite-size scaling (“linear dimensions $L$ scale with the correlation length $\xi$”) implies that the hyperscaling relation [@2] between critical exponents should hold [@45; @50], which is not the case for mean-field exponents (apart from $d=d^*=4$ dimensions). This problem already arises for Ising models with short-range interactions for $d>d^*$ [@51; @52; @53; @54; @55; @56; @57; @58; @59; @60; @61], and some disagreements between Monte Carlo results [@51; @52; @54] and theoretical predictions [@53; @60] have stimulated a long-standing debate (see [@61] for a detailed review).
Mean-field to Ising crossover
=============================
We consider the Hamiltonian [@5; @6] $${\mathcal H}/k_{\rm B}T = - \sum_i \sum_{j > i} K({\bf r}_i-{\bf r}_j) s_i s_j
- h_0 \sum_i s_i \;,
\label{eq:latham}$$ with $s_i = \pm 1$ and an interaction $K({\bf r}) \equiv cR^{-d}$ for $|{\bf
r}| \leq R$ and zero elsewhere. The critical behavior of this model on $d$-dimensional lattices can be studied efficiently with a new cluster algorithm adapted for long-range interactions [@62].
To analyze the crossover it is instructive to consider the associated Ginzburg–Landau field theory in continuous space, $$\begin{aligned}
\mathcal{H}(\phi)/k_{\rm B}T &=& - \int_{V} d{\bf r} \left\{ \frac{1}{2}
\int_{|{\bf r}-{\bf r}'| \leq R} d{\bf r}' \left[ \frac{c}{R^d} \phi({\bf r})
\phi({\bf r}') \right] - \frac{1}{2} v \phi^2({\bf r}) \right.\nonumber \\
&& \left. \phantom{-\int_{V} d{\bf r} \left\{
\vphantom{\int_{|{\bf r}-{\bf r}'|}\left[\frac{c}{R^d}\right]}
\right.}
- u_0 \phi^4({\bf r}) + h_0\phi({\bf r} \right\} \;,
\label{eq:hamil}\end{aligned}$$ where $\phi({\bf r})$ is the single-component order-parameter field, $v$ is a temperature-like parameter and $u_0$ is a constant. After Fourier transformation and suitable rescaling this can be rewritten as (here $N$ is the total number of lattice sites) $$\begin{aligned}
\bar{{\mathcal H}}/k_{\rm B}T &=& \frac{1}{2} \sum_{{\bf k}} \left[
k^2 + \frac{r_0}{R^2} \right]
\psi_{\bf k} \psi_{-{\bf k}} \nonumber \\
&& + \frac{u}{4 R^4 N}
\sum_{{\bf k}_1} \sum_{{\bf k}_2} \sum_{{\bf k}_3}
\psi_{{\bf k}_1} \psi_{{\bf k}_2} \psi_{{\bf k}_3}
\psi_{-{\bf k}_1-{\bf k}_2-{\bf k}_3} -
\frac{h}{R} \sqrt{\frac{N}{2}} \psi_{{\bf k}={\bf 0}} \;,
\label{eq:scal-hamil}\end{aligned}$$ where $u$ is related to $u_0$ and $h$ to $h_0$ [@6], and $r_0$ in mean-field theory is the deviation of the temperature from its critical-point value.
We are now interested in identifying the crossover scaling variable associated with the crossover from the Gaussian fixed point $u=0$ and $r_0=0$ to the nontrivial Ising fixed point (Fig. \[fig:1\]). Because of the trivial character of the Gaussian fixed point and the fact the crossover scaling description should hold all the way from the Ising fixed point to the Gaussian fixed point, one can infer the crossover length scale $l_0 = R^{4/(4-d)}$ exactly! This is done by considering a renormalization by a length scale $l$, such that the wavenumber changes from $k$ to $k'=kl$, the number of degrees of freedom is reduced from $N$ to $N'=Nl^{-d}$, and $\psi_k$ changes into $\psi'_{k'} = l^{-1}\psi_k$ to leave $\bar{{\mathcal H}}$ invariant. From inspection of the terms in the Hamiltonian one can conclude that the singular part of the free energy must satisfy the scaling relation $$\tilde{f}_{\rm s}\left( \frac{r_0}{R^2}, \frac{u}{R^4}, \frac{h}{R} \right)
= l^{-d}\tilde{f}_{\rm s}\left(\frac{r_0}{R^2}l^2,
\frac{u}{R^4}l^{4-d}, \frac{h}{R}l^{1+d/2}\right) \;.
\label{eq:enerscal1}$$ We see that a finite and nonzero value for the second argument of $\tilde{f}_{\rm s}$ is retained exactly when $l$ takes the value of the crossover scale $l_0$. Thus we conclude that the singular part of the free energy scales with $R$ as follows $$\tilde{f}_{\rm s} = R^{-4d/(4-d)} \hat{f}_{\rm s} \left(\tilde{r}_0
R^{2d/(4-d)}, \tilde{u}, h R^{3d/(4-d)} \right) \;,
\label{eq:enerscal}$$ where a natural choice of coordinates (Fig. \[fig:1\]) is to measure $\tilde{r}_0$ and $\tilde{u}_0$ as distances from the Ising fixed point, unlike in the original Hamiltonian, where $r_0$ and $u_0$ are distances from the Gaussian fixed point.
Equation (\[eq:enerscal\]) describes how the temperature distance $\tilde{r}_0$ from criticality and the magnetic field $h$ scale with the range of interaction $R$: Note that the crossover exponent is known exactly (unlike other cases of crossover, e.g., between the Ising and Heisenberg universality class in isotropic magnets with varying uniaxial anisotropy [@63]). The same result for the crossover exponent follows [@5], of course, from simple-minded arguments using the Ginzburg criterion. However, the location of the nontrivial fixed point $u^*$ (Fig. \[fig:1\]), the associated other exponents, and the explicit form of the scaling function $\tilde{f}_{\rm s}$ cannot be obtained exactly.
The calculation of the scaling function for the free-energy density or its derivatives, such as the susceptibility, is a nontrivial task for both renormalization-group and Monte Carlo calculations. This is demonstrated in Fig. \[fig:2\] where the effective critical exponent $\gamma^{+}_{\rm eff}$ of the susceptibility for $T>T_{\rm c}$ is plotted versus the thermal crossover scaling variable $t/G$, with $t=(T-T_{\rm c})/T_{\rm c}$ being the reduced temperature and $G=G_0 R^{-6}$ the Ginzburg number in $d=3$, for which $G_0
\approx 0.277$. Note that effective exponents are defined as $$\gamma^{\pm}_{\rm eff} \equiv -d \ln \hat{\chi} / d \ln |t| \;, \quad
\hat{\chi} \equiv k_{\rm B}T_{\rm c}(R) (\partial M /\partial h)_{T} \;,
\label{eq:gamma_eff}$$ where $\pm$ refers to $T \gtrless T_{\rm c}$, respectively, $M = \langle s
\rangle_{T, h}$, and the range $R$ is defined from the second moment of the interaction ($z$ being the effective coordination number) $$\begin{aligned}
R^2 &=& \sum_{j \neq i} |{\bf r}_i - {\bf r}_j|^2 K({\bf r}_i - {\bf r}_j) /
\sum_{j \neq i} K({\bf r}_i - {\bf r}_j) \nonumber \\
&=& \frac{1}{z} \sum_{j \neq i} |{\bf r}_i - {\bf r}_j |^2
\quad \mbox{with } |{\bf r}_i - {\bf r}_j | \leq R_m \;.\end{aligned}$$ Here the second equality holds only for a square-well potential and values $R_m^2 = 1, 2, 3, 4, 5, 6, 8, 12, 18, 28, 60, 100$, and $160$ were studied. From Fig. \[fig:2\] we see that the Monte Carlo results agree with all the theoretical calculations near the Gaussian fixed point, but do not yield the more rapid increase of $\gamma^{+}_{\rm eff}$ near the Ising fixed point. It is not clear what precise conclusions should be drawn from this discrepancy: All these theoretical treatments really rely on extrapolations of low-order renormalization-group expansions in $\varepsilon = 4-d$, and hence are perhaps rather inaccurate in $d=3$ dimensions. On the other hand, they clearly relate to the limit where $R \to \infty$ and $t \to 0$, with $tR^6$ fixed—a universal description of the crossover can only be expected in this limit. The Monte Carlo data shown in Fig. \[fig:2\] also include the range of small $R$, for which additional corrections to scaling present near the Ising fixed point (other than those attributable to the Ising–mean-field crossover) may come into play.
A rather successful description of the Monte Carlo data could be obtained by a fit to a function given by Anisimov [*et al.*]{} [@33]. Their description is also an interpolation formula based on low-order $\varepsilon$-expansions but contains a second parameter (in addition to $G$) describing a short-wavelength cutoff. However, one disturbing feature of this description is that one needs different amplitudes $G_0$ in the relation $G = G_0 R^{-6}$ above and below $T_{\rm c}$, and the ratio $G_0^{+}/G_0^{-}$ is an additional, ad hoc, parameter the significance of which is not understood [@33]. Thus we consider it an as yet unsettled problem as to just on which parameters the crossover scaling description should depend. In this context, we draw attention to the question whether the specific square-well form chosen for the exchange interaction matters. To answer this question, a more general form of $K({\bf r}_i-{\bf r}_j)$ was chosen (viz., a superposition of two square-well potentials which differ in range and strength but are chosen such that the same value for $R^2$ results [@10]. While $T_{\rm c}$ was shown not to be determined by $R$ alone, but depended on $K({\bf r}_i-{\bf r}_j)$ in a more detailed way, the same crossover scaling function resulted for all choices of the interaction profile studied [@10].
A particular merit of the description of Anisimov [*et al.*]{} [@33] is, however, that it can yield a non-monotonic variation of $\gamma^{-}_{\rm eff}$ with $t/G$: In $d=3$ a shallow minimum ($\gamma^{-}_{\rm eff} \approx 0.96 <
\gamma_{\rm MF} = 1$) occurs for $|t|R^6 = 10^2$ [@9] that can be fitted by this theory [@33]. Indeed, a very similar minimum has been observed in Ref. [@pelissetto98] from a mean-field expansion for Ising systems with medium-range interactions, see also Ref. [@pelissetto99] for a detailed review. In $d=2$ dimensions, such a minimum occurs as well and is much more pronounced than in $d=3$, while above $T_{\rm c}$ the variation of the effective exponent is still monotonic (Fig. \[fig:3\]). Note that the crossover is again spread out over many decades in the crossover variable $t/G$ ($G \propto R^{-2}$ in $d=2$), as in $d=3$, and that for $T<T_{\rm c}$ there are no analytical results whatsoever to compare our Monte Carlo results with! At this point, there is clearly still a gap in our knowledge about critical phenomena.
As a last point in this section, we add a few brief comments about the way in which the Monte Carlo results on effective exponents have been obtained. As is well known [@45; @46; @47; @48; @49], the Monte Carlo method converges to the exact statistical mechanics of a finite system only; the thermodynamic limit is never addressed directly. The typical situation is that one deals with a $L\times L$ or $L\times L\times L$ box with periodic boundary conditions. The critical singularities are rounded and shifted by the finite size of the system [@44; @45; @46; @47; @48; @49]. For the precise location of the critical point, a finite-size scaling analysis is required. The principle of finite-size scaling is that the linear dimension $L$ scales with the correlation length $\xi$. Therefore the $k$’th moment of the magnetization $m$ scales like: $$\left\langle |M|^k \right\rangle = L^{-k\beta/\nu} \tilde{M}_k(L/\xi) \;,
\label{eqn:8}$$ $\beta$ and $\nu$ being the critical exponents of the order parameter ($\langle
m \rangle \propto |t|^\beta$) and the correlation length ($\xi \propto
|t|^{-\nu}$), respectively, and $\tilde{M}_k$ being some scaling function. Therefore the straightforward observation results [@45] that these power law prefactors $L^{-k\beta/\nu}$ cancel out if one considers suitable ratios of moments, such as $$Q = \langle M^2\rangle^2/\langle M^4\rangle=\tilde{Q}(L/\xi) \;.
\label{eq:Q}$$ At $T_{\rm c}$ we have $\xi \to \infty$, of course, so $\tilde{Q}(0)$ is simply a constant, independent of the system size $L$. This justifies the simple recipe to record this ratio for different choices of $L$ and obtain $T_{\rm c}$ from the intersection point of these ratios [@45; @48; @49]. Note that the ordinate value of this intersection point is a universal constant (only depending on the shape of the system and on the boundary conditions, but not on $R$, for instance, provided one is in the asymptotic critical region).
However, this recipe so far ignores the crossover from one universality class to the other (as well as corrections to scaling). Nevertheles, it turns out that one can formulate a combined finite-size scaling and crossover scaling description for such problems [@5; @6; @7; @8; @9; @10; @11; @16; @17; @64]. A simplified description considers the variation of the correlation length, which is $\xi
\propto Rt^{-1/2}$ in the mean-field critical region, and $\xi \propto
(R^\kappa t)^{-\nu}$ in the Ising critical region \[the exponent $\kappa$ follows from the condition that for $t=t_{\rm cross} \propto R^{-2d/(4-d)}$ and the corresponding value of $\xi$, $\xi_{\rm cross}=\xi(t=t_{\rm cross})=l_0
\propto R^{4/(4-d)}$ a smooth crossover between both power laws occurs\]. Now it is of crucial importance to compare $L$ with the crossover length scale $\xi_{\rm cross}$: If $L$ is much less than $\xi_{\rm cross}$, then the finite size rounding occurs fully in the mean-field regime, before the crossover to Ising criticality has had a chance to come into play. Actually in this regime the correlation length $\xi$ is not the relevant length to describe the finite size rounding [@48; @51; @52], one rather needs the so-called “thermodynamic length” [@52], $\ell_T \propto |t|^{-2/d}$, as will be discussed in Sec. 4. In this regime ($L \ll \xi_{\rm cross}$) an accurate determination of $T_{\rm c}$ is clearly impossible. In order to accurately locate $T_{\rm c}$, we need to study the inverse regime, $L \gg \xi_{\rm cross}$: Only then can one see the mean-field critical behavior farther away from $T_{\rm c}$ crossing over to the Ising behavior at $t_{\rm cross}$ (remember that this crossover is spread out over several decades!) and the finite-size rounding sets in at a still much smaller value of $|t|$ (where $L\simeq\xi$). Since for large $R$, $\xi_{\rm cross}$ is also very large ($\xi_{\rm cross}\simeq l_0\propto
R^{4/(4-d)}$), one needs to simulate very large $L$ and hence such simulations are technically very difficult.
Thus it is not surprising that when this problem was first addressed with single-spin-flip Monte Carlo algorithms [@5] a satisfactory description of the full crossover could not be obtained, and the availability of an efficient cluster algorithm [@62] was crucial for obtaining meaningful results. In $d=2$, we could study $L$ up to 800 lattice units, and $R_m=100$ corresponding to $z=436$ interacting neighbors (Fig. \[fig:4\]). With these large lattices it is possible to follow the variation of $Q$ almost all the way from the mean-field limit at small $L$ to the Ising limit at large $L$, and in $\chi$ (Fig. \[fig:4\](a) the Ising asymptote (slope $3/4$ on the log–log plot) is nicely confirmed).
Since we know $T_{\rm c}$ very precisely and have data for such a wide range of $L$, it is also possible to carry out runs slightly away from $T_{\rm c}$, which are used to study the thermal crossover presented in Figs. \[fig:2\], \[fig:3\]. Only data not affected by the finite system size are used for the numerical derivative required in Eq. (\[eq:gamma\_eff\]).
First steps towards the study of crossover problems in polymer blends and solutions.
====================================================================================
As is well known [@3; @12; @13; @14], the Ginzburg–Landau–Wilson Hamiltonian for a symmetrical polymer mixture near its critical unmixing point can be mapped on to the Ising model with a medium range of interaction (in $d=3$ dimensions), $N^{1/2}$ (with $N$ being the chain length of the flexible macromolecule) playing the role of the interaction volume $R^3$. Qualitatively, this mapping is understood from the fact that a polymer coil has a random walk-like configuration. Its gyration radius $R_{gyr}$ scales as $R_{gyr}\simeq
a\sqrt{N/6}$, where $a$ is the size of the monomer. Thus the monomer density of one chain inside the volume that is occupies ($V\propto R^3_{gyr}$) is very small, $\rho=N/V\propto a^{-3}N^{-1/2}$. Hence in a dense melt ($\rho_{\rm melt}\simeq a^{-3}$) there are $N^{1/2}$ chains in the same volume, i.e. each chain interacts with $x=N^{1/2}$ “neighbors”. Thus as $N\to\infty$ one again expects a crossover from Ising-like critical behavior to mean-field like behavior, and this is verified experimentally [@18; @19] (though the corresponding prediction for the Ginzburg number $G\propto 1/N$ does not seem to work out.
First steps to study this crossover by computer simulation have been performed [@16; @17] using the bond fluctuation model of symmetrical polymer mixtures [@9; @65] applying a semi-grand canonical algorithm [@15] and histogram reweighting techniques [@66]. The model and methodology of these simulations have been extensively reviewed elsewhere [@3; @65] and hence we omit all the technical details here, and simply show an attempt to estimate the crossover scaling function of the order parameter [@17] (Fig. \[fig:5\]). Note that polymer are slowly relaxing objects and hence difficult to simulate—no counterpart to the cluster algorithm used for the Ising model [@62] is available, and hence the challenge remains to improve substantially the accuracy of studies such as shown in Fig. \[fig:5\] in order to be able to study the variation of effective exponents for this problem in analogy with Figs. \[fig:2\], \[fig:3\].
If one wishes to compare such simulations for polymer mixtures to experiments on real systems [@18; @19], an important complication that must be taken into account is the asymmetry in chain length, $N_A\ne N_B$. This leads to two very important technical complications: (i) While in the symmetrical case the coexistence curve (including the critical point) occurs at a chemical potential difference $\Delta\mu=0$, for $N_A\ne N_B$ phase coexistence occurs along a non trivial curve $\Delta\mu_{\rm coex}(T)$ in the ($\Delta\mu,T$) plane, and hence one has to search for the critical point ($\Delta\mu_{crit}=\Delta\mu_{\rm
coex}(T_{\rm c}),T_{\rm c}$) in a two dimensional variable space. Fig. \[fig:6\]. shows that this problem can also be overcome by finite-size scaling methods, utilizing the scaling behavior appropriate for first-order transitions [$\Delta\mu-\Delta\mu_{\rm coex}(T)\propto L^{-d}$ in $d$ dimensions [@48; @49]]{} in order to locate $\Delta\mu_{\rm coex}(T)$[@20]. (ii) Owing to the asymmetry, order parameter density and energy density become coupled, and this “field mixing” effect needs to be disentangled from the finite-size scaling analysis [@21]. This problem is well known from computer simulation of fluids and we shall not describe it here, but rather draw attention to a recent review [@67].
This field-mixing problem is particularly severe for the unmixing of polymers in solution beneath the $\Theta$-temperature (which formally can be considered as a limiting case of a polymer mixture where $N_B=N, N_A=1$[@3]), Fig. \[fig:7\]. However, by a suitable transformation of variables, one can construct from $\phi$ and the energy density $u$, an appropriate field $\mathcal{M}=(\phi-su)/(1-sr)$ (where $s,r$ are parameters that can be found from a suitable analysis of the simulations, see [@21; @67]), which then scales like the magnetization of the Ising ferromagnets. Figure \[fig:8\] shows that the distributions of this variable at criticality nicely coincides with the critical order parameter distribution of the Ising model (actually this mapping can be used as a method for precisely locating the critical point [@21; @67]).
From analyses of this kind it has been possible to obtain the critical parameters of the model as a function of chain length, see e.g. Fig. \[fig:8\]. The simulation results reproduce nicely the behavior $\rho_c\propto N^{-x}$ with $x\approx 0.37$ found also experimentally [@41; @42]. However, the simulations also show that the chains at the critical point are not yet partially collapsed, but are rather ideal, and hence rule out the interpretation of this exponent value (which differs from the classical results $x=1/2$ [@41]) as being due to the percolation of partially collapsed chains. Consequently, the physical interpretation of this exponent remains an open question [@34; @43].
Finite-size scaling above the upper critical dimension
======================================================
Remembering that the correlation length for $d>d^*=4$ has the mean-field critical behavior $\xi_b = \xi_0 t^{-1/2}$, the free-energy density can be written as [@68]
$$f_L=
L^{-d}\tilde{f}\left\{t\left(\frac{L}{\xi_0}\right)^2,
uL^{4-d}, hL^{1+d/2}\right\} \;.$$
Note that here exactly the same powers of $L$ appear as those for $l$ in Eq. (\[eq:enerscal1\]). For $d>d^*$ there is only the Gaussian fixed point to be considered. But although $u^*=0$ here and the power of $L$ in the term $uL^{4-d}$ is negative so that $uL^{4-d}\to 0$ for $L\to\infty$, the argument $uL^{4-d}$ must not be omitted: $u$ is a “dangerous irrelevant variable” [@69], so when we consider the zero-field susceptibility $\chi$ and the moment ratio $Q$ \[Eq. (\[eq:Q\])\], we find, using $u \propto \ell_0^{d-4}$,
$$\chi=\left(\frac{\partial ^2 f_L}{\partial h^2}\right)_T=L^2
P_\chi\left \{t\left
( \frac{L}{\xi_0}\right)^2,\left(\frac{L}{\ell_0}\right)^{4-d}\right\} \;,
\label{eq:pchi}$$
$$Q=P_Q\left \{ t\left
( \frac{L}{\xi_0}\right)^2,\left(\frac{L}{\ell_0}\right)^{4-d}\right\} \;.
\label{eq:pq}$$
Thus all scaling functions have two arguments, $t(L/\xi_0)^2$ and $(L/\ell_0)^{4-d}$. However, it turns out [@51] that a reduction to one-variable scaling occurs for $L\to\infty$, namely
$$\chi \to \lim_{L\to\infty}
L^{d/2}\tilde{P}_\chi\left(tL^{d/2}\xi_0^{-2}\ell_0^{(4-d)/2}\right) \;,$$
$$Q \to \lim_{L\to\infty}
\tilde{P}_Q\left(tL^{d/2}\xi_0^{-2}\ell_0^{(4-d)/2}\right) =
\tilde{P}_Q\{(L/\ell_T)^{d/2}\} \;,$$
where in the last step we have introduced the “thermodynamic length” $\ell_T\propto t^{-2/d}$ [@52], mentioned above.
Equations (13,14) can be understood from various arguments [@51; @52; @53]. Brézin and Zinn-Justin argue [@53] that in the initial Hamiltonian or the corresponding statistical weight, one can treat the contribution from the average magnetization $M$ separately,
$$\exp \left[-\mathcal{H} \{ s_i \}/k_BT \right] =
\exp\left\{-\frac{(M^2/M_b^2-1)^2}{8k_BT\chi_b/M_b^2 }L^d+\cdots \right\} \;,$$
where the dots stand for contributions with non-uniform magnetization, i.e. fluctuations. Here $M_b,\chi_b$ are the mean-field bulk magnetization and susceptibility, $M_b=\tilde{M_b}(-t)^{1/2}, \chi_b=\tilde{\chi_b}|t|^{-1}$. The zero-mode theory neglects these fluctuations altogether and there the distribution of the magnetization $P_L(M)$ scales as $$P_L(M)\propto L^{d/2}\exp\{ -[M^2/(\tilde{M}_b^2(-t))-1]^2(L/\ell_T)^d/8 \} \;.$$ From this result it is straightforward to derive the above scaling functions $\tilde{P}_\chi$ and $\tilde{P}_Q$ explicitly [@53].
Since this theory was proposed [@51; @52; @53] it has been a long-standing problem to verify the predictions by Monte Carlo simulation. In particular, when one plots the moment ratio $Q$ versus temperature deviation from criticality, one should find a universal intersection point at $T_{\rm c}$ at a value
$$\tilde{P}_Q(0)=8\pi^2/\Gamma^4(1/4)\simeq 0.456947 \;.$$
However, the Monte Carlo results for small systems seem to intersect at a different value $Q\simeq 0.52$ (Fig. \[fig:10\]). Also the temperature where this intersection occurs is a little off, but since one does not know $T_{\rm
c}$ beforehand, one could simply imagine that the abscissa in Fig. \[fig:10\] is mislabeled and $T_{\rm c}$ must be assigned differently.
Chen and Dohm [@59; @60] have recently criticized the whole approach sketched above and maintained that one must return to a finite-size scaling description in which both variables $t(L/\xi)^2$ and $(L/\ell_0)^{4-d}$ are kept separate, as in Eqs. (\[eq:pchi\]) and (\[eq:pq\]). They also obtained the scaling functions $P_Q$ and $P_\chi$ in a first-order loop expansion as a function of these variables. Indeed their result is qualitatively similar to the Monte Carlo data (Fig. \[fig:10\], broken curves), although in quantitative respects their treatment offers little improvement. This is seen, for instance, in a scaling plot of the susceptibility: The Chen–Dohm theory approaches the zero-mode results from above, while in the regime of interest the Monte Carlo data fall below the zero-mode result (Fig. \[fig:11\]). These discrepancies remain present for considerably larger $L$ than shown here [@61].
Thus we arrive at a rather disappointing state of affairs—although for the $\phi^4$ theory in $d=5$ dimensions all exponents are known, including those of the corrections to scaling, and in principle very complete analytical calculations are possible, the existing theories clearly are not so good. Perhaps the discrepancies result because the theory of Ref. [@60] is only one-loop order, perhaps because other corrections are missing. While presumably the zero-mode one-parameter scaling is true asymptotically for $L \to \infty$, the corrections to this limit disappear only rather slowly, as Fig. \[fig:10\](a) has demonstrated.
Concluding remarks
==================
While the estimates of the critical exponents for the $d=3$ Ising model are impressively accurate [@70; @71; @72] and analytical [@70] and Monte Carlo [@71; @72] estimates agree within very small error margins, the situation is different for the problems considered in the present paper: Analytical work is restricted to low-order $\varepsilon$-expansions or low-order loop-expansions and discrepancies between theory and simulation occur that are not fully understood. More work will be needed to clarify the situation. Note that the Ising to mean-field crossover considered here really is the simplest example of crossover phenomena, since the crossover exponent is rigorously known—crossover from one nontrivial fixed point to another is presumably more tricky to deal with. And for problems such as the critical point of polymer solutions, even the proper theoretical approach is controversial, and hence it is unclear whether the exponent $x \approx 0.37$ (Fig. \[fig:9\]) is a universal property at all [@34; @35; @36; @37; @38; @39; @40; @41; @42; @43].
Further problems appear when one is not concerned with bulk critical phenomena in ideal, homogeneous systems, but when one considers inhomogeneous systems, e.g., systems with random quenched disorder (e.g., Ising and Potts models exposed to random fields, spin glasses, etc. [@73]). For instance, for a Potts spin glass finite-size scaling is not even understood on the mean-field level, at least for cases where first-order transitions without latent heat occur [@74]. Also for systems with a regular inhomogeneity, e.g., Ising films with competing walls which allow for interface localization–delocalization transitions, one has fascinating critical behavior and crossover, of which the details still need to be unraveled [@75]. Thus the Monte Carlo investigation of phase transitions—both in equilibrium and in driven systems [@76]—will remain an active and challenging field.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank the HLRZ Jülich for access to a Cray-T3E where part of the computations have been performed. One of us (K.B.) thanks H.-P. Deutsch for a fruitful collaboration that led to the results shown in Fig. \[fig:5\].
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![Qualitative picture of the renormalization trajectory describing the crossover from the Gaussian fixed point $\mu_0^*=(0,0)$ to the Ising fixed point $\mu^*=(r_0^*,u^*)$.[]{data-label="fig:1"}](flow.eps){width="\figurewidth"}
![Effective susceptibility exponent $\gamma^*_{\rm eff}$ above $T_{\rm
c}$ for the three-dimensional Ising model with variable interaction range $R$ (numbers in the key) plotted vs. $t/G$, along with three theoretical calculations for this quantity; due to Refs. [@31] (BK), [@27] (BB), and [@25; @29] (SF), respectively. From Ref. [@9].[]{data-label="fig:2"}](gamma-ht2.eps){width="\figurewidth"}
![The effective susceptibility exponent $\gamma^*_{\rm eff}$ above $T_{\rm c}$ (a) and below $T_{\rm c}$ (b), for the two-dimensional Ising model with variable interaction range $R$ (numbers in the key), plotted vs. $tR^2$. From Ref. [@8].[]{data-label="fig:3"}](htchi-eff.eps){width="\figurewidth"}
![The effective susceptibility exponent $\gamma^*_{\rm eff}$ above $T_{\rm c}$ (a) and below $T_{\rm c}$ (b), for the two-dimensional Ising model with variable interaction range $R$ (numbers in the key), plotted vs. $tR^2$. From Ref. [@8].[]{data-label="fig:3"}](con-eff.eps){width="\figurewidth"}
![Finite-size crossover curve for the magnetic susceptibility $\chi$ divided by the system linear dimension (a) and the amplitude ratio $Q$ \[Eq. (\[eq:Q\])\] (b) for the two-dimensional Ising model at $K=K_c(R)$ plotted vs. the finite-size crossover scaling variable $L/R^2$ (note that $\xi_{\rm cross}=l_0 \propto R^2$ in $d=2$). In both quantities, range-dependent correction factors $C[\chi]$ and $C[Q]$ have been divided out to eliminate some corrections to scaling (see Ref. [@7] for a definition of these factors). From Ref. [@7].[]{data-label="fig:4"}](chi-collap.eps){width="\figurewidth"}
![Finite-size crossover curve for the magnetic susceptibility $\chi$ divided by the system linear dimension (a) and the amplitude ratio $Q$ \[Eq. (\[eq:Q\])\] (b) for the two-dimensional Ising model at $K=K_c(R)$ plotted vs. the finite-size crossover scaling variable $L/R^2$ (note that $\xi_{\rm cross}=l_0 \propto R^2$ in $d=2$). In both quantities, range-dependent correction factors $C[\chi]$ and $C[Q]$ have been divided out to eliminate some corrections to scaling (see Ref. [@7] for a definition of these factors). From Ref. [@7].[]{data-label="fig:4"}](q-collap.eps){width="\figurewidth"}
![Crossover scaling plot for the order parameter $\langle |m|\rangle =
\langle|\phi_A-\phi_B|\rangle / (\phi_A+\phi_B)$ of a binary polymer mixture (A,B) with symmetrical chain lengths $N_A=N_B=N$. $\phi_A, \phi_B$ are the volume fractions of $A$ and $B$ monomers, respectively. The points are simulation results for the bond-fluctuation model on a simple-cubic lattice, using concentration $\phi_v=0.5$ of vacant sites. Straight lines in this log–log plot indicate power laws with effective exponents, $\langle m \rangle
= \hat{B}_{\rm eff} t^{\beta_{\rm eff}}$, $t=1-T/T_{\rm c}$. The broken straight line shows the mean-field result, $\langle m \rangle
=\sqrt{3}t^{1/2}$, to which the data converge for $N \to \infty$. From Ref. [@17].[]{data-label="fig:5"}](fig5.eps){width="\figurewidth"}
![Finite-size scaling plot for the second moment of the order parameter of an asymmetric polymer mixture ($N_A=10$, $N_B=20$) at a temperature $T<T_{\rm c}$ ($\varepsilon=\varepsilon_{AB}/T=0.035$) as a function of the normalized chemical potential difference, in order to locate $\Delta \mu_{\rm
coex}(T)$ by optimizing the “data collapse” for the range of values of $L$ as indicated. From Ref. [@20].[]{data-label="fig:6"}](fig6.eps){width="\figurewidth"}
![Schematic phase diagram of a polymer solution using the temperature $T$ and the volume fraction $\phi$ taken by the effective monomers of the polymer chains as variables. The coexistence curve separates a dilute solution of collapsed chains (at $\phi_{\rm coex}^{(1)}$) from a semi-dilute solution of overlapping chains (at $\phi_{\rm coex}^{(2)}$). These two branches of the coexistence curve merge at a critical point $T_{\rm c}(N)$, $\phi_{\rm c}(N)$. For $N \to \infty$ this point merges with the $\Theta$-point of a polymer solution at infinite dilution ($\phi \to 0$).[]{data-label="fig:7"}](fig7.eps){width="\figurewidth"}
![Critical order-parameter distribution for a polymer solution with chain length $N=20$, modeled by the bond-fluctuation model on the simple-cubic lattice, for linear dimensions $L=40$ and $L=50$, open symbols, and compared to the order-parameter distribution of the Ising model (crosses). From Ref. [@43].[]{data-label="fig:8"}](poM_poly.eps){width="\figurewidth"}
![Estimates of the critical volume fraction $\phi_{\rm c}$ of monomers for a polymer solution (modeled by the bond-fluctuation model on the simple-cubic lattice, with an attractive energy between monomers at distances $r \leq \sqrt{6}$) as a function of the inverse chain length. The broken curven represents a fit of the form $\phi_{\rm c}=(1.1126+1.3N^{0.369})^{-1}$. From Ref. [@43].[]{data-label="fig:9"}](rhoc.eps){width="\figurewidth"}
![(a) Plot of $Q$ vs. $tL^2$ for the $d=5$ Ising model, demonstrating the occurrence of spurious intersections both in the Monte Carlo results [@61] and the Chen–Dohm theory [@60]. (b) Plot of $Q$ vs. the scaling variable $tL^{5/2}$; using parameters $\xi_0$, $\ell_0$ extracted from various limits of the susceptibility [@61], the Chen–Dohm theory can be evaluated without any adjustable parameter whatsoever. Note that for $L=12$ it is already graphically indistinguishable from the “zero-mode” theory. From Ref. [@61].[]{data-label="fig:10"}](qcross.eps){width="\figurewidth"}
![(a) Plot of $Q$ vs. $tL^2$ for the $d=5$ Ising model, demonstrating the occurrence of spurious intersections both in the Monte Carlo results [@61] and the Chen–Dohm theory [@60]. (b) Plot of $Q$ vs. the scaling variable $tL^{5/2}$; using parameters $\xi_0$, $\ell_0$ extracted from various limits of the susceptibility [@61], the Chen–Dohm theory can be evaluated without any adjustable parameter whatsoever. Note that for $L=12$ it is already graphically indistinguishable from the “zero-mode” theory. From Ref. [@61].[]{data-label="fig:10"}](qscale.eps){width="\figurewidth"}
![Plot of the scaled susceptibility $\chi L^{-5/2}$ vs. $tL^{5/2}$, including the “zero-mode” result of Ref. [@53], as well as the predictions of Chen and Dohm [@60] evaluated for the same values of $L$ as the Monte Carlo results shown. From Ref. [@61].[]{data-label="fig:11"}](chi_zoom.eps){width="\figurewidth"}
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Yositake Takane
title: 'Gauge-Invariant Cutoff for Dirac Electron Systems with a Vector Potential '
---
Introduction
============
In monolayer graphene, [@novoselov1; @novoselov2; @castro_neto] electron states near the band touching point are described by a massless Dirac model, or equivalently, a Weyl model; thus, they are called Dirac electrons. Such electron states have been shown to appear in various systems such as topological insulators, [@fu; @moore; @roy; @ARPES1; @ARPES2; @ARPES3] Weyl semimetals, [@shindou; @murakami; @huang1; @xu1] and $\alpha\rm{\mathchar`-}(BEDT{\mathchar`-}
TTF)_{2}I_{3}$. [@katayama; @kobayashi] These are referred to as Dirac electron systems. The study on a particular Dirac electron system is traced back to that on bismuth [@wolff; @fukuyama] if the massive case is included.
The electromagnetic response of Dirac electrons has been actively studied from various aspects. In the case of graphene, the studies on it have been extended to orbital magnetism, [@mcclure1; @sharma; @safran; @koshino1] the screening effect, [@gorbar; @ando1; @wunsch; @hwang] dynamical conductivity, [@ando2; @gusynin; @koshino2] and so on. Here, we focus on a fundamental difficulty that arises in the analysis based on a continuum Dirac model possessing an unbounded energy spectrum. When we analyze the response to a vector potential, the physical quantity under consideration diverges in some cases owing to the unbounded spectrum. To avoid such divergence, we usually introduce a cutoff in energy or wavenumber space. This cutoff gives rise to an unphysical result that breaks the gauge invariance with respect to a vector potential $\mib{A}$ and a scalar potential $\phi$. A typical example is that if a charge current density $\mib{j}$ is calculated in response to $\mib{A}$, we erroneously find that $\mib{j}$ becomes finite even when $\mib{A}$ is constant. [@principi; @polini; @katsnelson] Needless to say, a constant vector potential induces no effect on a physical system owing to the gauge invariance. Similar difficulties arise in the analysis of, for example, the superfluid density of Dirac electrons in the superconducting state [@kopnin; @mizoguchi] and the chiral magnetic effect in a Weyl semimetal. [@takane1; @takane2] Although the insufficiency of an energy or wavenumber cutoff has been recognized, [@principi; @polini] little attempt has been made to overcome this difficulty. [@juricic]
In this paper, we propose a modified energy cutoff procedure for general Dirac electron systems to improve the description of their electromagnetic response. Its original form is briefly reported in Ref. in an incomplete manner. The modified energy cutoff preserves the gauge invariance and removes a difficulty that arises in the analysis of the response to a vector potential. We use the modified energy cutoff procedure to calculate the charge and current densities induced by a vector potential in a two-dimensional (2D) massless Dirac electron system. We show that it enables us to describe the electromagnetic response in a gauge-invariant manner.
In the next section, we present a 2D massless Dirac model with a single valley, and derive the response functions for vector and scalar potentials by using an ordinary cutoff. The resulting response functions break the gauge invariance as well as the charge conservation relation. In Sect. 3, we propose a modified energy cutoff procedure and roughly show how it works. In Sect. 4, we derive the response functions by applying the modified energy cutoff procedure. The resulting response functions preserve the gauge invariance and satisfy the charge conservation relation. In Sect. 5, the modified energy cutoff procedure is justified in an accurate manner. The last section is devoted to a short summary. We set $\hbar = k_{\rm B} = 1$ throughout this paper.
Model, Formulation, and Known Results
=====================================
We introduce the 2D massless Dirac model with a single Dirac cone centered at $\mib{k}=(0,0)$: [@mcclure2; @slonczewski; @ajiki] $$\begin{aligned}
\label{eq:Hamiltonian}
H
& = \int d^{2}r
\psi^{\dagger}(\mib{r})
\Bigl[ v\left(\sigma_{x}\hat{k}_{x}+\sigma_{y}\hat{k}_{y}\right)
- \mu
\Bigr]
\psi(\mib{r}) ,\end{aligned}$$ where $\psi(\mib{r})$ represents the spinor field describing Dirac electrons, and $v$ and $\mu$ respectively denote the velocity and chemical potential. Here, $\sigma_{x}$ and $\sigma_{y}$ are the $x$- and $y$-components of the Pauli matrix, and $\hat{k}_{x}=-i\partial_{x}$ and $\hat{k}_{y}=-i\partial_{y}$. The eigenvalue of energy is determined as $$\begin{aligned}
E_{\eta}(\mib{k}) = \eta v|\mib{k}|-\mu ,\end{aligned}$$ where $\eta = +$ for the conduction band and $\eta = -$ for the valence band. The perturbations due to the vector potential $\mib{A}$ and the scalar potential $\phi$ are respectively expressed as $$\begin{aligned}
H_{A}
& = - \int d^{2}r\, \mib{j}(\mib{r})\cdot \mib{A}(\mib{r}) ,
\\
H_{\phi}
& = \int d^{2}r\, \rho(\mib{r})\phi(\mib{r}) .\end{aligned}$$ The charge current density $\mib{j}=(j_{x},j_{y})$ and the charge density $\rho$ are expressed as $$\begin{aligned}
\label{eq:def-current}
\mib{j} & = - ev \psi^{\dagger}(\mib{r})
\left(\sigma_{x},\sigma_{y}\right)\psi(\mib{r}) ,
\\
\label{eq:def-charge}
\rho & = - e \psi^{\dagger}(\mib{r})\psi(\mib{r}) .\end{aligned}$$
We consider the current and charge densities induced by the vector potential in the $x$-direction $$\begin{aligned}
\mib{A} = (A_{x}(\mib{q},\omega),0)e^{i\mib{q}\cdot\mib{r}-i\omega t}\end{aligned}$$ or the scalar potential $$\begin{aligned}
\phi = \phi(\mib{q},\omega)e^{i\mib{q}\cdot\mib{r}-i\omega t} .\end{aligned}$$ Within linear response theory, the average current and charge densities are expressed by the response functions $\chi_{\alpha \Gamma}$ with $\alpha = j$, $\rho$ and $\Gamma = A$, $\phi$. [@wunsch; @principi; @sabio; @stauber] They are defined so that the average current and charge densities are expressed as $$\begin{aligned}
\langle j_{x}(\mib{q},\omega) \rangle_{A}
& = -e^{2}v^{2}\chi_{j A}(\mib{q},\omega)A_{x}(\mib{q},\omega) ,
\\
\langle \rho(\mib{q},\omega) \rangle_{A}
& = -e^{2}v\chi_{\rho A}(\mib{q},\omega)
A_{x}(\mib{q},\omega) ,
\\
\langle j_{x}(\mib{q},\omega) \rangle_{\phi}
& = -e^{2}v\chi_{j \phi}(\mib{q},\omega)\phi(\mib{q},\omega) ,
\\
\langle \rho(\mib{q},\omega) \rangle_{\phi}
& = -e^{2}\chi_{\rho \phi}(\mib{q},\omega)
\phi(\mib{q},\omega) .\end{aligned}$$ The response functions are obtained by performing the analytic continuation of $i\nu \to \omega+i\delta$ from their Matsubara representation, $$\begin{aligned}
\label{eq:def-res-jj}
\Pi_{j A}(\mib{q},i\nu)
& = \int\frac{d^{2}k}{(2\pi)^{2}}T\sum_{\epsilon}
\nonumber \\
& \hspace{-2mm} \times
{\rm tr} \left\{\sigma_{x}G(\mib{k}+\mib{q},i\epsilon+i\nu)
\sigma_{x}G(\mib{k},i\epsilon)\right\} ,
\\
\label{eq:def-res-rj}
\Pi_{\rho A}(\mib{q},i\nu)
& = \int\frac{d^{2}k}{(2\pi)^{2}}T\sum_{\epsilon}
\nonumber \\
& \hspace{-2mm} \times
{\rm tr} \left\{G(\mib{k}+\mib{q},i\epsilon+i\nu)
\sigma_{x}G(\mib{k},i\epsilon)\right\} ,
\\
\label{eq:def-res-jr}
\Pi_{j \phi}(\mib{q},i\nu)
& = -\int\frac{d^{2}k}{(2\pi)^{2}}T\sum_{\epsilon}
\nonumber \\
& \hspace{-2mm} \times
{\rm tr} \left\{\sigma_{x}G(\mib{k}+\mib{q},i\epsilon+i\nu)
G(\mib{k},i\epsilon)\right\} ,
\\
\label{eq:def-res-rr}
\Pi_{\rho \phi}(\mib{q},i\nu)
& = -\int\frac{d^{2}k}{(2\pi)^{2}}T\sum_{\epsilon}
\nonumber \\
& \hspace{-2mm} \times
{\rm tr} \left\{G(\mib{k}+\mib{q},i\epsilon+i\nu)
G(\mib{k},i\epsilon)\right\} ,\end{aligned}$$ where $T$ is the temperature. Here, the thermal Green’s function is given by $$\begin{aligned}
G(\mib{k},i\epsilon)
= \frac{1}{2}\sum_{\eta = \pm}
\frac{1+ \eta\left(\sigma_{x}\cos\varphi_{\mib{k}}
+ \sigma_{y}\sin\varphi_{\mib{k}}\right)}
{i\epsilon - E_{\eta}(\mib{k})} .\end{aligned}$$ where $$\begin{aligned}
\cos\varphi_{\mib{k}} = \frac{k_{x}}{|\mib{k}|} ,
\hspace{5mm}
\sin\varphi_{\mib{k}} = \frac{k_{y}}{|\mib{k}|} .\end{aligned}$$ After performing the Matsubara summation, we find $$\begin{aligned}
\label{eq:Pi-jA}
\Pi_{j A}(\mib{q},i\nu)
& = \int\frac{d^{2}k}{(2\pi)^{2}}
\sum_{\eta,\eta'=\pm}
\frac{1+\eta\eta'\cos\left(\varphi_{\mib{k}}+\varphi_{\mib{k}+\mib{q}}
\right)}{2}
\nonumber \\
& \hspace{3mm} \times
\frac{f_{\rm FD}(E_{\eta}(\mib{k}))
- f_{\rm FD}(E_{\eta'}(\mib{k}+\mib{q}))}
{i\nu + E_{\eta}(\mib{k}) - E_{\eta'}(\mib{k}+\mib{q})} ,
\\
\label{eq:Pi-rA}
\Pi_{\rho A}(\mib{q},i\nu)
& = \int\frac{d^{2}k}{(2\pi)^{2}}
\sum_{\eta,\eta'=\pm}
\frac{\eta\cos\varphi_{\mib{k}} + \eta'\cos\varphi_{\mib{k}+\mib{q}}}{2}
\nonumber \\
& \hspace{3mm} \times
\frac{f_{\rm FD}(E_{\eta}(\mib{k}))
- f_{\rm FD}(E_{\eta'}(\mib{k}+\mib{q}))}
{i\nu + E_{\eta}(\mib{k}) - E_{\eta'}(\mib{k}+\mib{q})} ,
\\
\label{eq:Pi-rp}
\Pi_{\rho \phi}(\mib{q},i\nu)
& = -\int\frac{d^{2}k}{(2\pi)^{2}}
\sum_{\eta,\eta'=\pm}
\frac{1+\eta\eta'\cos\left(\varphi_{\mib{k}}-\varphi_{\mib{k}+\mib{q}}
\right)}{2}
\nonumber \\
& \hspace{3mm} \times
\frac{f_{\rm FD}(E_{\eta}(\mib{k}))
- f_{\rm FD}(E_{\eta'}(\mib{k}+\mib{q}))}
{i\nu + E_{\eta}(\mib{k}) - E_{\eta'}(\mib{k}+\mib{q})} ,
\\
\Pi_{j \phi}(\mib{q},i\nu)
& = - \Pi_{\rho A}(\mib{q},i\nu) ,\end{aligned}$$ where $f_{\rm FD}(E)$ represents the Fermi–Dirac function.
For simplicity, we hereafter focus on the case of $\mu = 0$ at $T = 0$. Equations (\[eq:Pi-jA\])–(\[eq:Pi-rp\]) indicate that $\Pi_{\alpha \Gamma}$ generally consists of the interband contribution arising from the terms with $\eta \neq \eta'$ and the intraband contribution arising from those with $\eta = \eta'$. In this case, only the interband terms contribute to the response functions, reflecting the fact that $f_{\rm FD}(E_{+}(\mib{k})) = 0$ and $f_{\rm FD}(E_{-}(\mib{k})) = 1$ for any $\mib{k}$. Let us consider the response to $\mib{A} = (A_{x}(\mib{q},\omega),0)e^{i\mib{q}\cdot\mib{r}-i\omega t}$. For this vector potential, we need to separately treat the transverse case with $\mib{q} = (0,q)$ and the longitudinal case with $\mib{q} = (q,0)$. Since the response functions describing $j_{x}$ diverge without a regularization, [@principi] we employ an ordinary energy cutoff at $E = -\varepsilon_{M}$ that restricts the integration over $\mib{k}$ by the condition of $|\mib{k}| < k_{M}$, where $k_{M} = \varepsilon_{M}/v$. The response function describing $\rho$ converges without a regularization. The response functions are given as $$\begin{aligned}
\label{eq:chi-jA-t_oc}
\chi_{j A}^{t,{\rm oc}}(q\hat{\mib{y}},\omega)
& = \frac{\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}{16v^{2}}
- \frac{\varepsilon_{M}}{4\pi v^{2}} ,
\\
\label{eq:chi-jA-l_oc}
\chi_{j A}^{l,{\rm oc}}(q\hat{\mib{x}},\omega)
& = - \frac{\omega^{2}}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}
- \frac{\varepsilon_{M}}{4\pi v^{2}} ,
\\
\chi_{\rho A}^{l}(q\hat{\mib{x}},\omega)
& = - \frac{\omega (vq)}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}} ,\end{aligned}$$ where $t$ and $l$ respectively represent the transverse and longitudinal cases, and ${\rm oc}$ indicates that the corresponding result is obtained by using the ordinary cutoff. Note that $\chi_{\rho A}^{t}(q\hat{\mib{y}},\omega) = 0$ as a transverse vector potential cannot induce a charge density. Equations (\[eq:chi-jA-t\_oc\]) and (\[eq:chi-jA-l\_oc\]) have been given in Ref. . Let us next consider the response to $\phi = \phi(\mib{q},\omega) e^{i\mib{q}\cdot\mib{r}-i\omega t}$. The response functions converge without a regularization, [@ando1; @wunsch] resulting in $$\begin{aligned}
\label{eq:chi-jp}
\chi_{j \phi}(q,\omega)
& = \frac{\omega (vq)}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}} ,
\\
\label{eq:chi-rp}
\chi_{\rho \phi}(q,\omega)
& = \frac{(vq)^{2}}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}} .\end{aligned}$$ Equation (\[eq:chi-rp\]) has been given in Ref. . For $(vq)^{2} < \omega^{2}$, the square roots in the above expressions should be read as $\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}
= -i\,{\rm sign}(\omega)\sqrt{\omega^{2}-(vq)^{2}}$.
It is easy to observe that these response functions break the gauge invariance with respect to $\mib{A}$ and $\phi$. [@principi; @polini] Owing to the gauge invariance, a static transverse vector potential can induce no electromagnetic response in the limit of $q \to 0$, while a static longitudinal vector potential cannot induce a response for any $q$. Contrary to this well-known fact, the term with $\varepsilon_{M}$ in $\chi_{j A}^{t,{\rm oc}}$ and $\chi_{j A}^{l,{\rm oc}}$ induces a finite charge current even when $\mib{A}$ is constant. Furthermore, the gauge invariance ensures that $\langle j_{x} \rangle_{A}$ induced by $A_{x}(q\hat{\mib{x}},\omega)$ must be identical to $\langle j_{x} \rangle_{\phi}$ induced by $\phi(q\hat{\mib{x}},\omega) \equiv (-\omega/q)A_{x}(q\hat{\mib{x}},\omega)$. However, they apparently differ from each other. Indeed, we find $$\begin{aligned}
\langle j_{x} \rangle_{A} - \langle j_{x} \rangle_{\phi}
= ev \frac{\varepsilon_{M}}{4\pi v}
eA_{x}(q\hat{\mib{x}},\omega) .\end{aligned}$$ Note that the response to a scalar potential satisfies the charge conservation relation $$\begin{aligned}
-\omega\langle \rho \rangle_{\phi} + q\langle j_{x} \rangle_{\phi} = 0 ,\end{aligned}$$ whereas the response to a vector potential breaks it as $$\begin{aligned}
-\omega\langle \rho \rangle_{A} + q\langle j_{x} \rangle_{A}
= e vq\frac{\varepsilon_{M}}{4\pi v} eA_{x}(q\hat{\mib{x}},\omega) .\end{aligned}$$ The above argument indicates that, although the response to a scalar potential is appropriate, we need to reconsider the response to a vector potential. It has been pointed out that this difficulty is caused by the ordinary cutoff, which breaks the gauge invariance. [@principi; @polini]
Gauge-Invariant Energy Cutoff
=============================
To overcome the difficulty associated with $\mib{A}$, we propose a modified energy cutoff that preserves the gauge invariance. This cutoff is implemented by two steps. Firstly, we replace the Fermi–Dirac function $f_{\rm FD}(E)$ in the expression for a response function with the modified distribution function $\tilde{f}_{\rm FD}(E)$ defined by $$\begin{aligned}
\tilde{f}_{\rm FD}(E) = f_{\rm FD}(E)\theta(E+\varepsilon_{M}) ,\end{aligned}$$ where $\theta(E)$ is the Heaviside step function. Secondly, we calculate the correction induced by this replacement in the zero-frequency limit of $\omega \to 0$. By adding the resulting correction to the main contribution given in Sect. 2, we obtain the final result \[see Eq. (\[eq:def-mc-t\]) as an example\].
The replacement of $f_{\rm FD}(E)$ with $\tilde{f}_{\rm FD}(E)$ directly results in the exclusion of the electron states with energy $E$ smaller than $-\varepsilon_{M}$. This is not equivalent to the restriction of $|\mib{k}| < k_{M}$ in the ordinary cutoff, as shown below. Indeed, it gives a new correction by which the gauge invariance is preserved. The zero-frequency limit is taken to pick up only the relevant correction in a selective manner. In other words, a spurious contribution is included without taking the limit. Accurate justification of this procedure is given in Sect. 5.
Now, we briefly point out an essential difference between the ordinary cutoff and the modified one proposed here. If the ordinary cutoff at $E = -\varepsilon_{M}$ is applied to the calculation of a response function, only an initial state is restricted to satisfy the condition of $-\varepsilon_{M} < E$. In other words, the restriction is not imposed on an intermediate state. By using the modified energy cutoff, we can thoroughly impose the restriction of $-\varepsilon_{M} < E$ both on initial and intermediate states. We here elucidate the importance of this difference by applying these two procedures to a simple problem.
Let us consider the variation in the total energy of electron states induced by a static vector potential $\mib{A} = (A_{x},0)e^{i\mib{q}\cdot\mib{x}}$ in the limit of $|\mib{q}| \to 0$. Obviously, as the resulting vector potential is constant, it never alters the total energy owing to the gauge invariance. We again focus on the case of $\mu = 0$ at $T = 0$, and calculate the variation $\delta U$ within a second-order perturbation theory with respect to $\mib{A}$. If the ordinary cutoff is applied, the variation arises only from interband processes and is expressed as $$\begin{aligned}
\delta U_{\rm inter}
& = \left(veA_{x}\right)^{2}\int\frac{d^{2}k}{(2\pi)^{2}}
\sin^{2}\left(\frac{\varphi_{\mib{k}}+\varphi_{\mib{k}+\mib{q}}}{2}\right)
\nonumber \\
& \hspace{5mm} \times
\frac{f_{\rm FD}(E_{-}(\mib{k}))
\left[1-f_{\rm FD}(E_{+}(\mib{k}+\mib{q}))\right]}
{E_{-}(\mib{k}) - E_{+}(\mib{k}+\mib{q})} ,\end{aligned}$$ where the integration over $\mib{k}$ is restricted by $k < k_{M}$ with $k = |\mib{k}|$. In the limit of $|\mib{q}| \to 0$, we find $$\begin{aligned}
\label{eq:inter-U}
\delta U_{\rm inter}
= - \left(eA_{x}\right)^{2}\frac{\varepsilon_{M}}{8\pi} ,\end{aligned}$$ which disagrees with the correct result, $\delta U = 0$, expected from the gauge invariance. This clearly indicates that the ordinary cutoff breaks the gauge invariance.
We show that the correct result is obtained if the modified energy cutoff is applied. [@comment0] The variation arises from not only interband processes but also intraband processes. The former contribution is identical to that given in Eq. (\[eq:inter-U\]). The latter contribution is expressed as $$\begin{aligned}
\delta U_{\rm intra}
& = \left(veA_{x}\right)^{2}\int\frac{d^{2}k}{(2\pi)^{2}}
\cos^{2}\left(\frac{\varphi_{\mib{k}}+\varphi_{\mib{k}+\mib{q}}}{2}\right)
\nonumber \\
& \hspace{5mm} \times
\frac{\tilde{f}_{\rm FD}(E_{-}(\mib{k}))
\left[1-\tilde{f}_{\rm FD}(E_{-}(\mib{k}+\mib{q}))\right]}
{E_{-}(\mib{k}) - E_{-}(\mib{k}+\mib{q})} .\end{aligned}$$ As a direct consequence of the restriction on the intermediate state with $E_{-}(\mib{k}+\mib{q})$, this gives a nonnegligible contribution when $E_{-}(\mib{k}+\mib{q}) < -\varepsilon_{M} < E_{-}(\mib{k})$. Approximating the fractional factor as $$\begin{aligned}
& \frac{\tilde{f}_{\rm FD}(E_{-}(\mib{k}))
\left[1-\tilde{f}_{\rm FD}(E_{-}(\mib{k}+\mib{q}))\right]}
{E_{-}(\mib{k}) - E_{-}(\mib{k}+\mib{q})}
\nonumber \\
& \hspace{10mm}
= \theta(E_{-}(\mib{k})+\varepsilon_{M})
\delta(E_{-}(\mib{k})+\varepsilon_{M}) ,\end{aligned}$$ we find $$\begin{aligned}
\label{eq:intra-U}
\delta U_{\rm intra}
= \left(eA_{x}\right)^{2}\frac{\varepsilon_{M}}{8\pi} ,\end{aligned}$$ which exactly cancels out $\delta U_{\rm inter}$. That is, the modified energy cutoff gives the correct result, $$\begin{aligned}
\delta U = \delta U_{\rm inter} + \delta U_{\rm intra} = 0 .\end{aligned}$$ This argument suggests that the modified energy cutoff is more suitable than the ordinary one in describing the response to a vector potential in Dirac electron systems.
The insufficiency of the ordinary cutoff is clearly explained from the fact that a constant vector potential $\mib{A} = (A_{x},0)$ only shifts the Dirac point from $\mib{k} = (0,0)$ to $(-eA_{x},0)$. In the absence of $\mib{A}$, the energy cutoff at $E = -\varepsilon_{M}$ is equivalent to restricting the integration over $\mib{k}$ by the condition of $k < k_{M}$. In the presence of $\mib{A}$, the energy cutoff is correctly carried out by modifying the condition as $\sqrt{(k_{x}+eA_{x})^{2}+k_{y}^{2}} < k_{M}$. The ordinary cutoff takes no account of such a modification; thus, it breaks the gauge invariance. Indeed, we can show that $\delta U_{\rm inter}$ is identical to the variation in the total energy under the ordinary cutoff: $$\begin{aligned}
\delta U_{\rm oc}
= \int\frac{d^{2}k}{(2\pi)^{2}}
\left( E_{-}(\mib{k}+e\mib{A}) - E_{-}(\mib{k}) \right) ,\end{aligned}$$ where the integration over $\mib{k}$ is restricted by $k < k_{M}$. In the modified energy cutoff, the modification of the condition is implicitly taken into account through $\tilde{f}_{\rm FD}(E)$.
Derivation of Response Functions
================================
By using the modified energy cutoff proposed in Sect. 3, we derive the response functions for $\mib{A}$ in the case of $\mu = 0$ at $T = 0$. The response functions $\chi_{j A}^{t,{\rm oc}}(q\hat{\mib{y}},\omega)$ and $\chi_{j A}^{l,{\rm oc}}(q\hat{\mib{x}},\omega)$ given in Sect. 2 are obtained by using the ordinary cutoff and consist of only the interband contribution arising from the terms with $\eta \neq \eta'$. Even though the modified energy cutoff is applied instead of the ordinary one, the interband contribution does not change. However, the intraband term with $\eta = \eta' = -$ gives an additional contribution. Hence, we derive this contribution $\delta\Pi_{j A}(\mib{q},i\nu)$ in the Matsubara representation. According to the procedure given in Sect. 3, the proper correction is obtained by taking the zero-frequency limit of $\omega \to 0$ after the analytic continuation of $i\nu \to \omega +i\delta$. For example, the correction to $\chi_{j A}^{t,{\rm oc}}(q\hat{\mib{y}},\omega)$ is given by $$\begin{aligned}
\label{eq:def-corre-t}
\delta\chi_{j A}^{t}(q\hat{\mib{y}},0)
= \lim_{\omega \to 0}
\left[ \left. \delta\Pi_{j A}(q\hat{\mib{y}},i\nu)
\right|_{i\nu \to \omega+i\delta}
\right] .\end{aligned}$$ The final result is expressed as $$\begin{aligned}
\label{eq:def-mc-t}
\chi_{j A}^{t,{\rm mc}}(q\hat{\mib{y}},\omega)
= \chi_{j A}^{t,{\rm oc}}(q\hat{\mib{y}},\omega)
+ \delta\chi_{j A}^{t}(q\hat{\mib{y}},0) ,\end{aligned}$$ where ${\rm mc}$ indicates that this is obtained by using the modified energy cutoff.
From Eq. (\[eq:Pi-jA\]), we find that the additional contribution is expressed as $$\begin{aligned}
\delta\Pi_{j A}(\mib{q},i\nu)
& = \int\frac{d^{2}k}{(2\pi)^{2}}
\frac{1+\cos\left(\varphi_{\mib{k}}+\varphi_{\mib{k}+\mib{q}}
\right)}{2}
\nonumber \\
& \hspace{-3mm} \times
\frac{\tilde{f}_{\rm FD}(E_{-}(\mib{k}))
- \tilde{f}_{\rm FD}(E_{-}(\mib{k}+\mib{q}))}
{i\nu + E_{-}(\mib{k}) - E_{-}(\mib{k}+\mib{q})} .\end{aligned}$$ A nonnegligible contribution arises from the cases of $E_{-}(\mib{k}+\mib{q}) < -\varepsilon_{M} < E_{-}(\mib{k})$ and $E_{-}(\mib{k}) < -\varepsilon_{M} < E_{-}(\mib{k}+\mib{q})$. This can be safely calculated by using the following approximation: $$\begin{aligned}
& \tilde{f}_{\rm FD}(E_{-}(\mib{k}))
-\tilde{f}_{\rm FD}(E_{-}(\mib{k}+\mib{q}))
\nonumber \\
& \hspace{5mm}
= -\frac{\partial \tilde{f}_{\rm FD}(E_{-}(\mib{k}))}
{\partial E_{-}(\mib{k})}
\left(-vq\sin\varphi_{\mib{k}}\right) ,\end{aligned}$$ for a transverse vector potential with $\mib{q} = (0,q)$. For a longitudinal vector potential with $\mib{q} = (q,0)$, the factor $-vq\sin\varphi_{\mib{k}}$ in the right-hand side should be replaced with $-vq\cos\varphi_{\mib{k}}$. The integration over $k$ yields $$\begin{aligned}
\label{eq:exp-deltaPi-t}
\delta\Pi_{j A}(q\hat{\mib{y}},i\nu)
= \frac{\varepsilon_{M}}{2\pi v^{2}}
\int_{0}^{2\pi}\frac{d\varphi}{2\pi}
\cos^{2}\varphi \frac{vq\sin\varphi}{i\nu+vq\sin\varphi}\end{aligned}$$ for a transverse vector potential. The correction to $\chi_{j A}^{t,{\rm oc}}(q\hat{\mib{y}},\omega)$ is obtained by using Eq. (\[eq:def-corre-t\]).
We find that the resulting corrections to $\chi_{j A}^{t,{\rm oc}}$ and $\chi_{j A}^{l,{\rm oc}}$ are equivalent and are given by $$\begin{aligned}
\delta\chi_{j A}^{t}(q\hat{\mib{y}},0)
= \delta\chi_{j A}^{l}(q\hat{\mib{x}},0)
= \frac{\varepsilon_{M}}{4\pi v^{2}} ,\end{aligned}$$ which exactly cancels out the second term of $\chi_{j A}^{t,{\rm oc}}$ and $\chi_{j A}^{l,{\rm oc}}$. No intraband correction appears in $\chi_{\rho A}^{l}$. By using the modified energy cutoff, we finally find that the response functions for a vector potential are given by $$\begin{aligned}
\chi_{j A}^{t,{\rm mc}}(q\hat{\mib{y}},\omega)
& = \frac{\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}{16v^{2}} ,
\\
\chi_{j A}^{l,{\rm mc}}(q\hat{\mib{x}},\omega)
& = - \frac{\omega^{2}}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}} ,
\\
\chi_{\rho A}^{l}(q\hat{\mib{x}},\omega)
& = - \frac{\omega (vq)}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}} .\end{aligned}$$ The electromagnetic response is described by these response functions together with those given in Eqs. (\[eq:chi-jp\]) and (\[eq:chi-rp\]).
In contrast to the results under the ordinary cutoff, the response functions $\chi_{j A}^{t,{\rm mc}}$ and $\chi_{j A}^{l,{\rm mc}}$ together with $\chi_{\rho A}^{l}$ satisfy the conditions required from the gauge invariance. Indeed, $\chi_{j A}^{t,{\rm mc}}(q\hat{\mib{y}},0) = 0$ in the limit of $q \to 0$ and $\chi_{j A}^{l,{\rm mc}}(q\hat{\mib{x}},0)
= \chi_{\rho A}^{l}(q\hat{\mib{x}},0) = 0$ for any $q$. In addition, we can show that $\langle j_{x} \rangle_{A}$ induced by $A_{x}(q\hat{\mib{x}},\omega)$ is identical to $\langle j_{x} \rangle_{\phi}$ induced by $\phi(q\hat{\mib{x}},\omega) \equiv (-\omega/q)A_{x}(q\hat{\mib{x}},\omega)$. We can also show that the charge conservation relation holds in the response to a longitudinal vector potential as $$\begin{aligned}
-\omega\langle \rho \rangle_{A} + q\langle j_{x} \rangle_{A} = 0 .\end{aligned}$$ The above argument indicates that the gauge invariance is preserved if we use the modified energy cutoff to calculate the response functions for $\mib{A}$.
The resulting response functions are equivalent to $\chi_{j A}^{t,{\rm oc}}$ and $\chi_{j A}^{l,{\rm oc}}$ obtained by using the ordinary cutoff if the term $-\varepsilon_{M}/(4\pi v^{2})$ is simply excluded. However, note that this exclusion has not been justified in a reliable manner. Indeed, it was claimed [@principi] that this term is physical in the limit of $q \to 0$ with $\omega \neq 0$.
Justification of the Modified Energy Cutoff
===========================================
In this section, we derive all the response functions by applying the modified energy cutoff procedure *without taking the zero-frequency limit of* $\omega \to 0$. The resulting response functions $\tilde{\chi}_{\alpha \Gamma}$ consist of two contributions: a relevant contribution that describes the actual response of Dirac electrons and an irrelevant contribution that reflects the effect of artificial excitations due to the cutoff. We show that the zero-frequency limit allows us to pick up only the relevant contribution, justifying the procedure given in Sect. 3.
We start with the expression of $\delta\Pi_{j A}$, given in Eq. (\[eq:exp-deltaPi-t\]), for a transverse vector potential. Performing the integration over $\varphi$, we find $$\begin{aligned}
\delta\Pi_{j A}(q\hat{\mib{y}},i\nu)
= \frac{\varepsilon_{M}}{4\pi v^{2}}
\left[ 1-\frac{2\nu\left(\sqrt{(vq)^{2}+\nu^{2}}-\nu\right)}
{(vq)^{2}}
\right] .\end{aligned}$$ The corresponding correction to $\chi_{j A}^{t,{\rm oc}}$ is obtained by performing the analytic continuation of $i\nu \to \omega +i\delta$. The result is written as $$\begin{aligned}
\label{eq:del-jAt}
& \delta\chi_{j A}^{t}(q\hat{\mib{y}},\omega)
\nonumber \\
& \hspace{0mm}
= \frac{\varepsilon_{M}}{4\pi v^{2}}
\left[ 1+\frac{2i\omega\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}{(vq)^{2}}
\Lambda(q,\omega)
\right] ,\end{aligned}$$ where $$\begin{aligned}
\Lambda(q,\omega)
= 1+\frac{i\omega}{\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}} .\end{aligned}$$ Performing calculations similar to this, we find that the corrections to the other response functions are $$\begin{aligned}
\label{eq:del-jAl}
\delta\chi_{j A}^{l}(q\hat{\mib{x}},\omega)
& = \frac{\varepsilon_{M}}{4\pi v^{2}}
\left[ 1+\frac{2 \omega^{2}}{(vq)^{2}}\Lambda(q,\omega) \right] ,
\\
\label{eq:del-rAl}
\delta\chi_{\rho A}^{l}(q\hat{\mib{x}},\omega)
& = \frac{\varepsilon_{M}}{2\pi v^{2}}
\frac{\omega}{vq}\Lambda(q,\omega) ,
\\
\label{eq:del-jp}
\delta\chi_{j \phi}(q,\omega)
& = - \frac{\varepsilon_{M}}{2\pi v^{2}}
\frac{\omega}{vq}\Lambda(q,\omega) ,
\\
\label{eq:del-rp}
\delta\chi_{\rho \phi}(q,\omega)
& = - \frac{\varepsilon_{M}}{2\pi v^{2}}\Lambda(q,\omega) .\end{aligned}$$
Adding each correction to the corresponding main contribution given in Sect. 2, we finally find the response functions $\tilde{\chi}_{\alpha \Gamma}$. The results are $$\begin{aligned}
\label{eq:tilde-jAt}
\tilde{\chi}_{j A}^{t}(q\hat{\mib{y}},\omega)
& = \frac{\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}{16v^{2}}
\nonumber \\
& \hspace{0mm}
+ \frac{\varepsilon_{M}}{2\pi v^{2}}
\frac{i\omega\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}{(vq)^{2}}
\Lambda(q,\omega) ,
\\
\label{eq:tilde-jAl}
\tilde{\chi}_{j A}^{l}(q\hat{\mib{x}},\omega)
& = - \frac{\omega^{2}}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}
\nonumber \\
& \hspace{0mm}
+ \frac{\varepsilon_{M}}{2\pi v^{2}}\frac{\omega^{2}}{(vq)^{2}}
\Lambda(q,\omega) ,
\\
\label{eq:tilde-rA}
\tilde{\chi}_{\rho A}^{l}(q\hat{\mib{x}},\omega)
& = - \frac{\omega (vq)}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}
\nonumber \\
& \hspace{0mm}
+ \frac{\varepsilon_{M}}{2\pi v^{2}}\frac{\omega}{vq}
\Lambda(q,\omega) ,
\\
\label{eq:tilde-jp}
\tilde{\chi}_{j \phi}(q,\omega)
& = \frac{\omega (vq)}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}
\nonumber \\
& \hspace{0mm}
- \frac{\varepsilon_{M}}{2\pi v^{2}}\frac{\omega}{vq}
\Lambda(q,\omega) ,
\\
\label{eq:tilde-rp}
\tilde{\chi}_{\rho \phi}(q,\omega)
& = \frac{(vq)^{2}}{16v^{2}\sqrt{(vq)^{2}-(\omega+i\delta)^{2}}}
- \frac{\varepsilon_{M}}{2\pi v^{2}}\Lambda(q,\omega) .\end{aligned}$$ It is easy to see that these response functions preserve the gauge invariance and satisfy the charge conservation relation.
Although Eqs. (\[eq:tilde-jAt\])–(\[eq:tilde-rp\]) satisfy the required conditions, we should not straightforwardly apply them to an actual physical system. The reason is that an irrelevant contribution is contained in the corrections, Eq. (\[eq:del-jAt\]) and Eqs. (\[eq:del-jAl\])–(\[eq:del-rp\]), and hence in the final results, Eqs. (\[eq:tilde-jAt\])–(\[eq:tilde-rp\]), as we show below. Note that $\varepsilon_{M}/(4\pi v^{2})$ in $\delta\chi_{j A}^{t}$ ($\delta\chi_{j A}^{l}$) cancels out the second term of $\chi_{j A}^{t,{\rm oc}}$ ($\chi_{j A}^{l,{\rm oc}}$), preserving the gauge invariance. Let us focus on the remaining terms with $\Lambda(q,\omega)$ in $\delta\chi_{j A}^{t}$ and $\delta\chi_{j A}^{l}$. Clearly, they represent the effect of electron excitations across the cutoff energy. Since such excitations are artificially allowed as a result of the modified energy cutoff, the terms with $\Lambda(q,\omega)$ are irrelevant in describing actual situations. That is, $\delta\chi_{j A}^{t}$ and $\delta\chi_{j A}^{l}$ consist of the relevant contribution preserving the gauge invariance and the irrelevant contribution describing the effect of artificial excitations. $\delta\chi_{\rho A}^{l}$ consists of only the irrelevant contribution. Note that the irrelevant contributions vanish in the zero-frequency limit of $\omega \to 0$. This is not accidental but is guaranteed by the gauge invariance. [@comment1]
We conclude that only the relevant contributions should be taken into account in calculating the response functions for a vector potential, and that the irrelevant contributions vanish in the zero-frequency limit. Hence, the relevant contributions are selectively picked up by taking the zero-frequency limit. This argument justifies the modified energy cutoff procedure proposed in Sect. 3.
In accordance with the argument given above, we show that the exclusion of the second terms in Eqs. (\[eq:tilde-jAt\])–(\[eq:tilde-rp\]) is reasonable in actual situations. Let us focus on the second term of $\tilde{\chi}_{\rho \phi}$. Owing to its presence, a finite charge density proportional to $\varepsilon_{M}$ is induced by $\phi$ despite the fact that the valence band is completely filled. Clearly, this should be regarded as an artifact induced by the artificial excitations across the cutoff energy. Hence, the second term should be excluded. Let us next focus on the second term of $\tilde{\chi}_{j \phi}$. Since this term is directly related with that of $\tilde{\chi}_{\rho \phi}$ through the charge conservation relation, it also describes a similar artifact and therefore should be excluded. The second term of $\tilde{\chi}_{j A}^{l}$ is directly related with that of $\tilde{\chi}_{j \phi}$ through the gauge invariance. Furthermore, the second term of $\tilde{\chi}_{j A}^{t}$ must be identical with that of $\tilde{\chi}_{j A}^{l}$ in the limit of $q \to 0$. Taking everything into consideration, we recognize that the second terms describe an artifact caused by the excitations across the cutoff energy; thus, they should be excluded. After the exclusion, $\tilde{\chi}_{j A}^{t}$, $\tilde{\chi}_{j A}^{l}$, and $\tilde{\chi}_{\rho A}^{l}$ are respectively reduced to $\chi_{j A}^{t,{\rm mc}}$, $\chi_{j A}^{l,{\rm mc}}$, and $\chi_{\rho A}^{l}$ obtained by using the modified energy cutoff in Sect. 4. Similarly, $\tilde{\chi}_{j \phi}$ and $\tilde{\chi}_{\rho \phi}$ are respectively reduced to $\chi_{j \phi}$ and $\chi_{\rho \phi}$ given in Sect. 2.
Summary
=======
We have proposed a modified energy cutoff procedure in terms of a modified distribution function for electrons in order to describe the electromagnetic response of Dirac electron systems in a gauge-invariant manner. We have shown that the response functions obtained by using this cutoff satisfy the necessary conditions that are required from the gauge invariance.
Although only the application to a 2D massless Dirac electron system is presented in this paper, the modified energy cutoff procedure can be used in various Dirac systems in any dimension regardless of the presence or absence of a mass gap. For example, it can be applied to the problem considered in Ref. , where the superfluid density in a superconducting state of three-dimensional massive Dirac electrons is calculated by using a continuum Dirac model under the ordinary cutoff. The resulting superfluid density does not vanish even in the normal state without an additional regularization. If this problem is analyzed by using the modified energy cutoff, the unphysical contribution \[Eq. (25) of Ref. \] is canceled out by the correction arising from the intraband term \[Eq. (21) of Ref. \].
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by JSPS KAKENHI Grant Number JP18K03460.
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Since the vector potential is static in this case, we need not take the zero-frequency limit in the second step.
Strictly speaking, the gauge invariance guarantees the vanishing of the irrelevant contribution in the zero-frequency limit only for a longitudinal vector potential. In the transverse case, we may need to employ much more careful treatment in some situations.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We address the nature of EUV waves through direct observations of the formation of a diffuse wave driven by the expansion of a coronal mass ejection (CME) and its subsequent separation from the CME front. The wave and the CME on 2011 June 7 were well observed by Atmospheric Imaging Assembly onboard Solar Dynamic Observatory. Following the solar eruption onset, marked by the beginning of the rapid increasing of the CME velocity and the X-ray flux of accompanying flare, the CME exhibits a strong lateral expansion. During this impulsive expansion phase, the expansion speed of the CME bubble increases from 100 km s$^{-1}$ to 450 km s$^{-1}$ in only six minutes. An important finding is that a diffuse wave front starts to separate from the front of the expanding bubble shortly after the lateral expansion slows down. Also a type-II burst is formed near the time of the separation. After the separation, two distinct fronts propagate with different kinematic properties. The diffuse front travels across the entire solar disk; while the sharp front rises up, forming the CME ejecta with the diffuse front ahead of it. These observations suggest that the previously termed EUV wave is a composite phenomenon and driven by the CME expansion. While the CME expansion is accelerating, the wave front is cospatial with the CME front, thus the two fronts are indiscernible. Following the end of the acceleration phase, the wave moves away from the CME front with gradually an increasing distance between them.'
author:
- 'X. Cheng, J. Zhang, O. Olmedo, A. Vourlidas, M. D. Ding, Y. Liu'
title: Investigation of the Formation and Separation of An EUV Wave from the Expansion of A Coronal Mass Ejection
---
Online-only material: animations, color figures
Introduction
============
Coronal mass ejections (CMEs) are large-scale eruptive phenomena from the Sun. They can carry large amounts of plasma and magnetic field energy into the interplanetary space, which may have severe effects on space environment and human technological systems around the Earth [@gosling93; @webb94]. A typical CME has a velocity of several hundred km s$^{-1}$, while the fastest one recorded is over 3000 km s$^{-1}$ [@Yashiro04]. The main acceleration of a CME usually occurs in the inner corona (e.g., $\leq$ 3.0 $R_\odot$) [@zhang01]. With the advent of the Atmospheric Imaging Assembly [AIA; @lemen11] onboard Solar Dynamics Observatory (*SDO*), details on the initiation and structural formation of CME starts to emerge. Recently, it was found that a twisted hot channel ($\sim$10 MK) as seen in AIA 131 [Å]{} passband starts to form before the flare onset, and its subsequent rise results in the CME and the accompanying flare; this hot channel has been argued to be the existence of magnetic flux rope prior the eruption [@zhang11]. Following the eruption onset, a CME rises up impulsively with a strong acceleration, and also expands quickly along the lateral direction, producing a plasma bubble [@pat09b; @cheng11]. The CME bubble is capable of driving a shock, which may generate the observable metric type II bursts [e.g., @liu09; @ma11].
One interesting phenomenon closely associated with CMEs is the globally-propagating bright feature in the corona dubbed the EIT or EUV wave since its discovery by *SOHO* spacecraft [@thompson98; @thompson99]. The physical nature of the moving front remains somewhat unclear. The front has been interpreted as a fast-mode magnetohydrodynamic (MHD) wave [@thompson99; @wang00; @warmuth01; @veronig08; @kienreich09; @gopa09; @pat09a; @pat09b; @liu11; @olmedo11], a soliton wave [@wills07], or slow-mode wave [@wang09]. Whereas, others believe that it is not at all a true wave. @chen02 [@chen05], @chen09, and @chen11 argued that the bright front results from the compression front driven by the successive stretching of the magnetic field overlying the erupting CME flux rope, although there should be a fast-mode wave ahead of the CME front. @attrill07 [@attrill09] and @dai10 claimed that the bright front is related to the magnetic reconnection between the outmost magnetic field of the CME and the magnetic loops from the quiet region. @delannee08 suggested that the current shell in their numerical simulation can also form the bright front. In order to bring together these opposing views, some authors recently tend to appeal for the hybrid model combining both wave and non-wave explanations [e.g., @cohen09; @liu10; @downs11]. Details of various EUV wave models can be found in recent review papers by @warmuth10, @gallagher11, and @zhukov11.
In this Letter, we investigate the physical nature of the EUV wave. From the observations, we find that the wave front has two distinct evolution stages, i.e., a compression front forming the CME front in the early stage and a stand-alone wave front separating from the CME front in the later stage. The data used are mainly from AIA onboard *SDO* and Sun Earth Connection Coronal and Heliospheric Investigation [SECCHI; @howard08] onboard Solar TErrestrial RElations Observatory (*STEREO*). Observations and results are presented in Section 2, followed by a summary and discussion in Section 3.
Observations and Results
========================
Overview of the CME Eruption
----------------------------
On 2011 June 7, the active region (AR) NOAA 11226 produced an M2.5 class flare at the location of S22$^{\circ}$W53$^{\circ}$, which started at 06:16 UT and peaked at 06:30 UT. Following the onset of the flare, the overlying magnetic field of the AR expanded rapidly and formed a plasma bubble with a sharp front. At $\sim$06:26 UT, the bubble clearly appeared in AIA 193 [Å]{} and 211 [Å]{} images (Figure\[f1\](a) and (b)), best seen in the running difference images in Figure \[f1\](c). After $\sim$3 minutes, the bubble front started to leave the FOV of AIA (Figure \[f1\](c)). Subsequently, it developed into CME ejecta as seen in the FOV of COR1 (Figure \[f1\](e) and (f)).
Rise and Early Expansion of the CME
-----------------------------------
The shape of the CME bubble and ejecta can be clearly seen and tracked, thanks to the high cadence high quality AIA observations. So we are able to study the structural and kinematic evolution of the CME with high precision. The CME bubble is fitted as a circle in the AIA FOV, from which the radius of the bubble is obtained. The top of the circle is regarded as the height of the bubble front. Figure \[f1\](c) and (d) display the fitting result, in which the bubble is represented by the blue dash-dotted lines. Once the CME entered the FOV of COR1 (Figure \[f1\](e) and (f)) and COR2 (not shown here), it appeared very similar to a flux rope structure. Thus, we use the graduated cylindrical shell (GCS) model [@thernisien06; @thernisien09] to model the 3D structure of the CME. The height-time variation of the CME front is shown in Figure \[f2\](a), along with the radius-time plot of the CME bubble in the AIA FOV.
Based on the height-time data, processed by the spline smoothing, we calculate the radial velocity of the CME front using a piece-wise numerical derivative method. The temporal evolution of the velocity is plotted in Figure \[f2\](b), in which we also plot the $GOES$ soft X-ray 1–8 Å flux. One can see that the CME accelerated during the rise phase of the flare; the radial velocity of the CME increased from $\sim$100 km s$^{-1}$ at 06:19 UT to $\sim$1200 km s$^{-1}$ at 06:35 UT. The average acceleration during this period is $\sim$1130 m s$^{-2}$. Moreover, we calculate the expansion velocity of the CME bubble based on the radius-time data (blue line in Figure \[f2\](b)). It is found that the expansion of the bubble experienced a different kinematic evolution. In the first seven minutes, the expansion velocity of the bubble quickly increased from $\sim$100 km s$^{-1}$ to $\sim$450 km s$^{-1}$ with an average acceleration of $\sim$830 m s$^{-2}$. Whereas, after $\sim$06:26 UT, the expansion of the bubble started to slow down although its front was still undergoing acceleration. Note that the uncertainty in the velocity calculation is mainly from the error in the height measurement, which is estimated to be 4 pixels (1700 km, 9400 km, and 44000 km for AIA, EUVI, and COR1 and COR2 respectively).
Separation of a Diffuse Wave from the CME Bubble
------------------------------------------------
One interesting finding from studying this event is that a diffuse wave is clearly seen to separate from the sharp bubble front. The separating wave front is mostly visible at the flank of the bubble in the FOV of AIA (Figure \[f1\](d)). The separation between the wave and the bubble is also clearly seen in the SECCHI observations (Figure \[f1\](e) and (f)). Inspecting the evolution of the wave and the bubble, we find that the diffuse wave front is always close to the bubble front at the top (or leading fronts along the radial direction). While, at the flanks of the bubble, the standoff distance between the two fronts gradually increases with time. At 06:40 UT, the diffuse front had propagated further away from the bubble front as shown in Figure \[f1\](e) and (f), from which we also find that the wave front above the limb coincides very well with that on the disk.
In order to investigate the detailed separation process of the wave front from the CME bubble, we transform the AIA images from the observed cartesian coordinates (x,y distance from sun center) into helio-projective coordinates (polar angle along the X-axis and projected heliocentric distance along the Y-axis). Figure \[f3\](a) shows the AIA 211 [Å]{} difference images in the transformed system. Note that the 211 [Å]{} difference images show the diffuse wave front best among all AIA passbands. One can see that the diffuse wave front can now be well distinguished from the sharp front of the CME bubble (see also the online movie for a better impression).
Next, from each transformed image we extract three horizontal slices located at heliocentric heights of 1.15, 1.05, and 0.95$R_\odot$, respectively. We then stack each slice vertically in a time sequence to make the slice-time plot. The results are shown in Figure \[f3\](b)-(d). From the stacking slice-time plots, one can see that the diffuse wave front has a different lateral evolution from the sharp bubble front. The wave front overlaps with the bubble front from $\sim$06:20 UT to $\sim$06:27 UT, i.e., the two fronts are exactly co-spatial, thus can not be discerned. After $\sim$06:27 UT, the wave front starts to separate from the bubble front, and the distance between the two fronts increases with time. The wave front continues to propagate, traversing through the AR in the north and the coronal hole in the south. On the other hand, the bubble’s lateral expansion slows down significantly and stops near the AR in the north and the coronal hole in the south (dotted lines in Figure \[f3\](b) and (c) respectively). Note that there is only one front that can be clearly identified in the slice-time plot of the slice within the solar disk, because of the increased complexity of features on the disk.
Through the slice-time plots, we can easily measure the lateral displacements of the CME bubble front and the wave front from a center position along the horizontal slice: the bubble front is denoted by the blue plus signs; and the wave front is shown by the red diamonds and squares (Figure \[f3\](b)-(d)). The resulting lateral propagation velocities are plotted in Figure \[f4\]. The uncertainty in the velocity values results from the uncertainty of the lateral distance measurement, which is about 2800 km. It is worth noting that the lateral propagation velocity of the bubble along a same heliocentric distance (or a horizontal slice in the transformed helio-projective image) differs from its intrinsic expansion velocity; the apparent lateral velocity is a combination of the intrinsic expansion velocity and a geometric velocity caused by the rising motion of the bubble. Also note that, the apparent lateral velocity of the wave along a slice is close to its real propagation velocity because the slice is almost perpendicular to the wave front.
From Figure \[f4\](a), we can see that, at the heliocentric height of 1.15 $R_\odot$, the CME bubble front reached an apparent lateral velocity of 960 km s$^{-1}$ at $\sim$06:27 UT, which then quickly decreased to almost zero about 9 minutes later. For the wave front, it also accelerated to the lateral velocity of 960 km s$^{-1}$ at $\sim$06:27 UT, but the lateral velocity only decreased to $\sim$600 km s$^{-1}$ in the next nine minutes. Moreover, the wave traversed the nearby AR and continued to propagate at the velocity of $\sim$600 km s$^{-1}$ (squares in Figure \[f4\](a)) [see also, @li11]. The apparent lateral propagation of the CME bubble and the wave at the heliocentric height of 1.05 $R_\odot$ is similar to that at 1.15 $R_\odot$. At $\sim$06:26 UT, both of them obtained the peak lateral velocity of $\sim$880 km s$^{-1}$. However, after $\sim$06:26 UT, the lateral velocity of the bubble front rapidly decreased, while the wave front only decreased to $\sim$600 km s$^{-1}$. Similarly, the wave front at the heliocentric height of 0.95 $R_\odot$ accelerated from 150 km s$^{-1}$ at $\sim$06:18 to 830 km s$^{-1}$ $\sim$06:25, and afterwards propagated freely with a velocity larger than $\sim$600 km s$^{-1}$.
Apparently, in the early evolution stage immediately following the eruption onset, the wave front can not be discerned from the CME bubble front, indicating that the wave front is still undergoing compression from the expanding bubble. The standoff distance between the two fronts is almost zero. Both of them obtain the maximum lateral velocity at the same time. When the CME bubble starts to decrease the velocity, the wave front starts to separate and propagate away from its driver. From Figure \[f4\], one can notice that the wave front has different peak lateral velocities at different heliocentric height. The wave’s lateral peak velocities increase with the heliocentric heights; the corresponding peak times of the velocities also delay with respect to the increasing height (Table \[tb\]). Such increase of the peak lateral velocity and its time delay are most likely from the combination effect of the intrinsic expansion motion and the fast rising motion of the CME bubble.
Summary and Discussion
======================
In this Letter, we focus on studying the separation process of two distinct fronts associated with a solar eruption that occurred on 2011 June 7. Following the eruption onset, the magnetic field of the source region quickly expands and forms a circular bubble. In the early stage of the eruption, the bubble expands strongly with an accelerating velocity. Afterwards, the apparent expansion velocity of the bubble close to the solar surface quickly decelerates to almost zero. In the meantime, a diffuse wave front starts to separate from the sharp bubble front. We conclude that the wave originates from the compression of the surrounding plasma by the impulsively expanding CME bubble. Due to a small standoff distance between the compression front and the driver front, the two fronts can not be distinguished during the early stage of the evolution when the driver is still undergoing acceleration. Through examining the radio data from CALLISTO radio spectrometer, we find that a type II radio burst started at $\sim$06:26 UT, almost at the same time (slightly earlier) as the wave started to separate from the driver. The radio observation suggests that a shock was generated at that time, indicating that the compression wave had just turned itself into a shock wave. Recent studies by @pat09a and @pat10 also noticed two fronts: the wave front and the bubble front. However, due to the relative low cadence of the *STEREO*-EUVI observations (5 minutes), the two distinct fronts were seen in only a few frames. Here, the AIA observations in every 12 seconds not only reveal the existence of two fronts but also the detailed separation process of the diffuse front from the sharp bubble front after the expansion of the bubble slows down. These observations clearly demonstrate that the separated diffuse front is a true MHD wave, driven by the early accelerating expansion of the CME bubble.
In conclusion, our observations help understand the physical nature of usually termed EUV waves. The EUV wave is actually a composite phenomenon, consisting of two distinct fronts, a wave front and a CME (compressed plasma) front. We further find that its evolution can be divided into two stages. The first stage takes place in the accelerating expansion phase of the CME bubble, which acts as the piston-driver of the MHD wave. During this stage, the wave front is coupled together with the compression front of the CME bubble. In the second stage, when the expanding velocity of the CME bubble slows down, the MHD wave front decouples from the compression bubble front, forms a distinct front, and propagates across the solar disk. The observational result for the coexistence of both wave and non-wave fronts are in general consistent with the models and numerical simulations for EUV waves by @chen02 [@chen05]. We believe that the previous dispute about the nature of EUV waves resides in, at least partly, different parts of the composite phenomenon that the authors may have observed. In fact, the duration of the wave and CME compression front coupling is different for each event, depending on the dynamics of the CME and the surrounding environment.
We thank P. F. Chen for many valuable comments that helped to improve the manuscript significantly. SDO is a mission of NASA’s Living With a Star Program. X.C., and M.D.D. are supported by NSFC under grants 10673004, 10828306, and 10933003 and NKBRSF under grant 2011CB811402. X.C. is also supported by the scholarship granted by the China Scholarship Council (CSC) under file No. 2010619071. J.Z. is supported by NSF grant ATM-0748003 and NASA grant NNG05GG19G. A.V. is supported by NASA contract S-136361-Y.
natexlab\#1[\#1]{}
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(An animation of this figure is available in the online journal)
(An animation of this figure is available in the online journal)
[ccc]{}\
Height & Peak velocity & Time$^{a}$\
$(R_\odot)$ & (km s$^{-1}$) & (UT)\
1.15 &960$\pm$48 &06:27:03\
1.05 &880$\pm$27 &06:26:24\
0.95 &830$\pm$43 &06:25:03\
\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Liouville equation differs from the von Neumann equation ‘only’ by a characteristic superoperator. We demonstrate this for Hamiltonian dynamics, in general, and for the Jaynes-Cummings model, in particular. – Employing superspace (instead of Hilbert space), we describe time evolution of density matrices in terms of path integrals which are formally identical for quantum and classical mechanics. They only differ by the interaction contributing to the action. This allows to import tools developed for Feynman path integrals, in order to deal with superoperators instead of quantum mechanical commutators in real time evolution. Perturbation theory is derived. Besides applications in classical statistical physics, the “classical path integral” and the parallel study of classical and quantum evolution indicate new aspects of (dynamically assisted) entanglement (generation). Our findings suggest to distinguish [*intra-*]{} from [*inter-space entanglement*]{}.'
address: |
Dipartimento di Fisica “Enrico Fermi”, Università di Pisa,\
Largo Pontecorvo 3, I-56127 Pisa, Italia\
[email protected]
author:
- 'HANS-THOMAS ELZE, GIOVANNI GAMBAROTTA AND FABIO VALLONE'
title: |
A PATH INTEGRAL FOR CLASSICAL DYNAMICS,\
ENTANGLEMENT, AND JAYNES-CUMMINGS MODEL\
AT THE QUANTUM-CLASSICAL DIVIDE
---
\#1[[\#1:]{}]{}
Introduction
============
The quantum-classical divide has been intensely studied in recent years with profound impact on various areas of research [@KieferEtAl; @Zurek; @Schlosshauer]. In particular, this concerns the foundations of quantum mechanics, new quantum technologies (quantum information processing, precision measurements, designer materials, etc.), recent observations of quantum coherent processes in biology, and, last not least, unresolved issues surrounding “quantum gravity” [^1].
Not surprisingly, these modern topics, which touch the foundations of quantum mechanics in one way or another, increase the impetus to try to reconstruct and to better understand the emergence of quantum mechanics from simpler dynamical structures beneath or more profound theoretical principles.
Indeed, there is a growing number of deterministic models of quantum mechanical objects which are based on conjectured fundamental information loss, coarse graining, or dissipation mechanisms [@tHooft06; @Elze05; @Blasone05; @Adler; @Smolin; @Vitiello01; @Isidro08; @Wetterich08]; see Refs.[@Elze09; @tHooft09] for most recent arguments. – We recall that ’t Hooft’s existence theorem [@tHooft07] shows that the evolution of all quantum mechanical objects that are characterized by a finite dimensional Hilbert space can be captured by a dissipative process. This holds also for objects that are described by a set of mutually commuting Hermitean operators [@Elze08].
However, a theory is lacking that would generally explain the emergence of quantum features of common objects, at the scales where they are observed.
In order to make progress in these matters, it may be useful to further examine the quantum-classical divide. Presently, we look more carefully into the common as well as the distinctive features of classical and quantum dynamics, as described by the Liouville and the von Neumann equations, respectively.
We will derive a new path integral representation of the propagator for density matrices in the classical theory. It is identical with the usual one at the kinematic level, employing the Feynman propagator of quantum mechanics [@Schulman]; this allows ‘external sources’ in the relevant action that are coupled to terms linear or quadratic in the generalized coordinates. Yet the interaction part differs in a characteristic way [^2]. The new formalism based on superoperators will be presented and illustrated here by perturbation theory applied to an anharmonic oscillator.
Similarly, as a case study, we will re-derive the Jaynes-Cummings model [@JaynesCummings] – the well-known benchmark model of quantum optics and cavity QED – based on classical dynamics described by a Liouville equation. Thus, when applied to the two-level dynamics of Rydberg atoms coupled to one mode of the photon field in a suitably tuned cavity, we find surprisingly that it is “almost classical”, with quantum and classical dynamics differing by a characteristic superoperator.
We conclude by pointing out some interesting problems, concerning the preparation of entangled states, in particular. Here, the quantum-classical divide shows new aspects, which may help to further unravel the hidden dynamics that must be involved when it is crossed – be it in the “classical limit” or following axiomatic “quantization rules”.
Hamiltonian dynamics and the Liouville superoperator
====================================================
To begin with, we will consider an object with a single continuous degree of freedom. We will treat an atom interacting with the electromagnetic field in a later chapter, while a relativistic field theory has been studied elsewhere [@Elze07].
Let us assume that there are only conservative forces and that Hamilton’s equations are determined by the generic Hamiltonian function: $$\label{HamiltonianF}
H(x,p):=\frac{1}{2}p^2+V(x)
\;\;,$$ defined in terms of generalized coordinate $x$ and momentum $p$ (a mass parameter will be inserted in due time, but is omitted here for simplicity), and where $V(x)$ denotes the potential. – An ensemble of such objects, for example, following trajectories with different initial conditions, is described by a distribution function $\rho$ in phase space, i.e., by the probability $\rho(x,p;t)\mbox{d}x\mbox{d}p$ to find a member of the ensemble in an infinitesimal volume at point $(x,p)$. This distribution evolves according to the [*Liouville equation*]{}: $$\label{LiouvilleEq}
-\partial_t\rho =\frac{\partial H}{\partial p}\cdot\frac{\partial \rho}{\partial_x}
-\frac{\partial H}{\partial x}\cdot\frac{\partial \rho}{\partial_p}
=\big\{p\partial_x-V'(x)\partial_p\big\}\rho
\;\;,$$ with $V'(x):=\mbox{d}V(x)/\mbox{d}x$. – We recall that the relative minus sign in the Poisson bracket, or between terms here, reflects the symplectic phase space symmetry. This will translate into the familiar commutator structure in Eq.(\[Schroed\]).
A Fourier transformation, $\rho (x,p;t)=\int\mbox{d}y\;\mbox{e}^{-ipy}\rho (x,y;t)$, replaces the Liouville equation by: $$\label{LFourier}
i\partial_t\rho =\big\{-\partial_y\partial_x+yV'(x)\big\}\rho
\;\;,$$ without changing the symbol for the distribution function, whenever changing variables. Thus, momentum is eliminated in favour of [*doubling*]{} the number of coordinates. Finally, with the transformation: $$\label{coordtrans}
Q:=x+y/2\;\;,\;\;\;q:=x-y/2
\;\;,$$ we obtain the Liouville equation in the form: $$\begin{aligned}
\label{Schroed}
i\partial_t\rho &=&\big\{ \hat H_Q-\hat H_q+{\cal E}(Q,q)\big\}\rho
\;\;, \\ [1ex] \label{HX}
\hat H_\chi &:=&-\frac{1}{2}\partial_\chi ^{\;2}+V(\chi )\;\;,
\;\;\;\mbox{for}\;\;\chi =Q,q
\;\;, \\ [1ex] \label{I}
{\cal E}(Q,q)&:=&(Q-q)V'(\frac{Q+q}{2})
-V(Q)+V(q)\;=\;-{\cal E}(q,Q)
\;\;. \end{aligned}$$ We remark that the presented reformulation of classical dynamics is rather independent of the number of degrees of freedom. It applies to matrix valued as well as to Grassmann valued variables, representing the “pseudoclassical” fermion fields introduced by Casalbuoni and by Berezin and Marinov. Field theories require a classical functional formalism [@Elze05; @Elze07].
Furthermore, the Eq.(\[Schroed\]) appears as the [*vonNeumann equation*]{} for a density operator $\hat \rho (t)$, considering $\rho (Q,q;t)$ as its matrix elements. We automatically recover the Hamiltonian operator $\hat H$ related to the Hamiltonian function, Eq.(\[HamiltonianF\]), as in quantum theory. However, an essential difference consists in the interaction ${\cal E}$ between [*bra-*]{} and [*ket-*]{} states. The Hilbert space and its dual here are coupled by a [*superoperator*]{} [^3].
Since the interaction ${\cal E}$ is antisymmetric under $Q\leftrightarrow q$, the complete (Liouville) operator on the right-hand side of Eq.(\[Schroed\]) has a symmetric spectrum with respect to zero and, in general, will not be bounded below. Therefore, with this coupling of the Hilbert space and its dual by the superoperator, corresponding to the absence of a stable ground state, our reformulation of Hamiltonian dynamics does not qualify as a quantum theory. Related observations were discussed, for example, in Refs. [@tHooft06; @Elze05; @Blasone05; @Vitiello01].
However, the following fact has been discussed in Refs.[@Elze09]: $$\label{Ezero}
{\cal E}\equiv 0\;\;\Longleftrightarrow\;\;
\mbox{potential}\;V(x)\;\mbox{is constant, linear, or harmonic}
\;,$$ with an eye on the possibility of having quantum phenomena emerge due to discrete spacetime structure. Analogously, the vanishing of ${\cal E}$ in a field theory is equivalent with having massive or massless free fields, with or without external sources, and with or without bilinear couplings. Generally, in these cases, anharmonic forces or interactions are absent.
In the following main parts of this work, we will study in more detail the classical Hamiltonian dynamics described by Eq.(\[Schroed\]), or by its appropriate generalizations, and pay particular attention to the presence of the superoperator ${\cal E}$, when comparing with the vonNeumann equation.
Concluding this introductory section, we recall relevant aspects of the interpretation of the density operator $\hat\rho (t)$, which we invoked here.
Expectations, operators and the Born rule
-----------------------------------------
We begin with the normalization of the classical probability distribution: $$\label{clnorm}
1\stackrel{!}{=}\int\frac{\mbox{d}x\mbox{d}p}{2\pi}\;\rho (x,p;t)=\int\mbox{d}Q\mbox{d}q\;
\delta (Q-q)\rho (Q,q;t)=:\mbox{Tr}\;\hat\rho (t)
\;\;,$$ incorporating the transformations of Section2. Consider a complete set of orthonormal eigenfunctions of the operator $\hat H_\chi $ of Eq.(\[HX\]), defined by $g_j(\chi ;t):=\mbox{exp}(-iE_jt)g_j(\chi )$ and $\hat H_\chi g_j(\chi )=E_jg_j(\chi )$, respectively, with a discrete spectrum, for simplicity. Then, we may expand $\rho$: $$\label{fexpans}
\rho (Q,q;t)=\sum_{j,k}\rho_{jk}(t)g_j(Q;t)g_k^*(q;t)
\;\;.$$ Employing this, the normalization condition (\[clnorm\]) can be stated as: $$\label{clnorm1}
1\stackrel{!}{=}\sum_{j,k}\rho_{jk}(t)\mbox{e}^{-i(E_j-E_k)t}\int\mbox{d}Q\;g_j(Q)g_k(Q)
=\sum_{j}\rho_{jj}(t)
\;\;.$$ Since the classical phase space distribution is real, the expansion coefficients form a Hermitean matrix, $\rho_{ij}=\rho^\ast_{ji}$, which we also denote by $\hat\rho$. The [*classical*]{} expectation values are calculated as follows: $$\begin{aligned}
\label{xexpect}
\langle x\rangle :=\int\frac{\mbox{d}x\mbox{d}p}{2\pi}\;x\rho (x,p;t)
&=&\int\mbox{d}Q\mbox{d}q\;\delta (Q-q)\frac{Q+q}{2}\rho (Q,q;t)
\;\;, \\ [1ex] \label{xexpect1}
&=:&\mbox{Tr}\;\big (\hat X\hat\rho (t)\big )
\;\;, \\ [1ex] \label{pexpect}
\langle p\rangle :=\int\frac{\mbox{d}x\mbox{d}p}{2\pi}\;p\rho (x,p;t)
&=&\int\mbox{d}Q\mbox{d}q\;\delta (Q-q)(-i)\frac{\partial_Q-\partial_q}{2}\rho (Q,q;t)
\;\;, \\ [1ex] \label{pexpect1}
&=:&\mbox{Tr}\;\big (\hat P\hat\rho (t)\big )
\;\;, \end{aligned}$$ introducing the operators $\hat X$ and $\hat P$, with matrix elements $X(q,Q)=\delta (Q-q)(Q+q)/2$ and $P(q,Q)=-i\big (\delta (Q-q)\stackrel{\rightharpoondown}{\partial}_Q-
\stackrel{\leftharpoondown}{\partial}_q\delta (Q-q)\big )$ (derivatives act left or right, as indicated). Eliminating one of the two integrations in the above equations with the help of the $\delta$-functions and suitable partial integrations, these operators are recognized as the coordinate and momentum operators of quantum theory.
Similarly, we find: $$\label{noncommops}
\int\frac{\mbox{d}x\mbox{d}p}{2\pi}\;xp\rho (x,p;t) =
\frac{1}{2}\mbox{Tr}\;\big ((\hat X\hat P+\hat P\hat X)\hat\rho (t)\big )
\;\;,$$ which constitutes an example of the symmetric Weyl ordering, when replacing classical phase space quantities by quantum operators. – However, we remark that Hilbert space operators appear here by rewriting classical statistical formulae and [*not*]{} by following a quantization rule.
The Eqs.(\[clnorm\]), (\[xexpect\])–(\[noncommops\]) are in accordance with the interpretation of $\rho (Q,q;t)$ as matrix elements of a density operator $\hat\rho (t)$. – However, there is an important [*caveat*]{}: The eigenvalues of normalized quantum mechanical density operators are usually constrained to lie between zero and one, corresponding to the interpretation as standard probabilities. This is not necessarily the case with the operator $\hat\rho $ obtained from a classical probability distribution. Similarly, the Wigner distribution – obtained from the matrix elements of a quantum mechanical density operator by applying the transformations leading from $\rho (x,p)$ to $\rho(Q,q)$ in reverse – generally, is not positive semi-definite on phase space, even though its marginal distributions are. Therefore, it does not qualify as a classical probability density.
As we have indicated before, there is clearly a dynamical feature missing, which governs the crossing of the quantum-classical divide, if not done ‘by hand’, as in any of the usual “quantization prescriptions”. Last not least, this must establish the [*Born rule*]{} by eliminating negative probabilities or by leading to their satisfactory interpretation.
From Hilbert space to superspace
================================
In this section, we reformulate the notions relevant for the dynamics of density operators, at which we arrived in the previous section, in a more convenient way, introducing the concept of [*superspace*]{} [^4], also called [*Liouville space*]{} – see Ref.[@superspace] for a concise presentation and numerous applications.
Considering a physical object characterized by the Hamiltonian $\hat{H}$, as in quantum theory, we introduce a complete set of basis states, $\{ |j\rangle\}$ ($j=1,\dots ,N$), assuming that the relevant Hilbert space is $N$-dimensional. Then, taking matrix elements of the vonNeumann equation, for example, we have: $$\label{vN}
i \partial_t \rho_{jk} = [(\hat{H}\hat{\rho})_{jk}- (\hat{\rho}\hat{H})_{jk}]\;\;, \;\;\;
j,k = 1,2,\dots, N
\;\;,$$ with a density matrix $\rho$ of $N^2$ elements. Which may be written as: $$i \partial_t \rho_{jk} = \sum_{l,m} {\cal L}_{jk,lm} \rho_{lm}
\;\;,$$ in terms of the [*Liouville superoperator*]{} $\hat {\cal L}$ defined by: $$\label{LvN}
\hat {\cal L}_{jk,lm}:= H_{jl} \delta_{km} - H_{km}^\ast \delta_{jl}
\;\;.$$ This definition suggests to introduce a space where the density operator is a vector, which is the Liouville space (or superspace). The dynamics of the density operator can then be more conveniently described, completely in parallel for classical and quantum mechanics, as we shall see.
Given the Hilbert space, as above, the density operator can be expanded as: $$\hat{\rho}= \sum_{j,k} \rho_{jk} |j \rangle \langle k|
\;\;.$$ We may think of the family of $N^2$ operators $|j\rangle \langle k|$, with $j,k=1,\cdots,N$, as a complete set of matrices, or vectors, and express the density operator as: $$|\rho \rangle= \sum_{j,k} \rho_{jk} |jk \rangle \rangle
\;\;,$$ where the “ket” $|jk \rangle \rangle$ denotes the Liouville space [*vector*]{} representing the Hilbert space [*operator*]{} $|j\rangle \langle k|$. Similarly, we introduce a “bra” vector $ \langle \langle jk|$ as the Hermitean conjugate of $|jk \rangle \rangle$.
In Liouville space, any operator $\hat{A}$ is represented by a vector and denoted by $|A \rangle \rangle $. It can be expanded as: $$\label{opex}
| A \rangle \rangle= \sum_{j,k} A_{jk} |jk \rangle \rangle
\;\;,$$ where $A_{jk}$ are the usual matrix elements $\langle j| \hat{A}| k \rangle$. – Furthermore, we can define a “bra” vector $ \langle \langle B |$, representing $\hat{B}^{\dag}$, and the scalar product of two operators: $$\langle \langle B| A \rangle \rangle :=\mbox{Tr} (\hat{B}^{\dag}\hat{A})
\;\;.$$ Then, the following orthonormality condition holds: $$\label{orthogonality}
\langle \langle jk|mn \rangle \rangle =\mbox{Tr}(|k\rangle \langle j| m \rangle \langle n|)=
\delta_{kn} \delta_{jm}
\;\;,$$ which is analogous to $\langle j|k\rangle = \delta_{jk}$. – Finally, consider the scalar product: $$\langle \langle jk| A \rangle \rangle =\mbox{Tr}(|k \rangle \langle j| \hat{A})=
\sum_{l} \langle l|k \rangle \langle j |\hat{A}| l \rangle= \langle j| \hat{A}|k \rangle \equiv A_{jk}
\;\;.$$ Upon substitution in Eq.(\[opex\]), this yields: $$|A \rangle \rangle = \sum_{j,k} |jk \rangle \rangle \langle \langle jk | A \rangle \rangle
\;\;.$$ This is consistent with the following completeness relation in Liouville space: $$\label{completeness}
\sum_{j,k} |jk \rangle \rangle \langle \langle jk | = {\mathbf 1}
\;\;.$$ Following these considerations, it can be verified that Liouville space is a linear space, in which the density operator $\hat{\rho}$ is a vector. In this space, a linear operator can be defined by: $$\hat{\cal F}:= \sum_{j,k,m,n} |jk \rangle \rangle \langle \langle jk |
\hat{\cal F}|mn \rangle \rangle \langle \langle mn| \equiv
\sum_{j,k,m,n}{\cal F}_{jk,mn}|jk \rangle \rangle \langle \langle mn|
\;\;,$$ i.e., in terms of appropriate matrix elements.
The importance of Liouville space for classical [*and*]{} quantum dynamics is that the Liouville and vonNeumann equations, both, can be written in the form: $$i \partial_t \hat{\rho} = \hat {\cal L} \hat{\rho}
\;\;,$$ with an appropriate superoperator $\hat {\cal L}$, cf. Eqs.(\[Schroed\])–(\[I\]) and (\[vN\])–(
\[LvN\]), respectively. Thus, there is a formal analogy (even isomorphism) between the structure of these equations and the Schrödinger equation, $i\partial_t \Psi = \hat{H} \Psi \;$.
These observations suggest that techniques or formal results concerning the solution of the Schrödinger equation can be transferred to the case of the Liouville or vonNeumann equations with the help of Liouville space notions. This concerns perturbation theory (and nonperturbative methods) as much as a path integral formulation, which we shall discuss in turn.
First of all, we introduce the Liouville space evolution operator $\hat{\cal U}$ satisfying: $$i \partial_t \hat{\cal U}(t,t_0) = \hat{\cal L}(t)\hat{\cal U}(t,t_0)
\;\;,$$ with the initial condition $\hat{\cal U}(t,t_0)=\mathbf{1}$. This implies: $$\label{Levol}
\hat{\rho}(t) = \hat{\cal U}(t,t_0) \hat{\rho}(t_0)
\;\;,$$ as the solution of the density operator equation of motion. For a time independent Liouville superoperator this yields:
$${\hat{\cal U}}(t,t_0)= \exp \big (-i {\hat{\cal L}}(t-t_0)\big )
\;\;.$$
Thus, time evolution of the density matrix is implemented by a superoperator in Liouville space, while in Hilbert space the evolution is described by:
$$\label{hevol}
{\hat{\rho}}(t)= \hat{U}(t,t_0) {\hat{\rho}}(t_0) \hat{U}^{\dag}(t,t_0)
\;\;,$$
with $\hat{U}(t,t_0):=\exp (-i\hat Ht)$. The Eqs.(\[Levol\]) and (\[hevol\]) represent the evolution of the same object, although in different spaces. – For a time dependent Hamiltonian, we have instead: $$\begin{aligned}
&\;&{\hat{\cal U}}(t,t_0)= \mbox{T}\exp\big (-i \int_{t_0}^t\mbox{d}\tau\; {\hat{\cal L}}(\tau) \big )
\\ [1ex]
&\;&\;\;\; :=
1 + \sum_{n=1}^{\infty} (-i)^n \int_{t_0}^t\mbox{d}\tau_n \int_{t_0}^{\tau_n}\mbox{d}\tau_{n-1}
\;\dots \int_{t_0}^{\tau_2}\mbox{d}\tau_1\;{\hat{\cal L}}(\tau_n){\hat{\cal L}}(\tau_{n-1})\dots {\hat{\cal L}}(\tau_1)
\;, \end{aligned}$$ in terms of the time-ordered exponential.
For later purposes, we also introduce the “interaction picture” in Liouville space. Considering a Liouville operator which consists of two parts: $${\hat{\cal L}}\equiv {\hat{\cal L}}_0(t)+ {\hat{\cal L}}'(t)
\;\;,$$ we obtain the evolution operator in the following form: $${\hat{\cal U}}(t,t_0) = {\hat{\cal U}}_0(t,t_0) {\hat{\cal U}}_I(t,t_0)
\;\;,$$ with: $${\hat{\cal U}}_0(t,t_0)=\mbox{T} \exp \big (-i \int_{t_0}^t\mbox{d}\tau\;{\hat{\cal L}}_0 (\tau) \big )
\;\;,$$ and: $$\label{UI}
{\hat{\cal U}}_I(t,t_0)=\mbox{T} \exp \big (-i \int_{t_0}^t\mbox{d}\tau\;{\hat{\cal L}}'_I(\tau) \big )
\;\;,$$ with $ {\hat{\cal L}}'_I (\tau) := {\hat{\cal U}}^{\dag}_0(\tau,t_0) {\hat{\cal L}}'(\tau) {\hat{\cal U}}_0 (\tau,t_0)$. For an operator ${\hat{\cal U}}_0$ that can be treated exactly, study of time evolution essentially concerns the operator ${\hat{\cal U}}_I$ – for this, Eq.(\[UI\]) presents the starting point of perturbation theory (expanding the exponential).
The quantum/classical path integral for the propagator of density matrices
==========================================================================
The technical ingredients needed for the Feynman path integral approach, for the derivation of quantum mechanical propagators in particular, are very well known. We import these here, especially from Ref.[@Schulman], in order to derive a path integral for the propagator of density matrices based on the Liouville space formulation of the preceding Section3.
Our derivation relies on the close formal similarity between the classical Liouville equation and the vonNeumann equation on one hand side and the Schrödinger equation on the other, in an appropriate representation, as we have discussed [^5].
Essentials of the Feynman path integral
---------------------------------------
We recall that the (forward propagating) operator Green’s function $\hat G$, $$\hat G(t,t_0)\equiv \theta (t-t_0)\exp \big (-i\hat H(t-t_0)/\hbar\big )
\;\;,$$ allows one to write the solution of the time dependent Schrödinger equation as: $|\psi (t)\rangle =\hat G(t,t_0)|\psi (t_0)\rangle$. – Correspondingly, for $t>t_0$, we have the coordinate space matrix elements: $$G(x,t;y,t_{0})=\langle x|\mbox{e}^{-i\hat H(t-t_0)/\hbar}|y\rangle
\;\;,$$ from which one derives the path integral representation of these amplitudes, for a generic [*Hamiltonian*]{} $\hat H=\hat p^2/2m+V(\hat x)$, describing a particle of mass $m$ in an external potential $V$, cf. Eq.(\[HamiltonianF\]), through the following steps [@Schulman]:
- Cut the time interval from $t_0$ to $t$ into a large number $N$ of equal pieces.
- Write the exponential of the Hamiltonian operator$\;\times\;$time as a product of identical factors, each factor representing the propagator for a small time interval $\propto 1/N$.
- Separate the kinetic and potential terms contributing to $\hat H$ in each one of these factors with the crucial help of the [*Trotter product formula*]{}.
- Alternatingly, insert complete sets of momentum and coordinate eigenstates, such as $\int\mbox{d}x\;|x\rangle\langle x |={\mathbf 1}$ (and correspondingly for momentum eigenstates) between the factors of exponentials involving either momentum or coordinate operators and evaluate the resulting Gaussian integrals over momentum variables.
- Realize that the obtained phases in the product of exponentials can be summed up to represent a discretized version of the [*classical action*]{} pertaining to the Hamiltonian function (corresponding to $\hat H$).
Taking the limit $N\rightarrow\infty$ in the end, one obtains the following Feynman path integral representation of the amplitudes in question: $$\begin{aligned}
\label{piprop}
&\;&G(x,t;y,t_{0})\;=
\\ [1ex]
&\;&\lim_{N\rightarrow\infty}(\frac{m}{2\pi i\hbar\epsilon})^{N/2}
\int\mbox{d}x_1\dots\mbox{d}x_{N-1}\;
\exp \Big (\frac{i\epsilon}{\hbar}
\sum_{j=0}^{N-1}\big (\frac{m}{2}(\frac{x_{j+1} -x_{j}}{\epsilon})^{2} -V(x_{j})
\big )\Big )
\nonumber \\ [1ex] \label{pathintegral1}
&=:&\int{\cal D}x\;\exp\Big (\frac{i}{\hbar}\int_{t_0}^t\mbox{d}\tau\;
\big (\frac{m}{2}\dot x^2 -V(x)\big )\Big)
\\ [1ex] \label{pathintegral2}
&\equiv&\int{\cal D}x\;\exp\big (\frac{i}{\hbar}S[\dot x,x]\big )
\;\;, \end{aligned}$$ where $\epsilon :=(t-t_0)/N$, $\dot x:=\mbox{d}x/\mbox{d}\tau$ and where $S$ denotes the relevant classical action, which is to be evaluated for each one of the paths contributing to the integral, with the boundary conditions $x(t)=x$ and $x(t_0)=y$.
The Liouville space propagator as a path integral
-------------------------------------------------
We are now ready to appreciate the economy of the Liouville space representation introduced in Section3. In particular, the formal solution of the classical Liouville equation as well as of the quantum mechanical vonNeumann equation, both, can be written in the form: $$|\rho(t)\rangle\rangle =\mbox{e}^{-i{\cal \hat H}(t-t_0)/\hbar}|\rho(t_0)\rangle\rangle
\;\;,$$ where ${\cal \hat H}$ is the appropriate super-Hamiltonian. Generally, we have: $$\langle\langle Q,q|{\cal \hat H}|Q',q'\rangle\rangle =
\delta (Q-Q')\delta (q-q')\big (\hat H(Q)-\hat H(q)+{\cal E}(Q,q))
\;\;,$$ where $\hat H$ denotes the appropriate Hamilton operator in coordinate representation, as indicated, which alone is relevant for the vonNeumann equation, while ${\cal E}$ represents the additional superoperator for classical dynamics, cf. Section2.
In order to solve the problem of time evolution in the present case, we need to know the (super)matrix elements entering the propagation equation: $$\label{rhoprop}
\langle\langle Q,q|\rho(t)\rangle\rangle =\int\mbox{d}Q'\mbox{d}q'\;
\langle\langle Q,q|\mbox{e}^{-i{\cal \hat H}(t-t_0)/\hbar}|Q',q'\rangle\rangle\langle\langle Q',q'|\rho(t_0)\rangle\rangle
\;\;,$$ which appears formally analogous to evolution of a state vector according the Schrödinger equation. Thus, not surprisingly, we go through the steps indicated in the preceding Section4.1, in order to construct the path integral representation of the propagator here.
However, in this derivation, we have to pay attention to the crucial role of the Trotter product formula. It turns out that it can be generalized for our purposes, where superoperators present the new feature, in a straightforward way; the relevant definitions and details of the proof will be given elsewhere [@FabThesis].
Rewriting the Eq.(\[rhoprop\]) as: $$\label{rhoprop1}
\rho (Q,q;t) = \int\mbox{d}Q'\mbox{d}q'\;{\cal G}(Q,q;t|Q',q';t_0)\rho (Q',q';t_0)
\;\;,$$ our interest is to know the superpropagator ${\cal G}$. Next, we will follow the recipe to arrive at a path integral representation, as summarized above, in Section4.1.
In particular, here we make use of suitably inserted complete sets of superspace vectors, such as: $$\label{supercomplete}
\int\mbox{d}Q\mbox{d}q\; |Q,q\rangle\rangle\langle\langle Q,q|={\mathbf 1}
\;\;,$$ and, correspondingly, for momentum space, cf. Eq.(\[completeness\]). Using the plane wave relation between coordinate and momentum eigenfunctions, we also employ: $$\label{planewaves}
\langle\langle P,p|=\frac{1}{2\pi\hbar }\int\mbox{d}Q\mbox{d}q\;
\exp\big (-\frac{i}{\hbar}(PQ-pq)\big )\langle\langle Q,q|
\;\;.$$ Furthermore, the orthogonality relation $\langle\langle Q,q|Q',q'\rangle\rangle =\delta (Q-Q')\delta(q-q')$, cf. Eq.(\[orthogonality\]), implies: $$\label{transf}
\langle\langle P,p|Q,q\rangle\rangle =\frac{1}{2\pi\hbar }
\exp\big (-\frac{i}{\hbar}(PQ-pq)\big )
\;\;.$$ Then, with all following steps of the derivation in parallel with the usual ones in quantum mechanics, it is straightforward to obtain in the present case [@FabThesis]: $$\label{superpropagator}
{\cal G}(Q_f,q_f;t|Q_i,q_i;t_0)=\int {\cal D}Q{\cal D}q
\;\exp\big (\frac{i}{\hbar}{\cal S}[\dot Q,Q;\dot q,q]\big )
\;\;,$$ with the boundary conditions $Q(t_i)=Q_i$, $q(t_i)=q_i$, $Q(t_f)=Q_f$, and $q(t_f)=q_f$, and where the superaction ${\cal S}$, corresponding to the super-Hamiltonian ${\cal H}'$ above, is defined as follows: $$\begin{aligned}
\label{superaction}
{\cal S}&\equiv&\int_{t_0}^t\mbox{d}\tau\;\big ({\cal T}(\dot Q,\dot q)-{\cal V}(Q,q)\big )
\\ [1ex] \label{superaction1}
&:=&\int_{t_0}^t\mbox{d}\tau\;\Big ({\textstyle \frac{m}{2}}\dot Q^2-V(Q)
-\big ({\textstyle \frac{m}{2}}\dot q^2-V(q)\big )
-{\cal E}(Q,q)\Big )
\;\;. \end{aligned}$$ We recall that ${\cal E}\equiv 0$ corresponds to evolution according to the vonNeumann equation, whereas ${\cal E}\neq 0$ represents classical dynamics in accordance with the Liouville equation, cf. Eqs.(\[Schroed\])–(\[I\]).
However simple this result may seem, the Eqs.(\[superpropagator\])–(\[superaction1\]) present a new approach to describe time evolution of the full density matrix, with the particular feature that classical and quantum mechanical motion are formally treated in parallel, differing only in the action entering the phase in the path integral [^6].
We emphasize that our derivations are not confined to one-dimensional or single-particle physics, but can be extended as well all the way to relativistic field theories. Various applications come to mind here, some of which will be discussed in the following and in the concluding section.
Perturbation theory and superpropagator Dyson equation
------------------------------------------------------
Considering the splitting of the superaction as in Eq.(\[superaction\]), the perturbation theory naturally departs from organizing contributions to the full superpropagator, Eq.(\[superpropagator\]), according to powers of the “perturbation” ${\cal V}$. Sometimes it may be advantageous to include parts of the perturbation into the “free” part ${\cal T}$. This must be familiar from quantum mechanics, which presents a special case of our general considerations here.
To begin with, if ${\cal V}(Q,q)\equiv V(Q)-V(q)$, corresponding to the superoperator related to the vonNeumann equation, then the path integrals in Eq.(\[superpropagator\]) factorize and we recover quantum mechanics.
In the absence of an external potential or other interactions (${\cal V}\equiv 0$), the zeroth order or [*free superpropagator*]{} ${\cal G}_0$ is obtained as: $$\begin{aligned}
\label{G0}
{\cal G}_0(Q_f,q_f;t|Q_i,q_i;t_0)&=&\int {\cal D}Q{\cal D}q
\;\exp\big (\frac{i}{\hbar}\int_{t_0}^t\mbox{d}\tau\;{\cal T}(\dot Q,\dot q)\big )
\\ [1ex] \label{G01}
&=&G_0(Q_f,t;Q_i,t_0)G_0^\ast (q_f,t;q_i,t_0)
\;\;, \end{aligned}$$ in terms of the well known free quantum mechanical propagator $G_0$, cf. Eqs.(\[piprop\]
)–(
\[pathintegral2\]), which is explicitly given by [@Schulman]: $$\label{G0QM}
G_0(x,t;y,t_0)\equiv G_0(x,y;T:=t-t_0)
=\big (\frac{m}{2\pi i\hbar T}\big )^{1/2}\exp\Big (\frac{im}{2\hbar T}(x-y)^2\Big )
\;\;,$$ for a free nonrelativistic particle of mass $m$. Remarkably, this zeroth order result is [*identical*]{} for classical and quantum mechanical propagation.
Despite the fact that the free propagator for the Schrödinger equation incorporates such phenomena as the quantum mechanical spreading of a wave packet, we learn here that it also describes the propagation of a [*free classical particle*]{}. It is straightforward to verify – following the transformations between Eqs.(\[HamiltonianF\]) and Eq.(\[I\]) – that a massive particle, initialized as $\rho (x,p,t_0):=2\pi \delta (x-x_0)\delta (p-p_0)$ is propagated to $\rho (x,p,t):=2\pi \delta (x-x_0-Tp/m)\delta (p-p_0)$, as expected.
For the perturbative expansion, we employ the standard formula: $$\begin{aligned}
&\;&\exp\big ( -\frac{i}{\hbar}\int_0^t\mbox{d}\tau\;{\cal V}(Q,q)_\tau \big )=
\sum_{n=0}^\infty\frac{1}{n!}\big (\frac{-i}{\hbar}
\int_0^t\mbox{d}\tau\;{\cal V}(Q,q)_\tau\big )^n
\\ [1ex]
&=&
\sum_{n=0}^\infty\big (\frac{-i}{\hbar}\big )^n
\int_0^t\mbox{d}\tau_1\;{\cal V}(Q,q)_{\tau_1}\;\dots\;\int_0^{\tau_{n-1}}\mbox{d}\tau_n\;
{\cal V}(Q,q)_{\tau_n}
\;\;, \end{aligned}$$ with ${\cal V}(Q,q)_{\tau_k}:= {\cal V}\big (Q(\tau_k),q(\tau_k)\big )$.
In order to analyze such terms at a given order, we make use of the important semigroup property of the (free) propagator and obtain to first order in the perturbation [@FabThesis]: $$\begin{aligned}
&\;&{\cal G}(Q,q;t|Q',q';t_0)={\cal G}_0(Q,q;t|Q',q';t_0)
\nonumber \\ [1ex] \label{firstorder}
&\;&-\frac{i}{\hbar}\int_{t_0}^t\mbox{d}\tau\int\mbox{d}x\mbox{d}y\;{\cal G}_0(Q,q;t|x,y;\tau )
{\cal V}(x,y){\cal G}_0(x,y;\tau |Q',q';t_0)+\mbox{O}({\cal V}^2)
\;, \end{aligned}$$ to be illustrated explicitly by the result for an anharmonic potential shortly.
We remark that on the right-hand side of Eq.(\[firstorder\]) the superpotential is preceded (and followed) by a zeroth order propagator. This observation, which similarly holds at every order of this expansion, leads to a recursion relation of the $k$-th order propagator in terms of the $(k-1)$-th order one. This allows us to resum the perturbation series in the form of a [*Dyson integral equation*]{} for the full superpropagator: $$\begin{aligned}
&\;&{\cal G}(Q,q;t|Q',q';t_0)={\cal G}_0(Q,q;t|Q',q';t_0)
\nonumber \\ [1ex] \label{Dyson}
&\;&-\frac{i}{\hbar}\int_{t_0}^t\mbox{d}\tau\int\mbox{d}x\mbox{d}y\;{\cal G}_0(Q,q;t|x,y;\tau )
{\cal V}(x,y){\cal G}(x,y;\tau |Q',q';t_0)
\;\;. \end{aligned}$$ The whole procedure follows the usual one in quantum mechanics, yet includes the case of classical mechanics, and possibly others, for a suitably chosen superaction.
Illustration: the case of an anharmonic potential
-------------------------------------------------
In order to make our general derivations more concrete and to extract some interesting general aspects, it may be useful to consider the example of a massive particle in an anharmonic potential, $V(x):=\lambda x^4$, where $\lambda$ is the coupling constant. – We recall that for constant, linear, or harmonic coupling terms there is no difference between classical and quantum dynamics, cf. (\[Ezero\]), in the representation that we have developed in this article.
The calculations evaluating the superpropagator to first order, here with: $$\label{anharmonic}
{\cal V}(x,y)\equiv (x-y)V'\big (\frac{x+y}{2}\big )
=\frac{\lambda}{2}\big (x^4-y^4+2(x^3y-xy^3)\big )
\;\;,$$ for classical dynamics (${\cal V}(x,y)\equiv V(x)-V(y)$ for quantum mechanics) consist in straightforward (if tedious) multiple Gaussian integrals, according to Eqs.(\[G0\])–(\[G0QM\]) and Eq.(\[firstorder\]). The final result is: $$\begin{aligned}
&\;&{\cal G}(Q,q;t|Q',q';t_0)\;=\;{\cal G}_0(Q,q;t|Q',q';t_0)
\nonumber \\ [1ex] \label{anharmonicprop1}
&\;&\;\;\cdot\;
\Big (1-\frac{i}{\hbar}\lambda\big [
C_1\Gamma_{\mbox{QM}}(Q,q;Q',q';T)
+C_2\Gamma_{\mbox{CL}}(Q,q;Q',q';T)\big ]\Big )
+\mbox{O}(\lambda^2)
\;, \end{aligned}$$ where $T:=t-t_0$, the coefficients for classical dynamics, $C_1:=1/2,\;C_2:=1/2$ ($C_1:=1,\;C_2:=0$ for quantum mechanics), and with the function:
$$\begin{aligned}
&\;&\Gamma_{\mbox{QM}}(Q,q;Q',q';T):=\frac{T}{5}\Big [
\frac{1}{2}\frac{i\hbar T}{m}(3Q^2+4Q{{Q'}}+3{{Q'}}^2)
\nonumber \\ [1ex] \label{GammaQM}
&\;&\;\;+Q^4+Q^3{{Q'}}+Q^2{{Q'}}^2+Q{{Q'}}^3+{{Q'}}^4
\Big ]
-\frac{T}{5}\Big [(Q,{{Q'}})\longleftrightarrow (q,{{q'}})\Big ]^\ast
\;\;, \end{aligned}$$
where the term is repeated, as indicated, with an exchange of variables and complex conjugation. Similarly: $$\begin{aligned}
&\;&\Gamma_{\mbox{CL}}(Q,q;Q',q';T):=\frac{T}{5}\Big (\;
\frac{i\hbar T}{m}\big (3Qq+2Q{{q'}}+2{{Q'}}q+3{{Q'}}{{q'}}\big )
\nonumber \\ [1ex]
&\;&\;\;+\frac{1}{2}\big [Q^3(4q+{{q'}})+Q^2{{Q'}}(3q+2{{q'}})+Q{{Q'}}^2(2q+3{{q'}})+{{Q'}}^3(q+4{{q'}})\big ]
\nonumber \\ [1ex] \label{GammaCL}
&\;&\;\;-\frac{1}{2}\big [(Q,{{Q'}})\longleftrightarrow (q,{{q'}})\big ]\;\Big )
\;\;. \end{aligned}$$
This result shows several interesting features. First of all, the perturbative expansion turns out to be a short-time expansion, with the overall scale of the first order correction set by $\lambda T$. Furthermore, different contributing terms differ by a scale set by $T/m$, i.e., by $T\;\times$Compton wavelength of the particle. Numerical studies visualizing the outcome here are presently underway [@FabThesis].
However, most interesting seem general similarities and differences between classical (“CL”) and quantum mechanical (“QM”) result in Eq.(\[anharmonicprop1\]). The CL result has the same zeroth order term as QM; at first order, CL has one term in common with QM which, however, is reduced by an overall factor 1/2. This obviously stems from the varied expressions for ${\cal V}$ between CL and QM, cf. Eq.(\[anharmonic\]). For the same reason, CL has additional terms, collected in $\Gamma_{\mbox{CL}}$, Eq.(\[GammaCL\]), which are absent in QM.
Intra- and inter-space entanglement
-----------------------------------
There is a [*qualitative difference between CL and QM*]{}, contained in $\Gamma_{\mbox{CL}}$ and based on the different superoperators that enter the full path integral, Eqs.(\[superpropagator\])–(\[superaction1\]). Equivalently, since the QM evolution is generated by a commutator of the Hamiltonian with the density operator $\hat\rho$, it superposes and, for multi-partite systems, generally, entangles underlying bra- and ket-states separately, $\propto H_{ij}\rho_{jk}-\rho_{ij}H_{jk}$. For a bi-partite system, it is revealing to write such terms more clearly as: $$\label{QMentangle}
[\hat H_{int},\hat\rho ]=\hat H_1\hat\rho_1\otimes\hat H_2\hat\rho_2-\hat\rho_1\hat H_1
\otimes\hat\rho_2\hat H_2
\;\;,$$ for an interaction $\propto \hat H_1\otimes\hat H_2$, with the factors acting on subsystems “1” and “2”, respectively, and where $\hat\rho =\hat\rho_1\otimes\hat\rho_2$, for a separable initial state. This has been called [*dynamically assisted entanglement generation*]{}, see, for example, Refs. [@Jacquod; @JacquodRev; @Hornberger].
It may come as a surprise that the CL evolution does this just as well, due to the contribution of $\Gamma_{\mbox{QM}}$ for the first two terms on the right-hand side of Eq.(\[anharmonic\]
) or, generally, due to the superoperator ${\cal E}$ of our earlier considerations. For polynomial interactions, for example, this superoperator [*always*]{} contains a contribution proportional to the usual QM terms.
However, the CL evolution produces additional correlations in $\hat\rho$, due to the generator $\propto {\cal L}_{ij;kl}\rho_{kl}$, which possibly [*entangles bra- and ket-states*]{}. – In comparison with Eq.(\[QMentangle\]), for example, such terms can have the unfamiliar structure: $$\label{CLentangle}
\hat H'_1\hat\rho_1\otimes\hat\rho_2\hat H'_2-\hat\rho_1\hat H'_1\otimes\hat H'_2\rho_2
\;\;,$$ which differs decidedly from a commutator. – This leads us to distinguish [*intra-*]{} (i.e., within given tensor product Hilbert space of subsystems “1” and “2”) and [*inter-space entanglement*]{} (i.e., between said Hilbert space and its dual).
For example, consider the anharmonic potential $V(x_1-x_2):=\lambda (x_1-x_2)^4$ for a bi-partite system consisting of particles “1” and “2”. Following and suitably generalizing our derivation in Section2, this leads to the interaction: $$\label{2particle}
{\cal V}(Q_1,Q_2;q_1,q_2)=
\frac{1}{2}\lambda\big (Q_1-q_1-(Q_2-q_2)\big )\big (Q_1+q_1-(Q_2+q_2)\big )^3
\;\;,$$ in terms of variables introduced previously, taking into account both subsystems; similarly as before, the $Q$ and $q$ variables refer to bra- and ket-states, respectively. Besides the separable terms, $\propto (Q_a-q_a)(Q_a+q_a)^3,\; a=1,2$, there are the terms which mix (and entangle) variables of both subsystems, as usual in QM. However, there are clearly additional terms that refer to Hilbert space and its dual simultaneously (and entangle corresponding states), for example, $\propto Q_aQ_bq_b^2,\; b\neq a$.
In retrospect, somehow, such difference between CL and QM evolution had to be expected: instead with superstates $|Q,q\rangle\rangle$, we could have worked with superstates $|x,p\rangle\rangle$, relating to coordinates and momenta of the classical theory. There, coordinates and momenta end up tightly correlated, due to Hamilton’s equations, and produce inter-space entanglement in an interacting bi-partite system.
Thus, we find that the confrontation of CL with QM, as in our side-by-side study, is quite revealing. In particular, we speculate that this opens new views on generating entanglement in multipartite systems, perhaps, by evolving through quasiclassical stages or by making use of decohered intermediary states [^7].
Concerning the quantum-classical divide, the present analysis shows that there is a deep formal similarity between CL and QM. However, this also demonstrates that what has been discussed in various ways as CL limit of QM – and which is similarly relevant for “emergent QM” – deserves more study.
While our work has been concerned mainly with the evolution of CL or QM objects, we recall that V.I.Man’ko and collaborators have pointed out that classical states may differ widely from what could be obtained as the “$\hbar\rightarrow 0$” limit of quantum mechanical ones. They show that all states can be classified by their ‘tomograms’ as [*either*]{} CL [*or*]{} QM, CL [*and*]{} QM, and [*neither*]{} CL [*nor*]{} CM [@Manko].
The classical limit might be a “[*F*]{}or[*A*]{}ll[*P*]{}ractical[*P*]{}urposes” limit, gradually approached through decoherence or “$\hbar\rightarrow 0$”. However, in order to bridge (if at all) the qualitative difference between intra- and inter-space entanglement that we find, and explain the “Man’ko classes of states”, some unknown dynamics beneath still awaits to be uncovered [^8].
The [*almost classical*]{} Jaynes-Cummings model
================================================
In this section, we apply our operator approach for the Liouville equation to a field theory, namely to a Rydberg atom interacting with the electromagnetic field. Following the approximations that lead to the quantum mechanical Jaynes-Cummings model [@JaynesCummings], we will show that the dynamics of this celebrated model is almost of classical character. As we shall see, if it were not for the anharmonic Coulomb interaction between electron and atom, the dynamics would be entirely classical.
The classical model
-------------------
We consider an electron (mass $m$) interacting electromagnetically with a positive charge (atom) fixed at the origin and with the radiation field. Thus, we depart from the classical Lagrangian: $$\label{L}
L:=\frac{m}{2}\dot x^2+\int\mbox{d}^3r\;\big\{\frac{1}{8\pi}(E^2-B^2)-\rho\phi +J\cdot A\big\}
\;\;,$$ where the electric and magnetic fields, respectively, are given by: $$\label{fields}
E=-\dot A-\nabla\phi \;\;,\;\;\; B=\nabla\times A
\;\;,$$ as usual, in terms of vector and scalar potential, $A$ and $\phi$, respectively. The charge and current densities, $\rho$ and $J$, respectively, are given by: $$\label{chargecurrent}
\rho (r)=-e\delta^3 (r-x)+\delta^3 (r)\;\;,\;\;\; J(r)=-e\dot x\delta^3 (r-x)
\;\;.$$
Next, we introduce Fourier modes of the fields, with $A(k)=A^*(-k)$ and $\phi (k)=\phi^*(-k)$, since the fields are real. We choose the Coulomb gauge by imposing $A_\parallel (k)=0$, which implies $\nabla\cdot A=0$. Correspondingly rewriting the Lagrangian, we determine the canonical momenta, in order to obtain the Hamiltonian of the classical model: $$\begin{aligned}
H&=&\frac{1}{2m}\Big (p+e\int\mbox{d}^3k\;
\big\{ A(k)\mbox{e}^{ik\cdot x}+A^*(k)\mbox{e}^{-ik\cdot x}\big\}\Big )^2-\frac{e^2}{|x|}
\nonumber \\ [1ex] \label{JCHamiltonian}
&\;&+\frac{1}{8\pi}\int\mbox{d}^3k\;\big\{\Pi^*(k)\cdot\Pi (k)+k^2A^*(k)\cdot A(k)\big\}
\\ [1ex] \label{JCHamiltonian1}
&\equiv&H(x,p;A,\Pi^*;A^*,\Pi )
\;\;, \end{aligned}$$ where we indicate the canonically conjugated pairs of variables of the Hamiltonian; the momentum integrations have to take into account that not all Fourier modes are independent, for real fields.
We are now in the position, cf. Section2, to describe this model in phase space. We proceed in four steps:
- First, we introduce the probability density (over phase space) $\rho (x,p;A,\Pi^*;A^*,\Pi )$, which will be interpreted, as before, as matrix element of a Hermitean density operator $\hat\rho$. We assume that the atom-electromagnetic-field system is confined to a cavity of finite volume $V$, thus replacing integrals by discrete mode sums, $\int\mbox{d}^3k\;g(k)\rightarrow V^{-1}\sum_k\;g_k$.
- Second, we obtain the Liouville equation, $-\partial_t\rho =\{ H,\rho\} =\dots\;$, evaluating the relevant Poisson bracket.
- Third, we replace momenta by coordinates via Fourier transformation, $p\rightarrow y$, $\Pi_k^*\rightarrow B_k$, $\Pi_k\rightarrow B_k^*$.
- Fourth, we perform the “Wigner rotations”, $Q:=x+y/2$, $q:=x-y/2$, $Q_k:=A_k+B_k/2$, and $q_k:=A_k-B_k/2$.
Details and the following derivations will be reported elsewhere [@GioThesis].
Introducing the following notation: $$\label{potentials}
V(\chi ):=-\frac{e^2}{|\chi |}\;\;,\;\mbox{for}\;\;\chi =Q,q\;\;;
\;\;\;{\cal E}(Q,q):=4e^2\frac{Q^2-q^2}{|Q+q|^3}-V(Q)+V(q)
\;\;,$$ where we suppress constant normalization factors etc., as before, we obtain the remarkable result that the [*classical*]{} evolution equation is: $$\begin{aligned}
\label{rhoevol}
i\partial_t\rho &=&\big\{\mbox{``von Neumann''}+{\cal E}+\Gamma +\Sigma \big\}\rho
\\ [1ex]
&\equiv&
\Big\{ -\frac{1}{2m}\partial_Q^{\; 2}+V(Q)-\big (-\frac{1}{2m}\partial_q^{\; 2}+V(q)\big )
\nonumber \\ [1ex]
&\;&\;+\frac{1}{8\pi}\sum_k\big [
-\partial_{Q_k}\cdot\partial_{Q_k^*}+\omega_k^{\; 2}Q_k\cdot Q_k^*
-\big (-\partial_{q_k}\cdot\partial_{q_k^*}+\omega_k^{\; 2}q_k\cdot q_k^*\big )\big ]
\nonumber \\ [1ex]
&\;&\;-i\frac{e}{m}\sum_k\big [
\mbox{e}^{ik\cdot Q}Q_k\cdot\partial_Q+\mbox{e}^{ik\cdot q}q_k\cdot\partial_q\big ]
\nonumber \\ [1ex]
&\;&\;+\frac{e^2}{2m}\sum_{k,k'}\big [
Q_k\cdot Q_{k'}^*\mbox{e}^{i(k-k')\cdot Q}-q_k\cdot q_{k'}^*\mbox{e}^{i(k-k')\cdot q}\big ]
\Big\}\rho
\nonumber \\ [1ex] \label{rhoevol1}
&\;&+\Big\{ {\cal E}(Q,q)+\Gamma +\Sigma \Big\}\rho
\;\;, \end{aligned}$$ with $\rho\equiv\rho (Q,Q_k,Q_k^*;q,q_k,q_k^*;t)$, where $k$ runs over all modes, $\omega_k:=|k|$, and where $\Gamma$ and $\Sigma$ denote rather complicated terms that involve all phase space variables; they are given explicitly in Ref. [@GioThesis]. While the last line of Eq.(\[rhoevol1\]) presents additional terms, in particular the superoperator ${\cal E}$, the previous terms represent exactly the terms of the [*quantum mechanical*]{} vonNeumann equation for the atom-field system under consideration; besides further interaction terms, due to minimal coupling, we find the contribution of the free electromagnetic field in the second and those of the electron interacting with the Coulomb potential of the Rydberg atom in the first line, respectively.
We anticipate that in the [*dipole approximation*]{} we have $\Gamma,\Sigma\rightarrow 0$. Therefore, we do not study further the impact of those terms here [@GioThesis].
Instead, we recall the well known additional approximations that turn the vonNeumann terms of Eqs.(\[rhoevol\])–(\[rhoevol1\]) into those of the [*Jaynes-Cummings model*]{} [@JaynesCummings]:
- The [*dipole approximation*]{}, assuming that $\tilde k\cdot l\ll 1$, where $\hbar\tilde k$ and $l$ denote a typical photon momentum and linear size of a Rydberg electron orbit, respectively.
- The restriction to [*one cavity photon mode*]{} with energy $\hbar\omega$. This yields the approximate Hamilton operator [@GioThesis]: $$\hat H=\sum_i\; \omega_i|i\rangle\langle i|+\omega (\hat a^\dagger \hat a+\frac{1}{2})
+i\sum_{i\neq j}d_{ij}(\hat a-\hat a^\dagger )|i\rangle\langle j| \;\;,$$ where the sums run over the Rydberg levels, with energies $\hbar\omega_i$, $\hat a^{(\dagger )}$ are photon annihilation (creation) operators, and where the last term involves the dipole transition amplitudes $d_{ij}$.
- The restriction to a [*two-level subspace*]{}, spanned by states $|g\rangle,|e\rangle$; the lower level energy is conveniently set to $\hbar\omega_g\equiv 0$, while the physical realizations considered, usually, have $\omega_e\approx\omega$, i.e., approximately resonant photon and excited electron states.
- The [*rotating wave approximation*]{}, which yields the ‘energy conserving’ dipole interaction term $\hat{\cal D}\propto\hat a|e\rangle\langle g|-|g\rangle\langle e|\hat a^\dagger$.
The resulting Jaynes-Cummings Hamiltonian is: $$\label{JCH}
\hat H_{\mbox{JC}}=\omega_e|e\rangle\langle e|+\omega (\hat a^\dagger \hat a+\frac{1}{2})
+id_{eg}\big (\hat a|e\rangle\langle g|-|g\rangle\langle e|\hat a^\dagger\big )
\;\;.$$ Then, following the above derivation, the evolution equation becomes: $$\label{JCevol}
i\partial_t\hat\rho = [\hat H_{\mbox{JC}},\hat\rho ]+\hat{\cal E}\rho
\;\;,$$ where we appropriately incorporated here the superoperator $\hat{\cal E}$. This term presents the [*only*]{} difference between the [*classical*]{} dynamics described by Eqs.(\[rhoevol\])–(\[rhoevol1\]) and the usual [*quantum mechanical*]{} one.
Thus, we find in this ‘standard model’ of quantum optics a detailed example for the similarity between CL and QM evolution laws.
Dipole interaction and Coulomb superoperator as perturbations
-------------------------------------------------------------
In order to illustrate our findings, we will briefly study the influence of the classical superoperator on the evolution described by the Jaynes-Cummings model in perturbation theory, while a more complete analysis will be presented in Ref. [@GioThesis].
Following Section4.3, cf. Eq.(\[Dyson\]), we presently treat the dipole interaction, $\hat{\cal D}\propto\hat a|e\rangle\langle g|-|g\rangle\langle e|\hat a^\dagger$, together with the superoperator $\hat{\cal E}$ of Eq.(\[JCevol\]) as perturbation. Correspondingly, we choose the Rydberg atom states and one-mode photon number states for the tensor product basis of the relevant Hilbert space.
Then, the density matrix evolves according to: $$\label{JCEvol}
\hat\rho (t)=\hat{\cal G}(t)\hat\rho (0)
=\hat{\cal G}_0(t)\hat\rho (0)-i\int_0^t\mbox{d}\tau\;
\hat{\cal G}_0(t-\tau )\big (\hat{\cal D}+\hat{\cal E}\big )\hat{\cal G}_0(\tau )\hat\rho (0)
\;\;,$$ to first order in $\hat{\cal D}+\hat{\cal E}$. While the dipole operator acts on atom and electomagnetic field states simultaneously, the superoperator acts only on the Rydberg states. Hence, the matrix elements of $\hat{\cal E}$ are defined by: $$\label{Ematrix}
{\cal E}_{ab,cd}:=\int\mbox{d}^3Q\mbox{d}^3q\;\psi^*_a(Q)\psi_b(q){\cal E}(Q,q)
\psi_c(Q)\psi^*_d(q)
\;\;,$$ where $\psi_{i\equiv (n,l,m)}$ denote standard hydrogen-like wave functions. Since these are eigenstates of parity, $P_i=\pm 1$, we find ${\cal E}_{ab,cd}=P_aP_bP_cP_d{\cal E}_{ab,cd}$, which implies the selection rule: ${\cal E}_{ab,cd}=0$, if $P_aP_bP_cP_d=-1$. Furthermore, we have: ${\cal E}_{ab,cd}=-{\cal E}_{ba,dc}^*$.
Consequently, the only nonzero matrix elements are ${\cal E}_{eg,eg}$, ${\cal E}_{ge,ge}$, ${\cal E}_{ee,gg}$, and ${\cal E}_{gg,ee}$. The latter two vanish for the specific ground and excited states used in cavity QED experiments [@CavityQED].
Taking matrix elements of Eq.(\[JCEvol\]), we find that the superoperator only affects the evolution of the matrix elements $\rho_{eg|nn'}$ and $\rho_{ge|nn'}=\rho_{eg|n'n}^*$ (by hermiticity); here the Fock states are labelled by photon numbers $n,n'$. Then, we find: $$\begin{aligned}
\rho_ {eg|nn'}(t)&=&\mbox{e}^{-it[\omega_e-\omega_g+\omega (n-n')]}
\Big ([1+it{\cal E}_{eg,eg}]\rho_{eg|nn'} (0)
\nonumber \\ [1ex] \label{rho1st}
&\;&+d_{eg} t\big [\sqrt{n+1}\rho_{gg}(0)\rho_{n+1\;n'}(0)
-\sqrt{n'}\rho_{ee}(0)\rho_{n\;n'-1}(0)\big ]\Big )
\;\;, \end{aligned}$$ where we assume that the initial state factorizes, for simplicity.
Thus, to first order in this perturbative expansion, we find that the superoperator competes with the dipole interaction, as far as the atom states are concerned; however, it does so without affecting the field states. Numerical estimates indicate that its matrix elements are not small compared to the ones of the dipole operator for cavity QED experiments (see Ref. [@GioThesis] for further details) [@CavityQED].
To summarize, [*almost*]{} all dynamical (operator) features of the Jaynes-Cummings model can be derived in the classical framework, as we have shown.
Nevertheless, there [*is*]{} a noticeable difference between classical and quantum evolution in the version of the Jaynes-Cummings model that is related to cavity QED experiments. This is solely due to the classical superoperator, which stems from the [*Coulomb interaction*]{} between electron and Rydberg ion.
In distinction, had we considered a charged particle trapped by a [*linear or harmonic potential*]{}, then the superoperator would vanish identically, cf. Section2, Eq.(\[Ezero\]). The correspondingly modified Jaynes-Cummings model could be seen as of entirely classical origin, despite its quantum mechanical appearance.
Conclusions
===========
Beginning with the Liouville equation of classical statistical mechanics, we have introduced a (super)operator formulation [@superspace], which brings it as close as possible to the vonNeumann equation of quantum mechanics, provided suitable coordinates are chosen in superspace [@Elze05; @ElzeAttractor]. Presently, we have concentrated on the similarities and differences between both evolution equations.
We have chosen the Jaynes-Cummings model [@JaynesCummings; @CavityQED], in particular, to illustrate both aspects and to show that this benchmark model of quantum optics and cavity QED can be interpreted to a large extent in terms of classical dynamics [@GioThesis]. Furthermore, this model serves as an example that our more general considerations apply not only to single- or few-particle systems, but to field theories as well.
While presently the relevant Hilbert space has been treated as tensor product space of the Rydberg electron single-particle and the photon Fock space, earlier also a functional approach combining fermion and boson fields has been discussed [@Elze07].
More generally, we discussed in parallel the formal solutions of the Liouville and the vonNeumann equations. Introducing suitable propagators, we derived a path integral representation for both cases side by side [@FabThesis].
The path integral for the propagator of the Liouville equation is new and may have interesting applications in classical physics. We derived and illustrated the related perturbation theory.
We discussed how these results and the action entering this path integral, in particular, hint at the possibility that a form of entanglement is also generated by classical dynamics, which has gone unnoticed before. It combines the quantum mechanical dynamically assisted entanglement generation [@Jacquod; @JacquodRev; @Hornberger] with a classical counterpart. We call the former [*intra-space entanglement*]{}, since it acts separately within the Hilbert spaces of bra- and ket-states. In distinction, the classical dynamics additionally produces [*inter-space entanglement*]{}, i.e., it correlates the Hilbert space and its adjoint in addition to what would, otherwise, be recognized as quantum entanglement.
If the relative strength of intra- and inter-space entanglement can be manipulated, for example, by driving a system dynamically between quantum and classical behaviour, this may open additional ways to influence the quantum mechanical entanglement, which is of central importance in research concerning quantum information and quantum foundations alike.
The close relation between classical and quantum mechanical dynamics that we uncovered may help to address in new ways problems related to the nature of classical or quantum states [@Zwitters; @Manko; @ElzeAttractor], to the pathways, if any, over the quantum-classical divide [@KieferEtAl; @Zurek; @tHooft06; @Elze05; @Vitiello01], or the measurement problem [@Schlosshauer; @Adler].
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to thank Nick Manton for several discussions of Liouville dynamics vs. QM, Vladimir Man’ko and Andreij Khrennikov for discussions of tomography of states and probabilistic formulations, and Marco Genovese for inviting H-TE to present this work at the 5th Workshop – ad memoriam of Carlo Novero – [*“Advances in Foundations of Quantum Mechanics and Quantum Information with Atoms and Photons”*]{} (Torino, May 2010).
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[^1]: I.e., the conflict between quantum mechanics necessitating an external time and diffeomorphism invariance in general relativity, for example, which defies its existence. Despite its great successes in describing the statistical aspects of experiments, quantum theory itself presents problems of interpretation, which are brought to the forefront in quantum cosmology. They arise from its indeterministic features and are clearly seen, for example, in the measurement problem.
[^2]: Another path integral for classical mechanics exists, which implements Hamilton’s equations as constraints, see Refs. [@Blasone05] and references there. In this approach, an analogy with quantum mechanics consists in the path integral as such, yet its integrand bears no resemblance.
[^3]: This superoperator is of a very specific form, which leads to the antisymmetry in Eq.(\[I\]). It differs from a Lindblad superoperator, often obtained as a symmetric double commutator structure, in the case of open quantum mechanical systems [@Diosi].
[^4]: The notion of [*superspace*]{} here, at first sight, has little in common and should not be confused with the corresponding term relating to supersymmetry.
[^5]: In this chapter, we reinstate $\hbar$ explicitly.
[^6]: We remark that a complementary approach, based on the effective action generating equal-time correlation functions for nonequilibrium statistical systems, has been presented in Ref.[@Wetterich], which results in evolution equations for a truncated set of correlation functions.
[^7]: Previous considerations of the semiclassical regime, such as in Refs. [@Jacquod; @JacquodRev], were motivated as suitable approximations of the quantum mechanical evolution, in particular, for studies of the different decoherence properties between classically regular and chaotic systems. Our results seem to show that crossing the quantum-classical divide may offer an additional resource for entanglement generation and related “truly quantum” phenomena. This might be related to the “underlying reality” of (CL and QM) physics, assumed in Refs.[@Zwitters; @Khrennikov], consisting in statistical correlations.
[^8]: A simple attractor model, motivated by assumptions about effects of fundamental spacetime discreteness [@Elze09], has been discussed in Ref. [@ElzeAttractor].
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title: 'Worksharing Tasks: an Efficient Way to Exploit Irregular and Fine-Grained Loop Parallelism'
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Acknowledgment {#acknowledgment .unnumbered}
==============
This work is supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (TIN2015-65316-P), by the Generalitat de Catalunya (2014-SGR-1051) and by the European Union’s Seventh Framework Programme (FP7/2007-2013) and the H2020 funding framework under grant agreement no. H2020-FETHPC-754304 (DEEP-EST).
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this paper we present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments, i.e., the phase variable is always between $-1$ and 1, at a point-wise level. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. Such an analysis technique could also be applied to a second order numerical scheme, in which the BDF temporal stencil is applied, the expansive term is updated by a second order Adams-Bashforth explicit extrapolation formula, and an artificial Douglas-Dupont regularization term is added to improve the stability property. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes.'
author:
- 'Wenbin Chen[^1]'
- 'Cheng Wang[^2]'
- 'Xiaoming Wang[^3]'
- 'Steven M. Wise[^4]'
bibliography:
- 'chlog\_3.bib'
title: 'Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential'
---
Cahn-Hilliard equation, logarithmic Flory Huggins energy potential, positivity preserving, energy stability, second order BDF scheme, optimal rate convergence analysis
35K35, 35K55, 49J40, 65K10, 65M06, 65M12
Introduction
============
The well-known Allen-Cahn (AC) [@allen79] and Cahn-Hilliard (CH) [@cahn58] equations are prototypical gradient flows with respect to a given free energy. We consider a bounded domain $\Omega \subset \mathbb{R}^d$ (with $d=2$ or $d=3$). For any $\phi \in H^1 (\Omega)$, with a point-wise bound, $-1 < \phi < 1$, the energy functional is given by $$\label{CH energy}
E(\phi)=\int_{\Omega}\left( ( 1+ \phi) \ln (1+\phi) + (1-\phi) \ln (1-\phi) - \frac{\theta_0}{2} \phi^2 +\frac{\varepsilon^2}{2}|\nabla \phi|^2\right) d {\bf x} ,$$ where $\varepsilon$, $\theta_0$ are positive constants associated with the diffuse interface width. See [@cahn1996; @elliott92a; @doi13; @elliott96b]. The AC and CH equations are precisely the $L^2$ (non-conserved) and $H^{-1}$ (conserved) gradient flows of the energy functional (\[CH energy\]), respectively, $$\partial_t \phi = - {\cal M} (\phi) \mu , \qquad \mbox{(Allen-Cahn)}
\label{AC equation-0}$$ and $$\partial_t\phi = \nabla \cdot ( {\cal M} (\phi) \nabla \mu ) , \qquad \mbox{(Cahn-Hilliard)}
\label{CH equation-0}$$ where $\mu$ is the chemical potential $$\mu := \delta_\phi E = \ln (1+\phi) - \ln (1-\phi) - \theta_0 \phi - \varepsilon^2 \Delta \phi ,
\label{CH-mu-0}$$ and ${\cal M} (\phi) >0$ is the mobility function. In a related example, Cahn, et al. [@cahn1996] have studied the Cahn-Hilliard equation with the fully degenerate mobility, $\mathcal{M} (\phi) = (1-\phi)(1+\phi)$, and have shown asymptotic convergence to a geometric model for motion by the surface Laplacian of mean curvature.
For simplicity of presentation, we suppose $\Omega$ is a cuboid and consider periodic boundary conditions. The case with homogeneous Neumann boundary condition could be analyzed in a similar manner. Due to the gradient structure of and , the following energy dissipation laws formally hold: $$\begin{aligned}
\frac{d}{dt} E(\phi(t))= & \ -\int_{\Omega} {\cal M} (\phi) |\mu|^2 d {\bf x} \le 0 \quad \mbox{(AC equation)} ,
\label{energy-decay-rate-AC}
\\
\frac{d}{dt} E(\phi(t))= & \ -\int_{\Omega} {\cal M} (\phi) |\nabla \mu |^2 d {\bf x} \le 0 \quad \mbox{(CH equation)} .
\label{energy-decay-rate-CH}
\end{aligned}$$ The free energy with the logarithmic potential is often considered to be more physically realistic than that with a polynomial free energy, because the former can be derived from regular or ideal solution theories [@doi13]. However, one well-known difficulty for the analysis of these models with logarithmic Flory Huggins energy potential – as it is called in the polymer science community [@doi13] – is associated with the singularity as the phase variable approaches $-1$ or $1$. PDE solutions are expected to satisfy a *positivity property*, specifically, $$0 < 1-\phi \quad \mbox{and} \quad 0 < 1+\phi .$$ In other words, the phase variable remains in the interval $(-1, 1)$, in a point-wise sense [@elliott96b]. However, it is a major challenge to create numerical schemes that mimic this property. To avoid such a subtle challenge, many efforts have been devoted to a polynomial approximation: $$E(\phi)=\int_{\Omega}\left( \frac14 (\phi^2 -1)^2 + \frac{\varepsilon^2}{2}|\nabla \phi|^2\right) d {\bf x} ,
\label{CH energy-polynomial-0}$$ which leads to the nonlinear, but non-singular, chemical potential $$\mu := \delta_\phi E = \phi^3 - \phi- \varepsilon^2 \Delta \phi .
\label{CH-polynomial-mu-0}$$ This model has a similar double-well structure as in the case and , but avoids the singularities as the phase variable approaches $-1$ or $1$. Meanwhile, the PDE solution may go beyond the given interval of $(-1, 1)$. There have been extensive numerical works for the Cahn-Hilliard equation with the polynomial approximation , ; see the related references [@christlieb14; @du91; @elliott93; @furihata01; @guillen14; @LiD2016b; @LiD2016a; @wu14], et cetera.
In this article we focus on the Cahn-Hilliard model with logarithmic Flory Huggins energy potential . At the PDE level, the positivity property (for both logarithmic arguments, $1 + \phi$ and $1 - \phi$) has been established in [@abels07; @debussche95; @elliott96b; @miranville04]. Moreover, in the 1-D and 2-D cases, the phase separation has also been justified at a theoretical level, i.e., a uniform distance between the phase variable and the singular limit values ($-1$ and $1$) have been derived, dependent on $\varepsilon$, $\theta_0$ and the initial data. The analysis for the degenerate mobility case could be found in [@barrett99; @elliott96b]. In addition, an improved analysis for the 2-D equation has been reported in a more recent work [@Giorgini17a]; also see the related references [@miranville11; @miranville12]. An extension to the Cahn-Hilliard model coupled with fluid flow is also discussed in [@abels09b; @Giorgini17].
At the level of numerical scheme design, the positivity preserving property is very challenging, due to the particularities of the spatial and temporal discretizations involved. There have been extensive numerical works for the CH model with Flory Huggins energy potential [@jeong16; @jeong15; @LiH2017; @LiX16; @peng17a; @peng17b; @yang17c], while a theoretical justification to assure the positivity of $1+\phi$ and $1-\phi$ has not been available (so that the numerical scheme is unconditionally well-defined). Among the existing literature, it is worth mentioning the numerical analysis to theoretically justify this issue in [@elliott92a]. The authors analyzed the implicit Euler scheme applied to the CH equation , , combined with the finite element approximation in space. In more details, the following result was proved: [*Under the condition that ${{\Delta t}}\le \frac{4 \varepsilon^2}{\theta_0^2}$, and the initial data satisfy $\frac{1}{| \Omega |} \left| \int_\Omega \, \phi_0 ({\bf x}) \, d {\bf x} \right| < 1 - \delta$, $\| \phi_0 \|_\infty \le 1$, then there is a unique numerical solution for the fully implicit Euler scheme, satisfying $\| \phi^n \|_\infty < 1$.*]{} An extension to the multi-component Cahn-Hilliard flow has also been reported in [@elliott96c].
Meanwhile, it is observed that, an energy stability property is not unconditionally available for the scheme studied in [@elliott92a], due to the implicit treatment of the concave diffusion term. Further, the time step constraint, ${{\Delta t}}\le \frac{4 \varepsilon^2}{\theta_0^2}$, could make the numerical implementation challenging for small $\varepsilon$ and large $\theta_0$. In this article, we propose and analyze an alternate numerical scheme, in which the implicit treatment for the concave diffusion term is replaced by an explicit one, combined with centered difference discretization in space. Again, the implicit treatment for the nonlinear logarithmic term requires a theoretical justification for the positivity of both $1+\phi$ and $1-\phi$, so that the numerical scheme is well-defined at a point-wise level. Instead of reconstructing an alternate energy functional to avoid the singularity for $\phi$ at $-1$ and $1$, as reported in [@elliott96c; @elliott92a], we use a new technique to theoretically justify the positivity of the numerical solution. First, the fully discrete numerical scheme corresponds to a minimization of a discrete energy function. And also, such an energy functional is strictly convex, as long as the phase variable stays within $(-1,1)$ at a point-wise level. Subsequently, to avoid a circular “chicken-and-the-egg" argument, we take a closed domain for the numerical solution variables, in which the limit bound values of $-1$ and $1$ are not reachable. In turn, the continuous energy function has to have a global minimum over this closed domain. Moreover, we make use of the following subtle fact: the singular nature of the logarithmic function prevents such a global minimum from being obtained at a boundary point (in terms of numerical solution variable domain), as long as the numerical solution stays bounded at the previous time step. As a result, since the global minimum could only possibly occur at an interior point in the numerical solution variable domain, we conclude that the numerical scheme has to be satisfied, so that the existence of the numerical solution is proved. In addition, due to the convex nature of the energy function, the uniqueness of the numerical solution becomes a direct consequence. As a further consequence, we observe that: as long as the numerical solution stays bounded at the previous time step, i.e., within $[-M, M]$ ($M >0$), not necessarily $(-1, 1)$, and its average stays between $-1$ and $1$, there must exist a unique numerical solution which stays within $(-1, 1)$ at the next time step. This leads to an interesting difference between the present results and those in [@elliott92a], where the requirement for the initial data, namely, $\| \phi_0 \|_\infty \le 1$, has to be imposed for the analysis to go through. On the other hand, the latter constraint is completely natural. Another new feature of the numerical analysis in this article is the theoretical justification of the energy stability. As a result of the unconditional energy stability, a uniform in time $H_h^1$ bound for the numerical solution could be derived. In addition, a detailed convergence analysis of the proposed numerical scheme could be derived, which gives an optimal rate error estimate in the $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0, T; H_h^1)$ norm. A key point in the analysis lies in the following subtle fact: since the nonlinear logarithmic term corresponds to a convex energy, the corresponding nonlinear error inner product is always non-negative. And also, the error estimate associated with the surface diffusion term indicates an $\ell^2 (0, T; H_h^1)$ convergence. Because of the explicit treatment for the expansive term, this convergence estimate does not require the time step constraint, in comparison with the existing results [@barrett95; @barrett96; @barrett01; @elliott96c; @elliott92a].
On the other hand, all these positivity-preserving schemes are only first order accurate in time, which is not satisfactory in the practical computations. In turn, a higher order accurate in time, positivity-preserving numerical scheme is highly desired. In this article, we propose and analyze a second order accurate scheme for the CH model with Flory Huggins energy potential , with unique solvability, positivity-preserving property and energy stability established. In more details, we apply the implicit backward differentiation formula (BDF) concept to derive second order temporal accuracy, while the expansive term is treated by a second order explicit extrapolation formula. An additional term $A {{\Delta t}}\Delta_h (\phi^{k+1} - \phi^k)$ is added, which represents a second order Douglas-Dupont-type regularization, and a careful calculation shows that energy stability is guaranteed, provided the mild condition $A \ge \frac{1}{16}$ is enforced. Moreover, the singular nature of the logarithmic term enables us to theoretically derive the positivity-preserving property of this second order numerical scheme, which is the first such result in this area. And also, an $H_h^{-1}$ inner product with the numerical error function leads to an optimal rate error estimate in the $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0, T; H_h^1)$ norm, with second order accuracy in both time and space.
The rest of the article is organized as follows. In Section \[sec:numerical scheme\] we propose the first order numerical scheme and state the corresponding theoretical results. The detailed proof for the positivity-preserving property of the numerical solution is provided in Section \[sec: proof\]. Subsequently, the energy stability analysis is established in Section \[sec:energy stability\], and the optimal rate convergence analysis is presented in Section \[sec:convergence\]. The second order BDF scheme is outlined and analyzed in Section \[sec:BDF2\]. Some numerical results are presented in Section \[sec:numerical results\], including a brief description of the 3-D multigrid solver. Finally, the concluding remarks are given in Section \[sec:conclusion\].
The first order numerical scheme {#sec:numerical scheme}
================================
In the spatial discretization, the centered finite difference approximation is applied. We recall some of the basics of this methodology.
Discretization of space {#subsec:finite difference}
-----------------------
We use the notation and results for some discrete functions and operators from [@guo16; @wise10; @wise09]. Let $\Omega = (0,L_x)\times(0,L_y)\times(0,L_z)$, where for simplicity, we assume $L_x =L_y=L_z =: L > 0$. Let $N\in\mathbb{N}$ be given, and define the grid spacing $h := \frac{L}{N}$. We will assume – but only for simplicity of notation, ultimately – that the mesh spacing in the $x$, $y$, and $z$-directions are the same. We define the following two uniform, infinite grids with grid spacing $h>0$: $$E := \{ p_{i+{\nicefrac{1}{2}}} \ |\ i\in {\mathbb{Z}}\}, \quad C := \{ p_i \ |\ i\in {\mathbb{Z}}\},$$ where $p_i = p(i) := (i-{\nicefrac{1}{2}})\cdot h$. Consider the following 3-D discrete $N^3$-periodic function spaces: $$\begin{aligned}
\begin{aligned}
{\mathcal C}_{\rm per} &:= \left\{\nu: C\times C
\times C\rightarrow {\mathbb{R}}\ \middle| \ \nu_{i,j,k} = \nu_{i+\alpha N,j+\beta N, k+\gamma N}, \ \forall \, i,j,k,\alpha,\beta,\gamma\in \mathbb{Z} \right\},
\\
{\mathcal E}^{\rm x}_{\rm per} &:=\left\{\nu: E\times C\times C\rightarrow {\mathbb{R}}\ \middle| \ \nu_{i+\frac12,j,k}= \nu_{i+\frac12+\alpha N,j+\beta N, k+\gamma N}, \ \forall \, i,j,k,\alpha,\beta,\gamma\in \mathbb{Z}\right\} .
\end{aligned}
\end{aligned}$$ Here we are using the identification $\nu_{i,j,k} = \nu(p_i,p_j,p_k)$, *et cetera*. The spaces ${\mathcal E}^{\rm y}_{\rm per}$ and ${\mathcal E}^{\rm z}_{\rm per}$ are analogously defined. The functions of ${\mathcal C}_{\rm per}$ are called [*cell centered functions*]{}. The functions of ${\mathcal E}^{\rm x}_{\rm per}$, ${\mathcal E}^{\rm y}_{\rm per}$, and ${\mathcal E}^{\rm z}_{\rm per}$, are called [*east-west*]{}, [*north-south*]{}, and [*up-down face-centered functions*]{}, respectively. We also define the mean zero space $$\mathring{\mathcal C}_{\rm per}:=\left\{\nu\in {\mathcal C}_{\rm per} \ \middle| 0 = \overline{\nu} := \frac{h^3}{| \Omega|} \sum_{i,j,k=1}^m \nu_{i,j,k} \right\} .$$ We define $\vec{\mathcal{E}}_{\rm per} := {\mathcal E}^{\rm x}_{\rm per}\times {\mathcal E}^{\rm y}_{\rm per}\times {\mathcal E}^{\rm z}_{\rm per}$.
We now introduce the important difference and average operators on the spaces: $$\begin{aligned}
&& A_x \nu_{i+{\nicefrac{1}{2}},j,k} := \frac{1}{2}\left(\nu_{i+1,j,k} + \nu_{i,j,k} \right), \quad D_x \nu_{i+{\nicefrac{1}{2}},j,k} := \frac{1}{h}\left(\nu_{i+1,j,k} - \nu_{i,j,k} \right),
\\
&& A_y \nu_{i,j+{\nicefrac{1}{2}},k} := \frac{1}{2}\left(\nu_{i,j+1,k} + \nu_{i,j,k} \right), \quad D_y \nu_{i,j+{\nicefrac{1}{2}},k} := \frac{1}{h}\left(\nu_{i,j+1,k} - \nu_{i,j,k} \right) ,
\\
&& A_z \nu_{i,j,k+{\nicefrac{1}{2}}} := \frac{1}{2}\left(\nu_{i,j,k+1} + \nu_{i,j,k} \right), \quad D_z \nu_{i,j,k+{\nicefrac{1}{2}}} := \frac{1}{h}\left(\nu_{i,j,k+1} - \nu_{i,j,k} \right) ,
\end{aligned}$$ with $A_x,\, D_x: {\mathcal C}_{\rm per}\rightarrow{\mathcal E}_{\rm per}^{\rm x}$, $A_y,\, D_y: {\mathcal C}_{\rm per}\rightarrow{\mathcal E}_{\rm per}^{\rm y}$, $A_z,\, D_z: {\mathcal C}_{\rm per}\rightarrow{\mathcal E}_{\rm per}^{\rm z}$. Likewise, $$\begin{aligned}
&& a_x \nu_{i, j, k} := \frac{1}{2}\left(\nu_{i+{\nicefrac{1}{2}}, j, k} + \nu_{i-{\nicefrac{1}{2}}, j, k} \right), \quad d_x \nu_{i, j, k} := \frac{1}{h}\left(\nu_{i+{\nicefrac{1}{2}}, j, k} - \nu_{i-{\nicefrac{1}{2}}, j, k} \right),
\\
&& a_y \nu_{i,j, k} := \frac{1}{2}\left(\nu_{i,j+{\nicefrac{1}{2}}, k} + \nu_{i,j-{\nicefrac{1}{2}}, k} \right), \quad d_y \nu_{i,j, k} := \frac{1}{h}\left(\nu_{i,j+{\nicefrac{1}{2}}, k} - \nu_{i,j-{\nicefrac{1}{2}}, k} \right),
\\
&& a_z \nu_{i,j,k} := \frac{1}{2}\left(\nu_{i, j,k+{\nicefrac{1}{2}}} + \nu_{i, j, k-{\nicefrac{1}{2}}} \right), \quad d_z \nu_{i,j, k} := \frac{1}{h}\left(\nu_{i, j,k+{\nicefrac{1}{2}}} - \nu_{i, j,k-{\nicefrac{1}{2}}} \right),
\end{aligned}$$ with $a_x,\, d_x : {\mathcal E}_{\rm per}^{\rm x}\rightarrow{\mathcal C}_{\rm per}$, $a_y,\, d_y : {\mathcal E}_{\rm per}^{\rm y}\rightarrow{\mathcal C}_{\rm per}$, and $a_z,\, d_z : {\mathcal E}_{\rm per}^{\rm z}\rightarrow{\mathcal C}_{\rm per}$. The discrete gradient ${\nabla_{\! h}}:{\mathcal C}_{\rm per}\rightarrow \vec{\mathcal{E}}_{\rm per}$ is defined via $${\nabla_{\! h}}\nu_{i,j,k} =\left( D_x\nu_{i+{\nicefrac{1}{2}}, j, k}, D_y\nu_{i, j+{\nicefrac{1}{2}}, k},D_z\nu_{i, j, k+{\nicefrac{1}{2}}}\right) ,$$ and the discrete divergence ${\nabla_{\! h}}\cdot :\vec{\mathcal{E}}_{\rm per} \rightarrow {\mathcal C}_{\rm per}$ is defined via $${\nabla_{\! h}}\cdot\vec{f}_{i,j,k} = d_x f^x_{i,j,k} + d_y f^y_{i,j,k} + d_z f^z_{i,j,k},$$ where $\vec{f} = (f^x,f^y,f^z)\in \vec{\mathcal{E}}_{\rm per}$. The standard 3-D discrete Laplacian, $\Delta_h : {\mathcal C}_{\rm per}\rightarrow{\mathcal C}_{\rm per}$, is given by $$\begin{aligned}
\Delta_h \nu_{i,j,k} := & \nabla_{h}\cdot\left(\nabla_{h}\phi\right)_{i,j,k} = d_x(D_x \nu)_{i,j,k} + d_y(D_y \nu)_{i,j,k}+d_z(D_z \nu)_{i,j,k}
\\
= & \ \frac{1}{h^2}\left( \nu_{i+1,j,k}+\nu_{i-1,j,k}+\nu_{i,j+1,k}+\nu_{i,j-1,k}+\nu_{i,j,k+1}+\nu_{i,j,k-1} - 6\nu_{i,j,k}\right).
\end{aligned}$$ More generally, if $\mathcal{D}$ is a periodic *scalar* function that is defined at all of the face center points and $\vec{f}\in\vec{\mathcal{E}}_{\rm per}$, then $\mathcal{D}\vec{f}\in\vec{\mathcal{E}}_{\rm per}$, assuming point-wise multiplication, and we may define $$\nabla_h\cdot \big(\mathcal{D} \vec{f} \big)_{i,j,k} = d_x\left(\mathcal{D}f^x\right)_{i,j,k} + d_y\left(\mathcal{D}f^y\right)_{i,j,k} + d_z\left(\mathcal{D}f^z\right)_{i,j,k} .$$ Specifically, if $\nu\in \mathcal{C}_{\rm per}$, then $\nabla_h \cdot\left(\mathcal{D} \nabla_h \ \ \right):\mathcal{C}_{\rm per} \rightarrow \mathcal{C}_{\rm per}$ is defined point-wise via $$\nabla_h\cdot \big(\mathcal{D} \nabla_h \nu \big)_{i,j,k} = d_x\left(\mathcal{D}D_x\nu\right)_{i,j,k} + d_y\left(\mathcal{D} D_y\nu\right)_{i,j,k} + d_z\left(\mathcal{D}D_z\nu\right)_{i,j,k} .$$
Now we are ready to define the following grid inner products: $$\begin{aligned}
{\left\langle \nu , \xi \right\rangle_\Omega} &:= h^3\sum_{i,j,k=1}^N \nu_{i,j,k}\, \xi_{i,j,k},\quad \nu,\, \xi\in {\mathcal C}_{\rm per},\quad
& {\left[ \nu , \xi \right]_{\rm x}} := {\left\langle a_x(\nu\xi) , 1 \right\rangle_\Omega} ,\quad \nu,\, \xi\in{\mathcal E}^{\rm x}_{\rm per},
\\
{\left[ \nu , \xi \right]_{\rm y}} &:= {\left\langle a_y(\nu\xi) , 1 \right\rangle_\Omega} ,\quad \nu,\, \xi\in{\mathcal E}^{\rm y}_{\rm per},\quad
&{\left[ \nu , \xi \right]_{\rm z}} := {\left\langle a_z(\nu\xi) , 1 \right\rangle_\Omega} ,\quad \nu,\, \xi\in{\mathcal E}^{\rm z}_{\rm per}.
\end{aligned}$$ $${\left[ \vec{f}_1 , \vec{f}_2 \right]_{\Omega}} : = {\left[ f_1^x , f_2^x \right]_{\rm x}} + {\left[ f_1^y , f_2^y \right]_{\rm y}} + {\left[ f_1^z , f_2^z \right]_{\rm z}}, \quad \vec{f}_i = (f_i^x,f_i^y,f_i^z) \in \vec{\mathcal{E}}_{\rm per}, \ i = 1,2.$$
We define the following norms for cell-centered functions. If $\nu\in {\mathcal C}_{\rm per}$, then ${\left\| \nu \right\|}_2^2 := {\left\langle \nu , \nu \right\rangle_\Omega}$; ${\left\| \nu \right\|}_p^p := {\left\langle |\nu|^p , 1 \right\rangle_\Omega}$, for $1\le p< \infty$, and ${\left\| \nu \right\|}_\infty := \max_{1\le i,j,k\le N}\left|\nu_{i,j,k}\right|$. We define norms of the gradient as follows: for $\nu\in{\mathcal C}_{\rm per}$, $${\left\| \nabla_h \nu \right\|}_2^2 : = {\left[ {\nabla_{\! h}}\nu , {\nabla_{\! h}}\nu \right]_{\Omega}} = {\left[ D_x\nu , D_x\nu \right]_{\rm x}} + {\left[ D_y\nu , D_y\nu \right]_{\rm y}} +{\left[ D_z\nu , D_z\nu \right]_{\rm z}},$$ and, more generally, for $1\le p<\infty$, $${\left\| \nabla_h \nu \right\|}_p := \left( {\left[ |D_x\nu|^p , 1 \right]_{\rm x}} + {\left[ |D_y\nu|^p , 1 \right]_{\rm y}} +{\left[ |D_z\nu|^p , 1 \right]_{\rm z}}\right)^{\frac1p} .$$ Higher order norms can be defined. For example, $${\left\| \nu \right\|}_{H_h^1}^2 : = {\left\| \nu \right\|}_2^2+ {\left\| \nabla_h \nu \right\|}_2^2, \quad {\left\| \nu \right\|}_{H_h^2}^2 : = {\left\| \nu \right\|}_{H_h^1}^2 + {\left\| \Delta_h \nu \right\|}_2^2.$$
\[lemma1\] Let $\mathcal{D}$ be an arbitrary periodic, scalar function defined on all of the face center points. For any $\psi, \nu \in {\mathcal C}_{\rm per}$ and any $\vec{f}\in\vec{\mathcal{E}}_{\rm per}$, the following summation by parts formulas are valid: $${\left\langle \psi , \nabla_h\cdot\vec{f} \right\rangle_\Omega} = - {\left[ \nabla_h \psi , \vec{f} \right]_{\Omega}}, \quad {\left\langle \psi , \nabla_h\cdot \left(\mathcal{D}\nabla_h\nu\right) \right\rangle_\Omega} = - {\left[ \nabla_h \psi , \mathcal{D}\nabla_h\nu \right]_{\Omega}} .
\label{lemma 1-0}$$
To facilitate the convergence analysis, we need to introduce a discrete analogue of the space $H_{per}^{-1}\left(\Omega\right)$, as outlined in [@wang11a]. Suppose that $\mathcal{D}$ is a positive, periodic scalar function defined at all of the face center points. For any $\phi\in{\mathcal C}_{\rm per}$, there exists a unique $\psi\in\mathring{\mathcal C}_{\rm per}$ that solves $$\begin{aligned}
\mathcal{L}_{\mathcal{D}}(\psi):= - \nabla_h \cdot\left(\mathcal{D}\nabla_h \psi\right) = \phi - \overline{\phi} ,
\end{aligned}$$ where, recall, $\overline{\phi} := |\Omega|^{-1}{\left\langle \phi , 1 \right\rangle_\Omega}$. We equip this space with a bilinear form: for any $\phi_1,\, \phi_2\in \mathring{\mathcal C}_{\rm per}$, define $${\left\langle \phi_1 , \phi_2 \right\rangle_{\mathcal{L}_{\mathcal{D}}^{-1}}} := {\left[ \mathcal{D}\nabla_h \psi_1 , \nabla_h \psi_2 \right]_{\Omega}},$$ where $\psi_i\in\mathring{\mathcal C}_{\rm per}$ is the unique solution to $$\mathcal{L}_{\mathcal{D}}(\psi_i):= - \nabla_h \cdot\left(\mathcal{D}\nabla_h \psi_i\right) = \phi_i, \quad i = 1, 2.$$ The following identity [@wang11a] is easy to prove via summation-by-parts: $${\left\langle \phi_1 , \phi_2 \right\rangle_{\mathcal{L}_{\mathcal{D}}^{-1}}} = {\left\langle \phi_1 , \mathcal{L}_{\mathcal{D}}^{-1} (\phi_2) \right\rangle_\Omega} = {\left\langle \mathcal{L}_{\mathcal{D}}^{-1} (\phi_1) , \phi_2 \right\rangle_\Omega},$$ and since $\mathcal{L}_{\mathcal{D}}$ is symmetric positive definite, ${\left\langle \ \cdot \ , \ \cdot \ \right\rangle_{\mathcal{L}_{\mathcal{D}}^{-1}}}$ is an inner product on $\mathring{\mathcal C}_{\rm per}$ [@wang11a]. When $\mathcal{D}\equiv 1$, we drop the subscript and write $\mathcal{L}_{1} = \mathcal{L}$, and in this case we usually write ${\left\langle \ \cdot \ , \ \cdot \ \right\rangle_{\mathcal{L}_{\mathcal{D}}^{-1}}} =: {\left\langle \ \cdot \ , \ \cdot \ \right\rangle_{-1,h}}$. In the gerneral setting, the norm associated to this inner product is denoted ${\left\| \phi \right\|}_{\mathcal{L}_{\mathcal{D}}^{-1}} := \sqrt{{\left\langle \phi , \phi \right\rangle_{\mathcal{L}_{\mathcal{D}}^{-1}}}}$, for all $\phi \in \mathring{\mathcal C}_{\rm per}$, but, if $\mathcal{D}\equiv 1$, we write ${\left\| \, \cdot \, \right\|}_{\mathcal{L}_{\mathcal{D}}^{-1}} =: {\left\| \, \cdot \, \right\|}_{-1,h}$.
The first order numerical scheme and the main theoretical results
-----------------------------------------------------------------
We follow the idea of convexity splitting and consider the following semi-implicit, fully discrete schemes: given $\phi^n\in \mathcal{C}_{\rm per}$, find $\phi^{n+1},\mu^{n+1}\in \mathcal{C}_{\rm per}$, such that $$\begin{aligned}
\frac{\phi^{n+1} - \phi^n}{{{\Delta t}}} = & \ - \hat{\cal M}^n \mu^{n+1} \quad \mbox{(AC equation)} ,
\label{scheme-AC_LOG-1}
\\
\frac{\phi^{n+1} - \phi^n}{{{\Delta t}}} = & \ \nabla_h \cdot ( \check{\cal M}^n \nabla_h \mu^{n+1} ) \quad \mbox{(CH equation)} ,
\label{scheme-CH_LOG-1}
\end{aligned}$$ where $$\mu^{n+1} = \ln (1+\phi^{n+1}) - \ln (1-\phi^{n+1}) - \theta_0 \phi^n - \varepsilon^2 \Delta_h \phi^{n+1} .
\label{scheme-mu-0}$$ The mobility approximations are defined as follows: for the Allen-Cahn approximation, $\hat{\cal M}^n = {\cal M}(\phi^n) \in \mathcal{C}_{\rm per}$, quite simply. For the Cahn-Hilliard approximation, we require that $\check{\cal M}^n$ is defined at all of the face center points. This is accomplished via $$\begin{aligned}
\label{scheme-CH_LOG-mobility-1}
\check{\cal M}_{i+{\nicefrac{1}{2}},j,k}^n = & \ {\cal M}(A_x \phi^n_{i+{\nicefrac{1}{2}},j,k}), \ \check{\cal M}_{i,j+{\nicefrac{1}{2}},k}^n = {\cal M}(A_y \phi^n_{i,j+{\nicefrac{1}{2}},k}),
\\
\check{\cal M}_{i,j,k+{\nicefrac{1}{2}}}^n = & \ {\cal M}(A_z \phi^n_{i,j,k+{\nicefrac{1}{2}}}).
\end{aligned}$$
Of course, a point-wise bound for the grid function $\phi^{n+1}$, namely, $-1 < \phi^{n+1}_{i,j,k} < 1$, is needed so that the numerical scheme is well-defined. The main theoretical results are stated below, which guarantee that there exist unique numerical solutions for and , so that the given bound is satisfied. In the first part, we assume that ${\cal M} (\phi) \equiv 1$; the non-constant mobility case will be analyzed in a later section.
For the Allen-Cahn equation, we have
\[AC-positivity\] Assume that ${\cal M} (\phi) \equiv 1$. Given $\phi^n\in\mathcal{C}_{\rm per}$, with ${\left\| \phi^n \right\|}_\infty \le M$, for some $M >0$, there exists a unique solution $\phi^{n+1} \in \mathcal{C}_{\rm per}$ to , with ${\left\| \phi^{n+1} \right\|}_\infty < 1$. Moreover, if the initial data satisfy ${\left\| \phi^0 \right\|}_\infty \le 1 - \delta_0$, there exists $\delta^\star\in (0,1)$, which depends upon $\delta_0$ but is independent of $\varepsilon$ and $n$, so that ${\left\| \phi^n \right\|}_\infty \le 1 - \delta^\star$, $\forall n \in \mathbb{N}$.
For the Cahn-Hilliard equation, we have
\[CH-positivity\] Assume that ${\cal M} (\phi) \equiv 1$. Given $\phi^n\in\mathcal{C}_{\rm per}$, with ${\left\| \phi^n \right\|}_\infty \le M$, for some $M >0$, and $\left|\overline{\phi^n}\right| < 1$, there exists a unique solution $\phi^{n+1}\in\mathcal{C}_{\rm per}$ to , with $\phi^{n+1}-\overline{\phi^n}\in\mathring{\mathcal{C}}_{\rm per}$ and ${\left\| \phi^{n+1} \right\|}_\infty < 1$.
Theoretical justification of the positivity-preserving properties {#sec: proof}
=================================================================
Proof of Theorem \[AC-positivity\]
----------------------------------
The analysis for the approximation to the Allen-Cahn equation is given first.
We observe that, the numerical solution of is equivalent to the minimization of the discrete energy functional $$\begin{aligned}
\mathcal{J}^n(\phi) &:=& \frac{1}{2 {{\Delta t}}} \| \phi - \phi^n \|_2^2 + {\left\langle 1+ \phi , \ln (1+\phi) \right\rangle_\Omega} + {\left\langle 1-\phi , \ln (1-\phi) \right\rangle_\Omega}
\nonumber
\\
&&
+ \frac{\varepsilon^2}{2} \| \nabla_h \phi \|_2^2 - \theta_0 {\left\langle \phi , \phi^n \right\rangle_\Omega},
\label{AC_LOG-positive-1}
\end{aligned}$$ over the compact, convex admissible set $A_h = \left\{ \phi \in \mathcal{C}_{\rm per} \ \middle| \ {\left\| \phi \right\|}_\infty \le 1 \right\} \subset \mathbb{R}^{N^3}$. We observe that $\mathcal{J}^n$ is a strictly convex function over this domain. We wish to prove that there exists a minimizer of $\mathcal{J}^n$ at an interior point of $A_h$. To this end, consider the following closed domain: for a given $\delta\in (0,\nicefrac{1}{2})$, $$A_{h,\delta} := \left\{ \phi \in \mathcal{C}_{\rm per} \ \middle| \ {\left\| \phi \right\|}_\infty \le 1 - \delta \right\} \subset A_h .
\label{AC_LOG-positive-2}$$ Since $A_{h,\delta}$ is a compact and convex set in $\mathbb{R}^{N^3}$, there exists a (not necessarily unique) minimizer of $\mathcal{J}^n$ over $A_{h,\delta}$. The key point of our positivity analysis is that such a minimizer could not occur on the boundary of $A_{h,\delta}$, if $\delta$ is small enough.
Assume a minimizer of $\mathcal{J}^n$ over $A_{h,\delta}$, denote it by $\phi^\star$, occurs at a boundary point. There is at least one grid point $\vec{\alpha}_0 = (i_0,j_0,k_0)$ such that $|\phi^\star_{\vec{\alpha}_0}| = 1 -\delta$. First, let us assume, that $\phi^\star_{\vec{\alpha}_0} = \delta - 1$, so that the grid function $\phi^\star$ has a global minimum at $\vec{\alpha}_0$. Since $\mathcal{J}^n$ is smooth over $A_{h,\delta}$, for all $\psi\in \mathcal{C}_{\rm per}$, the directional derivative is $$d_s \mathcal{J}^n(\phi^\star+s\psi)|_{s=0} = {\left\langle \frac{\phi^\star-\phi^n}{\Delta t}+\ln (1+\phi^\star) - \ln (1-\phi^\star) - \theta_0 \phi^n - \varepsilon^2 \Delta_h \phi^\star , \psi \right\rangle_\Omega}.$$ If the direction grid function is of the form $\psi_{i,j,k} = \delta_{i,i_0}\delta_{j,j_0}\delta_{k,k_0}$, where $\delta_{k,\ell}$ denotes the usual Kronecker delta function, $$\frac{1}{h^3} d_s \mathcal{J}^n(\phi^\star+s\psi)|_{s=0} = \ln \delta - \ln (2-\delta) - \theta_0 \phi^n_{\vec{\alpha}_0} - \varepsilon^2 \Delta_h \phi^\star_{\vec{\alpha}_0} + \frac{\delta -1 - \phi^n_{\vec{\alpha}_0}}{{{\Delta t}}} . \
\label{AC_LOG-positive-3}$$ Since $\phi^\star$ has a minimum at the grid point $\vec{\alpha}_0 = (i_0, j_0, k_0)$, it follows that $$\phi^\star_{\vec{\alpha}_0} = -1 + \delta \le \phi^\star_{i,j,k}, \quad \forall \ (i,j,k) \ne \vec{\alpha}_0, \quad \mbox{and} \quad \Delta_h \phi^\star_{\vec{\alpha}_0} \ge 0 .
\label{AC_LOG-positive-4}$$ The bound ${\left\| \phi^n \right\|}_\infty \le M$ and the fact that $\delta\in (0,\nicefrac{1}{2})$ imply that $$\delta -1 - \phi^n_{\vec{\alpha}_0} \le \delta -1 + M < M-\nicefrac{1}{2}.
\label{AC_LOG-positive-5}$$ Define the parameters $$\beta_0 := 2 \left( 1 + \exp \left\{ \theta_0 M + \frac{M - {\nicefrac{1}{2}}}{{{\Delta t}}} \right\} \right)^{-1}, \quad \beta := \min(\nicefrac{1}{2},\beta_0).$$ If $\delta \in (0,\beta)$, then $$\ln \delta - \ln (2-\delta) - \theta_0 \phi^n_{\vec{\alpha}_0} + \frac{ \delta - 1 - \phi^n_{\vec{\alpha}_0}}{{{\Delta t}}} < 0 .
\label{AC_LOG-positive-6}$$ Using the estimates – in reveals that, provided $0 < \delta < \beta$, $$\frac{1}{h^3} d_s \mathcal{J}^n(\phi^\star+s\psi)|_{s=0} < 0.
\label{AC_LOG-positive-7}$$ This yields a contradiction that $\mathcal{J}^n$ takes a global minimum at $\phi^\star$ over $A_{h,\delta}$, because the directional derivative at this boundary point is *negative* in a direction pointing into the interior of $A_{h,\delta}$. In other words, going in the direction of $\psi$, we are certain to find an interior point $\phi^\star+s\psi$, provided $s>0$ is sufficiently small, such that $\mathcal{J}^n(\phi^\star+s\psi) < \mathcal{J}^n(\phi^\star)$.
Using quite similar arguments, if $\phi^\star_{\vec{\alpha}_0}=1-\delta$, and $\delta\in (0,\beta)$, we would find that $$\frac{1}{h^3} d_s \mathcal{J}^n(\phi^\star+s\psi)|_{s=0} > 0.
\label{AC_LOG-positive-7-b}$$
A combination of these two facts shows that the global minimum of $\mathcal{J}^n$ over $A_{h,\delta}$ could only possibly occur at an interior point, when $\delta\in (0,\beta)$. We conclude that there must be a solution $\phi \in \left(A_{h,\delta}\right)^{\mathrm{o}}$, the interior region of $A_{h,\delta}$, so that for all $\psi\in\mathcal{C}_{\rm per}$, $$0 = d_s \mathcal{J}^n(\phi+s\psi)|_{s=0} .
\label{AC_LOG-positive-8}$$ which is equivalent to the numerical solution of , provided $\delta\in (0,\beta)$. The existence of a “positive" numerical solution is, therefore, established. In addition, since $\mathcal{J}^n$ is a strictly convex function over $A_h$, the uniqueness analysis for this numerical solution is straightforward.
For the second part of this theorem, let us make the *a priori* assumption that, for some $\delta_0\in (0,1)$, ${\left\| \phi^0 \right\|}_\infty = 1-\delta_0$. Furthermore, choose $\delta_1 \in (0, 1)$ so that $$\delta_1 < \frac{2}{ \exp(\theta_0 +1)}.$$ Define $\delta^\star = \min(\delta_0,\delta_1)$, and consider the space $A_{h,\delta^\star}$. Suppose that $\phi^{1,\star}$ is the minimizer of $J^0$ over $A_{h,\delta^\star}$. If we use an analysis similar to that of the first part, we can show that, if $\phi^{1,\star}$ is a boundary point of $A_{h,\delta^\star}$, we obtain a contradiction. Specifically, if at $\vec{\alpha}_0 = (i_0,j_0,k_0)$, $\phi^{\star,1}_{\vec{\alpha}_0} = \delta^\star - 1$ (a minimum point), then we find $$\begin{aligned}
0 = & \ \ln \delta^\star - \ln (2-\delta^\star) - \theta_0 \phi^0_{\vec{\alpha}_0} - \varepsilon^2 \Delta_h \phi^{1,\star}_{\vec{\alpha}_0} + \frac{\delta^\star -1 - \phi^0_{\vec{\alpha}_0}}{{{\Delta t}}}
\\
\le & \ \ln\delta^\star - \ln (2-\delta^\star) + \theta_0 < 0 .
\end{aligned}$$ Similarly, if at $\vec{\alpha}_0 = (i_0,j_0,k_0)$, $\phi^{\star,1}_{\vec{\alpha}_0} = 1-\delta^\star$ (a maximum point), then we likewise discover that $0>0$. This implies, ultimately, that the minimizer $\phi^1\in A_h$ of $J^0$ satisfies the bound $${\left\| \phi^1 \right\|}_\infty < 1- \delta^\star .$$ Clearly, $\delta^\star$ only depends on $\delta_0$ and $\theta_0$; it is independent of $\varepsilon$. This argument can be continued inductively, and we can conclude that, for any $n\in\mathbb{N}$, $${\left\| \phi^n \right\|}_\infty < 1- \delta^\star .$$ The proof of Theorem \[AC-positivity\] is complete.
Proof of Theorem \[CH-positivity\]
----------------------------------
If solutions to the Cahn-Hilliard scheme exist, it is clear that, for any $n\in\mathbb{N}$, $$\overline{\phi}_0 := |\Omega|^{-1}{\left\langle \phi^0 , 1 \right\rangle_\Omega} = |\Omega|^{-1}{\left\langle \phi^1 , 1 \right\rangle_\Omega} = \cdots = |\Omega|^{-1}{\left\langle \phi^n , 1 \right\rangle_\Omega} = \overline{\phi}_n,$$ with $|\overline{\phi}_n |< 1$. Thus we expect $\langle \phi^n - \overline{\phi}_0 , 1 \rangle_\Omega =0$. For the proof of Theorem \[CH-positivity\], we need the following technical lemma:
\[CH-positivity-Lem-0\] Suppose that $\phi_1$, $\phi_2 \in \mathcal{C}_{\rm per}$, with ${\left\langle \phi_1 - \phi_2 , 1 \right\rangle_\Omega} = 0$, that is, $\phi_1 - \phi_2\in \mathring{\mathcal{C}}_{\rm per}$, and assume that ${\left\| \phi_1 \right\|}_\infty < 1$, ${\left\| \phi_2 \right\|}_\infty \le M$. Then, we have the following estimate: $${\left\| \mathcal{L}^{-1} (\phi_1 - \phi_2) \right\|}_\infty \le C_1 ,
\label{CH_LOG-Lem-0}$$ where $C_1>0$ depends only upon $M$ and $\Omega$. In particular, $C_1$ is independent of the mesh spacing $h$.
Define $\psi := \phi_1 - \phi_2\in\mathring{\mathcal{C}}_{\rm per}$. Thus ${\left\| \psi \right\|}_{\infty} < M+1$. This fact implies that $${\left\| \psi \right\|}_2 = {\left\| \phi_1-\phi_2 \right\|}_2 \le (M+1) | \Omega |^{1/2} .
\label{CH_LOG-Lem-1}$$ Meanwhile, we denote $v = \mathcal{L}^{-1}(\psi)\in\mathring{\mathcal{C}}_{\rm per}$, so that $\mathcal{L}(v) = \psi$ with $\overline{v}=0$. Suppose that $N$ is odd, for simplicity, and $N=2K+1$. (The even case is handled in a very simliar manner.) Since $v\in \mathcal{C}_{\rm per}$ it has the discrete Fourier representation of the form $$v_{i,j,k} = \sum^{K}_{\ell,m,n=-K}
\hat{v}^N_{\ell,m,n} {\rm e}^{2 \pi i ( \ell p_{i} + m p_{j} + n p_{k} )/ L } ,
\label{def:Fourier-1}$$ where $p_{i} = (i-{\nicefrac{1}{2}})\cdot h$ and $\hat{v}^N_{\ell,m,n}$ are the discrete Fourier coefficients given by the discrete Fourier transform (DFT): $$\hat{v}^N_{i,j,k} := \frac{h^3}{L^3} \sum^{K}_{\ell,m,n=-K}
v_{\ell,m,n} {\rm e}^{-2 \pi i ( \ell p_{i} + m p_{j} + n p_{k} )/ L } .$$ Since $v\in\mathring{\mathcal{C}}_{\rm per}$, $\hat{v}_{0,0,0}^N = 0$. We define the Fourier interpolant of the grid function $v$ as $$\label{def:extension}
\mathsf{v}(x,y,z) := \sum^{K}_{\ell,m,n=-K} \hat{v}^N_{\ell,m,n} {\rm e}^{2 \pi i ( \ell x + m y + n z )/ L } , \quad x,y,z\in\mathbb{R} ,$$ and observe that $\mathsf{v}\in C_{\rm per}^\infty(\Omega)$. Parseval’s identity (at both the discrete and continuous levels) implies that $$\| v \|_2^2 = L^3 \sum^{K}_{\ell,m,n=-K}
|\hat{v}^N_{\ell,m,n}|^2 = {\left\| \mathsf{v} \right\|}_{L^2(\Omega)}^2 .$$ For the comparison between the discrete and continuous Laplacians, we start with the following Fourier expansions: $$\begin{aligned}
\Delta_h^x v_{i,j,k} &:=& \frac{v_{i+1,j,k}-2 v_{i,j,k}+v_{i-1,j,k}}{h^2}
\nonumber
\\
&=& \sum^{K}_{\ell,m,n=-K} \mu_{\ell} \hat{v}^N_{\ell,m,n} {\rm e}^{2 \pi i ( \ell x_{i} + m y_{j} + n z_{k} )/ L } ,
\\
\partial_x^2 \mathsf{v} (x,y,z) &=& \sum^{K}_{\ell,m,n=-K}
\nu_{\ell} \hat{v}^N_{\ell,m,n} {\rm e}^{2 \pi i
( \ell x + m y + n z )/ L } ,
\end{aligned}$$ with $$\mu_{\ell} = -\frac{4\sin^2 {\frac{\ell\pi h}{L}}}{h^2}, \quad
\nu_{\ell} = -\frac{4 \ell^2 \pi^2}{L^2}.$$ In turn, an application of Parseval’s identity yields $$\begin{aligned}
{\left\| \Delta_h^x v \right\|}^2_2 = L^3\sum^{K}_{\ell,m,n=-K}|\mu_{\ell}|^2|\hat{v}^N_{\ell,m,n}|^2, \\
{\left\| \partial_x^2 \mathsf{v} \right\|}^2_{L^2} =
L^3\sum^{K}_{\ell,m,n=-K}|\nu_{\ell}|^2|
\hat{v}^N_{\ell,m,n}|^2.\end{aligned}$$ The comparison of Fourier eigenvalues shows that $$\frac{4}{\pi^2} |\nu_{\ell}| \le |\mu_{\ell}| \le |\nu_{\ell}|,
\quad \mbox{for} \quad -K \le \ell \le K .$$ This indicates that $$\frac{4}{\pi^2} {\left\| \partial_x^2 \mathsf{v} \right\|}_{L^2} \le {\left\| \Delta_h^x v \right\|}_2 \le {\left\| \partial_x^2 \mathsf{v} \right\|}_{L^2}.$$ Similar estimates can be derived to reveal that $$\frac{4}{\pi^2}\|\Delta \mathsf{v}\|_{L^2} \le \|\Delta_{h} v \|_2 = \| \psi \|_2 \le \|\Delta \mathsf{v}\|_{L^2} ,
\label{CH_LOG-Lem-2}$$ which in turn yields that $\|\Delta \mathsf{v}\|_{L^2} \le \frac{(M+1) \pi^2 | \Omega |^{1/2}}{4}$.
Meanwhile, the following identity is obvious: $$\int_\Omega \mathsf{v} \, d {\bf x} = 0 , \quad \mbox{since} \quad \hat{v}_{0,0,0}^N = 0.
\label{CH_LOG-Lem-3}$$ Subsequently, an application of elliptic regularity implies that $$\| \mathsf{v} \|_{H^2} \le C \left( \left| \int_\Omega \mathsf{v} \, d {\bf x} \right| + \|\Delta \mathsf{v}\|_{L^2} \right) \le C_0 (M+1) | \Omega |^{1/2} ,
\label{CH_LOG-Lem-4}$$ for some constant $C_0>0$ that only depends upon $\Omega$. Since the grid function $v$ is the projection of the smooth function $\mathsf{v}$ into the cell-centered grid, the following discrete $\ell^\infty$ bound is clear: $$\| v \|_\infty \le \| \mathsf{v} \|_{L^\infty} \le C \| \mathsf{v} \|_{H^2} \le C_0 (M+1) | \Omega |^{1/2} ,
\label{CH_LOG-Lem-5}$$ in which the 3-D Sobolev embedding has been used in the second step. The proof of Lemma \[CH-positivity-Lem-0\] is completed by taking $C_1 := C_0 (M+1) | \Omega |^{1/2}$.
Now we proceed into the proof of Theorem \[CH-positivity\].
The numerical solution of is a minimizer of the following discrete energy functional: $$\begin{aligned}
\mathcal{J}^n (\phi) &:=& \frac{1}{2 {{\Delta t}}} {\left\| \phi - \phi^n \right\|}_{-1,h}^2 + {\left\langle 1+ \phi , \ln (1+\phi) \right\rangle_\Omega} + {\left\langle 1-\phi , \ln (1-\phi) \right\rangle_\Omega}
\nonumber
\\
&& + \frac{\varepsilon^2}{2} \| \nabla_h \phi \|_2^2 - \theta_0 {\left\langle \phi , \phi^n \right\rangle_\Omega},
\label{CH_LOG-positive-1}
\end{aligned}$$ over the admissible set $$A_h := \left\{ \phi \in \mathcal{C}_{\rm per} \ \middle| \ {\left\| \phi \right\|}_\infty \le 1, \quad {\left\langle \phi-\overline{\phi}_0 , 1 \right\rangle_\Omega}=0 \right\} \subset \mathbb{R}^{N^3}.$$ Observe that $\mathcal{J}^n$ is a strictly convex function over this domain.
To facilitate the analysis below, we transform the minimization problem into an equivalent one. Consider the functional $$\begin{aligned}
\mathcal{F}^n (\varphi) &:=& \mathcal{J}^n (\varphi + \overline{\phi}_0)
\\
&=& \frac{1}{2 {{\Delta t}}} {\left\| \varphi + \overline{\phi}_0 - \phi^n \right\|}_{-1,h}^2 + {\left\langle 1+ \varphi + \overline{\phi}_0 , \ln ( 1 + \varphi + \overline{\phi}_0 ) \right\rangle_\Omega}
\nonumber
\\
&& + {\left\langle 1- \varphi - \overline{\phi}_0 , \ln (1-\varphi - \overline{\phi}_0) \right\rangle_\Omega} + \frac{\varepsilon^2}{2} {\left\| \nabla_h \varphi \right\|}_2^2 - \theta_0 {\left\langle \varphi + \overline{\phi}_0 , \phi^n \right\rangle_\Omega} ,
\label{CH_LOG-positive-1-b}
\end{aligned}$$ defined on the set $$\mathring{A}_h := \left\{ \varphi \in \mathring{\mathcal{C}}_{\rm per} \ \middle| \ -1-\overline{\phi}_0 \le \varphi \le 1-\overline{\phi}_0 \right\} \subset \mathbb{R}^{N^3}.$$ If $\varphi\in \mathring{A}_h$ minimizes $\mathcal{F}^n$, then $\phi := \varphi + \overline{\phi}_0\in A_h$ minimizes $\mathcal{J}^n$, and *vice versa*. Next, we prove that there exists a minimizer of $\mathcal{F}^n$ over the domain $\mathring{A}_h$. Similar to our previous arguments, we consider the following closed domain: for $\delta\in (0,{\nicefrac{1}{2}})$, $$\mathring{A}_{h,\delta} := \left\{ \varphi \in \mathring{\mathcal{C}}_{\rm per} \ \middle| \ \delta -1-\overline{\phi}_0 \le \varphi \le 1-\delta-\overline{\phi}_0 \right\} \subset \mathbb{R}^{N^3}.
\label{CH_LOG-positive-2}$$ Since $\mathring{A}_{h,\delta}$ is a bounded, compact, and convex set in the subspace $\mathring{\mathcal{C}}_{\rm per}$, there exists a (not necessarily unique) minimizer of $\mathcal{F}^n$ over $\mathring{A}_{h, \delta}$. The key point of the positivity analysis is that such a minimizer could not occur on the boundary of $\mathring{A}_{h,\delta}$, if $\delta$ is sufficiently small. To be more explicit, by the boundary of $\mathring{A}_{h, \delta}$, we mean the locus of points $\psi\in\mathring{A}_{h, \delta}$ such that ${\left\| \psi+\overline{\phi}_0 \right\|}_\infty = 1-\delta$, precisely.
To get a contradiction, suppose that the minimizer of $\mathcal{F}^n$, call it $\varphi^\star$ occurs at a boundary point of $\mathring{A}_{h,\delta}$. There is at least one grid point $\vec{\alpha}_0 = (i_0,j_0,k_0)$ such that $|\varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0| = 1 -\delta$. First, let us assume, that $\varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0 = \delta - 1$, so that the grid function $\varphi^\star$ has a global minimum at $\vec{\alpha}_0$. Suppose that $\vec{\alpha}_1 = (i_1,j_1,k_1)$ is a grid point at which $\varphi^\star$ achieves its maximum. By the fact that $\overline{\varphi^\star} = 0$, it is obvious that $$1-\delta \ge \varphi^\star_{\vec{\alpha}_1}+\overline{\phi}_0 \ge \overline{\phi}_0.$$ Since $\mathcal{F}^n$ is smooth over $\mathring{A}_{h,\delta}$, for all $\psi\in \mathring{\mathcal{C}}_{\rm per}$, the directional derivative is $$\begin{aligned}
d_s \mathcal{F}^n(\varphi^\star+s\psi)|_{s=0} = & \ {\left\langle \ln (1+\varphi^\star+\overline{\phi}_0) - \ln (1-\varphi^\star-\overline{\phi}_0) , \psi \right\rangle_\Omega}
\\
& - {\left\langle \theta_0 \phi^n + \varepsilon^2 \Delta_h \varphi^\star , \psi \right\rangle_\Omega}
\\
& + \frac{1}{\Delta t}{\left\langle (-\Delta_h)^{-1}\left(\varphi^\star-\phi^n+\overline{\phi}_0 \right) , \psi \right\rangle_\Omega}.
\end{aligned}$$ This time, let us pick the direction $\psi \in \mathring{\mathcal{C}}_{\rm per}$, such that $$\psi_{i,j,k} = \delta_{i,i_0}\delta_{j,j_0}\delta_{k,k_0} - \delta_{i,i_1}\delta_{j,j_1}\delta_{k,k_1} .$$ Then the derivative may be expressed as $$\begin{aligned}
\frac{1}{h^3} d_s \mathcal{F}^n(\varphi^\star+s\psi)|_{s=0} &=& \ln (1+ \varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0) - \ln (1-\varphi^\star_{\vec{\alpha}_0}-\overline{\phi}_0)
\nonumber
\\
&& - \ln (1+ \varphi^\star_{\vec{\alpha}_1}+\overline{\phi}_0) + \ln (1-\varphi^\star_{\vec{\alpha}_1}-\overline{\phi}_0)
\nonumber
\\
&& - \theta_0 ( \phi^n_{\vec{\alpha}_0} - \phi^n_{\vec{\alpha}_1} ) - \varepsilon^2 ( \Delta_h \varphi^\star_{\vec{\alpha}_0} - \Delta_h \varphi^\star_{\vec{\alpha}_1} )
\nonumber
\\
&& + \frac{1}{{{\Delta t}}} (-\Delta_h)^{-1} ( \varphi^\star - \phi^n +\overline{\phi}_0 )_{\vec{\alpha}_0}
\nonumber
\\
&& - \frac{1}{{{\Delta t}}}(-\Delta_h)^{-1} ( \varphi^\star - \phi^n + \overline{\phi}_0)_{\vec{\alpha}_1} . \label{CH_LOG-positive-4}
\end{aligned}$$ For simplicity, now let us write $\phi^\star := \varphi^\star +\overline{\phi}_0$. Since $\phi^\star_{\vec{\alpha}_0} = -1 + \delta$ and $\phi^\star_{\vec{\alpha}_1} \ge \overline{\phi}_0$, we have $$\ln (1+ \phi^\star_{\vec{\alpha}_0}) - \ln (1-\phi^\star_{\vec{\alpha}_0}) - \ln (1+ \phi^\star_{\vec{\alpha}_1}) + \ln (1-\phi^\star_{\vec{\alpha}_1}) \le \ln \frac{\delta}{2 - \delta} - \ln \frac{1+\overline{\phi}_0}{1-\overline{\phi}_0} .
\label{CH_LOG-positive-5}$$ Since $\phi^\star$ takes a minimum at the grid point $\vec{\alpha}_0$, with $\phi^\star_{\vec{\alpha}_0} = -1 + \delta \le \phi^\star_{i,j,k}$, for any $(i,j,k)$, and a maximum at the grid point $\vec{\alpha}_1$, with $\phi^\star_{\vec{\alpha}_1} \ge \phi^\star_{i,j,k}$, for any $(i,j,k)$, $$\Delta_h \phi^\star_{\vec{\alpha}_0} \ge 0 , \quad \Delta_h \phi^\star_{\vec{\alpha}_1} \le 0 .
\label{CH_LOG-positive-6}$$ For the numerical solution $\phi^n$ at the previous time step, the *a priori* assumption ${\left\| \phi^n \right\|}_\infty \le M$ indicates that $$-2 M \le \phi^n_{\vec{\alpha}_0} - \phi^n_{\vec{\alpha}_1} \le 2M .
\label{CH_LOG-positive-8}$$ For the last two terms appearing in (\[CH\_LOG-positive-4\]), we apply Lemma \[CH-positivity-Lem-0\] and obtain $$- 2 C_1 \le (-\Delta_h)^{-1} ( \phi^\star - \phi^n )_{\vec{\alpha}_0} - (-\Delta_h)^{-1} ( \phi^\star - \phi^n )_{\vec{\alpha}_1} \le 2 C_1 .
\label{CH_LOG-positive-9}$$ Consequently, a substitution of (\[CH\_LOG-positive-5\]) – (\[CH\_LOG-positive-9\]) into (\[CH\_LOG-positive-4\]) yields the following bound on the directional derivative: $$\frac{1}{h^3} d_s \mathcal{F}^n(\varphi^\star+s\psi)|_{s=0} \le \ln \frac{\delta}{2 - \delta} - \ln \frac{1+\overline{\phi}_0}{1-\overline{\phi}_0} + 2M \theta_0 + 2 C_1 {{\Delta t}}^{-1} .
\label{CH_LOG-positive-10}$$ We denote $C_2 = 2M \theta_0 + 2 C_1 {{\Delta t}}^{-1}$. Note that $C_2$ is a constant for a fixed ${{\Delta t}}$, though it becomes singular as ${{\Delta t}}\to 0$. However, for any fixed ${{\Delta t}}$, we may choose $\delta\in(0,{\nicefrac{1}{2}})$ sufficiently small so that $$\ln \frac{\delta}{2 - \delta} - \ln \frac{1+\overline{\phi}_0}{1-\overline{\phi}_0} + C_2 < 0 . \label{CH_LOG-positive-11}$$ This in turn shows that, provided $\delta$ satisfies (\[CH\_LOG-positive-11\]), $$\frac{1}{h^3} d_s \mathcal{F}^n(\varphi^\star+s\psi)|_{s=0} < 0 .
\label{CH_LOG-positive-12}$$ As before, this contradicts the assumption that $\mathcal{F}^n$ has a minimum at $\varphi^\star$, since the directional derivative is negative in a direction pointing into the interior of $\mathring{A}_{h,\delta}$.
Using very similar arguments, we can also prove that the global minimum of $\mathcal{F}^n$ over $\mathring{A}_{h,\delta}$ could not occur at a boundary point $\varphi^\star$ such that $\varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0 = 1-\delta$, for some $\vec{\alpha}_0$, so that the grid function $\varphi^\star$ has a global maximum at $\vec{\alpha}_0$. The details are left to interested readers.
A combination of these two facts shows that, the global minimum of $\mathcal{F}^n$ over $\mathring{A}_{h,\delta}$ could only possibly occur at interior point $\varphi\in (\mathring{A}_{h,\delta})^{\rm o}\subset (\mathring{A}_h)^{\rm o}$. We conclude that there must be a solution $\phi = \varphi+\overline{\phi}_0\in A_h$ that minimizes $\mathcal{J}^n$ over $A_h$, which is equivalent to the numerical solution of (\[scheme-CH\_LOG-1\]), (\[scheme-mu-0\]). The existence of the numerical solution is established.
In addition, since $\mathcal{J}^n$ is a strictly convex function over $A_h$, the uniqueness analysis for this numerical solution is straightforward. The proof of Theorem \[CH-positivity\] is complete.
The positivity-preserving analysis is based on a key fact that the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values. As a result, the point-wise positivity for the logarithmic arguments could be derived as long as the numerical solution at the previous time step stays bounded between $-M$ and $M$ (even if $M >1$), and the initial average stays between $-1$ and 1. This is a modest improvement to the results in [@elliott92a], in which the authors constructed a cut-off energy functional to avoid the singularity.
The proof of Theorem \[AC-positivity\] follows a standard maximum principle type argument; that is the key reason why we are able to obtain a uniform separation bound for the numerical solution (\[scheme-AC\_LOG-1\]): ${\left\| \phi^n \right\|}_\infty \le 1 - \delta^\star$, if the initial data satisfy a similar condition. For the Cahh-Hilliard flow, such a uniform bound is not available for the corresponding numerical solution (\[scheme-CH\_LOG-1\]), (\[scheme-mu-0\]) any more, since the maximum principle could not be directly applied to an $H^{-1}$ gradient flow. In addition, the mass conservation constraint has made the corresponding analysis more involved.
For the Cahn-Hilliard flow, lack of maximum principle has been an essential mathematical challenge. To overcome this difficulty, we have to obtain the point-wise bound for the linear chemical potential part. With the help of the *a priori* $\ell^\infty$ bound of the numerical solution we are investigating, an $O ({{\Delta t}}^{-1})$ estimate is derived for such a bound, which is contained in the form of $C_2$. Such a bound is a fixed constant for a fixed ${{\Delta t}}$, while it becomes singular as ${{\Delta t}}\to 0$.
Another key idea of this analysis should also be mentioned: although the nonlinear term contains a singular limit as $\phi$ approaches either $-1$ or 1, the convexity of this nonlinear potential has greatly aided in the positivity analysis.
In addition to the positivity-preserving property, the semi-implicit nature of our proposed scheme: implicit treatment for the logarithmic terms and the surface diffusion term, combined with an explicit treatment for the linear stretching/expansive term, ensures the unique solvability. In comparison, for the fully implicit scheme analyzed in [@elliott92a], the unique solvability is only available under a time step constraint: ${{\Delta t}}\le \frac{4 \varepsilon^2}{\theta_0^2}$. In fact, the existence of the positivity-preserving numerical solution could also be established for the fully implicit Euler scheme, using the same idea presented in this section. Only the uniqueness analysis of the numerical solution requires such a time step constraint.
For simplicity of presentation, we only analyze the finite difference scheme over a rectangular domain in this article. The idea of this positivity analysis could be similarly extended to the finite element and pseudo-spectral spatial approximations, as well as the case of a general domain. The details may be considered in the future works.
The positivity preserving property in the non-constant mobility case
--------------------------------------------------------------------
In this subsection we look at the numerical scheme (\[scheme-CH\_LOG-1\]), (\[scheme-mu-0\]) with a nonconstant mobility, but with the strict positivity assumption that ${\cal M} (x) \ge {\cal M}_0 >0 $, for all $x\in[-1,1]$. Then, for any $\phi \in\mathring{\mathcal C}_{\Omega}$, there exists a unique $\psi\in\mathring{\mathcal C}_{\Omega}$ that solves $$\mathcal{L}_{\check{\cal M}^n} (\psi):= - \nabla_h \cdot ( \check{\cal M}^n \nabla_h \psi ) = \phi .
\label{CH-mobility-0}$$ In turn, the following norm may be introduced: $${\left\| \phi \right\|}_{\mathcal{L}_{\check{\cal M}^n}^{-1} } = \sqrt{{\left\langle \phi , \mathcal{L}_{\check{\cal M}^n}^{-1} (\phi) \right\rangle_\Omega}} .$$ Similar to Lemma \[CH-positivity-Lem-0\], the following estimate is needed in the positivity analysis.
\[CH-mobility-positivity-Lem-0\] Suppose that $\phi_1$, $\phi_2 \in \mathcal{C}_{\rm per}$, with ${\left\langle \phi_1 - \phi_2 , 1 \right\rangle_\Omega} = 0$, that is, $\phi_1 - \phi_2\in \mathring{\mathcal{C}}_{\rm per}$, and assume that ${\left\| \phi_1 \right\|}_\infty < 1$, ${\left\| \phi_2 \right\|}_\infty \le M$. Then, we have the following estimate: $${\left\| \mathcal{L}_{\check{\cal M}^n}^{-1} (\phi_1 - \phi_2) \right\|}_\infty \le C_4 := C_3 \mathcal{M}_0^{-1} h^{-1/2} ,
\label{CH_LOG-mobility-Lem-0}$$ where $C_3>0$ depends only upon $M$ and $\Omega$.
Define $\psi :=\phi_1 - \phi_2$ and $v:= \mathcal{L}_{\check{\mathcal{M}}^n}^{-1}(\psi)$ Similar to the estimate , we get $$\begin{aligned}
\| \psi \|_2 = \| \phi_1 - \phi_2 \|_2 \le (M+1) | \Omega |^{1/2} . \label{CH_LOG-mobility-Lem-1}\end{aligned}$$ To obtain a bound for $v\in\mathring{\mathcal{C}}_{\rm per}$, observe that, by summation-by-parts, $$\mathcal{M}_0{\left\| \nabla_h v \right\|}_2^2 \le {\left[ \check{\cal M}^n \nabla_h v , \nabla_h v \right]_{\Omega}} = {\left\langle \psi , v \right\rangle_\Omega} \le \| \psi \|_2 \cdot \| v \|_2 \le C_{\rm P} \| \psi \|_2 \cdot \| \nabla_h v \|_2 ,
\label{CH_LOG-mobility-Lem-2}$$ in which the discrete Poincaré inequality, $$\| \psi \|_2 \le C_{\rm P} \| \nabla_h \psi \|_2, \quad \forall \psi \in \mathring{\mathcal{C}}_{\rm per},$$ has been applied in the last step. Therefore $${\left\| \nabla_h v \right\|}_2 \le C_{\rm P} \mathcal{M}_0^{-1} {\left\| \psi \right\|}_2 .
\label{CH_LOG-mobility-Lem-3}$$ Subsequently, an application of a 3-D inverse inequality, for $v\in \mathring{\mathcal{C}}_{\rm per}$, leads to $$\begin{aligned}
{\left\| v \right\|}_\infty &\le& C_{\rm I} h^{-1/2} {\left\| \nabla_h v \right\|}_2 \le C_{\rm I} h^{-1/2} C_{\rm P} \mathcal{M}_0^{-1} {\left\| \psi \right\|}_2
\nonumber
\\
&\le & C_{\rm I} h^{-1/2} C_{\rm P} \mathcal{M}_0^{-1} (M+1) | \Omega |^{1/2} ,
\label{CH_LOG-mobility-Lem-4}
\end{aligned}$$ where the constant in the inverse inequality, $C_{\rm I}>0$, is independent of $h$. Therefore, is valid, with $C_3 := C_{\rm I} C_{\rm P} (M+1) | \Omega |^{1/2}$. This completes the proof.
The positivity-preserving property of the numerical scheme (\[scheme-CH\_LOG-1\]), (\[scheme-mu-0\]) for the non-constant mobility case is stated below.
\[CH-mobility-positivity\] Assume that ${\cal M} (x) \ge {\cal M}_0 >0 $, for all $x\in[-1,1]$. Given $\phi^n\in\mathcal{C}_{\rm per}$, with ${\left\| \phi^n \right\|} \le M$, for some $M >0$, and $\left|\overline{\phi^n}\right| < 1$, there exists a unique solution $\phi^{n+1}\in\mathcal{C}_{\rm per}$ to , with $\phi^{n+1}-\overline{\phi^n}\in\mathring{\mathcal{C}}_{\rm per}$ and ${\left\| \phi^{n+1} \right\|}_\infty < 1$.
The proof of this theorem follows the same ideas as in that of Theorem \[CH-positivity\]; we just provide a brief outline. Similar to , the numerical solution of (\[scheme-CH\_LOG-1\]) is equivalent to the minimization of the following discrete energy functional: $$\begin{aligned}
\mathcal{J}^n (\phi) &=& \frac{1}{2 {{\Delta t}}} {\left\| \phi - \phi^n \right\|}_{ \mathcal{L}_{\check{\cal M}^n}^{-1}}^2 + {\left\langle 1+ \phi , \ln (1+\phi) \right\rangle_\Omega} + {\left\langle 1-\phi , \ln (1-\phi) \right\rangle_\Omega}
\nonumber
\\
& & + \frac{\varepsilon^2}{2} {\left\| \nabla_h \phi \right\|}_2^2 - \theta_0 {\left\langle \phi , \phi^n \right\rangle_\Omega} ,
\label{CH_LOG-mobility-positive-1}
\end{aligned}$$ over the admissible set $$A_h := \left\{ \phi \in \mathcal{C}_{\rm per} \ \middle| \ {\left\| \phi \right\|}_\infty \le 1, \ {\left\langle \phi-\overline{\phi}_0 , 1 \right\rangle_\Omega}=0 \right\} .$$ The equivalent minimization problem is similar to previous one: find a minimizer $\varphi\in \mathring{A}_h$ the functional $$\mathcal{F}^n (\varphi) := \mathcal{J}^n (\varphi + \overline{\phi}_0), \quad \mbox{with} \quad
\mathring{A}_h := \left\{ \varphi \in \mathring{\mathcal{C}}_{\rm per} \ \middle| \ -1-\overline{\phi}_0 \le \varphi \le 1-\overline{\phi}_0 \right\} \subset \mathbb{R}^{N^3}.$$ There exists a (not necessarily unique) minimizer of $\mathcal{F}^n$ over the restricted set $\mathring{A}_{h, \delta}$, defined in , where $\delta\in(0,{\nicefrac{1}{2}})$. To get a contradiction, suppose that the minimizer of $\mathcal{F}^n$, call it $\varphi^\star$, occurs at a boundary point of $\mathring{A}_{h,\delta}$. There is at least one grid point $\vec{\alpha}_0 = (i_0,j_0,k_0)$ such that $|\varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0| = 1 -\delta$. As before, we first assume that $\varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0 = \delta - 1$, so that the grid function $\varphi^\star$ has a global minimum at $\vec{\alpha}_0$. Suppose that $\vec{\alpha}_1 = (i_1,j_1,k_1)$ is a grid point at which $\varphi^\star$ achieves its maximum.
The directional derivative, in the direction $$\psi_{i,j,k} = \delta_{i,i_0}\delta_{j,j_0}\delta_{k,k_0} - \delta_{i,i_1}\delta_{j,j_1}\delta_{k,k_1} ,$$ satisfies $$\begin{aligned}
\frac{1}{h^3} d_s \mathcal{F}^n(\varphi^\star+s\psi)|_{s=0} &=& \ln (1+ \varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0) - \ln (1-\varphi^\star_{\vec{\alpha}_0}-\overline{\phi}_0)
\nonumber
\\
&& - \ln (1+ \varphi^\star_{\vec{\alpha}_1}+\overline{\phi}_0) + \ln (1-\varphi^\star_{\vec{\alpha}_1}-\overline{\phi}_0)
\nonumber
\\
&& - \theta_0 ( \phi^n_{\vec{\alpha}_0} - \phi^n_{\vec{\alpha}_1} ) - \varepsilon^2 ( \Delta_h \varphi^\star_{\vec{\alpha}_0} - \Delta_h \varphi^\star_{\vec{\alpha}_1} )
\nonumber
\\
&& + \frac{1}{{{\Delta t}}} \mathcal{L}_{\check{\mathcal{M}}^n}^{-1} ( \varphi^\star - \phi^n +\overline{\phi}_0 )_{\vec{\alpha}_0} - \frac{1}{{{\Delta t}}}\mathcal{L}_{\check{\mathcal{M}}^n}^{-1} ( \varphi^\star - \phi^n + \overline{\phi}_0)_{\vec{\alpha}_1} .
\nonumber
\\
&&
\label{CH_LOG-mobility-positive-4}
\end{aligned}$$ We now apply Lemma \[CH-mobility-positivity-Lem-0\] to obtain (keeping in mind that $\phi^* = \varphi^* + \bar{\phi}_0$) $$- 2 C_4 \le \mathcal{L}_{\check{\cal M}^n}^{-1} ( \phi^\star - \phi^n )_{\vec{\alpha}_0} - ( \mathcal{L}_{\check{\cal M}^n}^{-1} ( \phi^\star - \phi^n )_{\vec{\alpha}_1} \le 2 C_4 .
\label{CH_LOG-mobility-positive-9}$$ This, together with some other estimates, obtained as in the proof of Theorem \[CH-positivity\], yields $$\frac{1}{h^3} d_s \mathcal{F}^n(\varphi^\star+s\psi)|_{s=0} \le \ln \frac{\delta}{2 - \delta} - \ln \frac{1+\overline{\phi}_0}{1-\overline{\phi}_0} + C_5.
\label{CH_LOG-mobility-positive-10}$$ where $C_5:= 2M\theta_0 + 2 C_4 {{\Delta t}}^{-1}$. For $\delta\in(0,{\nicefrac{1}{2}})$ sufficiently small, the right hand side is strictly less than $0$. The rest of the analysis follows the proof of Theorem \[CH-positivity\]; the details are left to the interested readers.
In the proof of Theorem \[CH-mobility-positivity\], the point-wise positivity of the mobility, $\check{\cal M}^n \ge \mathcal{M}_0 > 0$, is assumed for the convenience of the analysis. However, at the PDE level, the CH flow with a degenerate mobility has been analyzed in [@barrett99; @elliott96b]. The numerical scheme for the degenerate mobility equation will also be considered in the authors’ future works. In fact, our assumption could be relaxed to allow for certain mobilities satisfying ${\cal M} (\phi^n) > 0$ at a point-wise level; the technical details are left to interested readers. In particular, for the case of the standard symmetric degenerate mobility, $\mathcal{M} (\phi) = (1-\phi)(1+\phi)$, the PDE analyses for which were undertaken by [@cahn1996; @elliott96b], our analysis would go through, with the help of a subtle fact that $\mathcal{M} (\phi)$ only degenerates at $\phi=-1$ and 1, combined with the positivity-preserving result at the previous time step.
Unconditional energy stability and uniform in time $H_h^1$ bound {#sec:energy stability}
================================================================
The discrete energy is defined as $$\begin{aligned}
E_h (\phi) = {\left\langle 1+ \phi , \ln (1+\phi) \right\rangle_\Omega} + {\left\langle 1-\phi , \ln (1-\phi) \right\rangle_\Omega} + \frac{\varepsilon^2}{2} {\left\| \nabla_h \phi \right\|}_2^2 - \frac{\theta_0}{2} {\left\| \phi \right\|}_2^2 .
\label{CH-discrete energy}
\end{aligned}$$ For the numerical scheme for the Cahn-Hilliard equation (\[scheme-CH\_LOG-1\]), (\[scheme-mu-0\]), the existence and unique solvability (so that the numerical solution stays within $(-1,1)$ at a point-wise level) have been established in Theorem \[CH-mobility-positivity\]. Because the scheme uses a convex-concave decomposition, it is unconditionally energy stability. This result is stated in the following theorem, whose proof is omitted for the sake of brevity and also because it is standard:
\[CH-mobility-energy stability\] For simplicity, suppose that $N=2K+1$, and let $\mathcal{P}_N:C_{\rm per}(\Omega)\to \mathcal{B}_K(\Omega)$ denote the Fourier projection operator, where $\mathcal{B}_K$ is space of $\Omega$-periodic (complex) trigonometric polynomials of degree up to and including $K$. By $\mathcal{P}_h:C_{\rm per}(\Omega)\to \mathcal{C}_{\rm per}$ denote the canonical grid projection operator. Suppose that $\phi^0:= \mathcal{P}_h(\mathcal{P}_N\Phi)$, where $\Phi\in C^6_{\rm per}(\Omega)$ and ${\left\| \Phi \right\|}_{L^\infty}<1$. Then $(\Phi,1)_{L^2} = {\left\langle \phi^0 , 1 \right\rangle_\Omega}$, and, for any ${{\Delta t}}>0$, $h >0$, and $m\in\mathbb{N}$, $$E_h(\phi^m) + {\left[ \check{\mathcal{M}}^{m-1}\nabla_h\mu^m , \nabla_h\mu^m \right]_{\Omega}} \le E_h(\phi^{m-1}),$$ so that $E_h(\phi^m) \le E_h(\phi^0) \le C_6$, with $C_6>0$ independent of $h$. Therefore, since $- \frac{\theta_0}{2} | \Omega |
+ \frac{\varepsilon^2}{2} \| \nabla_h \phi^m \|_2^2 \le E_h (\phi^m)$, we have $${\left\| \nabla_h \phi^m \right\|}_2 \le \sqrt{2 C_6 + \theta_0 | \Omega | } \varepsilon^{-1} =:C_7 , \quad \forall m\in\mathbb{N} .
\label{CH-mobility-H1-2-3}$$
The unconditional energy stability of the proposed scheme (\[scheme-CH\_LOG-1\]), (\[scheme-mu-0\]) follows from the convex-concave decomposition of the energy, an idea popularized in Eyre’s work [@eyre98]. The method has been applied to the phase field crystal (PFC) equation and the modified version [@wang11a; @wise09]; epitaxial thin film growth models [@chen12; @wang10a]; non-local gradient model [@guan14a]; the Cahn-Hilliard model coupled with fluid flow [@chen16; @diegel15a; @feng12; @LiuY17; @wise10]; *et cetera*. Second order accurate energy stable schemes have also been reported in recent years, based on either a secant/Crank-Nicolson or BDF approach. See, for example, [@baskaran13a; @baskaran13b; @chen14; @diegel16; @diegel17; @guo16; @han15; @hu09; @shen12; @guan14b; @yan17]. In particular, for the multi-component Cahn-Hilliard model, the related works could also be found in [@barrett97; @barrett98].
For the CH model with Flory Huggins energy potential, there have been some works to address the energy stability in the existing literature [@jeong16; @LiH2017; @LiX16; @peng17b; @yang17c]. However, the positivity-preserving property has not been theoretically justified for these numerical works, so that the existence of the numerical solutions in these works is not available at a theoretical level.
Optimal rate convergence analysis in $\ell^\infty (0,T; H^{-1}) \cap \ell^2 (0,T; H^1)$ {#sec:convergence}
=======================================================================================
For simplicity of presentation, we assume ${\cal M} \equiv 1$ in this section; the convergence analysis for the non-constant mobility case will be considered in future works.
Let $\Phi$ be the exact solution for the Cahn-Hilliard flow – . With initial data with sufficient regularity, we could assume that the exact solution has regularity of class $\mathcal{R}$: $$\Phi \in \mathcal{R} := H^2 \left(0,T; C_{\rm per}(\Omega)\right) \cap H^1 \left(0,T; C^2_{\rm per}(\Omega)\right) \cap L^\infty \left(0,T; C^6_{\rm per}(\Omega)\right).
\label{assumption:regularity.1}$$ Define $\Phi_N (\, \cdot \, ,t) := {\cal P}_N \Phi (\, \cdot \, ,t)$, the (spatial) Fourier projection of the exact solution into ${\cal B}^K$, the space of trigonometric polynomials of degree to and including $K$. The following projection approximation is standard: if $\Phi\in L^\infty(0,T;H^\ell_{\rm per}(\Omega))$, $${\left\| \Phi_N - \Phi \right\|}_{L^\infty(0,T;H^k)}
\le C h^{\ell-k} {\left\| \Phi \right\|}_{L^\infty(0,T;H^\ell)}, \quad \forall \ 0 \le k \le \ell .
\label{projection-est-0}$$ By $\Phi_N^m$, $\Phi^m$ we denote $\Phi_N(\, \cdot \, , t_m)$ and $\Phi(\, \cdot \, , t_m)$, respectively, with $T_m = m\cdot {{\Delta t}}$. Since $\Phi_N \in {\cal B}^K$, the mass conservative property is available at the discrete level: $$\overline{\Phi_N^m} = \frac{1}{|\Omega|}\int_\Omega \, \Phi_N ( \cdot, t_m) \, d {\bf x} = \frac{1}{|\Omega|}\int_\Omega \, \Phi_N ( \cdot, t_{m-1}) \, d {\bf x} = \overline{\Phi_N^{m-1}} , \quad \forall \ m \in\mathbb{N}.
\label{mass conserv-1}$$ On the other hand, the solution of (\[scheme-CH\_LOG-1\]), (\[scheme-mu-0\]) is also mass conservative at the discrete level: $$\overline{\phi^m} = \overline{\phi^{m-1}} , \quad \forall \ m \in \mathbb{N} .
\label{mass conserv-2}$$ As indicated before, we use the mass conservative projection for the initial data: $\phi^0 = {\mathcal P}_h \Phi_N (\, \cdot \, , t=0)$, that is $$\phi^0_{i,j,k} := \Phi_N (p_i, p_j, p_k, t=0) ,
\label{initial data-0}$$ The error grid function is defined as $$\tilde{\phi}^m := \mathcal{P}_h \Phi_N^m - \phi^m , \quad \forall \ m \in \left\{ 0 ,1 , 2, 3, \cdots \right\} .
\label{CH_LOG-error function-1}$$ Therefore, it follows that $\overline{\tilde{\phi}^m} =0$, for any $m \in \left\{ 0 ,1 , 2, 3, \cdots \right\}$, so that the discrete norm ${\left\| \, \cdot \, \right\|}_{-1,h}$ is well defined for the error grid function.
\[thm:convergence\] Given initial data $\Phi(\, \cdot \, ,t=0) \in C^6_{\rm per}(\Omega)$, suppose the exact solution for Cahn-Hilliard equation - is of regularity class $\mathcal{R}$. Then, provided ${{\Delta t}}$ and $h$ are sufficiently small, for all positive integers $n$, such that $t_n \le T$, we have $$\| \tilde{\phi}^n \|_{-1,h} + \left( \varepsilon^2 {{\Delta t}}\sum_{m=1}^{n} \| \nabla_h \tilde{\phi}^m \|_2^2 \right)^{1/2} \le C ( {{\Delta t}}+ h^2 ),
\label{CH_LOG-convergence-0}$$ where $C>0$ is independent of $n$, ${{\Delta t}}$, and $h$.
A careful consistency analysis indicates the following truncation error estimate: $$\frac{\Phi_N^{n+1} - \Phi_N^n}{{{\Delta t}}} = \Delta_h
\left( \ln (1+\Phi_N^{n+1}) - \ln (1-\Phi_N^{n+1}) - \theta_0 \Phi_N^n - \varepsilon^2 \Delta_h \Phi_N^{n+1} \right) + \tau^n ,
\label{CH_LOG-consistency-1}$$ with $\| \tau^n \|_{-1,h} \le C ({{\Delta t}}+ h^2)$. Observe that in equation , and from this point forward, we drop the operator $\mathcal{P}_h$, which should appear in front of $\Phi_N$, for simplicity.
Subtracting the numerical scheme (\[scheme-CH\_LOG-1\]) from (\[CH\_LOG-consistency-1\]) gives $$\begin{aligned}
\frac{\tilde{\phi}^{n+1} - \tilde{\phi}^n}{{{\Delta t}}} &=& \Delta_h
\Bigl( ( \ln (1+\Phi_N^{n+1}) - \ln (1+\phi^{n+1})) - ( \ln (1-\Phi_N^{n+1}) - \ln (1-\phi^{n+1})) \nonumber
\\
&& \quad
- \theta_0 \tilde{\phi}^{n+1} - \varepsilon^2 \Delta_h \tilde{\phi}^{n+1} \Bigr)
+ \tau^n .
\label{CH_LOG-consistency-2}
\end{aligned}$$
Since the numerical error function has zero-mean, we see that $(-\Delta_h)^{-1} \tilde{\phi}^m$ is well-defined, for any $k \ge 0$. Taking a discrete inner product with (\[CH\_LOG-consistency-2\]) by $2 (-\Delta_h)^{-1} \tilde{\phi}^{n+1}$ yields $$\begin{aligned}
\| \tilde{\phi}^{n+1} \|_{-1,h}^2 &-& \| \tilde{\phi}^n \|_{-1,h}^2 + \| \tilde{\phi}^{n+1} - \tilde{\phi}^n \|_{-1,h}^2 - 2 \varepsilon^2 {{\Delta t}}{\left\langle \tilde{\phi}^{n+1} , \Delta_h \tilde{\phi}^{n+1} \right\rangle_\Omega}
\nonumber
\\
&+& 2 {{\Delta t}}{\left\langle \ln (1+\Phi_N^{n+1}) - \ln (1+\phi^{n+1}) , \tilde{\phi}^{n+1} \right\rangle_\Omega}
\nonumber
\\
&-& 2 {{\Delta t}}{\left\langle \ln (1-\Phi_N^{n+1}) - \ln (1-\phi^{n+1}) , \tilde{\phi}^{n+1} \right\rangle_\Omega}
\nonumber
\\
& & \hspace{0.25in}= 2 \theta_0 {{\Delta t}}{\left\langle \tilde{\phi}^n , \tilde{\phi}^{n+1} \right\rangle_\Omega} + 2 {{\Delta t}}{\left\langle \tau^n , \tilde{\phi}^{n+1} \right\rangle_\Omega}.
\label{CH_LOG-convergence-1}
\end{aligned}$$ The estimate for the term associated with the surface diffusion is straightforward: $$\begin{aligned}
- \langle \tilde{\phi}^{n+1} , \Delta_h \tilde{\phi}^{n+1} \rangle = \| \nabla_h \tilde{\phi}^{n+1} \|_2^2 . \label{CH_LOG-convergence-2}\end{aligned}$$ For the nonlinear inner product, the fact that $-1 < \phi^{n+1} < 1$, $-1 < \Phi^{n+1} < 1$ (at a point-wise level) yields the following result: $$\begin{aligned}
{\left\langle \ln (1+\Phi_N^{n+1}) - \ln (1+\phi^{n+1}) , \tilde{\phi}^{n+1} \right\rangle_\Omega} &\ge& 0 ,
\label{CH_LOG-convergence-3-1}
\\
- {\left\langle \ln (1-\Phi_N^{n+1}) - \ln (1-\phi^{n+1}) , \tilde{\phi}^{n+1} \right\rangle_\Omega} &\ge& 0,
\label{CH_LOG-convergence-3-2}
\end{aligned}$$ due to the fact that $\ln$ is an increasing function. In other words, the convexity of the nonlinear term plays an essential role in this analysis. For the inner product associated with the concave part, the following estimate is derived: $$\begin{aligned}
2 \theta_0 {\left\langle \tilde{\phi}^n , \tilde{\phi}^{n+1} \right\rangle_\Omega} &\le& 2 \theta_0 \| \tilde{\phi}^n \|_{-1,h} \| \nabla_h \tilde{\phi}^{n+1} \|_2
\nonumber
\\
& \le & \theta_0^2 \varepsilon^{-2} \| \tilde{\phi}^n \|_{-1,h}^2 + \varepsilon^2 \| \nabla_h \tilde{\phi}^{n+1} \|_2 .
\label{CH_LOG-convergence-4}
\end{aligned}$$ The term associated with the truncation error can be controlled in a standard way: $$2 {\left\langle \tau^n , \tilde{\phi}^{n+1} \right\rangle_\Omega} \le 2 \| \tau^n \|_{-1,h} \| \nabla_h \tilde{\phi}^{n+1} \|_2 \le 2 \varepsilon^{-2} \| \tau^n \|_{-1,h}^2 + \frac{\varepsilon^2}{2} \| \nabla_h \tilde{\phi}^{n+1} \|_2^2 .
\label{CH_LOG-convergence-5}$$ Using estimates (\[CH\_LOG-convergence-2\]) – (\[CH\_LOG-convergence-5\]) in (\[CH\_LOG-convergence-1\]) yields $$\begin{aligned}
\| \tilde{\phi}^{n+1} \|_{-1,h}^2 - \| \tilde{\phi}^n \|_{-1,h}^2 & +& \frac{\varepsilon^2}{2} {{\Delta t}}\| \nabla_h \tilde{\phi}^{n+1} \|_2^2
\nonumber
\\
&\le& \theta_0^2 \varepsilon^{-2} {{\Delta t}}\| \tilde{\phi}^n \|_{-1,h}^2 + 2 \varepsilon^{-2} {{\Delta t}}\| \tau^n \|_{-1,h}^2 .
\label{CH_LOG-convergence-6}
\end{aligned}$$ Finally, an application of a discrete Gronwall inequality results in the desired convergence estimate: $$\| \tilde{\phi}^{n+1} \|_{-1,h} + \left( \varepsilon^2 {{\Delta t}}\sum_{k=0}^{n+1} \| \nabla_h \tilde{\phi}^m \|_2^2 \right)^{1/2} \le C ( {{\Delta t}}+ h^2) ,
\label{CH_LOG-convergence-7}$$ where $C>0$ is independent of ${{\Delta t}}$, $h$, and $n$. This completes the proof of the theorem.
For the Cahn-Hilliard equation with logarithmic potential, there have been some existing works of error estimate [@barrett95; @barrett96; @barrett01] in the framework of finite element analysis, with implicit Euler method in the temporal discretization. Again, the time step constraint ${{\Delta t}}\le \frac{4 \varepsilon^2}{\theta_0^2}$ has to be imposed to ensure the positivity-preserving property of the numerical scheme, while no constraint is needed in the convergence analysis of our proposed scheme.
The second order numerical scheme {#sec:BDF2}
=================================
We propose the following second order scheme for the CH equation -: given $\phi^n, \phi^{n-1} \in \mathcal{C}_{\rm per}$, find $\phi^{n+1},\mu^{n+1}\in \mathcal{C}_{\rm per}$, such that $$\frac{\frac32 \phi^{n+1} - 2 \phi^n + \frac12 \phi^{n-1}}{{{\Delta t}}} = \nabla_h \cdot ( \widehat{\cal M}^{n+1} \nabla_h \mu^{n+1} ) ,
\label{BDF2-CH_LOG-1}$$ where $$\begin{aligned}
\label{scheme-mu-BDF2-1}
\mu^{n+1} = & \ln (1+\phi^{n+1}) - \ln (1-\phi^{n+1}) - \theta_0 \check{\phi}^{n+1} - A {{\Delta t}}\Delta_h ( \phi^{n+1} - \phi^n ) - \varepsilon^2 \Delta_h \phi^{n+1} ,
\\
\check{\phi}^{n+1} = & 2 \phi^n - \phi^{n-1} , \nonumber
\end{aligned}$$ and the discrete mobility function is defined at the face center points in a similar way as in : $\widehat{\cal M}_{i+{\nicefrac{1}{2}},j,k}^{n+1} = {\cal M}(A_x \check{\phi}^{n+1}_{i+{\nicefrac{1}{2}},j,k})$, $\widehat{\cal M}_{i,j+{\nicefrac{1}{2}},k}^{n+1} = {\cal M}(A_y \check{\phi}^{n+1}_{i,j+{\nicefrac{1}{2}},k})$, $\widehat{\cal M}_{i,j,k+{\nicefrac{1}{2}}}^{n+1} = {\cal M}(A_z \check{\phi}^{n+1}_{i,j,k+{\nicefrac{1}{2}}})$.
In the case of constant mobility ${\cal M} (\phi) \equiv 1$, the positivity-preserving property is established in the following theorem.
\[CH-BDF2-positivity\] Assume that ${\cal M} (\phi) \equiv 1$. Given $\phi^k \in\mathcal{C}_{\rm per}$, with ${\left\| \phi^k \right\|}_\infty \le M$, $k=n, n-1$, for some $M >0$, and $\left|\overline{\phi^k}\right| = \left|\overline{\phi^{n-1}} \right| < 1$, there exists a unique solution $\phi^{n+1}\in\mathcal{C}_{\rm per}$ to , with $\phi^{n+1}-\overline{\phi^n}\in\mathring{\mathcal{C}}_{\rm per}$ and ${\left\| \phi^{n+1} \right\|}_\infty < 1$.
We follow the notations in the proof of Theorem \[CH-positivity\]. The numerical solution of is a minimizer of the following discrete energy functional over the admissible set $A_h$: $$\begin{aligned}
\mathcal{J}^{n, (2)} (\phi) &:=& \frac{1}{3 {{\Delta t}}} {\left\| \frac32 \phi - 2 \phi^n + \frac12 \phi^{n-1} \right\|}_{-1,h}^2 \nonumber
\\
&&
+ {\left\langle 1+ \phi , \ln (1+\phi) \right\rangle_\Omega} + {\left\langle 1-\phi , \ln (1-\phi) \right\rangle_\Omega} \nonumber
\\
&&
+ \frac{\varepsilon^2 + A {{\Delta t}}}{2} \| \nabla_h \phi \|_2^2 + {\left\langle \phi , A{{\Delta t}}\Delta_h \phi^n - \theta_0\check{\phi}^{n+1} \right\rangle_\Omega} .
\label{CH_LOG-BDF2-positive-1}
\end{aligned}$$ Of course, $\mathcal{J}^{n, (2)}$ is strictly convex over $A_h$. Again, such a minimization problem is equivalent to the following transformed functional over $\mathring{A}_h$: $$\begin{aligned}
\mathcal{F}^{n, (2)} (\varphi) &:=& \mathcal{J}^{n, (2)} (\varphi + \overline{\phi}_0)
\nonumber
\\
&=& \frac{1}{3 {{\Delta t}}} {\left\| \frac32 (\varphi + \overline{\phi}_0) - 2 \phi^n + \frac12 \phi^{n-1} \right\|}_{-1,h}^2 \nonumber
\\
&&
+ {\left\langle 1+ \varphi + \overline{\phi}_0 , \ln ( 1 + \varphi + \overline{\phi}_0 ) \right\rangle_\Omega}
+ {\left\langle 1- \varphi - \overline{\phi}_0 , \ln (1-\varphi - \overline{\phi}_0) \right\rangle_\Omega} \nonumber
\\
&& + \frac{\varepsilon^2 + A {{\Delta t}}}{2} {\left\| \nabla_h \varphi \right\|}_2^2
+ {\left\langle \varphi + \overline{\phi}_0 , A{{\Delta t}}\Delta_h \phi^n - \theta_0\check{\phi}^{n+1} \right\rangle_\Omega} .
\label{CH_LOG-BDF2-positive-1-b}
\end{aligned}$$ To obtain the existence of a minimizer for $\mathcal{F}^{n, (2)}$ over $\mathring{A}_h$, we consider the closed domain $\mathring{A}_{h,\delta}$ for $0 < \delta < \frac12$, as defined by . There exists a (not necessarily unique) minimizer of $\mathcal{F}^{n, (2)}$ over $\mathring{A}_{h, \delta}$, and we have to prove such a minimizer could not occur on the boundary of $\mathring{A}_{h,\delta}$, if $\delta$ is sufficiently small. To get a contradiction, suppose that the minimizer of $\mathcal{F}^{n, (2)}$, call it $\varphi^\star$ occurs at a boundary point of $\mathring{A}_{h,\delta}$. There is at least one grid point $\vec{\alpha}_0 = (i_0,j_0,k_0)$ such that $|\varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0| = 1 -\delta$. Similarly, we assume that $\varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0 = \delta - 1$, so that the grid function $\varphi^\star$ has a global minimum at $\vec{\alpha}_0$, and $\vec{\alpha}_1 = (i_1,j_1,k_1)$ is a grid point at which $\varphi^\star$ achieves its maximum. Meanwhile, for all $\psi\in \mathring{\mathcal{C}}_{\rm per}$, the directional derivative becomes $$\begin{aligned}
d_s \mathcal{F}^{n, (2)} (\varphi^\star+s\psi)|_{s=0} = & \ {\left\langle \ln (1+\varphi^\star+\overline{\phi}_0) - \ln (1-\varphi^\star-\overline{\phi}_0) , \psi \right\rangle_\Omega}
\\
& + {\left\langle A {{\Delta t}}\Delta_h \phi^n - \theta_0 \check{\phi}^{n+1} , \psi \right\rangle_\Omega}
- (\varepsilon^2 + A {{\Delta t}}) {\left\langle \Delta_h \varphi^\star , \psi \right\rangle_\Omega}
\\
& + \frac{1}{\Delta t}{\left\langle (-\Delta_h)^{-1}\left( \frac32 ( \varphi^\star +\overline{\phi}_0 ) - 2 \phi^n + \frac12 \phi^{n-1} \right) , \psi \right\rangle_\Omega}.
\end{aligned}$$ In more details, this derivative may be expressed as $$\begin{aligned}
\frac{1}{h^3} d_s \mathcal{F}^{n, (2)} (\varphi^\star+s\psi)|_{s=0} &=& \ln (1+ \varphi^\star_{\vec{\alpha}_0}+\overline{\phi}_0) - \ln (1-\varphi^\star_{\vec{\alpha}_0}-\overline{\phi}_0)
\nonumber
\\
&& - \ln (1+ \varphi^\star_{\vec{\alpha}_1}+\overline{\phi}_0) + \ln (1-\varphi^\star_{\vec{\alpha}_1}-\overline{\phi}_0)
\nonumber
\\
&& - \theta_0 ( \check{\phi}^{n+1}_{\vec{\alpha}_0} - \check{\phi}^{n+1}_{\vec{\alpha}_1} ) + A {{\Delta t}}( \Delta_h \phi^n_{\vec{\alpha}_0} - \Delta_h \phi^n_{\vec{\alpha}_1} ) \nonumber
\\
&& - ( \varepsilon^2 + A {{\Delta t}}) ( \Delta_h \varphi^\star_{\vec{\alpha}_0} - \Delta_h \varphi^\star_{\vec{\alpha}_1} )
\nonumber
\\
&& + \frac{1}{{{\Delta t}}} (-\Delta_h)^{-1} ( \frac32 ( \varphi^\star +\overline{\phi}_0 ) - 2 \phi^n + \frac12 \phi^{n-1} )_{\vec{\alpha}_0}
\nonumber
\\
&& - \frac{1}{{{\Delta t}}}(-\Delta_h)^{-1} ( \frac32 ( \varphi^\star +\overline{\phi}_0 ) - 2 \phi^n + \frac12 \phi^{n-1} )_{\vec{\alpha}_1} .
\label{CH_LOG-BDF2-positive-4}
\end{aligned}$$ Furthermore, the following estimates are derived $$\begin{aligned}
\Delta_h \phi^\star_{\vec{\alpha}_0} &\ge& 0 , \quad \Delta_h \phi^\star_{\vec{\alpha}_1} \le 0 , \label{CH_LOG-BDF2-positive-6-1}
\\
-6 M &\le& \check{\phi}^{n+1}_{\vec{\alpha}_0} - \check{\phi}^{n+1}_{\vec{\alpha}_1} \le 6M , \label{CH_LOG-BDF2-positive-6-2}
\\
\Delta_h \phi^n_{\vec{\alpha}_0} &\le& \frac{12 M}{h^2} , \quad
\Delta_h \phi^n_{\vec{\alpha}_1} \ge - \frac{12 M}{h^2} ,
\label{CH_LOG-BDF2-positive-6-3}
\\
- 5 C_1 &\le& (-\Delta_h)^{-1} ( \frac32 ( \varphi^\star +\overline{\phi}_0 ) - 2 \phi^n + \frac12 \phi^{n-1} )_{\vec{\alpha}_0} \nonumber
\\
&&
- \frac{1}{{{\Delta t}}}(-\Delta_h)^{-1} ( \frac32 ( \varphi^\star +\overline{\phi}_0 ) - 2 \phi^n + \frac12 \phi^{n-1} )_{\vec{\alpha}_1} \le 5 C_1 ,
\label{CH_LOG-BDF2-positive-6-4}
\end{aligned}$$ in which we have repeatedly made use of the fact that $\| \phi^k \|_\infty \le M$, $k=n, n-1$, as well as the application of Lemma \[CH-positivity-Lem-0\]. Subsequently, a substitution of (\[CH\_LOG-BDF2-positive-6-1\]) – (\[CH\_LOG-BDF2-positive-6-4\]) and into (\[CH\_LOG-BDF2-positive-4\]) yields the following bound: $$\frac{1}{h^3} d_s \mathcal{F}^{n, (2)} (\varphi^\star+s\psi)|_{s=0} \le \ln \frac{\delta}{2 - \delta} - \ln \frac{1+\overline{\phi}_0}{1-\overline{\phi}_0} + 6 M \theta_0 + 12 M {{\Delta t}}h^{-2} + 10 C_1 {{\Delta t}}^{-1} .
\label{CH_LOG-BDF2-positive-10}$$ The rest analysis follows the same arguments as in the proof of Theorem \[CH-positivity\]; the details are left to interested readers.
Again, for the second order scheme, a careful calculation implies that $C_8 = O ({{\Delta t}}^{-1} + {{\Delta t}}h^{-2})$, which becomes singular as ${{\Delta t}}, h\to 0$. Even so, since the values of $h$ and ${{\Delta t}}$ are fixed, a $\delta\in(0,{\nicefrac{1}{2}})$ exists so that the size of $C_8$ is not an issue.
The non-constant mobility case could be analyzed in the same fashion; we state the result below, and the technical details are left to interested readers.
\[CH-mobility-BDF2-positivity\] Assume that ${\cal M} (x) \ge {\cal M}_0 >0 $, for all $x\in[-1,1]$. Given $\phi^k \in\mathcal{C}_{\rm per}$, with ${\left\| \phi^k \right\|} \le M$, $k=n, n-1$, for some $M >0$, and $\left|\overline{\phi^n}\right| = \left|\overline{\phi^{n-1}}\right| < 1$, there exists a unique solution $\phi^{n+1}\in\mathcal{C}_{\rm per}$ to , with $\phi^{n+1}-\overline{\phi^n}\in\mathring{\mathcal{C}}_{\rm per}$ and ${\left\| \phi^{n+1} \right\|}_\infty < 1$.
In the case of constant mobility ${\cal M} (\phi) \equiv 1$, a modified energy stability is available for the second order BDF scheme , provided that $A \ge \frac{1}{16}$.
\[CH-BDF2-energy stability\] Suppose ${\cal M} (\phi) \equiv 1$. With the same assumptions as in Theorem \[CH-mobility-energy stability\], we have the stability analysis of the following modified energy functional for the proposed numerical scheme : $$\begin{aligned}
&&
\tilde{E}_h (\phi^{n+1}, \phi^n) \le \tilde{E}_h (\phi^n, \phi^{n-1}) , \quad
\mbox{with} \label{CH-BDF2-stability-0}
\\
&&
\tilde{E}_h (\phi^{n+1}, \phi^n) = E_h (\phi^{n+1})
+ \frac{1}{4 {{\Delta t}}} \| \phi^{n+1} - \phi^n \|_{-1,h}^2
+ \frac12 \| \phi^{n+1} - \phi^n \|_2^2 , \label{mod energy-BDF2-1}\end{aligned}$$ for any ${{\Delta t}}, h >0$, provided that $A \ge \frac{1}{16}$.
By taking an inner product with by $(-\Delta_h)^{-1} (\phi^{n+1} - \phi^n)$, we could derive the following inequalities: $$\begin{aligned}
&&
\left\langle \frac{\frac32 \phi^{n+1} - 2 \phi^n + \frac12 \phi^{n-1}}{{{\Delta t}}} ,
(-\Delta_h)^{-1} (\phi^{n+1} - \phi^n) \right\rangle_\Omega
\nonumber
\\
&& \qquad
= \frac{3}{2 {{\Delta t}}} \| \phi^{n+1} - \phi^n \| _{-1,h}^2
- \frac12 \langle \phi^{n+1} - \phi^n , \phi^n - \phi^{n-1} \rangle_{-1, h} \nonumber
\\
&& \qquad
\ge \frac{1}{{{\Delta t}}} \left(
\frac54 \| \phi^{n+1} - \phi^n \|_{-1, h}^2
- \frac14 \| \phi^n - \phi^{n-1} \|_{-1, h}^2 \right) ,
\label{CH-BDF2-stability-1}
\\
&&
\left\langle - \Delta_h ( \ln (1+\phi^{n+1}) ) ,
(-\Delta_h)^{-1} (\phi^{n+1} - \phi^n) \right\rangle_\Omega
= \left\langle \ln (1+\phi^{n+1}) , \phi^{n+1} - \phi^n \right\rangle_\Omega \nonumber
\\
&& \qquad
\ge {\left\langle 1+ \phi^{n+1} , \ln (1+\phi^{n+1}) \right\rangle_\Omega}
- {\left\langle 1+ \phi^n , \ln (1+\phi^n) \right\rangle_\Omega} ,
\label{CH-BDF-stability-2-1}
\\
&&
\left\langle \Delta_h ( \ln (1-\phi^{n+1}) ) ,
(-\Delta_h)^{-1} (\phi^{n+1} - \phi^n) \right\rangle_\Omega
= - \left\langle \ln (1-\phi^{n+1}) , \phi^{n+1} - \phi^n \right\rangle_\Omega \nonumber
\\
&& \qquad
\ge - {\left\langle 1- \phi^{n+1} , \ln (1-\phi^{n+1}) \right\rangle_\Omega}
+ {\left\langle 1- \phi^n , \ln (1-\phi^n) \right\rangle_\Omega} ,
\label{CH-BDF-stability-2-2}
\\
&&
\left\langle \Delta_h^2 \phi^{n+1} ,
(-\Delta_h)^{-1} (\phi^{n+1} - \phi^n) \right\rangle_\Omega
= \left\langle \nabla_h \phi^{n+1} ,
\nabla_h (\phi^{n+1} - \phi^n) \right\rangle_\Omega \nonumber
\\
&& \qquad
= \frac12 \left( \| \nabla_h \phi^{n+1} \|_2^2
- \| \nabla_h \phi^n \|_2^2
+ \| \nabla_h ( \phi^{n+1} - \phi^n ) \|_2^2 \right) ,
\label{CH-BDF2-stability-3}
\\
&&
{{\Delta t}}\left\langle \Delta_h^2 ( \phi^{n+1} - \phi^n ) ,
(-\Delta_h)^{-1} (\phi^{n+1} - \phi^n) \right\rangle_\Omega
= {{\Delta t}}\| \nabla_h ( \phi^{n+1} - \phi^n ) \|_2^2 ,
\label{CH-BDF2-stability-4}
\\
&&
\left\langle \Delta_h ( 2 \phi^n - \phi^{n-1}) ,
(-\Delta_h)^{-1} (\phi^{n+1} - \phi^n) \right\rangle_\Omega
= - \left\langle 2 \phi^n - \phi^{n-1} ,
\phi^{n+1} - \phi^n) \right\rangle_\Omega \nonumber
\\
&& \qquad
\ge - \frac12 \left( \| \phi^{n+1} \|_2^2 - \| \phi^n \|_2^2 \right)
- \frac12 \| \phi^n - \phi^{n-1} \|_2^2 ,
\label{CH-BDF2-stability-5}\end{aligned}$$ in which , are based on the convexity of $(1 + \phi) \ln (1+ \phi)$, $(1 - \phi) \ln (1- \phi)$, respectively. Meanwhile, an application of Cauchy inequality indicates the following estimate: $$\frac{1}{{{\Delta t}}} \| \phi^{n+1} - \phi^n \|_{-1,h}^2
+ A {{\Delta t}}\| \nabla_h ( \phi^{n+1} - \phi^n ) \|_2^2
\ge 2 A^{1/2} \| \phi^{n+1} - \phi^n \|_2^2 .
\label{CH-BDF2-stability-6}$$ Therefore, a combination of - and yields $$\begin{aligned}
&&
E_h (\phi^{n+1}) - E_h (\phi^n)
+ \frac{1}{4 {{\Delta t}}} \left( \| \phi^{n+1} - \phi^n \|_{-1, h}^2
- \| \phi^n - \phi^{n-1} \|_{-1, h}^2 \right) \nonumber
\\
&&
+ \frac12 \left( \| \phi^{n+1} - \phi^n \|_2^2 - \| \phi^n - \phi^{n-1} \|_2^2 \right)
\le ( - 2 A^{1/2} + \frac12 ) \| \phi^{n+1} - \phi^n \|_2^2 \le 0 ,
\label{scheme-BDF-stability-7}\end{aligned}$$ provided that $A \ge \frac{1}{16}$. Therefore, by denoting a modified energy as given by , we get the energy estimate . This completes the proof of Theorem \[CH-BDF2-energy stability\].
With the same assumption that ${\cal M} (\phi) \equiv 1$, the convergence result is stated in the following theorem.
\[thm:BDF2-convergence\] Given initial data $\Phi(\, \cdot \, ,t=0) \in C^6_{\rm per}(\Omega)$, suppose the exact solution for Cahn-Hilliard equation - is of regularity class $\mathcal{R}_2 := H^3 \left(0,T; C_{\rm per}(\Omega)\right) \cap H^3 \left(0,T; C^2_{\rm per}(\Omega)\right) \cap L^\infty \left(0,T; C^6_{\rm per}(\Omega)\right)$. Then, provided ${{\Delta t}}$ and $h$ are sufficiently small, for all positive integers $n$, such that $t_n \le T$, we have the following convergence estimate for the numerical solution $$\| \tilde{\phi}^n \|_{-1,h} + \left( \varepsilon^2 {{\Delta t}}\sum_{m=1}^{n} \| \nabla_h \tilde{\phi}^m \|_2^2 \right)^{1/2} \le C ( {{\Delta t}}^2 + h^2 ),
\label{CH_LOG-BDF2-convergence-0}$$ where $C>0$ is independent of $n$, ${{\Delta t}}$, and $h$.
A careful consistency analysis indicates the following truncation error estimate: $$\begin{aligned}
\frac{\frac32 \Phi_N^{n+1} - 2 \Phi_N^n + \frac12 \Phi_N^{n-1}}{{{\Delta t}}} &=& \Delta_h
\Bigl( \ln (1+\Phi_N^{n+1}) - \ln (1-\Phi_N^{n+1}) - \theta_0 \check{\Phi}_N^{n+1} - \varepsilon^2 \Delta_h \Phi_N^{n+1} \nonumber
\\
&&
- A {{\Delta t}}\Delta_h ( \Phi_N^{n+1} - \Phi_N^n ) \Bigr) + \tau^n ,
\label{CH_LOG-BDF2-consistency-1}
\end{aligned}$$ with $\check{\Phi}_N^n = 2 \Phi_N^n - \Phi_N^{n-1}$, $\| \tau^n \|_{-1,h} \le C ({{\Delta t}}^2 + h^2)$. In turn, subtracting the numerical scheme from gives $$\begin{aligned}
\frac{\frac32 \tilde{\phi}^{n+1} - 2 \tilde{\phi}^n + \frac12 \tilde{\phi}^{n-1}}{{{\Delta t}}} &=&
\Delta_h
\Bigl( ( \ln (1+\Phi_N^{n+1}) - \ln (1+\phi^{n+1})) \nonumber
\\
&& \quad
- ( \ln (1-\Phi_N^{n+1}) - \ln (1-\phi^{n+1}))
- \theta_0 \tilde{\check{\phi}}^{n+1} \nonumber
\\
&& \quad
- \varepsilon^2 \Delta_h \tilde{\phi}^{n+1}
- A {{\Delta t}}\Delta_h ( \tilde{\phi}^{n+1} - \tilde{\phi}^n ) \Bigr)
+ \tau^n ,
\label{CH_LOG-BDF2-consistency-2}
\end{aligned}$$ with $\tilde{\check{\phi}}^{n+1} = 2 \tilde{\phi}^n - \tilde{\phi}^{n-1}$. Taking a discrete inner product with by $2 (-\Delta_h)^{-1} \tilde{\phi}^{n+1}$ yields $$\begin{aligned}
&&
\left\langle 3 \tilde{\phi}^{n+1} - 4 \tilde{\phi}^n + \tilde{\phi}^{n-1} ,
\tilde{\phi}^{n+1} \right\rangle_{-1, h}
- 2 \varepsilon^2 {{\Delta t}}{\left\langle \tilde{\phi}^{n+1} , \Delta_h \tilde{\phi}^{n+1} \right\rangle_\Omega}
\nonumber
\\
&+& 2 {{\Delta t}}{\left\langle \ln (1+\Phi_N^{n+1}) - \ln (1+\phi^{n+1}) , \tilde{\phi}^{n+1} \right\rangle_\Omega}
\nonumber
\\
&-&
2 {{\Delta t}}{\left\langle \ln (1-\Phi_N^{n+1}) - \ln (1-\phi^{n+1}) , \tilde{\phi}^{n+1} \right\rangle_\Omega}
\nonumber
\\
&-&
2 A {{\Delta t}}\langle \Delta_h ( \tilde{\phi}^{n+1} - \tilde{\phi}^n ) , \tilde{\phi}^{n+1} \rangle_\Omega
= 2 \theta_0 {{\Delta t}}{\left\langle \tilde{\phi}^n , \tilde{\phi}^{n+1} \right\rangle_\Omega} + 2 {{\Delta t}}{\left\langle \tau^n , \tilde{\phi}^{n+1} \right\rangle_\Omega}.
\label{CH_LOG-BDF2-convergence-1}
\end{aligned}$$ For the temporal derivative stencil, the following identity is valid: $$\begin{aligned}
\left\langle 3 \tilde{\phi}^{n+1} - 4 \tilde{\phi}^n + \tilde{\phi}^{n-1} ,
\tilde{\phi}^{n+1} \right\rangle_{-1, h}
&=& \frac12 \Bigl( \| \tilde{\phi}^{n+1} \|_{-1, h}^2 - \| \tilde{\phi}^n \|_{-1, h}^2
\nonumber
\\
&&
+ \| 2 \tilde{\phi}^{n+1} - \tilde{\phi}^n \|_{-1, h}^2
- \| 2 \tilde{\phi}^n - \tilde{\phi}^{n-1} \|_{-1, h}^2 \nonumber
\\
&&
+ \| \tilde{\phi}^{n+1} - 2 \tilde{\phi}^n + \tilde{\phi}^{n-1} \|_{-1, h}^2 \Bigr) .
\label{CH_LOG-BDF2-convergence-2}
\end{aligned}$$ The estimates for the terms associated with the surface diffusion, the nonlinear product and the truncation error follow exactly the same way as in , , , , respectively. For the concave expansive error term, a similar inequality is available: $$\begin{aligned}
2 \theta_0 {\left\langle \tilde{\check{\phi}}^{n+1} , \tilde{\phi}^{n+1} \right\rangle_\Omega} &\le& 2 \theta_0 \| \tilde{\check{\phi}}^{n+1} \|_{-1,h} \| \nabla_h \tilde{\phi}^{n+1} \|_2
\le \theta_0^2 \varepsilon^{-2} \| \tilde{\check{\phi}}^{n+1} \|_{-1,h}^2 + \varepsilon^2 \| \nabla_h \tilde{\phi}^{n+1} \|_2
\nonumber
\\
& \le & \theta_0^2 \varepsilon^{-2} ( 8 \| \tilde{\phi}^n \|_{-1,h}^2
+ 2 \| \tilde{\phi}^{n-1} \|_{-1,h}^2) + \varepsilon^2 \| \nabla_h \tilde{\phi}^{n+1} \|_2 .
\label{CH_LOG-BDF2-convergence-4}
\end{aligned}$$ In addition, the following identity could be derived for the artificial diffusion term: $$\begin{aligned}
&&
- 2 \langle \Delta_h ( \tilde{\phi}^{n+1} - \tilde{\phi}^n ) , \tilde{\phi}^{n+1} \rangle_\Omega
= 2 \langle \nabla_h ( \tilde{\phi}^{n+1} - \tilde{\phi}^n ) , \nabla_h \tilde{\phi}^{n+1} \rangle_\Omega \nonumber
\\
&=&
\| \nabla_h \tilde{\phi}^{n+1} \|_2^2 - \| \nabla_h \tilde{\phi}^n \|_2^2
+ \| \nabla_h ( \tilde{\phi}^{n+1} - \| \nabla_h \tilde{\phi}^n ) \|_2^2 .
\label{CH_LOG-BDF2-convergence-5}
\end{aligned}$$ Subsequently, a substitution of (\[CH\_LOG-BDF2-convergence-2\]) – (\[CH\_LOG-BDF2-convergence-5\]), , , and into (\[CH\_LOG-BDF2-convergence-1\]) yields $$\begin{aligned}
&&
\| \tilde{\phi}^{n+1} \|_{-1,h}^2 - \| \tilde{\phi}^n \|_{-1,h}^2
+ \| 2 \tilde{\phi}^{n+1} - \tilde{\phi}^n \|_{-1, h}^2
- \| 2 \tilde{\phi}^n - \tilde{\phi}^{n-1} \|_{-1, h}^2 \nonumber
\\
&&
+ A {{\Delta t}}( \| \nabla_h \tilde{\phi}^{n+1} \|_2^2 - \| \nabla_h \tilde{\phi}^n \|_2^2 )
+ \frac{\varepsilon^2}{2} {{\Delta t}}\| \nabla_h \tilde{\phi}^{n+1} \|_2^2
\nonumber
\\
&\le& 4 \theta_0^2 \varepsilon^{-2} ( 4 \| \tilde{\phi}^n \|_{-1,h}^2
+ \| \tilde{\phi}^{n-1} \|_{-1,h}^2) + 4 \varepsilon^{-2} {{\Delta t}}\| \tau^n \|_{-1,h}^2 .
\label{CH_LOG-BDF2-convergence-6}
\end{aligned}$$ Finally, an application of a discrete Gronwall inequality results in the desired convergence estimate: $$\| \tilde{\phi}^{n+1} \|_{-1,h} + \Bigl( \varepsilon^2 {{\Delta t}}\sum_{k=0}^{n+1} \| \nabla_h \tilde{\phi}^m \|_2^2 \Bigr)^{1/2} \le C ( {{\Delta t}}^2 + h^2) ,
\label{CH_LOG-BDF2-convergence-7}$$ where $C>0$ is independent of ${{\Delta t}}$, $h$, and $n$. This completes the proof of the Theorem \[thm:BDF2-convergence\].
Numerical results {#sec:numerical results}
=================
In this section we describe a simple multigrid solver for the proposed schemes, and we provided some tests that show the efficiency of the solver and the accuracy of the scheme. We demonstrate, in particular, the positivity of the solutions to the proposed Cahn-Hilliard scheme.
For the discussion of the numerical computations, we use a slightly different formulation of the Cahn-Hilliard equation, one that allows for a comparison with the so-called obstacle potential. Specifically, we will use the standard Ginzburg-Landau free energy $E[\phi] = \int_\Omega \left\{f(\phi) +\frac{\varepsilon^2}{2}|\nabla\phi|^2 \right\} d\mathbf{x}$, where $f(\phi) = f_c(\phi) - f_e(\phi)$ and $$f_c(\phi) = \frac{1}{2\theta_0}\left[(1-\phi)\ln(1-\phi) +(1+\phi)\ln(1+\phi)\right] , \quad f_e(\phi) = \frac{1}{2}(\phi-1)(\phi+1) .$$ Importantly, as $\theta_0 \to \infty$, $f$ tends to the obstacle potential $$f_{\rm obs}(\theta) = \left\{
\begin{array}{ccc}
\frac{1}{2}(\phi-1)(\phi+1) & \mbox{if} & -1 < \phi < 1
\\
\infty & \mbox{if} & |\phi|\ge 1
\end{array}
\right. ,$$ which has been investigated elsewhere [@blowey91; @blowey92]. While we are only interested in the case of finite values of $\theta_0$, it is interesting to explore the effects of increasing $\theta_0$. For finite $\theta_0$, clearly $f_e'(\phi) = \phi$ and $$f'_c(\phi) = \frac{1}{2\theta_0} \left[ \ln(1+\phi) - \ln(1-\phi)\right].$$ The Cahn-Hillard equation still takes the form , but with the chemical potential expressed as $$\mu = f_c'(\phi)-f_e'(\phi) - \varepsilon^2\Delta\phi.$$ As before, we assume that the mobility satisfies ${\cal M} (x) \ge \mathcal{M}_0 >0$, for all $x\in[-1,1]$, for some $\mathcal{M}_0$, though as we have remarked, this can be relaxed.
Multgrid solver {#subsec:solver}
---------------
In this subsection, we describe a nonlinear full approximation storage (FAS) multigrid solver for the convex-concave decomposition scheme for the Cahn-Hilliard equation with logarithmic potential. The solver for the Allen-Cahn equation is simpler, and we omit its description. Our solver is similar in style to the one presented in [@jeong16], and it can be extended to the case of multi-component systems as in the article. For an alternative approach to the one taken here and in [@jeong16], see, for example, [@graser15].
For our solver implementation, we will need to regularize $f'_c$. This is due to the fact that our multigrid solver is not designed to guarantee the boundedness of the solution for arbitrary multigrid iterations, as we discuss below. Our solver will, however, converge to the correct bounded solution, provided the regularization is sufficiently small. We show this in our tests.
To effect the desired regularization, we modify the logarithm as follows: for a given $\delta\in (0,\nicefrac{1}{4})$ we define $$\ln_\delta (\phi) = \left\{
\begin{array}{ccc}
\ln(\phi) & \mbox{if} & \delta < \phi
\\
\ln(\delta) + \frac{\phi-\delta}{\delta} & \mbox{if} & \phi \le \delta
\end{array}
\right. .$$ The regularized logarithm, $\ln_\delta$ is defined for all values of $\phi$. Using this function, we define $$f'_{c,\delta}(\phi) = \frac{1}{2\theta_0} \left[ \ln_\delta(1+\phi) - \ln_\delta(1-\phi)\right].$$ We then observe that $f'_c(\phi) = f'_{c,\delta}(\phi)$, for all $-1+\delta \le \phi \le 1-\delta$. Consequently, we can always take the value of $\delta$ to be small enough such that the theoretical solution to our scheme lies in this range of equivalence.
The first-order convex-concave decomposition (CS1) scheme in 2-D is equivalent to the following: find $\phi, \mu\in\mathcal{C}_{\rm per}$ whose components satisfy $$\begin{aligned}
\phi_{i,j} - {{\Delta t}}\, d_x\left({\cal M} \left(A_x\phi^m\right) D_x\mu \right)_{i,j} - {{\Delta t}}\, d_y\left( {\cal M} \left(A_y\phi^m\right) D_y\mu\right)_{i,j} &=& \phi_{i,j}^m ,
\label{disc-ch-1}
\\
\mu_{i,j} - f_{c,\delta}'\left(\phi_{i,j}\right) +\epsilon^2\Delta_h\phi_{i,j} &=& -\phi_{i,j}^m ,
\label{disc-ch-2}
\end{aligned}$$ where we have dropped the time superscripts $m+1$ on the unknowns. The 3-D equations are similar, and they are omitted for simplicity. For the sake of comparison, the standard backward Euler scheme (BE) is $$\begin{aligned}
\phi_{i,j} - {{\Delta t}}\, d_x\left({\cal M} \left(A_x\phi \right) D_x\mu \right)_{i,j} - {{\Delta t}}\, d_y\left( {\cal M} \left(A_y\phi \right) D_y\mu\right)_{i,j} &=& \phi_{i,j}^m ,
\label{disc-ch-be-1}
\\
\mu_{i,j} - f_{c,\delta}'\left(\phi_{i,j}\right) +\phi_{i,j} +\epsilon^2\Delta_h\phi_{i,j} &=& 0 .
\label{disc-ch-be-2}
\end{aligned}$$ We note that, for solvability and stability considerations, the sign of the linear term ($\phi$) in the chemical potential equation is problematical. However, this scheme is solvable with a mild time step restriction.
The energy stable BDF2 (BDF2\_ES) scheme is expressed in 2D as $$\begin{aligned}
\phi_{i,j} - \frac{2{{\Delta t}}}{3}\, d_x\left({\cal M} \left(A_x\check\phi^{m+1}\right) D_x\mu \right)_{i,j} &
\nonumber
\\
- \frac{2{{\Delta t}}}{3}\, d_y\left( {\cal M} \left(A_y\check\phi^{m+1}\right) D_y\mu\right)_{i,j} &=& \frac{4}{3}\phi_{i,j}^m - \frac{1}{3}\phi_{i,j}^{m-1} ,
\label{disc-ch-bdf2-cs-1}
\\
\mu_{i,j} - f_{c,\delta}'\left(\phi_{i,j}\right) +\epsilon^2\Delta_h\phi_{i,j} +A {{\Delta t}}\Delta_h \phi_{i,j} &=& A {{\Delta t}}\Delta_h \phi_{i,j}^m -\check\phi_{i,j}^{m+1} ,
\label{disc-ch-bdf2-cs-2}
\end{aligned}$$ where $$\check{\phi}^{m+1}_{i,j} = 2 \phi^m_{i,j} - \phi^{m-1}_{i,j} .$$ The standard BDF2 scheme is $$\begin{aligned}
\phi_{i,j} - \frac{2{{\Delta t}}}{3}\, d_x\left({\cal M} \left(A_x \phi \right) D_x\mu \right)_{i,j} &
\nonumber
\\
- \frac{2{{\Delta t}}}{3}\, d_y\left( {\cal M} \left(A_y \phi \right) D_y\mu\right)_{i,j} &=& \frac{4}{3}\phi_{i,j}^m - \frac{1}{3}\phi_{i,j}^{m-1} ,
\label{disc-ch-bdf2-1}
\\
\mu_{i,j} - f_{c,\delta}'\left(\phi_{i,j}\right) + \phi_{i,j} +\epsilon^2\Delta_h\phi_{i,j} &=& 0 .
\label{disc-ch-bdf2-2}
\end{aligned}$$ As for the backward Euler scheme, solvability and stability are not unconditionally guaranteed for this scheme.
We use a nonlinear FAS multigrid method to solve all of the schemes efficiently. We give the details only for the (CS1) scheme, equations (\[disc-ch-1\]) – (\[disc-ch-2\]). The details for the other methods are quite similar. Our solver requires defining operator and source terms, which we do as follows. Let ${\mbox{\boldmath$\phi$}}= \left(\phi,\mu\right)^T$. Define the nonlinear operator ${\bf N} = (N^{(1)},N^{(2)})^T$ as $$\begin{aligned}
N^{(1)}_{i,j}\left({\mbox{\boldmath$\phi$}}\right) &=& \phi_{i,j} - {{\Delta t}}\, d_x\left(M\left(A_x\phi^m\right) D_x\mu \right)_{i,j} - {{\Delta t}}\, d_y\left(M\left(A_y\phi^m\right) D_y\mu \right)_{i,j} ,
\label{op-disc-pfc-1}
\\
N^{(2)}_{i,j}\left({\mbox{\boldmath$\phi$}}\right) &=& \mu_{i,j} - f_{c,\delta}'\left(\phi_{i,j}\right) +\epsilon^2\Delta_h\phi_{i,j} ,
\label{op-disc-pfc-2}
\end{aligned}$$ and the source ${\bf S}= (S^{(1)},S^{(2)})^T$ as $$S^{(1)}_{i,j}\left({\mbox{\boldmath$\phi$}}\right) = \phi_{i,j} \ , \qquad S^{(2)}_{i,j}\left({\mbox{\boldmath$\phi$}}\right) = - \phi_{i,j} .
\label{source-disc-pfc}$$ Then, of course, Equations (\[disc-ch-1\]) – (\[disc-ch-2\]) are equivalent to ${\bf N}({\mbox{\boldmath$\phi$}}^{m+1}) = {\bf S}({\mbox{\boldmath$\phi$}}^m)$. Notice that the operator ${\bf N}$ depends upon the time step $m$, because its definition involves the solution $\phi^m$.
We mention that for the backward Euler (BE) scheme, the only difference in this decomposition is that $$N^{(2)}_{i,j}\left({\mbox{\boldmath$\phi$}}\right) = \mu_{i,j} - f_{c,\delta}'\left(\phi_{i,j}\right) + \phi_{i,j} +\epsilon^2\Delta_h\phi_{i,j} , \quad S^{(2)}_{i,j}\left({\mbox{\boldmath$\phi$}}\right) = 0.$$ The BDF2\_ES and BDF2 schemes are handled using similar considerations.
We will describe a somewhat standard nonlinear FAS multigrid scheme for solving the vector equation ${\bf N}({\mbox{\boldmath$\phi$}}^{m+1}) = {\bf S}({\mbox{\boldmath$\phi$}}^m)$. Here we will sketch only the important points of the algorithm; the reader is referred to Trottenberg *et al.* [@trottenberg01 Sec. 5.3] and our paper [@wise10] for complete details. For this issue, we need to discuss a smoothing operator for generating *smoothed* approximate solutions of ${\bf N}({\mbox{\boldmath$\phi$}}) = {\bf S}$. The action of this operator is represented as $$\widetilde{\mbox{\boldmath$\phi$}}= \mbox{Smooth} \left(\lambda,{\mbox{\boldmath$\phi$}},{\bf N},{\bf S}
\right),
\label{smooth-operator}$$ where ${\mbox{\boldmath$\phi$}}$ is an approximate solution prior to smoothing, $\bar{\mbox{\boldmath$\phi$}}$ is the smoothed approximation, and $\lambda$ is the number of smoothing sweeps. For smoothing we use a nonlinear Gauss-Seidel method with Red-Black ordering. In what follows, to simplify the discussion, we give the details of the relaxation using the simpler lexicographic ordering. Let $\ell$ be the index for the lexicographic Gauss-Seidel. (Note that the smoothing index $\ell$ in the following should not be confused with the time step index $m$.) Now we set $$\begin{aligned}
M^{\rm ew}_{i+{\nicefrac{1}{2}},j} := \mathcal{M}\left( A_x\phi^m_{i+{\nicefrac{1}{2}},j} \right) \ , \ \quad && M^{\rm ns}_{i,j+{\nicefrac{1}{2}}} := \mathcal{M}\left( A_y\phi^m_{i,j+{\nicefrac{1}{2}}}\right) \ .
\nonumber
\end{aligned}$$ The Gauss-Seidel smoothing is as follows: for every $(i,j)$, stepping lexicographically from $(1,1)$ to $(N,N)$, find $\phi^{\ell+1}_{i,j}$, and $\mu^{\ell+1}_{i,j}$ that solve $$\begin{aligned}
&&\hspace{-.2in} \phi^{\ell+1}_{i,j}+\frac{{{\Delta t}}}{h^2}\left(M_{i+{\nicefrac{1}{2}},j}^{\rm ew} + M_{i-{\nicefrac{1}{2}},j}^{\rm ew} +M_{i,j+{\nicefrac{1}{2}}}^{\rm ns} +M_{i,j-{\nicefrac{1}{2}}}^{\rm ns}\right)\mu^{\ell+1}_{i,j}
\nonumber
\\
&=& S^{(1)}_{i,j}\left({\mbox{\boldmath$\phi$}}^m \right)
\nonumber
\\
& &+ \frac{{{\Delta t}}}{h^2}\Big(M_{i+{\nicefrac{1}{2}},j}^{\rm ew}\mu^{\ell}_{i+1,j} + M_{i-{\nicefrac{1}{2}},j}^{\rm ew}\mu^{\ell+1}_{i-1,j} +M_{i,j+{\nicefrac{1}{2}}}^{\rm ns}\mu^{\ell}_{i,j+1} +M_{i,j-{\nicefrac{1}{2}}}^{\rm ns}\mu^{\ell+1}_{i,j-1}\Big) , \quad
\label{smooth-1}
\\
&&\hspace{-0.2in}\left(-f_{c,\delta}''\left(\phi^{\ell}_{i,j}\right)-\frac{4\epsilon^2}{h^2}\right)\phi^{\ell+1}_{i,j} + \mu^{\ell+1}_{i,j}
\nonumber
\\
&=& S^{(2)}_{i,j}\left({\mbox{\boldmath$\phi$}}^m \right) + f_{c,\delta}'\left( \phi_{i,j}^\ell\right)- \phi_{i,j}^\ell f_{c,\delta}''\left( \phi_{i,j}^\ell\right)
\nonumber
\\
&&-\frac{\epsilon^2}{h^2}\left(\phi_{i+1,j}^{\ell}+\phi_{i-1,j}^{\ell+1}+\phi_{i,j+1}^{\ell}+\phi_{i,j-1}^{\ell+1} \right) .
\label{smooth-2}
\end{aligned}$$
Note that we have linearized the logarithmic term using a local Newton approximation, but otherwise this is a standard vector application of block Gauss-Seidel. The $2\times 2$ linear system defined by (\[smooth-1\]) – (\[smooth-2\]) is unconditionally solvable (the determinant of the coefficient matrix is always positive in this case). We use Cramer’s Rule to obtain $\phi^{\ell+1}_{i,j}$ and $\mu^{\ell+1}_{i,j}$. However, we observe that it is not guaranteed that $-1<\phi_{i,j}^{\ell+1}<1$ for an arbitrary smoothing step.
The only difference for the backward Euler (BE) scheme is that second equation in the block smoother is replaced by $$\begin{aligned}
&&\hspace{-0.2in}\left(-f_{c,\delta}''\left(\phi^{\ell}_{i,j}\right)-\frac{4\epsilon^2}{h^2}\right)\phi^{\ell+1}_{i,j} + \mu^{\ell+1}_{i,j}
\nonumber
\\
&=& S^{(2)}_{i,j}\left({\mbox{\boldmath$\phi$}}^m \right) + f_{c,\delta}'\left( \phi_{i,j}^\ell\right) - \phi_{i,j}^\ell - \phi_{i,j}^\ell f_{c,\delta}''\left( \phi_{i,j}^\ell\right)
\nonumber
\\
&&-\frac{\epsilon^2}{h^2}\left(\phi_{i+1,j}^{\ell}+\phi_{i-1,j}^{\ell+1}+\phi_{i,j+1}^{\ell}+\phi_{i,j-1}^{\ell+1} \right) .
\end{aligned}$$
One full block Gauss-Seidel sweep has concluded when we have stepped lexicographically through all the grid points, from $(1,1)$ to $(N,N)$. When $\lambda$ full smoothing sweeps has completed the vector result is labeled $\widetilde{{\mbox{\boldmath$\phi$}}}$, as in Eq. (\[smooth-operator\]), and the action of the smoothing operator in is complete.
Multigrid works on a hierarchy of grids. We denote the grid level by the index $n$, where $n_{\min} \le n \le n_{\max}$, $n_{\max}$ is the index for the finest grid, and $n_{\min}$ is the index for the coarsest grid. We need operators for communicating information from coarse levels to fine levels, and *vice versa*. By ${\bf I}_n^{n-1}$ we denote the restriction operator, which transfers fine grid functions, with grid index $n$, to the coarse grid, indexed by $n-1$. By ${\bf I}_{n-1}^n$ we denote the prolongation operator, which transfers coarse grid functions (level $n-1$) to the fine grid (level $n$). Here we work on cell-centered grids. The restriction operator is defined by cell-center averaging; for the prolongation operator we use piece-wise constant interpolation [@trottenberg01 Sec. 2.8.4]. The rest of the details of the nonlinear multigrid solver are similar to those given in [@wise10]. The details are omitted for the sake of brevity.
Regularization of the logarithm in the multigrid solver {#subsec:delta-parameter}
-------------------------------------------------------
We now give a very brief descussion on how to choose the regularization parameter $\delta$ for the logarithm. The regularization parameter must be chosen small enough so that the computed numerical solutions satisfy $-1+\delta < \phi_{i,j}^m < 1-\delta$, for any $i,j,m$. To understand the issue, consider the following simulation set-up: the common parameters are taken to be $\varepsilon = 5.0\times 10^{-3}$, ${\mathcal M}\equiv 1$, $T=1.0$, $L=1.0$, $h= 1/256$, ${{\Delta t}}= 1.0\times 10^{-3}$, $\tau = 1.0\times 10^{-9}$ (multigrid stopping tolerance). The initial conditions are $\phi_{i,j}^0 = 0.2+r_{i,j}$, where $r_{i,j} \in[-0.05,0.05]$ is a uniformly distributed random variable. We choose various values of the quench parameter $\theta_0$, the smallest being $\theta_0 = 2.0$ and the largest, $\theta_0 = 3.5$. As $\theta_0\to\infty$, the maxima and minima will tend to $1$ and $-1$, respectively, as the singular potential approaches the the obstacle potential.
We compute the maxima and minima of $\phi_{i,j}^m$ and report the values in Table \[tab01\]. Observe that for modest values of $\theta_0$, $\delta\in (0,1)$ can always be chosen so that ${\left\| \phi^m \right\|}_\infty < 1-\delta$. We point out, in particular, the case for which $\theta_0 = 3.0$. We have taken two different values of $\delta$, $1.0\times 10^{-3}$ and $1.0\times 10^{-5}$. The computed solutions – as well as the energies (not shown), which are decreasing at each time step – for these two cases are the same up to round-off errors.
To be safe, in all of the computed solutions that follow, we use the smaller regularization parameter $\delta = 1.0\times 10^{-5}$. The same considerations are applied when picking the regularization parameter for 3-D simulations. In Figure \[fig01\], we show spinodal decomposition simulation using the parameters given in the caption. For $\delta$ sufficiently small, the computed solution stays well inside the interval $(-1+\delta, 1-\delta)$.
![Three-dimensional simulation. The parameters are $L = L_x = L_y = L_z = 1.0$; $\varepsilon = 5\times 10^{-3}$; $\theta_0 = 3.0$; $\delta = 10^{-5}$; $T = 5.0$; $\tau = 10^{-8}$; ${{\Delta t}}= 10^{-3}$, $h = \frac{1}{256}$. For initial data, $\phi^0_{i,j,k} = r_{i,j,k}$, where $r_{i,j,k}$ is a uniformly distributed random variable from the interval $[-0.05,0.05]$. The computed solution stays in the interval $[-0.996,0.996]$, well inside the interval $(-1+\delta, 1-\delta)$. This computation is done using the first-order convex-concave decomposition (CS1) scheme.[]{data-label="fig01"}](fig01.jpg){width="\textwidth"}
$\theta_0$ $\delta$ $\lambda$ ${\displaystyle \max_{i,j,k}\phi_{i,j}^k}$ ${\displaystyle \min_{i,j,k}\phi_{i,j}^k}$
------------ --------------------- ----------- -------------------------------------------- --------------------------------------------
$2.0$ $1.0\times 10^{-3}$ 2 $0.958159539817000$ $-0.969040263101000$
$2.5$ $1.0\times 10^{-3}$ 2 $0.986118743476000$ $-0.990903230905000$
$3.0$ $1.0\times 10^{-3}$ 2 $0.995203610902000$ $-0.997255351479000$
$3.0$ $1.0\times 10^{-5}$ 2 $0.995203610606000$ $-0.997255351459000$
$3.2$ $1.0\times 10^{-5}$ 3 $0.996851091247000$ $-0.998305411243000$
$3.5$ $1.0\times 10^{-5}$ 3 $0.998298616883000$ $-0.999144402772000$
: Maximum and minimum values of $\phi_{i,j}^k$ during spinodal decomposition, computed using the first-order convex-concave decomposition (CS1) scheme. The common parameters are $\varepsilon = 5.0\times 10^{-3}$, $T=1.0$, $L=1.0$, $h= 1/256$, $s = 1.0\times 10^{-3}$, $\tau = 1.0\times 10^{-9}$. The initial conditions are $\phi_{i,j}^0 = 0.2+r_{i,j}$, where $r_{i,j} \in[-0.05,0.05]$ is a uniformly distributed random variable. As $\theta_0$ becomes larger, the potential $f$ approaches the so-called obstacle potential, and the maxima and minima approach $+1$ and $-1$, respectively. But, observe that the computed values stay well within the range $(-1+\delta, 1-\delta)$.[]{data-label="tab01"}
Asymptotic (${{\Delta t}}, h \to 0$) convergence test {#subsec-convergence}
-----------------------------------------------------
Here we give a convergence test for the first-order convex-concave decomposition (CS1) scheme method in 2-D. The initial condition for our convergence test is given by $$\phi(x,y,0) = 1.8\left(\frac{1-\cos\left(\frac{4x\pi}{3.2}\right)}{2}\right) \left(\frac{1-\cos\left(\frac{2 y\pi}{3.2}\right)}{2}\right)-0.9 .
\label{eqn:init}$$ The other parameters are as follows: (domain size) $L = L_x = L_y = 3.2$; (interfacial parameter) $\varepsilon = 0.2$; (mobility) ${\mathcal M}\equiv 1$; (quench parameter) $\theta_0 = 3.0$; ($\ln$ regularization parameter) $\delta = 1\times 10^{-5}$; (final time) $T = 0.4$; (solver stopping tolerance) $\tau = 10^{-9}$; (refinement path) ${{\Delta t}}= 0.4 h^2$. The test results are given in Table \[tab02\] and confirm the predicted accuracy: first order in time and second order in space. The other scheme are expected to exhibit optimal convergence rates, but the tests are not reported here for the sake of brevity.
$h_c$ $h_{f}$ ${\left\| \delta_\phi \right\|}_{2}$ Rate
------------------- ------------------- -------------------------------------- -------
$\frac{3.2}{16}$ $\frac{3.2}{32}$ $5.6689\times 10^{-2}$ –
$\frac{3.2}{32}$ $\frac{3.2}{64}$ $1.6071\times 10^{-2}$ 1.819
$\frac{3.2}{64}$ $\frac{3.2}{128}$ $4.1541\times 10^{-3}$ 1.952
$\frac{3.2}{128}$ $\frac{3.2}{256}$ $1.0472\times 10^{-3}$ 1.988
: Errors and convergence rates. The parameters are (domain size) $L = L_x = L_y = 3.2$; (interfacial parameter) $\varepsilon = 0.2$; (mobility) ${\mathcal M}\equiv 1$; (quench parameter) $\theta_0 = 3.0$; ($\ln$ regularization parameter) $\delta = 10^{-5}$; (final time) $T = 0.4$; (solver stopping tolerance) $\tau = 10^{-9}$; (refinement path) ${{\Delta t}}= 0.4 h^2$. The test results confirm the predicted accuracy: first order in time and second order in space.[]{data-label="tab02"}
Algebraic convergence tests for the multigrid solver {#subsec-multigrid-convergence}
----------------------------------------------------
In this next test, we give some evidence that our multigrid solver for the first-order convex-concave decomposition (CS1) scheme has optimal or nearly optimal complexity. The solvers for the other schemes have similar, near-optimal performance. We use the same test as in Section \[subsec-convergence\]. The only difference is that for this test, we use a fixed time step size, ${{\Delta t}}=10^{-1}$ for all runs. We plot on a semi-log scale of the residual ${\left\| r^n \right\|}_{2}$ with respect to the multigrid iteration count $n$ at the 10th and final time step, *i.e.*, $t=T=1.0$. The initial condition is defined in , and the other parameters are as follows: $L = L_x = L_y = 3.2$; $\varepsilon = 0.2$; ${\mathcal M}\equiv 1$; $\delta = 10^{-5}$. The quench parameter is varied, $\theta_0 = 3.5$, 3.0, and 2.0. The number of multigrid smoothing sweeps is held fixed at $\lambda = 2$. The multigrid stopping tolerance is taken to be $\tau = 10^{-9}$. The tests, reported in Figure \[fig02\], indicate that the residual is reduced by nearly the same amount for each multigrid iteration. This is solid evidence for optimal or nearly optimal complexity. We do observe some minor degradation for larger values of $\theta_0$, which is expected, since the problem becomes increasingly stiff for larger values of $\theta_0$. In particular, the potential is approaching the super-singular obstacle potential in this limit.
![Solver convergence (complexity) test for the problem defined in Section \[subsec-convergence\]. We use a fixed time step size, ${{\Delta t}}=10^{-1}$ for all runs. We plot on a semi-log scale of the residual ${\left\| r^n \right\|}_{\ell^2}$ with respect to the multigrid iteration count $n$ at the 10th and final time step, *i.e.*, $t=T=1.0$. The initial data is defined in , and the other parameters are as follows: $L = L_x = L_y = 3.2$; $\varepsilon = 0.2$; ${\mathcal M}\equiv 1$; $\delta = 10^{-5}$. The quench parameter is varied $\theta_0 = 3.5$, 3.0, and 2.0. The number of multigrid smoothing sweeps is held fixed at $\lambda = 2$. The multigrid stopping tolerance is taken to be $\tau = 10^{-9}$. We observe that the residual is decreasing by a nearly constant factor for each iteration. More iterations are required for larger values of $\theta_0$, as expected.[]{data-label="fig02"}](fig02.pdf){width="\textwidth"}
![Initial data and high-resolution approximate solutions at $t=0.5$ and $t=1.0$. A high-resolution solution is computed using the BDF2 scheme – with the initial data shown in the figure ($t=0$). The parameters for the high-resolution approximation are ${{\Delta t}}= 1.0\times 10^{-5}$ and $h = 1.0/256$. The other parameters are $\Omega = (0,1.0)\times (0,1.0)$ and $\varepsilon = 5.0\times 10^{-3}$, $\theta_0 = 3.0$, $\delta = 1.0\times 10^{-5}$. Significant coarsening occurs between $t=0$ and $t=1.0$. In the simulation, we observe that, for the approximate solution, $0.99672 \ge \phi \ge -0.99821$.[]{data-label="fig:compare"}](fig03.pdf){width="\textwidth"}
Scheme Error $t=0.1$ Error $t=0.5$ Error $t=1$ Ave. Itr. $ \displaystyle{\max_{i,j,k}\phi^k_{i,j}}$
---------- -------------------- -------------------- -------------------- ----------- --------------------------------------------
BDF2 $2.2496{\rm e}-04$ $9.3172{\rm e}-04$ $5.2566{\rm e}-04$ 4.6237 0.99671
BDF2\_ES $3.9485{\rm e}-02$ $1.9105{\rm e}-01$ $2.6703{\rm e}-01$ 3.7373 0.99890
$5.8446{\rm e}-03$ $1.9113{\rm e}-02$ $1.4204{\rm e}-02$ 3.5390 0.99913
BE $2.7285{\rm e}-03$ $8.4211{\rm e}-03$ $5.7435{\rm e}-03$ 6.5645 0.99668
CS1 $3.5965{\rm e}-01$ $5.6166{\rm e}-01$ $7.5356{\rm e}-01$ 4.0003 0.99621
: The errors, average V-cycle iteration numbers for the FAS multigrid solvers, and the maximum values of $\phi$ for the various schemes with fixed time and space step sizes ${{\Delta t}}= 1.0\times 10^{-4}$ and $h = 1.0/256$. The other parameters are $\Omega = (0,1.0)\times (0,1.0)$ and $\varepsilon = 5.0\times 10^{-3}$, $\theta = 3.0$, $\delta = 1.0\times 10^{-5}$. The “errors,” which are reported at times $t=0.1$, $t=0.5$ and $t=1.0$, are precisely the differences between the comparison approximations and the high-resolution target approximation computed using the BDF2 with the much smaller time step size ${{\Delta t}}= 5\times 10^{-6}$. See Figure \[fig:compare\].[]{data-label="tab:comparison-1"}
Scheme Error $t=0.1$ Error $t=0.5$ Error $t=1$ Ave. Itr. $ \displaystyle{\max_{i,j,k}\phi^k_{i,j}}$
---------- -------------------- -------------------- -------------------- ----------- --------------------------------------------
BDF2 $5.7762{\rm e}-05$ $2.3749{\rm e}-04$ $1.3267{\rm e}-04$ 3.49560 0.99671
BDF2\_ES $1.0079{\rm e}-02$ $1.1464{\rm e}-02$ $7.6568{\rm e}-03$ 2.70300 0.99668
$1.6392{\rm e}-03$ $5.1465{\rm e}-03$ $4.0927{\rm e}-03$ 2.6401 0.99671
BE $1.3510{\rm e}-03$ $4.1560{\rm e}-03$ $2.8182{\rm e}-03$ 3.63475 0.99670
CS1 $1.4975{\rm e}-01$ $2.7690{\rm e}-01$ $3.5650{\rm e}-01$ 2.78145 0.99628
: The errors, average V-cycle iteration numbers for the FAS multigrid solvers, and the maximum values of $\phi$ for the various scheme with fixed time and space step sizes ${{\Delta t}}= 5.0\times 10^{-5}$ and $h = 1.0/256$. The other parameters are $\Omega = (0,1.0)\times (0,1.0)$ and $\varepsilon = 5.0\times 10^{-3}$, $\theta = 3.0$, $\delta = 1.0\times 10^{-5}$. The “errors,” which are reported at times $t=0.1$, $t=0.5$ and $t=1.0$, are precisely the differences between the comparison approximations and the high-resolution target approximation computed using the BDF2 with the much smaller time step size ${{\Delta t}}= 5\times 10^{-6}$. See Figure \[fig:compare\].[]{data-label="tab:comparison-2"}
### Initial data and a high-resolution approximate solution at $t=1$
A high-resolution solution is computed using the BDF2 scheme – with the initial data shown in Figure \[fig:compare\] ($t=0$). The parameters for the approximation are ${{\Delta t}}= 5\times 10^{-6}$ and $h = 1.0/256$. The physical parameters are $\Omega = (0,1)^2$, $\varepsilon = 5.0\times 10^{-3}$, ${\mathcal M}\equiv 1$; $\theta_0 = 3.0$, and $\delta = 1.0\times 10^{-5}$. Note that the time step size ${{\Delta t}}= 5\times 10^{-6}$ is 10 times smaller than what will be used in the comparison tests, and we will treat the approximation obtained here as the target solution. We point out that computing the target solution with the slightly larger time step of ${{\Delta t}}= 1\times 10^{-5}$ does not change the results presented in Tables \[tab:comparison-1\] and \[tab:comparison-2\] in any significant way.
### Comparison results
For the comparison computations we use the same parameters as above – $h = 1.0/256$, $\Omega = (0,1)^2$, $\varepsilon = 5.0\times 10^{-3}$, ${\mathcal M}\equiv 1$; $\theta_0 = 3.0$, $\delta = 1.0\times 10^{-5}$ – but we use larger time step sizes: ${{\Delta t}}= 1.0\times 10^{-4}$ (Table \[tab:comparison-1\]) and ${{\Delta t}}= 5.0\times 10^{-5}$ (Table \[tab:comparison-2\]). To solve all of the schemes, we employ the FAS multigrid methods detailed above. The results of the tests are reported in Tables \[tab:comparison-1\] and \[tab:comparison-2\], and they paint a complicated picture. The BDF2 scheme shows excellent accuracy and efficiency. Based on our experience, this method is the most accurate of the four that have been test, which is why it is used to generate our target solution. Our new BDF2\_ES scheme is slightly more efficient, but not nearly as accurate. When the stabilization parameter is set to zero ($A = 0$), its accuracy increases significantly, but its provable stability is lost.
The first-order convex-concave decomposition scheme is the worst in the tests for accuracy, but the second best in efficiency per step. The worst in efficiency per time step is the backward Euler scheme; like the BDF2 scheme, it does not have a convex structure. But, like pure BDF2, the fully implicit backward Euler has very good accuracy, better than the energy stabilized BDF2 scheme with the stabilization parameter set to zero.
All of the schemes are positivity preserving, as long as they are solvable. Even though we did not prove this claim for the fully implicit schemes, such a fact can be established in our theory, though the details are significantly more complicated and are skipped in this presentation.
Conclusion remarks {#sec:conclusion}
==================
In this paper we have presented and analyzed two positivity preserving, energy stable finite difference schemes for the Allen Cahn/Cahn-Hilliard model with a logarithmic Flory Huggins energy potential, including both the first and second order temporal accuracy. In particular, the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution from reaching these singular values, and this subtle fact indicates that the proposed numerical algorithm has a unique solution with preserved positivity for the logarithmic arguments. In turn, the numerical scheme is always well-defined, as long as the numerical solution stays bounded at the previous time step, which is natural. And also, an unconditional energy stability has been theoretically justified; in particular, an artificial Douglas-Dupont regularization term is added in the second order BDF scheme to ensure the energy stability. In addition, an optimal rate convergence in the $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm has been established for both the first and second order accurate schemes. An efficient multigrid solver is applied in the practical implementation, and some numerical results are presented, which demonstrate the robustness and efficiency of the numerical solver.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is supported in part by the grants NSFC 11671098, 11331004, 91630309, a 111 project B08018 (W. Chen), NSF DMS-1418689 (C. Wang), NSF DMS-1715504 and Fudan University start-up (X. Wang) and NSF DMS-1719854 (S. Wise). C. Wang also thanks the Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, and Shanghai Center for Mathematical Sciences, for support during his visit. During the finalization of the manuscript, S. Wise was partially supported by the Techniche Universität, Dresden (TUD), as a senior Dresden Fellow and by Oak Ridge National Laboratory (ORNL) while this work was being completed. S. Wise thanks TUD and Prof. Axel Voigt for the generous support and hospitality and thank Cory Hauck (ORNL) for support and discussions on this and related topics.
[^1]: Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences; Fudan University, Shanghai, China 200433 ([[email protected]]{})
[^2]: Mathematics Department; University of Massachusetts; North Dartmouth, MA 02747, USA ([corresponding author: [email protected]]{})
[^3]: Fudan University, Shanghai, China 200433, and Florida State University, Tallahassee, FL 32306, USA ([[email protected]]{})
[^4]: Mathematics Department; University of Tennessee; Knoxville, TN 37996, USA ([[email protected]]{})
|
{
"pile_set_name": "ArXiv"
}
|
---
address:
- 'A. Yu. Karlovich: Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829–516 Caparica, Portugal.'
- 'Yu. I. Karlovich: Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos, México.'
- 'A. B. Lebre: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049–001 Lisboa, Portugal.'
author:
- 'Alexei Yu. Karlovich, Yuri I. Karlovich, and Amarino B. Lebre'
title: |
Necessary Conditions for Fredholmness\
of Singular Integral Operators with Shifts\
and Slowly Oscillating Data
---
Introduction
============
A bounded linear operator on a Banach space is said to be Fredholm if its image is closed and the dimensions of its kernel and the kernel of its adjoint operator are finite. Let ${\mathbb{R}}_+:=(0,+\infty)$. A bounded and continuous function $f$ on ${\mathbb{R}}$ is called slowly oscillating (at $0$ and $\infty$) if for each (equivalently, for some) $\lambda\in(0,1)$, $$\lim_{r\to s}\sup_{t,\tau\in[\lambda r,r]}|f(t)-f(\tau)|=0
\quad (s\in\{0,\infty\}).$$ The set $SO({\mathbb{R}}_+)$ of all slowly oscillating functions forms a $C^*$-algebra. This algebra properly contains $C(\overline{{\mathbb{R}}}_+)$, the $C^*$-algebra of all continuous functions on $\overline{{\mathbb{R}}}_+:=[0,+\infty]$. Suppose $\alpha$ is an orientation-preserving diffeomorphism of ${\mathbb{R}}_+$ onto itself, which has only two fixed points $0$ and $\infty$. We say that $\alpha$ is a slowly oscillating shift if $\log\alpha'$ is bounded and $\alpha'\in SO({\mathbb{R}}_+)$. The set of all slowly oscillating shifts is denoted by $SOS({\mathbb{R}}_+)$.
Through the paper we suppose that $1<p<\infty$ and $1/p+1/q=1$. It is easy to see that if $\alpha\in SOS({\mathbb{R}}_+)$, then the shift operator $W_\alpha$ defined by $W_\alpha f=f\circ\alpha$ is bounded and invertible on all spaces $L^p({\mathbb{R}}_+)$ and its inverse is given by $W_\alpha^{-1}=W_\beta$, where $\beta:=\alpha_{-1}$ is the inverse function to $\alpha$. It is well known that the Cauchy singular integral operator $S$ given by $$(Sf)(t):=\lim_{\varepsilon\to 0} \frac{1}{\pi i}
\int_{{\mathbb{R}}_+\setminus(t-\varepsilon,t+\varepsilon)}
\frac{f(\tau)}{\tau-t}\:d\tau\quad(t\in{\mathbb{R}}_+)$$ is bounded on all Lebesgue spaces $L^p({\mathbb{R}}_+)$ for $1<p<\infty$. Put $P_\pm:=(I\pm S)/2$.
By $M(\mathfrak{A})$ denote the maximal ideal space of a unital commutative Banach algebra $\mathfrak{A}$. Identifying the points $t\in\overline{{\mathbb{R}}}_+$ with the evaluation functionals $t(f)=f(t)$ for $f\in C(\overline{{\mathbb{R}}}_+)$, we get $M(C(\overline{{\mathbb{R}}}_+))=\overline{{\mathbb{R}}}_+$. Consider the fibers $$M_s(SO({\mathbb{R}}_+)):=\big\{\xi\in M(SO({\mathbb{R}}_+)):\xi|_{C(\overline{{\mathbb{R}}}_+)}=s\big\}$$ of the maximal ideal space $M(SO({\mathbb{R}}_+))$ over the points $s\in\{0,\infty\}$. By [@K08 Proposition 2.1], the set $$\Delta:=M_0(SO({\mathbb{R}}_+))\cup M_\infty(SO({\mathbb{R}}_+))$$ coincides with $\operatorname{clos}_{SO^*}{\mathbb{R}}_+\setminus{\mathbb{R}}_+$ where $\operatorname{clos}_{SO^*}{\mathbb{R}}_+$ is the weak-star closure of ${\mathbb{R}}_+$ in the dual space of $SO({\mathbb{R}}_+)$. Then $M(SO({\mathbb{R}}_+))=\Delta\cup{\mathbb{R}}_+$. In what follows we write $a(\xi):=\xi(a)$ for every $a\in SO({\mathbb{R}}_+)$ and every $\xi\in\Delta$. This paper is a continuation of our work [@KKLsufficiency], where the following result was proved.
\[th:sufficiency\] Let $a,b,c,d\in SO({\mathbb{R}}_+)$ and $\alpha\in SOS({\mathbb{R}}_+)$. The singular integral operator $$\label{eq:def-N}
N:= (aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$$ with the shift $\alpha$ is Fredholm on the space $L^p({\mathbb{R}}_+)$ if the following two conditions are fulfilled:
1. the functional operators $A_+:=a I-bW_\alpha$ and $A_-:=cI-dW_\alpha$ are invertible on the space $L^p({\mathbb{R}}_+)$;
2. for every pair $(\xi,x)\in\Delta\times{\mathbb{R}}$, $$\begin{aligned}
n_\xi(x)
&:=
\Big[a(\xi)-b(\xi)e^{i\omega(\xi)(x+i/p)}\Big]
\frac{1+\coth[\pi(x+i/p)]}{2}
\nonumber
\\
&\quad+
\Big[c(\xi)-d(\xi)e^{i\omega(\xi)(x+i/p)}\Big]
\frac{1-\coth[\pi(x+i/p)]}{2}\ne 0,
\label{eq:def-n}\end{aligned}$$ where $\omega(t):=\log[\alpha(t)/t]\in SO({\mathbb{R}}_+)$.
It turns out that the sufficient conditions for the Fredholmness of the operator $N$ contained in Theorem \[th:sufficiency\] are also necessary.
\[th:main\] Suppose $a,b,c,d\in SO({\mathbb{R}}_+)$ and $\alpha\in SOS({\mathbb{R}}_+)$. If the operator $N$ given by is Fredholm on $L^p({\mathbb{R}}_+)$, then conditions [(i)]{} and [(ii)]{} of Theorem [\[th:sufficiency\]]{} are fulfilled.
The proof of Theorem \[th:main\] is based on the method of limit operators, which was essentially developed by V. S. Rabinovich (see, e.g., [@BKR00; @RRS04; @L06] and the references therein), and on the Allan-Douglas localization (see [@BS06]). The paper is organized as follows. In Section \[sec:SO-SOS\] we collect properties of slowly oscillating functions and slowly oscillating shifts. In Section \[sec:Mellin-convolution\] we recall properties of Mellin convolution operators with piecewise continuous and semi-almost periodic symbols. In Section \[sec:LO\] we recall that if an $A$ operator is invertible modulo some ideal ${\mathfrak{J}}$ and the limit operators for all operators in this ideal vanish, then the limit operator of $A$ is invertible whenever it exists. Further we calculate the limit operators of the operator $N$ with respect to two different systems of pseudoisometries (dilations and modulations). Let ${\mathcal{K}}$ be the ideal of all compact operators on $L^p({\mathbb{R}}_+)$. In [@KKLsufficiency] we introduced the algebra ${\mathcal{Z}}$ generated by the ideal ${\mathcal{K}}$, the operators $I,S$, and $cR$, where $c\in SO({\mathbb{R}}_+)$ and $R$ is the operator with fixed singularities at $0$ and $\infty$ given by $$(R f)(t):=\frac{1}{\pi i}\int_0^\infty\frac{f(\tau)}{\tau+t}\quad (t\in{\mathbb{R}}_+).$$ It turns out that the algebra $\Lambda$ of all operators commuting with the elements of ${\mathcal{Z}}$ modulo the ideal ${\mathcal{K}}$ contains the operator $N$. In Section \[sec:localization\] we state a consequence of the Allan-Douglas local principle for $A\in\Lambda$, which was obtained in [@KKLsufficiency]. In Section \[sec:FO\] we formulate an invertibility criterion for $aI-bW_\alpha$ with slowly oscillating data (Theorem \[th:FO\]) and prove two auxiliary statements: a corollary of Theorem \[th:FO\] related to the existence of infinite dimensional kernel or cokernel for $aI-bW_\alpha$ and a criterion for the invertibility of $aI-bW_\alpha$ with multiplicative shift $\alpha$. In the proof of the latter result we use limit operators with respect to a specially chosen system of modulations, so that the limit operator of $W_\alpha$ is equal to $W_\alpha$. Section \[sec:necessity\] is devoted to the proof of Theorem \[th:main\]. First we observe that the limit operators with respect to dilations are $$\label{eq:LO-introduction}
\big(a(\xi)I-b(\xi)W_{\alpha_\xi}\big)P_++
\big(c(\xi)I-d(\xi)W_{\alpha_\xi}\big)P_-,$$ where $\xi\in\Delta$ and $\alpha_\xi(t)=e^{\omega(\xi)}t$ is a multiplicative shift. Since $N$ is Fredholm, all limit operators are invertible for $\xi\in\Delta$. Applying the results of Sections \[sec:localization\] and \[sec:FO\], we prove that then the operators $a(\xi)I-b(\xi)W_{\alpha_\xi}$ and $c(\xi)I-d(\xi)W_{\alpha_\xi}$ are invertible for all $\xi\in\Delta$. Since the fibers $M_0(SO({\mathbb{R}}_+))$ and $M_\infty(SO({\mathbb{R}}_+))$ are connected, from the above observation and Theorem \[th:FO\] it follows that the operators $aI-bW_\alpha$ and $cI-dW_\alpha$ are invertible, and this is condition (i) of Theorem \[th:sufficiency\]. On the other hand, the (invertible) limit operators are similar to the Mellin convolution operators with the semi-almost periodic symbols $n_\xi$. Applying the invertibility criterion for such operators (Theorem \[th:invertibility-convolution\]), we arrive at condition (ii) of Theorem \[th:sufficiency\].
Slowly oscillating functions and shifts {#sec:SO-SOS}
=======================================
Properties of slowly oscillating functions
------------------------------------------
The following two lemmas give important properties of the fibers $M_0(SO({\mathbb{R}}_+))$ and $M_\infty(SO({\mathbb{R}}_+))$.
\[le:SO-fundamental-property\] Let $\{a_k\}_{k=1}^\infty$ be a countable subset of the space $SO({\mathbb{R}}_+)$ and $s\in\{0,\infty\}$. For each $\xi\in M_s(SO({\mathbb{R}}_+))$ there exists a sequence $\{t_n\}\subset{\mathbb{R}}_+$ such that $t_n\to s$ as $n\to\infty$ and $$\label{eq:SO-fundamental-property}
\xi(a_k)=\lim_{n\to\infty}a_k(t_n)\quad\mbox{for all}\quad k\in{\mathbb{N}}.$$ Conversely, if $\{t_n\}\subset{\mathbb{R}}_+$ is a sequence such that $t_n\to s$ as $n\to\infty$, then there exists a functional $\xi\in M_s(SO({\mathbb{R}}_+))$ such that holds.
\[le:connected-fibers\] The fibers $M_0(SO({\mathbb{R}}_+))$ and $M_\infty(SO({\mathbb{R}}_+))$ are connected compact Hausdorff spaces.
Fix $s\in\{0,\infty\}$. Since $M_s(SO({\mathbb{R}}_+))$ is a closed subset of the compact Hausdorff space $M(SO({\mathbb{R}}_+))$, we conclude that $M_s(SO({\mathbb{R}}_+))$ also is a compact Hausdorff space. Suppose the fiber $M_s(SO({\mathbb{R}}_+))$ is disconnected. Then there exist two disjoint closed subsets $X_1$ and $X_2$ such that $M_s(SO({\mathbb{R}}_+))=X_1\cup X_2$. Take a continuous function $\widehat{a}$ on $M_s(SO({\mathbb{R}}_+))$ such that $\widehat{a}(X_1)\subset[0,1/3]$ and $\widehat{a}(X_2)\subset[2/3,1]$. By the Tietze extension theorem (see e.g. [@RS80 Theorem IV.11]), the function $\widehat{a}$ is extended to a continuous function on the whole compact space $M(SO({\mathbb{R}}_+))$. We denote this extension again by $\widehat{a}$. Because $SO({\mathbb{R}}_+)$ is a $C^*$-algebra, the function $\widehat{a}\in C(M(SO({\mathbb{R}}_+)))$ is the Gelfand transform of a function $a\in SO({\mathbb{R}}_+)$. Then in view of Lemma \[le:SO-fundamental-property\] there are sequences $t_n',t_n''\to s$ such that there exist $\lim\limits_{n\to\infty}a(t_n')\in[0,1/3]$ and $\lim\limits_{n\to\infty}a(t_n'')\in[2/3,1]$. Since $a\in SO({\mathbb{R}}_+)$ is continuous on ${\mathbb{R}}_+$, there are points $t_n$ between $t_n'$ and $t_n''$ such that $a(t_n)=1/2$. Then $t_n\to s$, $\displaystyle\lim_{n\to\infty}
a(t_n)=1/2$, and hence $1/2\in\widehat{a}(X_1)\cup\widehat{a}(X_2)$, a contradiction. Thus, $M_s(SO({\mathbb{R}}_+))$ is a connected set.
Repeating literally the proofs of [@KKL03 Proposition 3.3] and [@KKL03 Lemma 3.5], we obtain the following two statements.
\[le:SO-nec\] Suppose $\varphi\in C^1({\mathbb{R}}_+)$ and put $\psi(t):=t\varphi'(t)$ for $t\in{\mathbb{R}}_+$. If $\varphi,\,\psi\in SO({\mathbb{R}}_+)$, then $\lim\limits_{t\to s}\psi(t)=0$ for $s\in\{0,\infty\}$.
\[le:SO-uniform\] Let $a\in SO({\mathbb{R}}_+)$. Suppose continuous functions $f_j:{\mathbb{R}}_+\to{\mathbb{R}}_+$ $(j=1,2)$ and ${{\mathcal{F}}}:{\mathbb{R}}_+\times{\mathbb{R}}_+\to{\mathbb{R}}_+$ satisfy the relation $$xf_1(y)\le{{\mathcal{F}}}(x,y)\le xf_2(y), \quad x,y\in{\mathbb{R}}_+.$$ If for some sequence $t=\{t_n\}_{n=1}^\infty$ tending to $s\in\{0,\infty\}$ the limit $$\lim_{n\to\infty}a(t_n)=:a_t$$ exists, then for every $y\in{\mathbb{R}}_+$ the limit $\lim\limits_{n\to\infty}a({{\mathcal{F}}}(t_n,y))$ also exists. Moreover, $$\lim_{n\to\infty}a({{\mathcal{F}}}(t_n,y))=a_t,$$ and the convergence is uniform on every segment $J\subset{\mathbb{R}}_+$.
Properties of slowly oscillating shifts
---------------------------------------
In this subsection we list necessary properties of slowly oscillating shifts.
\[le:exp-repr\] An orientation-preserving non-Carleman shift $\alpha:{\mathbb{R}}_+\to{\mathbb{R}}_+$ belongs to $SOS({\mathbb{R}}_+)$ if and only if $$\label{eq:exp-repr-1}
\alpha(t)=te^{\omega (t)},\quad t\in {\mathbb{R}}_+,$$ for some real-valued function $\omega\in SO({\mathbb{R}}_+)\cap C^1({\mathbb{R}}_+)$ such that the function $t\mapsto t\omega^\prime(t)$ also belongs to $SO({\mathbb{R}}_+)$ and $\inf\limits_{t\in{\mathbb{R}}_+}\big(1+t\omega'(t)\big)>0$.
The function $\omega$ in is referred to as the exponent function of $\alpha$.
\[le:continuous-SOS\] Suppose $c\in SO({\mathbb{R}}_+)$ and $\alpha\in SOS({\mathbb{R}}_+)$. Then $c\circ\alpha\in SO({\mathbb{R}}_+)$.
\[le:SOS-inverse\] If $\alpha\in SOS({\mathbb{R}}_+)$, then $\beta\in SOS({\mathbb{R}}_+)$.
\[le:SOS-derivative\] Suppose $\alpha\in SOS({\mathbb{R}}_+)$ and $\omega$ is the exponent function of $\alpha$. Then $\alpha'(\xi)=e^{\omega(\xi)}$ for all $\xi\in\Delta$.
By Lemma \[le:exp-repr\], $\alpha(\tau)=\tau e^{\omega(\tau)}$ for $\tau\in{\mathbb{R}}_+$ with $\omega\in SO({\mathbb{R}}_+)\cap C^1({\mathbb{R}}_+)$ and $\psi(\tau)=\tau\omega'(\tau)\in SO({\mathbb{R}}_+)$. From Lemma \[le:SO-nec\] it follows that $$\label{eq:SOS-derivative-1}
\lim_{\tau\to s}\tau\omega'(\tau)
=0
\quad\mbox{for}\quad s\in\{0,\infty\}.$$ Fix $s\in\{0,\infty\}$. By Lemma \[le:SO-fundamental-property\], for a given $\xi\in M_s(SO({\mathbb{R}}_+))$ there exists a sequence $t_n\to s$ as $n\to\infty$ such that $$\label{eq:SOS-derivative-2}
\omega(\xi) =\lim_{n\to\infty}\omega(t_n),
\quad
\alpha'(\xi)=\lim_{n\to\infty}\alpha'(t_n)$$ (recall that $\alpha'\in SO({\mathbb{R}}_+)$). Clearly, $\alpha'(\tau)=(1+t\omega'(\tau))e^{\omega(\tau)}$ for $\tau\in{\mathbb{R}}_+$. Combining this relation with –, we get $\alpha'(\xi)=e^{\omega(\xi)}$.
Convolution operators {#sec:Mellin-convolution}
=====================
Fourier convolution operators
-----------------------------
For a Banach space $X$, let ${\mathcal{B}}(X)$ be the Banach algebra of all bounded linear operators on $X$ and let ${\mathcal{K}}(X)$ be the closed two-sided ideal of all compact operators on $X$.
Let $F:L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ denote the Fourier transform, $$(Ff)(x):=\int_{\mathbb{R}}f(y)e^{-ixy}dy\quad (x\in{\mathbb{R}}),$$ and let $F^{-1}:L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ be the inverse of $F$. A function $a\in L^\infty({\mathbb{R}})$ is called a Fourier multiplier if the map $f\mapsto F^{-1}aFf$ maps $L^2({\mathbb{R}})\cap L^p({\mathbb{R}})$ onto itself and extends to a bounded operator on $L^p({\mathbb{R}})$. The latter operator is then denoted by $W^0(a)$. We let $M_p({\mathbb{R}})$ stand for the set of all Fourier multipliers on $L^p({\mathbb{R}})$. One can show that $M_p({\mathbb{R}})$ is a Banach algebra under the norm $$\|a\|_{M_p({\mathbb{R}})}:=\|W^0(a)\|_{{\mathcal{B}}(L^p({\mathbb{R}}))}.$$
Mellin convolution operators
----------------------------
Let $d\mu(t)=dt/t$ be the (normalized) invariant measure on ${\mathbb{R}}_+$. Consider the Fourier transform on $L^2(\mathbb{R}_+,d\mu)$, which is usually referred to as the Mellin transform and is defined by $$M:L^2({\mathbb{R}}_+,d\mu)\to L^2({\mathbb{R}}),
\quad
(Mf)(x)=\int_{{\mathbb{R}}_+} f(t) t^{-ix}\,\frac{dt}{t}.$$ It is an invertible operator, with inverse given by $${M^{-1}}:L^2({\mathbb{R}})\to L^2({\mathbb{R}}_{+},d\mu),
\quad
({M^{-1}}g)(t)= \frac{1}{2\pi}\int_{{\mathbb{R}}}
g(x)t^{ix}\,dx.$$ Let $E$ be the isometric isomorphism $$\label{eq:def-E}
E:L^p({\mathbb{R}}_+,d\mu)\to L^p({\mathbb{R}}),
\quad
(Ef)(x):=f(e^x)\quad (x\in{\mathbb{R}}).$$ Then the map $A\mapsto E^{-1}AE$ transforms the Fourier convolution operator given by $W^0(a)=F^{-1}aF$ to the Mellin convolution operator $$\operatorname{Co}(a):=M^{-1}aM$$ with the same symbol $a$. Hence the class of Fourier multipliers on $L^p({\mathbb{R}})$ coincides with the class of Mellin multipliers on $L^p({\mathbb{R}}_+,d\mu)$.
Piecewise continuous multipliers
--------------------------------
We denote by $PC$ the $C^*$-algebra of all bounded piecewise continuous functions on $\dot{{\mathbb{R}}}ª={\mathbb{R}}\cup\{\infty\}$. By definition, $a\in PC$ if and only if $a\in L^\infty({\mathbb{R}})$ and the one-sided limits $$a(x_0-0):=\lim_{x\to x_0-0}a(x),
\quad
a(x_0+0):=\lim_{x\to x_0+0}a(x)$$ exist for each $x_0\in\dot{{\mathbb{R}}}$. If a function $a$ is given everywhere on ${\mathbb{R}}$, then its total variation of $a$ is defined by $$V(a):=\sup\sum_{k=1}^n|a(x_k)-a(x_{k-1})|,$$ where the supremum is taken over all $n\in{\mathbb{N}}$ and $$-\infty<x_0<x_1<\dots<x_n<+\infty.$$ If $a$ has a finite total variation, then it has finite one-sided limits $a(x-0)$ and $a(x+0)$ for all $x\in\dot{{\mathbb{R}}}$, that is, $a\in PC$. If $a$ is an absolutely continuous function of finite total variation on ${\mathbb{R}}$, then $a'\in L^1({\mathbb{R}})$ and $$V(a)=\int_{\mathbb{R}}|a'(x)|dx$$ (see, e.g., [@N55 Chap. VIII, Sections 3 and 9; Chap. XI, Section 4]).
The following theorem gives an important subset of $M_p({\mathbb{R}})$. Its proof can be found, e.g., in [@BKS02 Theorem 17.1].
\[th:Stechkin\] If $a\in PC$ has finite total variation $V(a)$, then $a\in M_p({\mathbb{R}})$ and $$\|a\|_{M_p({\mathbb{R}})}\le\|S_{\mathbb{R}}\|_{{\mathcal{B}}(L^p({\mathbb{R}}))}\big(\|a\|_{L^\infty({\mathbb{R}})}+V(a)\big),$$ where $S_{\mathbb{R}}$ is the Cauchy singular integral operator on ${\mathbb{R}}$.
According to [@BKS02 p. 325], let $PC_p$ be the closure in $M_p({\mathbb{R}})$ of the set of all functions $a\in PC$ with finite total variation on ${\mathbb{R}}$. Following [@BKS02 p. 331], put $$C_p(\overline{{\mathbb{R}}}):=PC_p\cap C({\mathbb{R}}),\quad \overline{{\mathbb{R}}}:=[-\infty,+\infty].$$
Algebra generated by the Cauchy singular integral operator
----------------------------------------------------------
Suppose $\mathfrak{A}$ is a Banach algebra and $\mathfrak{S}$ is a subset of $\mathfrak{A}$. Let ${\operatorname{alg}}_\mathfrak{A}\mathfrak{S}$ denote the smallest closed subalgebra of $\mathfrak{A}$ containing $\mathfrak{S}$ and let $\operatorname{id}_\mathfrak{A}\mathfrak{S}$ denote the smallest closed two-sided ideal of $\mathfrak{A}$ containing $\mathfrak{S}$.
Let ${\mathcal{B}}:={\mathcal{B}}(L^p({\mathbb{R}}_+))$, ${\mathcal{K}}:={\mathcal{K}}(L^p({\mathbb{R}}_+))$, and ${\mathcal{A}}:={\operatorname{alg}}_{\mathcal{B}}\{I,S\}$. Consider the isometric isomorphism $$\label{eq:def-Phi}
\Phi:L^p({\mathbb{R}}_+)\to L^p({\mathbb{R}}_+,d\mu),
\quad
(\Phi f)(t):=t^{1/p}f(t)\quad(t\in{\mathbb{R}}_+).$$ The following fact is well known (see, e.g., [@RS90 Section 2]).
\[th:algebra-A\] The algebra ${\mathcal{A}}$ is the smallest closed subalgebra of ${\mathcal{B}}$ that contains the operators $\Phi^{-1}\operatorname{Co}(a)\Phi$ with $a\in C_p(\overline{{\mathbb{R}}})$. The functions $$s_p(x):=\coth[\pi(x+i/p)]
\quad
r_p(x):=1/\sinh[\pi(x+i/p)]
\quad(x\in{\mathbb{R}})$$ belong to $C_p(\overline{{\mathbb{R}}})$ and the operators $S$ and $R$ are similar to the Mellin convolution operators: $$\Phi S\Phi^{-1}=\operatorname{Co}(s_p),
\quad
\Phi R\Phi^{-1}=\operatorname{Co}(r_p).$$
From $s_p^2-r_p^2=1$ and Theorem \[th:algebra-A\] it follows that $$\label{eq:S-R-relation}
4P_+P_-=4P_-P_+=I-S^2=-R^2.$$
\[th:compactness-commutators\] If $a\in SO({\mathbb{R}}_+)$ and $\alpha\in SOS({\mathbb{R}}_+)$, then for every $A\in{\mathcal{A}}$ the operators $aA-AaI$ and $W_\alpha A-AW_\alpha$ are compact.
Semi-almost periodic multipliers
--------------------------------
The following simple statement motivates us to enlarge the class of piecewise continuous multipliers.
\[le:mult-shift-convolution\] Let $\alpha:{\mathbb{R}}_+\to{\mathbb{R}}_+$ be a multiplicative shift given by $\alpha(t)=kt$ for all $t\in{\mathbb{R}}_+$ with some $k\in{\mathbb{R}}_+$. Then $\Phi W_\alpha \Phi^{-1}=\operatorname{Co}(m)$ with $m(x):=e^{i(x+i/p)\log k}$ for $x\in{\mathbb{R}}$.
The proof is a matter of a direct calculation.
A function $p:{\mathbb{R}}\to{\mathbb{C}}$ of the form $p(x)=\sum_{\lambda\in\Omega} r_\lambda e^{i\lambda x}$, where $r_\lambda\in{\mathbb{C}}$, $\lambda\in{\mathbb{R}}$, and $\Omega$ is a finite subset of ${\mathbb{R}}$, is called an almost periodic polynomial. The set of all almost periodic polynomials is denoted by $AP_0$. From Lemma \[le:mult-shift-convolution\] it follows that $AP_0\subset M_p({\mathbb{R}})$. According to [@BKS02 p. 372], $AP_p$ denotes the closure of the set of all almost periodic polynomials in the norm of $M_p({\mathbb{R}})$ and $SAP_p$ denotes the smallest closed subalgebra of $M_p({\mathbb{R}})$ that contains $C_p(\overline{{\mathbb{R}}})$ and $AP_p$.
Applying the inverse closedness of the algebra $SAP_p$ in $L^\infty({\mathbb{R}})$ (see [@BKS02 Proposition 19.4]), we immediately get the following.
\[th:invertibility-convolution\] Suppose $a\in SAP_p$. The Mellin convolution operator $\operatorname{Co}(a)$ is invertible on the space $L^p({\mathbb{R}}_+,d\mu)$ if and only if $\inf\limits_{x\in{\mathbb{R}}}|a(x)|>0$.
Limit operators {#sec:LO}
===============
Abstract approach
-----------------
In our previous work [@KKL03] the techniques of limit operators (see, e.g., [@BKR00; @L06; @RRS04]) was successfully used to study the invertibility of binomial functional operators that are now the coefficients of the singular integral operator with shift $N$ given by . In what follows we make use of such techniques to obtain a necessary condition for the Fredholmness of the operator $N$. Let us recall the abstract version of such techniques.
Let $X$ be a Banach space and let $X^*$ be its dual space. We say that an operator $U\in{\mathcal{B}}(X)$ is a *pseudoisometry* if $U$ is invertible in ${\mathcal{B}}(X)$ and $$\|U\|_{{\mathcal{B}}(X)}=1/\|U^{-1}\|_{{\mathcal{B}}(X)}.$$ Let $A\in{\mathcal{B}}(X)$ and ${\mathcal{U}}=\{U_n\}_{n=1}^\infty$ be a sequence of pseudoisometries. If the strong limits $$\label{eq:LO-defi}
\begin{split}
A_{\mathcal{U}}:=\operatornamewithlimits{s-lim}_{n\to\infty}
(U_n^{-1}AU_n)\;\mbox{ in }\; {\mathcal{B}}(X),
\quad
A_{{\mathcal{U}}^*} :=\operatornamewithlimits{s-lim}_{n\to\infty}
(U_n^{-1}AU_n)^* \;\mbox{ in }\;
{\mathcal{B}}(X^*)
\end{split}$$ exist, then always $(A_{\mathcal{U}})^*=A_{{{\mathcal{U}}}^*}$, and we will refer the operator $A_{\mathcal{U}}$ to as a [*limit operator*]{} for the operator $A$ with respect to the sequence ${\mathcal{U}}$. Note that usually the limit operator $A_{\mathcal{U}}$ is defined independently of the existence of the strong limit $A_{{{\mathcal{U}}}^*}$ (see, e.g., [@BKR00; @RRS04]), while we need the existence of the both limits for our purposes. If the limit operator $A_{\mathcal{U}}$ exists, then it is uniquely determined by $A$ and ${\mathcal{U}}$, which justifies the notation $A_{\mathcal{U}}$.
In the next statement we collect basic properties of limit operators.
\[le:LO-properties\] Suppose ${\mathcal{U}}=\{U_n\}_{n=1}^\infty\subset{\mathcal{B}}(X)$ is a sequence of pseudoisometries.
1. If $A\in{\mathcal{B}}(X)$ and $A_{\mathcal{U}}$ exists, then $\|A_{\mathcal{U}}\|_{{\mathcal{B}}(X)}\le\|A\|_{{\mathcal{B}}(X)}$.
2. If $A,B\in{\mathcal{B}}(X)$, $\alpha\in{\mathbb{C}}$, and if the limit operators $A_{\mathcal{U}}$, $B_{\mathcal{U}}$ exist, then the limit operators $(\alpha A)_{\mathcal{U}}$, $(A+B)_{\mathcal{U}}$, $(AB)_{\mathcal{U}}$ also exist and $$(\alpha A)_{\mathcal{U}}=\alpha A_{\mathcal{U}}, \quad
(A+B)_{\mathcal{U}}=A_{\mathcal{U}}+B_{\mathcal{U}}, \quad
(AB)_{\mathcal{U}}=A_{\mathcal{U}}B_{\mathcal{U}}.$$
3. If $A\in{\mathcal{B}}(X)$ and if $\{A_m\}_{m=1}^\infty\subset{\mathcal{B}}(X)$ is such that the limit operators $(A_m)_{\mathcal{U}}$ exist for all $m\in{\mathbb{N}}$ and $\|A-A_m\|_{{\mathcal{B}}(X)}\to 0$ as $m\to\infty$, then the limit operator $A_{\mathcal{U}}$ exists and $\|A_{\mathcal{U}}-(A_m)_{\mathcal{U}}\|_{{\mathcal{B}}(X)}\to 0$ as $m\to \infty$.
The proofs of the above results can be found in [@L06 Proposition 3.4] or [@RRS04 Proposition 1.2.2].
\[th:inv-quotient-algebra\] Let $X$ be a Banach space, let ${\mathfrak{A}}$ be a closed subalgebra of ${\mathcal{B}}(X)$, and let ${\mathfrak{J}}$ be a closed two-sided ideal of ${\mathfrak{A}}$. Suppose $A\in{\mathfrak{A}}$ and ${\mathcal{U}}=\{U_n\}_{n=1}^\infty\subset{\mathcal{B}}(X)$ is a sequence of pseudoisometries such that the limit operator $A_{\mathcal{U}}$ exists and the limit operators $J_{\mathcal{U}}$ exist and are equal to zero for all $J\in{\mathfrak{J}}$. If the coset $A+{\mathfrak{J}}$ is invertible in the quotient algebra ${\mathfrak{A}}/{\mathfrak{J}}$, then the limit operator $A_{\mathcal{U}}$ is invertible.
The proof is developed by analogy with [@RRS04 Proposition 1.2.9].
Strong convergence of shift operators
-------------------------------------
To calculate limit operators for the shift operator $W_\alpha$, we need a result on the strong convergence of shift operators.
\[le:strong-shift\] Let $\alpha_n:{\mathbb{R}}_+\to{\mathbb{R}}_+$ for $n\in{\mathbb{N}}\cup\{0\}$ be orientation-preserving diffeomorphisms having only two fixed points $0$ and $\infty$, and $\beta_n$ be their inverses. If $\log\alpha_n'\in L^\infty({\mathbb{R}}_+)$ for all $n\in{\mathbb{N}}\cup\{0\}$ and
- $\displaystyle\sup_{n\in{\mathbb{N}}\cup\{0\}}\|\beta_n'\|_{L^\infty({\mathbb{R}}_+)}<\infty,$
- $\alpha_n\to\alpha_0$ pointwise on ${\mathbb{R}}_+$ as $n\to\infty$;
then the sequence of shift operators $W_{\alpha_n}\in{\mathcal{B}}$ converges strongly to the shift operator $W_{\alpha_0}\in{\mathcal{B}}$.
The idea of the proof is borrowed from [@DS98 Theorem 1]. Let $\chi_E$ denote the characteristic function of a set $E\subset{\mathbb{R}}_+$. Since the linear span of the set $\{\chi_{[0,\tau]}:\tau\in{\mathbb{R}}_+\}$ is dense in the space $L^p({\mathbb{R}}_+)$ and the operators $W_{\alpha_n}$ are uniformly bounded on $L^p({\mathbb{R}}_+)$ in view of (i), it is sufficient to prove that $$\label{eq:strong-shift-1}
\lim_{n\to\infty}
\big\|W_{\alpha_n}\chi_{[0,\tau]}-W_{\alpha_0}\chi_{[0,\tau]}\big\|_{L^p({\mathbb{R}}_+)}=0
\quad\text{for all}\quad\tau\in{\mathbb{R}}_+.$$ It is easy to see that $$\begin{aligned}
\big\|W_{\alpha_n}\chi_{[0,\tau]} &-W_{\alpha_0}\chi_{[0,\tau]}\big\|_{L^p({\mathbb{R}}_+)}^p
=
\int_{{\mathbb{R}}_+}\big|\chi_{[0,\tau]}(\alpha_n(t))-\chi_{[0,\tau]}(\alpha_0(t))\big|^p\,dt
\nonumber
\\
&=
\int_{{\mathbb{R}}_+}\big|\chi_{[0,\beta_n(\tau)]}(t)-\chi_{[0,\beta_0(\tau)]}(t)\big|^p\,dt
=
\big|\beta_n(\tau)-\beta_0(\tau)\big|.
\label{eq:strong-shift-2}\end{aligned}$$ On the other hand, $$\begin{aligned}
\big|\beta_n(\tau)-\beta_0(\tau)\big|
&=
\Big|
\beta_n \big[\alpha_0(\beta_0(\tau))\big]-\beta_n\big[\alpha_n(\beta_0(\tau))\big]\Big|
\nonumber
\\
&\le
\sup_{n\in{\mathbb{N}}}\big\|\beta_n'\big\|_{L^\infty({\mathbb{R}}_+)}
\big|\alpha_0(\beta_0(\tau))-\alpha_n(\beta_0(\tau))\big|.
\label{eq:strong-shift-3}\end{aligned}$$ From and the hypotheses of the lemma it follows that $$|\beta_n(\tau)-\beta_0(\tau)|=o(1)\quad\mbox{as}\quad n\to\infty$$ for every $\tau\in{\mathbb{R}}_+$. Combining this with , we arrive at .
Realization with dilations
--------------------------
For $x\in{\mathbb{R}}_+$, consider the dilation operator $V_x$ defined on $L^p({\mathbb{R}}_+)$ by $$(V_x f)(t):=f(t/x)\quad (t\in{\mathbb{R}}_+).$$ It is easy to see that $V_x$ is invertible on the space $L^p({\mathbb{R}}_+)$ and $V_x^{-1}=V_{1/x}$. Moreover, $\|V_x\|_{{\mathcal{B}}}=x^{1/p}$ and hence $V_x$ is a pseudoisometry for every $x\in{\mathbb{R}}_+$.
Fix $s\in\{0,\infty\}$. We say that a sequence $h:=\{h_n\}_{n=1}^\infty\subset{\mathbb{R}}_+$ is a test sequence relative to the point $s$ if $$\lim_{n\to\infty}h_n=s.$$ With each test sequence $h$ relative to the point $s$ we associate the sequence of pseudoisometries ${\mathcal{V}}_h^s:=\{V_{h_n}\}_{n=1}^\infty\subset{\mathcal{B}}$.
\[le:LO-compact-dilations\] Let $h:=\{h_n\}_{n=1}^\infty\subset{\mathbb{R}}_+$ be a test sequence relative to $s\in\{0,\infty\}$. For any operator $K\in{\mathcal{K}}$, the limit operator $K_{{\mathcal{V}}_h^s}$ with respect to the sequence of pseudoisometries ${\mathcal{V}}_h^s:=\{V_{h_n}\}_{n=1}^\infty\subset{\mathcal{B}}$ exists and is the zero operator.
Consider the isometric isomorphism $E\Phi:L^p({\mathbb{R}}_+)\to L^p({\mathbb{R}})$, where $E$ is defined by and $\Phi$ is defined by . Then $\widetilde{K}:=E\Phi K\Phi^{-1}E^{-1}$ is compact on $L^p({\mathbb{R}})$ for every $K\in{\mathcal{K}}(L^p({\mathbb{R}}_+))$, and for every $x\in{\mathbb{R}}_+$, $E\Phi V_x\Phi^{-1}E^{-1}=\widetilde{V}_x$, where $\widetilde{V}_x\in{\mathcal{B}}(L^p({\mathbb{R}}))$ and $(\widetilde{V}_xf)(y)=x^{1/p}f(y-\log x)$ for all $y\in{\mathbb{R}}$. By [@BKS02 Lemma 18.9], $\operatornamewithlimits{s-lim}\limits_{n\to\infty}\widetilde{V}_{h_n}^{-1}
\widetilde{K}\widetilde{V}_{h_n}I=0$ on $L^p({\mathbb{R}})$ for every test sequence $h=\{h_n\}_{n=1}^\infty\subset{\mathbb{R}}_+$. Therefore $$\operatornamewithlimits{s-lim}_{n\to\infty}
V_{h_n}^{-1}KV_{h_n}=
\operatornamewithlimits{s-lim}_{n\to\infty}
\Phi^{-1}E^{-1}\widetilde{V}_{h_n}^{-1}\widetilde{K}\widetilde{V}_{h_n}E\Phi=0
\quad\mbox{on}\quad L^p({\mathbb{R}}_+).$$ Analogously, $(V_{h_n}^{-1}KV_{h_n})^*=V_{h_n}^{-1}K^*V_{h_n}$ converges strongly to zero on the space $L^q({\mathbb{R}}_+)$. Thus, the limit operator $K_{{\mathcal{V}}_h^s}$ exists and is equal to zero.
\[le:LO-dilations\] Suppose $a,b,c,d\in SO({\mathbb{R}}_+)$, $\alpha\in SOS({\mathbb{R}}_+)$, and the operator $N$ is given by . Let $s\in\{0,\infty\}$. For every functional $\xi\in M_s(SO({\mathbb{R}}_+))$ there exists a test sequence $h^\xi=\{h_n^\xi\}_{n=1}^\infty\subset{\mathbb{R}}_+$ relative to the point $s$ such that the limit operator $N_{{\mathcal{V}}_{h^\xi}^s}$ with respect to the sequence of pseudoisometries ${\mathcal{V}}_{h^\xi}^s:=\{V_{h_n^\xi}\}_{n=1}^\infty\subset{\mathcal{B}}$ exists and $$\label{eq:LO-dilations-1}
N_{{\mathcal{V}}_{h^\xi}^s}=
\big(a(\xi)I-b(\xi)W_{\alpha_\xi}\big)P_+ +
\big(c(\xi)I-d(\xi)W_{\alpha_\xi}\big)P_-,$$ where $\alpha_\xi(t):=e^{\omega(\xi)}t$ and $\omega(t):=\log[\alpha(t)/t]$ for $t\in{\mathbb{R}}_+$.
Fix $s\in\{0,\infty\}$ and $\xi\in M_s(SO({\mathbb{R}}_+))$. From Lemma \[le:SOS-inverse\] it follows that $\beta:=\alpha_{-1}$ is a slowly oscillating shift. Then, by Lemma \[le:exp-repr\], $\alpha',\beta'\in SO({\mathbb{R}}_+)$ and the functions $\omega$ and $\zeta(t):=\log[\beta(t)/t]$ are real-valued slowly oscillating functions. Clearly, $\overline{a},\overline{b}, \overline{c},\overline{d}\in SO({\mathbb{R}}_+)$. Let $${\mathcal{G}}:=\{a,b,c,d,\alpha',\beta',\omega,\zeta\}
\subset SO({\mathbb{R}}_+).$$ By Lemma \[le:SO-fundamental-property\], there exists a test sequence $h^\xi=\{h_n^\xi\}_{n=1}^\infty\subset{\mathbb{R}}_+$ relative to the point $s$ such that the limit $$\label{eq:LO-dilations-2}
g(\xi)=\xi(g)=\lim_{n\to\infty}g(h_n^\xi)$$ exists for every function $g\in{\mathcal{G}}$. Lemma \[le:SO-uniform\] implies that for every $t\in{\mathbb{R}}_+$, $$\label{eq:LO-dilations-3}
\lim_{n\to\infty}|g(h_n^\xi t)-g(h_n^\xi)|=0,$$ and the convergence is uniform in $t$ on every segment $J\subset{\mathbb{R}}_+$. Since $$V_{h_n^\xi}^{-1}(gI)V_{h_n^\xi}=g_nI,
\quad
(V_{h_n^\xi}(gI)V_{h_n^\xi})^*=\overline{g_n}I,$$ where $g_n(t):=g(h_n^\xi t)$ for all $t\in{\mathbb{R}}_+$, we infer from and that for the multiplication operator on $L^p({\mathbb{R}}_+)$ and its adjoint on $L^q({\mathbb{R}}_+)$, $$\begin{split}
&
\operatornamewithlimits{s-lim}_{n\to\infty}
V_{h_n^\xi}^{-1}(gI)V_{h_n^\xi}
=
\operatornamewithlimits{s-lim}_{n\to\infty}g_nI
=
\lim_{n\to\infty}g(h_n^\xi)I=g(\xi)I,
\\
&
\operatornamewithlimits{s-lim}_{n\to\infty}
(V_{h_n^\xi}^{-1}(gI)V_{h_n^\xi})^*
=
\operatornamewithlimits{s-lim}_{n\to\infty}\overline{g_n}I
=
\lim_{n\to\infty}\overline{g(h_n^\xi)}I=\overline{g(\xi)}I.
\end{split}$$ Hence $$\label{eq:LO-dilations-4}
(gI)_{{\mathcal{V}}_{h^\xi}^s}=g(\xi)I
\quad\mbox{for}\quad g\in\{a,b,c,d\}.$$ From Lemma \[le:exp-repr\] it follows that $\alpha(t)=te^{\omega(t)}$. Therefore, for all $n\in{\mathbb{N}}$, $$\label{eq:LO-dilations-5}
V_{h_n^\xi}^{-1}W_\alpha V_{h_n^\xi}=W_{\alpha_\xi^{(n)}},$$ where $\alpha_\xi^{(n)}(t):=te^{\omega(h_n^\xi t)}$ for $t\in{\mathbb{R}}_+$. From – we conclude that $$\label{eq:LO-dilations-6}
\lim_{n\to\infty}\alpha_\xi^{(n)}(t)=te^{\omega(\xi)}=\alpha_\xi(t),
\quad t\in{\mathbb{R}}_+.$$ Since $\log\alpha'$ is bounded, we have $$\label{eq:SOS-derivative*}
0<m_\alpha:=\inf_{y\in{\mathbb{R}}_+}\alpha'(y).$$ Let $\beta_\xi^{(n)}$ be the inverse shift to $\alpha_\xi^{(n)}$. It is easy to see that for all $n\in{\mathbb{N}}$ and $t\in{\mathbb{R}}_+$, $$(\beta_\xi^{(n)})'(t)
=
\frac{1}{(\alpha_\xi^{(n)})'[\beta_\xi^{(n)}(t)]}
=
\frac{1}{\alpha'[h_n^\xi\beta_\xi^{(n)}(t)]}.$$ From this equality and it follows that for all $n\in{\mathbb{N}}$ and $t\in{\mathbb{R}}_+$, $$\label{eq:LO-dilations-7}
(\beta_\xi^{(n)})'(t)\le 1/m_\alpha<+\infty.$$ Moreover, the derivative of the inverse shift to $\alpha_\xi$ is constant. Thus, combining – and Lemma \[le:strong-shift\], we see that $$\label{eq:LO-dilations-8}
\operatornamewithlimits{s-lim}_{n\to\infty}
V_{h_n^\xi}^{-1}W_\alpha V_{h_n^\xi}=W_{\alpha_\xi}
\quad\mbox{on}\quad L^p({\mathbb{R}}_+).$$ It is easy to see that $(W_\alpha)^*=\beta'W_\beta$. We have already proved that $$\label{eq:LO-dilations-9}
\operatornamewithlimits{s-lim}_{n\to\infty}
V_{h_n^\xi}^{-1}(\beta' I)V_{h_n^\xi}=\beta'(\xi)I
\quad\mbox{on}\quad L^q({\mathbb{R}}_+).$$ Analogously to one can show that $$\label{eq:LO-dilations-10}
\operatornamewithlimits{s-lim}_{n\to\infty}
V_{h_n^\xi}^{-1}W_\beta V_{h_n^\xi}=W_{\beta_\xi}
\quad\mbox{on}\quad L^q({\mathbb{R}}_+),$$ where $\beta_\xi(t)=te^{\zeta(\xi)}$. From – we obtain $$\begin{aligned}
\operatornamewithlimits{s-lim}_{n\to\infty}\left(V_{h_n^\xi}^{-1} W_\alpha V_{h_n^\xi}\right)^*
&=
\left(\operatornamewithlimits{s-lim}_{n\to\infty}V_{h_n^\xi}^{-1} (\beta'I) V_{h_n^\xi}\right)
\left(\operatornamewithlimits{s-lim}_{n\to\infty} V_{h_n^\xi}^{-1} W_\beta V_{h_n^\xi}\right)\nonumber
\\
&=
\beta'(\xi)W_{\beta_\xi}.
\label{eq:LO-dilations-12}\end{aligned}$$ Equalities and imply that $$\label{eq:LO-dilations-13}
(W_\alpha)_{{\mathcal{V}}_{h^\xi}^s}=W_{\alpha_\xi}.$$ It is easy to see that $S^*=S$ and $V_{h_n^\xi}^{-1}SV_{h_n^\xi}=S$. Hence $$\label{eq:LO-dilations-14}
(P_+)_{{\mathcal{V}}_h^\xi}=P_+,
\quad
(P_-)_{{\mathcal{V}}_h^\xi}=P_-.$$ Combining , , and with Lemma \[le:LO-properties\](b), we see that the limit operator $N_{{\mathcal{V}}_{h^\xi}^s}$ exists and is calculated by .
Realization with modulations
----------------------------
For $x\in{\mathbb{R}}$, consider the modulation operator $E_x$ defined on $L^p({\mathbb{R}}_+)$ by $$(E_xf)(t):=t^{ix}f(t)\quad (t\in{\mathbb{R}}_+).$$ It is clear that $E_x$ is invertible on the space $L^p({\mathbb{R}}_+)$ and $E_x^{-1}=E_{-x}$. Moreover, $\|E_x\|_{{\mathcal{B}}}=1$ and hence $E_x$ is a pseudoisometry for each $x\in{\mathbb{R}}$.
Fix $s\in\{-\infty,+\infty\}$. We say that a sequence $\mu:=\{\mu_n\}_{n=1}^\infty\subset{\mathbb{R}}$ is a test sequence relative to the point $s$ if $$\lim_{n\to\infty}\mu_n=s.$$ With each test sequence $\mu$ relative to the point $s$ we associate the sequence of pseudoisometries ${\mathcal{E}}_\mu^s:=\{E_{\mu_n}\}_{n=1}^\infty\subset{\mathcal{B}}$.
\[le:LO-compact-modulations\] Suppose $\mu:=\{\mu_n\}_{n=1}^\infty\subset{\mathbb{R}}$ is a test sequence relative to a point $s\in\{-\infty,+\infty\}$. For any operator $K\in{\mathcal{K}}$, the limit operator $K_{{\mathcal{E}}_\mu^s}$ with respect to the sequence of pseudoisometries ${\mathcal{E}}_\mu^s:=\{E_{\mu_n}\}_{n=1}^\infty\subset{\mathcal{B}}$ exists and is the zero operator.
Following the proof of Lemma \[le:LO-compact-dilations\], consider the isometric isomorphism $E\Phi:L^p({\mathbb{R}}_+)\to L^p({\mathbb{R}})$ and the operator $\widetilde{K}:=E\Phi K\Phi^{-1}E^{-1}\in{\mathcal{K}}(L^p({\mathbb{R}}))$, where $K\in{\mathcal{K}}(L^p({\mathbb{R}}_+))$. For every $x\in{\mathbb{R}}$ we get $E\Phi E_x\Phi^{-1}E^{-1}=e_xI,$ where $e_x(y):=e^{ixy}$ for $y\in{\mathbb{R}}$. It was shown in [@KL08 Lemma 3.8] (see also [@BKS02 Lemma 10.1] for $p=2$) that $\operatornamewithlimits{s-lim}\limits_{n\to\infty}e_{-\mu_n}\widetilde{K}e_{\mu_n}I=0$ on $L^p({\mathbb{R}})$ for every sequence $\mu_n$ tending to $+\infty$ or to $-\infty$. Therefore $$\operatornamewithlimits{s-lim}_{n\to\infty}
E_{\mu_n}^{-1}KE_{\mu_n}=
\operatornamewithlimits{s-lim}_{n\to\infty}
\Phi^{-1}E^{-1}e_{-\mu_n}\widetilde{K}e_{\mu_n}E\Phi=0
\quad\mbox{on}\quad L^p({\mathbb{R}}_+).$$ Analogously, $(E_{\mu_n}^{-1}KE_{\mu_n})^*=E_{\mu_n}^{-1}K^*E_{\mu_n}$ converges strongly to zero on the space $L^q({\mathbb{R}}_+)$. Thus, for every sequence $\mu=\{\mu_n\}_{n=1}^\infty\subset{\mathbb{R}}$ converging to $s\in\{-\infty,+\infty\}$, we have $K_{{\mathcal{E}}_\mu^s}=0$.
\[le:LO-modulations\] Suppose $g\in SO({\mathbb{R}}_+)$. Let $\mu=\{\mu_n\}_{n=1}^\infty\subset{\mathbb{R}}$ be a test sequence relative to $s\in\{-\infty,+\infty\}$. Then the limit operators $(gI)_{{\mathcal{E}}_\mu^s}$ and $S_{{\mathcal{E}}_\mu^s}$ with respect to the sequence of pseudoisometries ${\mathcal{E}}_\mu^s:=\{E_{\mu_n}\}_{n=1}^\infty\subset{\mathcal{B}}$ exist and are given by $$(gI)_{{\mathcal{E}}_\mu^{\pm\infty}}=gI,
\quad
S_{{\mathcal{E}}_\mu^{\pm\infty}}=\pm I.$$
Let $s=+\infty$. It is easy to see that $$E_{\mu_n}^{-1}(gI)E_{\mu_n}=gI,
\quad
(E_{\mu_n}^{-1}(gI)E_{\mu_n})^*=(gI)^*$$ for all $n\in{\mathbb{N}}$. Hence $(gI)_{{\mathcal{E}}_\mu^{+\infty}}=gI$.
Let us show that $S_{{\mathcal{E}}_\mu^{+\infty}}=I$. From Theorem \[th:algebra-A\] it follows that $$E_{\mu_n}^{-1}SE_{\mu_n}=\Phi^{-1}\operatorname{Co}(s_{p,\mu_n})\Phi,$$ where $s_{p,\mu_n}(x):=s_p(x+\mu_n)$ for $x\in{\mathbb{R}}$. Hence we must show that $$\label{eq:LO-modulations-1}
\lim_{n\to\infty}\|\operatorname{Co}(s_{p,\mu_n}-1)\psi\|_{L^p({\mathbb{R}}_+,d\mu)}=0$$ for $\psi\in L^p({\mathbb{R}}_+,d\mu)$.
According to [@S70 Chap. III, Section 2.2], for every $f\in L^p({\mathbb{R}})$ and every $\varphi\in L^1({\mathbb{R}})$ with $\int_{\mathbb{R}}\varphi(x)dx=1$, we have $$\label{eq:LO-modulations-2}
\lim_{{\varepsilon}\to 0}\|f*\varphi_{\varepsilon}-f\|_{L^p({\mathbb{R}})}=0,$$ where $\varphi_{\varepsilon}(x):={\varepsilon}^{-1}\varphi(x/{\varepsilon})$ for $x\in{\mathbb{R}}$ and ${\varepsilon}>0$. Choosing now rapidly decreasing functions $\varphi$ in the Schwarz space $\mathcal{S}({\mathbb{R}})$ whose Fourier transforms $F\varphi$ have compact supports in ${\mathbb{R}}$, we derive from that the set $Y$ of the functions $f\in L^2({\mathbb{R}})\cap L^p({\mathbb{R}})$, for which $Ff$ has compact support in ${\mathbb{R}}$, is dense in $L^p({\mathbb{R}})$. Hence the set $D$ of all functions $\psi\in L^2({\mathbb{R}}_+,d\mu)\cap L^p({\mathbb{R}}_+,d\mu)$, for which the Mellin transform $M\psi$ has compact support in ${\mathbb{R}}$, is dense in $L^p({\mathbb{R}}_+,d\mu)$. Obviously, it is sufficient to prove for all $\psi\in D$.
Fix $\psi\in D$. Since the support of $M\psi$ is compact, there exists a function $\chi\in C_0^\infty({\mathbb{R}})$ with a compact support $K$ such that $$\label{eq:LO-modulations-3}
\operatorname{Co}(s_{p,\mu_n}-1)\psi=M^{-1}\chi(s_{p,\mu_n}-1)M\psi.$$ From Theorem \[th:Stechkin\] and it follows that $$\label{eq:LO-modulations-4}
\|M^{-1}\chi(s_{p,\mu_n}-1)M\psi\|_{L^p({\mathbb{R}}_+,d\mu)}
\le
c_p
\|\psi\|_{L^p({\mathbb{R}}_+,d\mu)}
\|\chi(s_{p,\mu_n}-1)\|_V,$$ where $c_p:=\|S\|_{{\mathcal{B}}(L^p({\mathbb{R}}))}$ and $\|\cdot\|_V:=\|\cdot\|_{L^\infty({\mathbb{R}})}+V(\cdot)$. It remains to show that $$\label{eq:LO-modulations-5}
\|\chi(s_{p,\mu_n}-1)\|_V
=
\|\chi(s_{p,\mu_n}-1)\|_{L^\infty({\mathbb{R}})}+V\big(\chi(s_{p,\mu_n}-1)\big)
\to 0$$ as $n\to\infty$. We have $$\label{eq:LO-modulations-6}
\|\chi(s_{p,\mu_n}-1)\|_{L^\infty({\mathbb{R}})}
\le
\|\chi\|_{L^\infty({\mathbb{R}})}\sup_{x\in K}|s_p(x+\mu_n)-1|$$ and $$\begin{aligned}
V\big(&\chi(s_{p,\mu_n}-1)\big)
=
\int_{\mathbb{R}}\left|\frac{d}{dx}\big[\chi(x)(s_p(x+\mu_n)-1)\big]\right|dx
\nonumber
\\
&\le
\int_{\mathbb{R}}|\chi(x)|\,|s_p'(x+\mu_n)|dx
+
\int_{\mathbb{R}}|\chi'(x)|\,|s_p(x+\mu_n)-1|dx
\nonumber
\\
&\le
C\sup_{x\in K}|s_p'(x+\mu_n)|+
C'\sup_{x\in K}|s_p(x+\mu_n)-1|,
\label{eq:LO-modulations-7}\end{aligned}$$ where $$C:=\int_{\mathbb{R}}|\chi(x)|dx<\infty,\quad
C':=\int_{\mathbb{R}}|\chi'(x)|dx<\infty.$$ Taking into account that $\mu_n\to+\infty$ and $$s_p(x+\mu_n)=\coth[\pi(x+\mu_n+i/p)],
\
s_p'(x+\mu_n)=-\frac{\pi}{\sinh^2[\pi(x+\mu_n+i/p)]},$$ we see that $$\label{eq:LO-modulations-8}
\lim_{n\to\infty}\sup_{x\in K}|s_p(x+\mu_n)-1|=0,
\quad
\lim_{n\to\infty}\sup_{x\in K}|s_p'(x+\mu_n)|=0.$$ Combining –, we arrive at . From – we obtain . Since $(E_{\mu_n}^{-1}SE_{\mu_n})^*=E_{\mu_n}^{-1}SE_{\mu_n}$, this completes the proof for $s=+\infty$.
The proof for $s=-\infty$ is analogous.
Localization {#sec:localization}
============
Algebras ${\mathcal{Z}}$, ${\mathcal{F}}$, and $\Lambda$
--------------------------------------------------------
Let us consider $$\begin{split}
{\mathcal{Z}}&:={\operatorname{alg}}_{\mathcal{B}}\big\{I,S,cR,K:\ c\in SO({\mathbb{R}}_+),\ K\in{\mathcal{K}}\big\},
\\
{\mathcal{F}}&:={\operatorname{alg}}_{\mathcal{B}}\big\{aI,S,W_\alpha,W_\alpha^{-1}:a\in SO({\mathbb{R}}_+)\big\},
\\
\Lambda &:=\big\{A\in{\mathcal{B}}:\ AC-CA\in{\mathcal{K}}\text{ for all }C\in{\mathcal{Z}}\big\}.
\end{split}$$ It is easy to see that $\Lambda$ is a closed unital subalgebra of ${\mathcal{B}}$.
\[th:embeddings\] We have ${\mathcal{K}}\subset{\mathcal{Z}}\subset{\mathcal{F}}\subset\Lambda$.
\[le:alg-Lambda\] An operator $A\in\Lambda$ is Fredholm if and only if the coset $A^\pi:=A+{\mathcal{K}}$ is invertible in the quotient algebra $\Lambda^\pi:=\Lambda/{\mathcal{K}}$.
The proof is straightforward.
Fredholmness of operators in the algebra $\Lambda$
--------------------------------------------------
By [@KKLsufficiency Theorem 6.11], the maximal ideal space $M({\mathcal{Z}}^\pi)$ of the commutative Banach algebra ${\mathcal{Z}}^\pi:={\mathcal{Z}}/{\mathcal{K}}$ is homeomorphic to the set $\{-\infty,+\infty\}\cup(\Delta\times{\mathbb{R}})$. Let $$\begin{aligned}
{\mathcal{I}}_{\pm\infty}^\pi&:=\operatorname{id}_{{\mathcal{Z}}^\pi}
\big\{P_\mp^\pi,(gR)^\pi\ : \ g\in SO({\mathbb{R}}_+)\big\},
\\
{\mathcal{I}}_{\xi,x}^\pi&:=\big\{Z^\pi\in{\mathcal{Z}}^\pi:\
(Z^\pi)\widehat{\hspace{2mm}}(\xi,x)=0\big\}\quad\text{for }\;(\xi,x)\in\Delta\times{\mathbb{R}},\end{aligned}$$ where $(Z^\pi)\widehat{\hspace{2mm}}$ is the Gelfand transform of $Z^\pi$, which was explicitly given in [@KKLsufficiency Section 6]. Further, let ${\mathcal{J}}_{\pm\infty}^\pi$ and ${\mathcal{J}}_{\xi,x}^\pi$ be the closed two-sided ideals of the Banach algebra $\Lambda^\pi$ generated by the ideals ${\mathcal{I}}^\pi_{\pm\infty}$ and ${\mathcal{I}}^\pi_{\xi,x}$ of the algebra ${\mathcal{Z}}^\pi$, respectively, and put $$\Lambda^\pi_{\pm\infty}:=\Lambda^\pi/{\mathcal{J}}^\pi_{\pm\infty},
\quad
\Lambda^\pi_{\xi,x}:=\Lambda^\pi/{\mathcal{J}}^\pi_{\xi,x}$$ for the corresponding quotient algebras.
\[th:localization-realization\] An operator $A\in\Lambda$ is Fredholm on the space $L^p({\mathbb{R}}_+)$ if and only if the following two conditions are fulfilled:
- the cosets $A^\pi+{\mathcal{J}}_{\pm\infty}^\pi$ are invertible in the quotient algebras ${\Lambda}_{\pm\infty}^\pi$, respectively;
- for every $(\xi,x)\in\Delta\times{\mathbb{R}}$, the coset $A^\pi+{\mathcal{J}}_{\xi,x}^\pi$ is invertible in the quotient algebra ${\Lambda}_{\xi,x}^\pi$.
Quotient algebras $\Lambda_{+\infty}$ and $\Lambda_{-\infty}$
-------------------------------------------------------------
Let ${\mathcal{J}}_{\pm\infty}$ be the closed two-sided ideal of the algebra $\Lambda$ generated by the operator $P_{\mp}$ and the ideal ${\mathcal{K}}$. By $\Lambda_{\pm\infty}$ denote the quotient algebra $\Lambda/{\mathcal{J}}_{\pm\infty}$.
It is not difficult to see that $R\in\operatorname{id}_{\mathcal{A}}\{P_-\}\cap\operatorname{id}_{\mathcal{A}}\{P_+\}$. Hence the ideals ${\mathcal{J}}_{\pm\infty}^\pi$ of the quotient algebra $\Lambda^\pi$ can be also represented in the form ${\mathcal{J}}_{\pm\infty}^\pi=\operatorname{id}_{\Lambda^\pi}\{P_\mp^\pi\}$, and therefore, respectively, $$\label{id-form}
{\mathcal{J}}_{\pm\infty}^\pi=\{A^\pi:\ A\in{\mathcal{J}}_{\pm\infty}\}.$$
\[le:lifting\] Suppose $C_-,C_+\in\Lambda$.
1. The invertibility of the coset $(C_+P_++C_-P_-)^\pi+{\mathcal{J}}_{+\infty}^\pi$ in the quotient algebra $\Lambda_{+\infty}^\pi$ is equivalent to the invertibility of the coset $C_++{\mathcal{J}}_{+\infty}$ in the quotient algebra $\Lambda_{+\infty}$.
2. The invertibility of the coset $(C_+P_++C_-P_-)^\pi+{\mathcal{J}}_{-\infty}^\pi$ in the quotient algebra $\Lambda_{-\infty}^\pi$ is equivalent to the invertibility of the coset $C_-+{\mathcal{J}}_{-\infty}$ in the quotient algebra $\Lambda_{-\infty}$.
\(a) Consider the mapping $\varphi:\Lambda/{\mathcal{J}}_{+\infty}\to\Lambda^\pi/{\mathcal{J}}_{+\infty}^\pi$ given by $$A+{\mathcal{J}}_{+\infty}\mapsto A^\pi+{\mathcal{J}}_{+\infty}^\pi\quad(A\in\Lambda).$$ Obviously, $\varphi$ is a homomorphism of $\Lambda/{\mathcal{J}}_{+\infty}$ onto $\Lambda^\pi/{\mathcal{J}}_{+\infty}^\pi$. If $A^\pi\in{\mathcal{J}}_{+\infty}^\pi$, then from it follows that $A\in{\mathcal{J}}_{+\infty}$, and therefore $\varphi$ is injective. So, $\varphi$ is an isomorphism. Then $C_++{\mathcal{J}}_{+\infty}=\varphi^{-1}(C_+^\pi+{\mathcal{J}}_{+\infty}^\pi)$ is invertible in $\Lambda/{\mathcal{J}}_{+\infty}$ if and only if $C_+^\pi+{\mathcal{J}}_{+\infty}^\pi$ is invertible in $\Lambda^\pi/{\mathcal{J}}_{+\infty}^\pi$. By the definition of the ideal ${\mathcal{J}}_{+\infty}^\pi$, we have $(C_\pm P_-)^\pi\in{\mathcal{J}}_{+\infty}^\pi$. From this observation and $P_++P_-=I$ it follows that $$(C_+P_++C_-P_-)^\pi+{\mathcal{J}}_{+\infty}^\pi=C_+^\pi+{\mathcal{J}}_{+\infty}^\pi,$$ which finishes the proof of part (a). The proof of part (b) is analogous.
Invertibility of binomial functional operators {#sec:FO}
==============================================
Invertibility of functional operators with slowly oscillating data
------------------------------------------------------------------
For $s\in\{0,\infty\}$, $a,b\in SO({\mathbb{R}}_+)$, and $\alpha\in SOS({\mathbb{R}}_+)$, put $$\begin{aligned}
L_*(s;a,b,\alpha)
&:=
\liminf\limits_{t\to s}
\left( |a(t)|-|b(t)|\big(\alpha'(t)\big)^{-1/p}\right),
\\
L^*(s;a,b,\alpha)
&:=
\limsup\limits_{t\to s}
\left(|a(t)|-|b(t)|\big(\alpha'(t)\big)^{-1/p}\right).\end{aligned}$$
\[th:FO\] Suppose $a,b\in SO({\mathbb{R}}_+)$ and $\alpha\in SOS({\mathbb{R}}_+)$. The functional operator $aI-bW_\alpha$ is invertible on the Lebesgue space $L^p({\mathbb{R}}_+)$ if and only if either $$\label{eq:FO-1}
\inf\limits_{t\in{\mathbb{R}}_+}|a(t)|>0,
\quad
L_*(0;a,b,\alpha)>0,
\quad
L_*(\infty;a,b,\alpha)>0;$$ or $$\label{eq:FO-2}
\inf\limits_{t\in{\mathbb{R}}_+}|b(t)|>0,
\quad
L^*(0;a,b,\alpha)<0,
\quad
L^*(\infty;a,b,\alpha)<0.$$ If holds, then $$(aI-bW_\alpha)^{-1}=\sum_{n=0}^\infty (a^{-1}bW_\alpha)^n a^{-1}I.$$ If holds, then $$(aI-bW_\alpha)^{-1}=-W_\alpha^{-1}\sum_{n=0}^\infty (b^{-1}aW_\alpha^{-1})^n b^{-1}I.$$
Invertibility of auxiliary binomial functional operators {#subsec:FO-aux}
--------------------------------------------------------
Suppose $\alpha_0(t):=t$ and $\alpha_n(t):=\alpha[\alpha_{n-1}(t)]$ for $n\in{\mathbb{Z}}$ and $t\in{\mathbb{R}}_+$. Fix a point $\tau\in{\mathbb{R}}_+$ and put $$\tau_-:=\lim_{n\to-\infty}\alpha_n(\tau),
\quad
\tau_+:=\lim_{n\to+\infty}\alpha_n(\tau).$$ Then either $\tau_-=0$ and $\tau_+=\infty$, or $\tau_-=\infty$ and $\tau_+=0$. Let $\gamma$ be a segment of ${\mathbb{R}}_+$ with endpoints $\tau$ and $\alpha(\tau)$. Suppose $\chi_\gamma$ is the characteristic function of $\gamma$ and $\widetilde{\chi}_\gamma$ is an arbitrary function in $C({\mathbb{R}}_+)$ with nonempty support in $\gamma$. Consider the half-open intervals $$\gamma_-:=\bigcup_{k=1}^\infty\alpha_{-k}(\gamma),
\quad
\gamma_+:=\bigcup_{k=1}^\infty\alpha_k(\gamma).$$ Let $\widetilde{\tau}_\pm$ denote the endpoint of the half-open interval $\gamma\cup\gamma_\pm$ such that $\widetilde{\tau}_\pm\ne\tau_\pm$, respectively. Consider functions $\chi_\pm\in C(\overline{{\mathbb{R}}}_+)$ such that $\chi_-(t)=1$ for all $t\in\gamma_-$, $\chi_+(t)=1$ for all $t\in\gamma_+$, and $\chi_-(t)+\chi_+(t)=1$ for all $t\in{\mathbb{R}}_+$.
\[le:FO-aux\] Let $A=aI-bW_\alpha$ where $a,b\in SO({\mathbb{R}}_+)$ and $\alpha\in SOS({\mathbb{R}}_+)$.
1. Suppose $$\label{eq:FO-aux-1}
L_*(\tau_-;a,b,\alpha)>0>L^*(\tau_+;a,b,\alpha),$$ $$\label{eq:FO-aux-2}
\inf_{t\in\gamma\cup\gamma_-}|a(t)|>0,
\quad
\inf_{t\in\gamma\cup\gamma_+}|b(t)|>0,$$ and put $$\label{eq:FO-aux-3}
\widetilde{a}(t):=
\left\{\begin{array}{ll}
a(t) & \text{for } t\in\gamma\cup\gamma_-,
\\
a(\widetilde{\tau}_-) & \text{otherwise},
\end{array}\right.
\
\widetilde{b}(t):=\left\{\begin{array}{lll}
b(t) & \text{for } t\in\gamma\cup\gamma_+,
\\
b(\widetilde{\tau}_+) & \text{otherwise}.
\end{array}\right.$$ Then the operators $$\label{eq:FO-aux-4}
A_{1,\chi_-}:=\widetilde{a}I-b\chi_-W_\alpha,
\quad
A_{2,\chi_+}:=a\chi_+I-\widetilde{b}W_\alpha$$ are invertible on the space $L^p({\mathbb{R}}_+)$ and $$\label{eq:FO-aux-5}
A\Pi_r=0,\quad
\widetilde{\chi}_\gamma\Pi_r=\widetilde{\chi}_\gamma I
\quad{for}\quad
\Pi_r:=(A_{1,\chi_-}^{-1}-A_{2,\chi_+}^{-1})a\chi_\gamma I.$$
2. Suppose $$\label{eq:FO-aux-6}
L^*(\tau_-;a,b,\alpha)<0<L_*(\tau_+;a,b,\alpha),$$ $$\inf_{t\in\gamma\cup\gamma_+}|a(t)|>0,
\quad
\inf_{t\in\gamma\cup\gamma_-}|b(t)|>0,$$ and put $$\widetilde{a}(t):=
\left\{\begin{array}{ll}
a(t) & \text{for } t\in\gamma\cup\gamma_+,
\\
a(\widetilde{\tau}_+) & \text{otherwise},
\end{array}\right.
\
\widetilde{b}(t):=\left\{\begin{array}{lll}
b(t) & \text{for } t\in\gamma\cup\gamma_-,
\\
b(\widetilde{\tau}_-) & \text{otherwise}.
\end{array}\right.$$ Then the operators the operators $$A_{1,\chi_+\circ\alpha}:=\widetilde{a}I-b(\chi_+\circ\alpha)W_\alpha,
\quad
A_{2,\chi_-}:=a\chi_-I-\widetilde{b}W_\alpha,$$ are invertible on the space $L^p({\mathbb{R}}_+)$ and $$\Pi_l A=0,
\quad
\Pi_l\widetilde{\chi}_\gamma I=\widetilde{\chi}_\gamma I
\quad{for}\quad
\Pi_l:=\chi_\gamma a(A_{1,\chi_+\circ\alpha}^{-1}-A_{2,\chi_-}^{-1}).$$
\(a) The idea of the proof is borrowed from [@K84 Lemma 3]. Clearly, the functions defined by belong to $SO({\mathbb{R}}_+)$. From – it follows that $$\inf_{t\in{\mathbb{R}}_+}|\widetilde{a}(t)|>0,
\quad
\inf_{t\in{\mathbb{R}}_+}|\widetilde{b}(t)|>0,
\quad
L_*(\tau_\pm;\widetilde{a},b\chi_-,\alpha)>0,
\quad
L^*(\tau_\pm;a\chi_+,\widetilde{b},\alpha)<0.$$ By Theorem \[th:FO\], the operators are invertible on the space $L^p({\mathbb{R}}_+)$, and $$\label{eq:FO-aux-8}
A_{1,\chi_-}^{-1}=\sum_{n=0}^\infty (\widetilde{a}^{-1}b\chi_-W_\alpha)^n
\widetilde{a}^{-1}I,
\
A_{2,\chi_+}^{-1}=-W_\alpha^{-1}\sum_{n=0}^\infty (\widetilde{b}^{-1}a\chi_+
W_\alpha^{-1})^n \widetilde{b}^{-1}I.$$ Further, in view of , we get the relations $$\label{eq:FO-aux-9}
\begin{aligned}
&
(aI-bW_\alpha)W_\alpha^n\chi_\gamma I=(\widetilde{a}I-b\chi_-W_\alpha)
W_\alpha^n\chi_\gamma I & (n\in{\mathbb{N}}\cup\{0\}),
\\
&
(aI-bW_\alpha)(W_\alpha^{-1})^n\chi_\gamma I=(a\chi_+I-\widetilde{b}W_\alpha)
(W_\alpha^{-1})^n \chi_\gamma I & (n\in{\mathbb{N}}).
\end{aligned}$$ Applying and we infer that $$\begin{aligned}
AA_{1,\chi_-}^{-1}a\chi_\gamma I
&=A_{1,\chi_-}A_{1,\chi_-}^{-1}a\chi_\gamma I=a\chi_\gamma I,
\\
AA_{2,\chi_+}^{-1}a\chi_\gamma I
&=A_{2,\chi_+}A_{2,\chi_+}^{-1}a\chi_\gamma I=a\chi_\gamma I,\end{aligned}$$ whence $A\Pi_r=A(A_{1,\chi_-}^{-1}-A_{2,\chi_+}^{-1})a\chi_\gamma I=0$. On the other hand, since $$\widetilde{\chi}_\gamma W_\alpha^n\chi_\gamma I=0,\quad
\widetilde{\chi}_\gamma(W_\alpha^{-1})^n\chi_\gamma I=0\quad(n\in{\mathbb{N}}),$$ we deduce from that $$\begin{aligned}
\widetilde{\chi}_\gamma\Pi_r
&=
\widetilde{\chi}_\gamma\sum_{n=0}^\infty
(\widetilde{a}^{-1}b\chi_-W_\alpha)^n\widetilde{a}^{-1}a\chi_\gamma I +\widetilde{\chi}_\gamma W_\alpha^{-1}
\sum_{n=0}^\infty(\widetilde{b}^{-1}a\chi_+W_\alpha^{-1})^n \widetilde{b}^{-1}a\chi_\gamma I
\\
&=
\widetilde{\chi}_\gamma\chi_\gamma I=\widetilde{\chi}_\gamma I,\end{aligned}$$ which completes the proof of . Part (a) is proved.
The proof of part (b) is similar and therefore is omitted.
Invertibility of functional operators with multiplicative shifts
----------------------------------------------------------------
\[le:binom-mult\] Let $a,b\in SO({\mathbb{R}}_+)$ and $\alpha:{\mathbb{R}}_+\to{\mathbb{R}}_+$ be a multiplicative shift given by $\alpha(t)=kt$ for all $t\in{\mathbb{R}}_+$ with some $k\in{\mathbb{R}}_+$. The following statements are equivalent:
1. the functional operator $aI-bW_\alpha$ is invertible on the space $L^p({\mathbb{R}}_+)$;
2. the coset $aI-bW_\alpha+{\mathcal{J}}_{+\infty}$ is invertible in the quotient algebra $\Lambda_{+\infty}$;
3. the coset $aI-bW_\alpha+{\mathcal{J}}_{-\infty}$ is invertible in the quotient algebra $\Lambda_{-\infty}$.
\(i) $\Rightarrow$ (ii). Clearly, if $A:=aI-bW_\alpha$ is invertible on $L^p({\mathbb{R}}_+)$, then it is Fredholm. Moreover, $A\in\Lambda$ by Theorem \[th:embeddings\]. Then from Lemma \[le:alg-Lambda\] we see that there exists $B\in\Lambda$ such that $AB-I\in{\mathcal{K}}$ and $BA-I\in{\mathcal{K}}$. Since the ideal ${\mathcal{J}}_{+\infty}$ contains ${\mathcal{K}}$, the latter relations imply that the coset $A+{\mathcal{J}}_{+\infty}$ is invertible in the quotient algebra $\Lambda_{+\infty}$ and $B+{\mathcal{J}}_{+\infty}$ is its inverse.
\(ii) $\Rightarrow$ (i). Consider the test sequence $\nu:=\{\nu_n\}_{n=1}^\infty\subset{\mathbb{R}}$ relative to $+\infty$ and given by $$\nu_n:=\left\{\begin{array}{lll}
2\pi n|\log k|^{-1} &\mbox{if} &k\ne 1, \\
2\pi n &\mbox{if} & k=1
\end{array}\right.
\quad(n\in{\mathbb{N}}).$$ Then for every $n\in{\mathbb{N}}$, $f\in L^p({\mathbb{R}}_+)$, and $t\in{\mathbb{R}}_+$, $$(E_{\nu_n}^{-1}W_\alpha E_{\nu_n}f)(t)
=
t^{-i\nu_n}(kt)^{i\nu_n}f(kt)=k^{i\nu_n}f(kt)=(W_\alpha f)(t)$$ because $k^{i\nu_n}=1$. That is, $E_{\nu_n}^{-1}W_\alpha E_{\nu_n}=W_\alpha$. Thus, the limit operator $(W_\alpha)_{{\mathcal{E}}_\nu^{+\infty}}$ with respect to the sequence of pseudoisometries ${\mathcal{E}}_\nu^{+\infty}:=\{E_{\nu_n}\}_{n=1}^\infty$ exists and $$\label{eq:binom-mult-1}
(W_\alpha)_{{\mathcal{E}}_\nu^{+\infty}}=W_\alpha.$$ From Lemma \[le:LO-modulations\] we obtain $$\label{eq:binom-mult-2}
(aI)_{{\mathcal{E}}_\nu^{+\infty}}=aI,
\quad
(bI)_{{\mathcal{E}}_\nu^{+\infty}}=bI,
\quad
S_{{\mathcal{E}}_\nu^{+\infty}}=I.$$ On the other hand, by Lemma \[le:LO-compact-modulations\], $$\label{eq:binom-mult-3}
K_{{\mathcal{E}}_\nu^{+\infty}}=0
\quad\mbox{for every}\quad K\in{\mathcal{K}}.$$ Combining – with Lemma \[le:LO-properties\](b)–(c), we see that $$\label{eq:binom-mult-4}
(aI-bW_\alpha)_{{\mathcal{E}}_\nu^{+\infty}}=aI-bW_\alpha$$ and $J_{{\mathcal{E}}_\nu^{+\infty}}=0$ for all $J\in{\mathcal{J}}_{+\infty}$ (recall that the ideal ${\mathcal{J}}_{+\infty}$ is generated by ${\mathcal{K}}$ and $P_-=(I-S)/2$, so $(P_-)_{{\mathcal{E}}_\nu^{+\infty}}=0$).
Since the coset $aI-bW_\alpha+{\mathcal{J}}_{+\infty}$ is invertible in the quotient algebra $\Lambda_{+\infty}$, we deduce from Theorem \[th:inv-quotient-algebra\] that the limit operator is invertible. This finishes the proof of the implication (ii) $\Rightarrow$ (i). Thus, the equivalence (i) $\Leftrightarrow$ (ii) is proved.
The proof of the equivalence (i) $\Leftrightarrow$ (iii) is analogous.
Necessary conditions for Fredholmness {#sec:necessity}
=====================================
Necessity of condition (i)
--------------------------
In this subsection we prove that condition (i) in Theorem \[th:sufficiency\] is necessary for the Fredholmness of the operator $N$. We start with the following auxiliary result.
\[le:compactness\] Let the functions $\chi_\gamma$ and $\widetilde{\chi}_\gamma$ be as in Section [\[subsec:FO-aux\]]{}, and $n\in{\mathbb{Z}}$.
1. The operators $(\chi_\gamma\circ\alpha_n) P_- P_+$ and $P_- P_+(\chi_\gamma\circ\alpha_n)I$ are compact.
2. The operators $\widetilde{\chi}_\gamma P_\pm$ and $P_\pm\widetilde{\chi}_\gamma I$ are not compact.
\(a) In view of , we have $P_- P_+\in\operatorname{id}_{\mathcal{A}}\{R\}$. Let $c$ be a continuous function equal $1$ on the support of $\chi_\gamma\circ\alpha_n$ and vanishing at $0$ and $\infty$. Then the operator $(c\circ\alpha_n)P_- P_+$ is compact by [@KKLsufficiency Corollary 6.6]. Therefore the operator $(\chi_\gamma\circ\alpha_n)P_- P_+=(\chi_\gamma\circ\alpha_n)(c\circ\alpha_n)P_- P_+$ is also compact. The compactness of $P_- P_+(\chi_\gamma\circ\alpha_n)I$ is proved analogously with the aid of Theorem \[th:compactness-commutators\]. Part (a) is proved. Part (b) follows from [@RS90 Theorem 4.1(c)].
\[le:nec-1\] Suppose $a,b,c,d\in SO({\mathbb{R}}_+)$, $\alpha\in SOS({\mathbb{R}}_+)$, and the operator $N$ is given by .
1. If $N$ is Fredholm, then either $$\label{eq:nec-1-1}
L_*(0;a,b,\alpha)>0,
\quad
L_*(\infty;a,b,\alpha)>0;$$ or $$\label{eq:nec-1-2}
L^*(0;a,b,\alpha)<0,
\quad
L^*(\infty;a,b,\alpha)<0.$$
2. If $N$ is Fredholm, then either $$\label{eq:nec-1-3}
L_*(0;c,d,\alpha)>0,
\quad
L_*(\infty;c,d,\alpha)>0;$$ or $$\label{eq:nec-1-4}
L^*(0;c,d,\alpha)<0,
\quad
L^*(\infty;c,d,\alpha)<0.$$
\(a) Fix $s\in\{0,\infty\}$ and $\xi\in M_s(SO({\mathbb{R}}_+))$. By Lemma \[le:LO-dilations\], there exists a test sequence $h^\xi=\{h_n^\xi\}_{n=1}^\infty\subset{\mathbb{R}}_+$ relative to the point $s$ such that the limit operator $N_{{\mathcal{V}}_{h^\xi}^s}$ with respect to the sequence of pseudoisometries ${\mathcal{V}}_{h^\xi}^s:=\{V_{h_n^\xi}\}_{n=1}^\infty\subset{\mathcal{B}}$ exists and $$\label{eq:def-limit-N}
N_{{\mathcal{V}}_{h^\xi}^s}= \big(a(\xi)I-b(\xi)W_{\alpha_\xi}\big)P_+ +
\big(c(\xi)I-d(\xi)W_{\alpha_\xi}\big)P_-,$$ where $\alpha_\xi(t)=e^{\omega(\xi)}t$ is a multiplicative shift and $\omega(t)=\log[\alpha(t)/t]$ belongs to $SO({\mathbb{R}}_+)$ in view of Lemma \[le:exp-repr\]. From Lemma \[le:LO-compact-dilations\] it follows that $K_{{\mathcal{V}}_{h^\xi}^s}=0$ for every $K\in{\mathcal{K}}$. Since the operator $N$ is Fredholm, the coset $N^\pi=N+{\mathcal{K}}$ is invertible in the quotient algebra $\Lambda^\pi$ in view of Lemma \[le:alg-Lambda\]. Applying Theorem \[th:inv-quotient-algebra\] with ${\mathfrak{A}}=\Lambda$, ${\mathfrak{J}}={\mathcal{K}}$, $A=N$, and ${\mathcal{U}}={\mathcal{V}}_{h^\xi}^s$, we conclude that the operator $N_{{\mathcal{V}}_ {h^\xi}^s}$ is invertible.
Obviously, $N_{{\mathcal{V}}_{h^\xi}^s}$ is Fredholm. Then from Theorem \[th:localization-realization\] it follows that the coset $(N_{{\mathcal{V}}_{h^\xi}^s})^\pi+{\mathcal{J}}_{+\infty}^\pi$ is invertible in the quotient algebra $\Lambda_{+\infty}^\pi$. By Lemma \[le:lifting\], this is equivalent to the invertibility of the coset $a(\xi)I-b(\xi)W_{\alpha_\xi}+{\mathcal{J}}_{+\infty}$ in the quotient algebra $\Lambda_{+\infty}$. Since $\alpha_\xi(t)=e^{\omega(\xi)}t$ is a multiplicative shift, from Lemma \[le:binom-mult\] it follows that the above condition is equivalent to the invertibility of the operator $a(\xi)I-b(\xi)W_{\alpha_\xi}$. Applying Theorem \[th:FO\] to this operator and taking into account Lemma \[le:SOS-derivative\], we obtain either $$\label{eq:nec-1-5}
|a(\xi)|-|b(\xi)|\big(\alpha'(\xi)\big)^{-1/p}=
|a(\xi)|-|b(\xi)|\big(e^{\omega(\xi)}\big)^{-1/p}>0$$ or $$\label{eq:nec-1-6}
|a(\xi)|-|b(\xi)|\big(\alpha'(\xi)\big)^{-1/p}=
|a(\xi)|-|b(\xi)|\big(e^{\omega(\xi)}\big)^{-1/p}<0.$$ The fibers $M_s(SO({\mathbb{R}}_+))$ are connected compact Hausdorff spaces by Lemma \[le:connected-fibers\]. Since $a(\xi),b(\xi)$, and $\alpha'(\xi)$ depend continuously on $\xi\in M_s(SO({\mathbb{R}}_+))$, we deduce that if $N$ is Fredholm, then for every $s\in\{0,\infty\}$ either holds for all $\xi\in M_s(SO({\mathbb{R}}_+))$ or holds for all $\xi\in M_s(SO({\mathbb{R}}_+))$. Hence we conclude from Lemma \[le:SO-fundamental-property\] that for each $s\in\{0,\infty\}$ either $L_*(s;a,b,\alpha)>0$ or $L^*(s;a,b,\alpha)<0$.
It remains to prove that actually either or is fulfilled, that is, to show that $L^*(0;a,b,\alpha)<0<L_*(\infty;a,b,\alpha)$ or $L^*(\infty;a,b,\alpha)<0<L_*(0;a,b,\alpha)$ are impossible. Since either $\tau_-=0$ and $\tau_+=\infty$, or $\tau_-=\infty$ and $\tau_+=0$, the latter inequalities take either the form , or the form .
On the contrary, suppose is fulfilled. Then there are open neighborhoods $u(\tau_\pm)\subset{\mathbb{R}}_+$ of $\tau_\pm$ such that $|a|$ is separated from zero on $u(\tau_-)$ and $|b|$ is separated from zero on $u(\tau_+)$. Take a segment $\gamma\subset
{\mathbb{R}}_+\setminus\overline{u(\tau_-)\cup u(\tau_+)}$ with endpoints $\tau$ and $\alpha(\tau)$. Since $N$ is Fredholm, by a small perturbation of coefficients $a,b$ in $C\big({\mathbb{R}}_+
\setminus(u(\tau_-)\cup u(\tau_+))\big)$ we can achieve the fulfillment of for perturbed coefficients keeping the operator $N$ Fredholm. Notice that inequalities remain valid for perturbed coefficients. Let us save notation $a,b$ for perturbed coefficients. Then in virtue of Lemma \[le:FO-aux\](a) we obtain the operator $\Pi_r$ given by . Setting now $A_+:=aI-bW_\alpha$ and $A_-:=cI-dW_\alpha$ and taking into account Theorem \[th:compactness-commutators\], we get $$\label{eq:nec-1-7}
NP_+=(A_+P_++A_-P_-)P_+\simeq P_+A_++(A_--A_+)P_-P_+,$$ where $C\simeq D$ means that $C-D$ is a compact operator. Recall that since $N$ is Fredholm, there is an operator $N^{(-1)}\in{\mathcal{B}}$, called a regularizer of $N$, such that $NN^{(-1)}\simeq N^{(-1)}N\simeq I$. Then, applying $\Pi_r$ and $N^{(-1)}$, we infer from that $$\label{eq:nec-1-8}
P_+\Pi_r\simeq N^{(-1)}NP_+\Pi_r\simeq N^{(-1)}P_+A_+\Pi_r+N^{(-1)}
(A_--A_+)P_-P_+\Pi_r.$$ From Lemma \[le:compactness\](a) and $$\Pi_r=\sum_{n=0}^\infty(\chi_\gamma\circ\alpha_n)
(a^{-1}b\chi_-W_\alpha)^n +(\chi_\gamma\circ\alpha_{-n-1})W_\alpha^{-1}
\sum_{n=0}^\infty(b^{-1}a\chi_+W_\alpha^{-1})^n b^{-1}aI$$ we get $P_-P_+\Pi_r\simeq 0$. On the other hand, by with $A=A_+$, we obtain $A_+\Pi_r=0$. The latter two relations imply in view of that $P_+\Pi_r\simeq 0$. Let $\widetilde{\chi}_\gamma$ be as in Section \[subsec:FO-aux\]. From and Theorem \[th:compactness-commutators\] we get $$P_+\widetilde{\chi}_\gamma I=P_+\widetilde{\chi}_\gamma\Pi_r
\simeq\widetilde{\chi}_\gamma P_+\Pi_r\simeq 0.$$ Hence $P_+\widetilde{\chi}_\gamma I$ is a compact operator, which is impossible due to Lemma \[le:compactness\](b).
Analogously, if holds, then applying Lemma \[le:FO-aux\](b) we conclude that $$\Pi_l P_+\simeq \Pi_l P_+ N N^{(-1)}\simeq \Pi_l(A_+P_++P_-P_+(A_--A_+))
N^{(-1)}\simeq 0,$$ and hence $$\widetilde{\chi}_\gamma P_+=\Pi_l\widetilde{\chi}_\gamma P_+
\simeq \Pi_l P_+\widetilde{\chi}_\gamma I\simeq 0,$$ which again is impossible. Thus, either or holds, and hence part (a) is proved. The proof of part (b) is analogous.
\[le:nec-2\] Suppose $a,b,c,d\in SO({\mathbb{R}}_+)$, $\alpha\in SOS({\mathbb{R}}_+)$, and the operator $N$ is given by .
1. If $N$ is Fredholm and is fulfilled, then $\inf\limits_{t\in{\mathbb{R}}_+}|a(t)|>0$.
2. If $N$ is Fredholm and is fulfilled, then $\inf\limits_{t\in{\mathbb{R}}_+}|b(t)|>0$.
3. If $N$ is Fredholm and is fulfilled, then $\inf\limits_{t\in{\mathbb{R}}_+}|c(t)|>0$.
4. If $N$ is Fredholm and is fulfilled, then $\inf\limits_{t\in{\mathbb{R}}_+}|d(t)|>0$.
\(a) Assume the contrary, that is, $\inf\limits_{t\in{\mathbb{R}}_+}|a(t)|=0$. From it follows that there exist numbers $0<m<M<\infty$ such that the function $a$ is bounded away from zero on $(0,m]\cup[M,\infty)$. Hence there is a point $t_0\in(m,M)$ such that $a(t_0)=0$. Fix some $\tau$ such that $t_0$ belongs to the interior of the segment $\gamma$ with the endpoints $\tau$ and $\alpha(\tau)$. Choose $m$ and $M$ such that $\gamma\subset[m,M]$. Suppose $u=u(t_0)$ is a closed neighborhood of the point $t_0$ that is contained in $\gamma$ and whose endpoints do not coincide with $\tau$ and $\alpha(\tau)$. Then there exists a continuous function $\chi_u$ supported in $u$ and such that $\chi_u(t_0)=1$.
Let $\varphi,\psi\in SO({\mathbb{R}}_+)$ be functions such that
1. $a(t)=\varphi(t)=\psi(t)$ for $t\in(0,m]\cup[M,\infty)$;
2. $\varphi(t)=0$ for $t\in u$ and $\varphi(t)\ne 0$ for $t\in{\mathbb{R}}\setminus u$;
3. $\psi(t)\ne 0$ for $t\in{\mathbb{R}}$;
4. $\varphi(t)=\psi(t)$ for $t\in{\mathbb{R}}\setminus\gamma$.
Consider the operator $$\widetilde{N}=(\varphi I-bW_\alpha)P_++(cI-dW_\alpha)P_-.$$ It is clear that $\|\widetilde{N}-N\|_{\mathcal{B}}=O(\|\varphi-a\|_{L^\infty({\mathbb{R}}_+)})$. Since $\varphi$ can be chosen arbitrarily close to $a$ in the norm of $L^\infty({\mathbb{R}}_+)$ and $N$ is Fredholm, we can guarantee that $\widetilde{N}$ is also Fredholm for some $\varphi$ as above.
By Theorem \[th:localization-realization\], the coset $\widetilde{N}^\pi+{\mathcal{J}}_{+\infty}^\pi$ is invertible in the quotient algebra $\Lambda_{+\infty}^\pi$. Therefore, the coset $\varphi I-bW_\alpha+{\mathcal{J}}_{+\infty}$ is invertible in the quotient algebra $\Lambda_{+\infty}$ in view of Lemma \[le:lifting\](a). Then there exists an operator $B\in\Lambda$ such that $$\label{eq:nec-2-1}
(\varphi I-bW_\alpha+{\mathcal{J}}_{+\infty})(B+{\mathcal{J}}_{+\infty})=I+{\mathcal{J}}_{+\infty}.$$ On the other hand, $\inf\limits_{t\in{\mathbb{R}}_+}|\psi(t)|>0$, and by we have for $s\in\{0,\infty\}$, $$L_*(s;\psi,b,\alpha)=L_*(s;a,b,\alpha)>0.$$ By Theorem \[th:FO\], the operator $\psi I-bW_\alpha$ is invertible and $$(\psi I-bW_\alpha)^{-1}
=
\sum_{n=0}^\infty\left(\frac{b}{\psi}W_\alpha\right)^n\frac{1}{\psi}I
=
\frac{1}{\psi}\sum_{n=0}^\infty\left(\frac{b}{\psi\circ\alpha}W_\alpha\right)^n.$$ Let $$C:=\chi_u\psi(\psi I-bW_\alpha)^{-1}.$$ From Theorem \[th:embeddings\] we see that $C\in{\mathcal{F}}\subset\Lambda$. From the choice of $\varphi$ and $\psi$ it follows that $\chi_u\varphi=0$ and $\chi_u(\varphi\circ\alpha_k)=\chi_u(\psi\circ\alpha_k)$ for all $k\in{\mathbb{N}}$. Therefore, $$C(\varphi I-bW_\alpha)=
\chi_u\left(\sum_{n=0}^\infty\frac{b}{\psi\circ\alpha}W_\alpha\right)
(\varphi I-bW_\alpha)=\chi_u\varphi I=0.$$ Hence $$\label{eq:nec-2-2}
(C+{\mathcal{J}}_{+\infty})(\varphi I-bW_\alpha+{\mathcal{J}}_{+\infty})={\mathcal{J}}_{+\infty}.$$ Multiplying from the right by $B+{\mathcal{J}}_{+\infty}$ and taking into account , we obtain $C+{\mathcal{J}}_{+\infty}={\mathcal{J}}_{+\infty}$. Then $C\in{\mathcal{J}}_{+\infty}$.
It is clear that $\chi_u\circ\alpha_k=0$ for $k\in{\mathbb{N}}$. Then $$C\chi_u I=\chi_u\sum_{n=0}^\infty\left(\frac{b}{\psi\circ\alpha}W_\alpha\right)^n\chi_u I
=\chi_u^2 I\in{\mathcal{J}}_{+\infty}.$$ From Lemmas \[le:LO-compact-modulations\], \[le:LO-modulations\], and \[le:LO-properties\] it follows that for an arbitrary sequence of pseudoisometries ${\mathcal{E}}_\mu^{+\infty}=\{E_{\mu_n}\}_{n=1}^\infty\subset{\mathcal{B}}$, the limit operators for all operators $J\in{\mathcal{J}}_{+\infty}$ are equal to zero. In particular, then $(\chi_u^2 I)_{{\mathcal{E}}_\mu^{+\infty}}=0$. On the other hand, since $\chi_u^2\in SO({\mathbb{R}}_+)$, from Lemma \[le:LO-modulations\] it also follows that $(\chi_u^2 I)_{{\mathcal{E}}_\mu^{+\infty}}=\chi_u^2I\ne 0$. This contradiction shows that $\inf\limits_{t\in{\mathbb{R}}_+}|a(t)|>0$. Part (a) is proved.
\(b) If the operator $N$ is Fredholm, then the operator $$\begin{aligned}
-W_\alpha^{-1}[(aI &-bW_\alpha)P_++(cI-dW_\alpha)P_-]
\nonumber
\\
&=
[(b\circ\beta)I-(a\circ\beta)W_\beta]P_+
+[(d\circ\beta)I-(c\circ\beta)W_\beta]P_-
\label{eq:nec-2-3}\end{aligned}$$ is also Fredholm. Recall that $\beta\in SOS({\mathbb{R}}_+)$ by Lemma \[le:SOS-inverse\]. Then from Lemma \[le:continuous-SOS\] we see that $a\circ\beta,b\circ\beta\in SO({\mathbb{R}}_+)$. Since $\beta$ preserves the orientation, has only two fixed points $0$ and $\infty$, and $\log\alpha'$ is bounded, we obtain for $s\in\{0,\infty\}$, $$\begin{split}
L^*(s;a,b,\alpha)
&=-\liminf_{t\to s}
\big((\alpha'\circ\beta)(t)\big)^{-1/p}
\Big(
|(b\circ\beta)(t)|
-
|(a\circ\beta)(t)|(\beta'(t))^{-1/p}
\Big)
\\
&\ge
-\sup_{t\in{\mathbb{R}}_+}
\big((\alpha'\circ\beta)(t)\big)^{-1/p}
L_*(s;b\circ\beta,a\circ\beta,\beta).
\end{split}$$ Hence implies that $L_*(s;b\circ\beta,a\circ\beta,\beta)>0$ for $s\in\{0,\infty\}$. Applying part (a) to the operator , we obtain $$0<\inf_{t\in{\mathbb{R}}_+}|(b\circ\beta)(t)|=\inf_{t\in{\mathbb{R}}_+}|b(t)|.$$ Part (b) is proved. Parts (c) and (d) are proved by analogy with parts (a) and (b), respectively.
Combining Lemmas \[le:nec-1\]–\[le:nec-2\] and Theorem \[th:FO\], we arrive at the following part of Theorem \[th:main\].
\[th:nec-i\] Suppose $a,b,c,d\in SO({\mathbb{R}}_+)$, $\alpha\in SOS({\mathbb{R}}_+)$, and the operator $N$ is given by . If the operator $N$ is Fredholm on the space $L^p({\mathbb{R}}_+)$, then the functional operators $A_+:=aI-bW_\alpha$ and $A_-:=cI-dW_\alpha$ are invertible on the space $L^p({\mathbb{R}}_+)$.
Necessity of condition (ii)
---------------------------
To finish the proof of Theorem \[th:main\], it remains to prove the following.
\[th:nec-ii\] Suppose $a,b,c,d\in SO({\mathbb{R}}_+)$, $\alpha\in SOS({\mathbb{R}}_+)$, the operator $N$ is given by , and for every $\xi\in\Delta$ the function $n_\xi$ is defined by . If the operator $N$ is Fredholm on the space $L^p({\mathbb{R}}_+)$, then $n_\xi(x)\ne 0$ for every pair $(\xi,x)\in\Delta\times{\mathbb{R}}$.
Fix $\xi\in\Delta$. In the proof of Lemma \[le:nec-1\] it was shown that if $N$ is Fredholm, then the operator $N_{{\mathcal{V}}_{h^\xi}^s}$ given by with $\alpha_\xi(t)=e^{\omega(\xi)}t$ is invertible. On the other hand, taking into account Theorem \[th:algebra-A\] and Lemma \[le:mult-shift-convolution\], we see that $N_{{\mathcal{V}}_{h^\xi}^s}=\Phi^{-1} \operatorname{Co}(n_\xi)\Phi$, where $n_\xi\in SAP_p$ is given by . Hence, $\operatorname{Co}(n_\xi)$ is invertible on the space $L^p({\mathbb{R}}_+,d\mu)$. Then from Theorem \[th:invertibility-convolution\] we deduce that $\inf\limits_{x\in{\mathbb{R}}}|n_\xi(x)|>0$. Since $s\in\{0,\infty\}$ and $\xi
\in M_s(SO({\mathbb{R}}_+))$ were chosen arbitrarily, we conclude that $n_\xi(x)\ne 0$ for all $(\xi,x)\in\Delta\times{\mathbb{R}}$.
Acknowledgment {#acknowledgment .unnumbered}
--------------
This work is partially supported by “Centro de Análise Funcional e Aplicações" at Instituto Superior Técnico (Lisboa, Portugal), which is financed by FCT (Portugal). The second author is also supported by the SEP-CONACYT Project No. 25564 (México) and by PROMEP (México) via “Proyecto de Redes".
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|
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"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.'
author:
- Hemanth Saratchandran
title: A four dimensional hyperbolic link complement in a standard $S^4$
---
=0.2cm =0.0cm
Introduction
============
In the 1970’s W. Thurston initiated a grand study of the topology of 3-manifolds motivated by the realisation that many 3-manifolds admitted homogeneous Riemannian metrics. Thurston’s insight was that one could use this geometry to study the topology of a 3-manifold, the ultimate goal being a unification of the worlds of geometry and topology in dimension three. Two worlds thought to be completely distinct before Thurston’s time. Part of Thurston’s study involved a detailed understanding of how various link complements in 3-manifolds could admit homogeneous geometries. He was able to show that many link complements admitted a hyperbolic geometry thereby throwing the world of hyperbolic geometry to the forefront (see [@thurston_1] chap.3, p.27 and [@thurston_2] cor.2.5, p.360). This grand vision of Thurston came to be known as the “geometrisation conjecture” and was finally proved by G. Perelman in 2003. The geometrisation theorem together with the classification of surfaces leads to a complete understanding of the worlds of two and three dimensional manifolds.
The quest for a complete understanding of the world of 4-manifolds, analogous to the two and three dimensional cases, is a futile quest. The main reason being that, every finitely presented group can arise as the fundamental group of a compact 4-manifold. As there is no algorithm to tell whether two finitely presented groups are isomorphic, there is no algorithm to tell if two 4-manifolds have the same fundamental groups. Motivated by Thurston’s work on which link complements, in dimension 3, admit hyperbolic geometries we can restrict our attention to the class of hyperbolic 4-manifolds and try to understand which 4-manifolds have link complements that admit a hyperbolic geometry. The main problem here is that hyperbolic 4-manifolds are characteristically different from hyperbolic 3-manifolds, and the techniques used to show that a 3-manifold admits a link complement that is hyperbolic simply do not have four dimensional counter parts. Due to this, the quest to understand which 4-manifolds admit link complements that are hyperbolic takes on a very different feel right from the start.
In the last decade there have been quite a few constructions of hyperbolic 4-manifolds. One of the simplest constructions was given by J. Ratcliffe and S. Tschantz in their paper [@ratcliffe]. Many of the 4-manifolds they construct are non-compact and have ends of the form $E \times [0, \infty)$, with $E$ a closed flat 3-manifold that is a circle bundle over a surface. We can take these ends, chop them off at a certain point, and produce a compact 4-manifold $M_0$ with boundary a certain number of flat closed 3-manifolds $E_i$. The region near the boundary is a copy of $E_i \times [0, t]$, for some $t > 0$. These boundary 3-manifolds, being circle bundles, bound a 4-manifold $F_i$, the associated disk bundle. We can then glue the manifold $F_i$ to $M_0$ by identifying the boundary components via the identity, we will call this a “filling” of $M_0$.
{width="8cm" height="8cm"}
It could be the case that the 3-manifold $E$ fibres over a surface in more than one way, hence the filling process will depend on the choice of $S^1$-fibre. Carrying out this gluing procedure for each boundary component of $M_0$ produces a closed 4-manifold $\widetilde{M}$. The original hyperbolic 4-manifold can then be seen to be a codimension two link complement inside of the closed 4-manifold $\widetilde{M}$. At this point one of the most basic questions we can ask is, can we explicitly identify the manifold $\widetilde{M}$? This question is related to the above problem of finding 4-manifolds that have a hyperbolic link complement in that the ability to classify such an $\widetilde{M}$ would then result in an explicit example of a four dimensional hyperbolic link complement. However, one has to be very careful in using the word “classify”. The main reason being that the world of 4-manifolds is a truly wild world, there are 4-manifolds out there that do not admit a single smooth structure, others out there that admit countably many smooth structures, and some even admitting uncountably many smooth structures! This exotic behaviour of 4-manifolds shows the mathematician that the problem of characterising a 4-manifold up to homeomorphism is a very different problem than to characterise it up to diffeomorphism, something one does not witness when restricting to manifolds of dimensions two and three.
In general the identification of the topological/smooth type of the manifold $\widetilde{M}$ can prove to be an impossible task. However, if for example the filled in manifolds one obtains are simply connected then there is certainly hope. A good way to try and smoothly identify a simply connected 4-manifold is to resort to a “calculus of links” developed by R. Kirby towards the end of the 70’s (see [@kirby]). Kirby showed that given a handle decomposition of a closed 4-manifold, the one and two handles were really what one had to worry about in trying to understand the manifold. The one and two handle structure of such a manifold can be neatly encoded in a link diagram in $S^3$, which we can view as ${\ensuremath{\mathbb R}}^3 \cup \{\infty\}$. Therefore, the one and two-handle structure of such a manifold could be explicitly visualised by a link diagram in ${\ensuremath{\mathbb R}}^3$, which we now call a Kirby diagram. Kirby was able to prove that if one applied certain elementary moves to the link diagram, obtaining a new link diagram, the 4-manifold that corresponded to this new link diagram would be diffeomorphic to the original 4-manifold. In this way Kirby set up a “calculus” that one could appeal to in order to simplify their link diagram but without any compensation being paid on the diffeomorphism type of their closed 4-manifold.
The aim of this paper is to use the theory of Kirby calculus to construct an explicit example of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere. Our approach begins with the work we started in our paper [@sarat]. In that paper we show how to construct a Kirby diagram for any one of the Ratcliffe-Tschantz hyperbolic 4-manifolds. Using this construction we will show how to understand a filling on the level of the Kirby diagram. We will then show, using methods from the theory of Kirby calculus, how to reduce the Kirby diagram of the filling and obtain a closed 4-manifold that has the same Kirby diagram of the standard 4-sphere. This will then tell us that our original filling is diffeomorphic to the standard 4-sphere, in turn producing a four dimensional hyperbolic link complement in a standard 4-sphere. The main theorem takes the form:
(Theorem \[mainthm\_1\]) There exists a collection $L$ of five linked 2-tori embedded in a smooth 4-manifold $X$ such that $X$ is diffeomorphic to a standard $S^4$, and $X-L$ admits a finite volume hyperbolic geometry.
We would like to mention that D. Ivan$\check{s}$i$\acute{c}$ proves in his paper [@ivansic] (see thm.4.3, p.18) that there exists a system of five linked tori embedded in a smooth manifold $X$ that is homeomorphic to $S^4$ such that $X-L$ admits a finite volume hyperbolic geometry. The manifold he uses to construct $X$ is the same manifold we are going to use. However, he does not prove that $X$ is diffeomorphic to $S^4$. Furthermore, D. Ivan$\check{s}$i$\acute{c}$, J. Ratcliffe and S. Tschantz in their paper [@tschantz] construct several more examples of hyperbolic link complements in 4-manifolds that are homeomorphic to $S^4$, they do not prove that any of the fillings they construct are diffeomorphic to $S^4$. We were unaware of the existence of these two papers during the undertaking of this work, in fact we were only made aware of these two papers quite recently. In a future paper we will show how to use the methods of this paper and [@sarat] to construct a hyperbolic link complement in 4-manifold that is diffeomorphic to a standard $S^2 \times S^2$.
Unfortunately we have not been able to make this paper as self contained as we would have liked. The main reason being that it would have become far too long. The constructions and techniques used in this paper heavily depend on those outlined in [@sarat] and we recommend the reader look at that paper to get an idea of what is going on. We also outline, in that paper, the very basic properties of the hyperbolic 24-cell that we will need, and give some preliminary information on the construction of the Ratcliffe-Tschantz manifolds.
Acknowledgements {#acknowledgements .unnumbered}
================
The author wishes to thank Marc Lackenby for various discussions to do with this work and the several comments/corrections he gave on earlier drafts of this work. We would also like to thank Andras Juhasz, Panos Papazoglou and John Parker for the comments and corrections they gave on an earlier draft.
Parabolic transformations and the Euclidean structure of a cusp {#parabolics}
===============================================================
In this section we are going to explain how to compute parabolic transformations associated to each cusp component of a Ratcliffe-Tschantz hyperbolic 4-manifold. Each of their manifolds has the hyperbolic 24-cell as a fundamental domain. The 24-cell is a self dual 4-dimensional ideal hyperbolic polyhedron. For the basic construction of the hyperbolic 24-cell that we will use we refer the reader to section two of [@sarat], for more background information on various properties of the 24-cell we recommend the reader consult [@cox] Ch.4 and [@kerckhoff] Sect.3.
Recall that the 24-cell $P$ has twenty four ideal vertices, eight of the form $(\pm 1,0,0,0)$, $(0,\pm 1,0,0)$, $(0,0,\pm 1, 0)$, $(0,0,0,\pm 1)$ and sixteen of the form $(\pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2)$. When we apply any group of side pairing transformations defining any one of the 5-cusped Ratcliffe-Tschantz manifolds we find that the ideal vertices split into five equivalence classes. We have four of the form: $$\{(1,0,0,0), (-1,0,0,0)\}, \{(0,1,0,0), (0,-1,0,0)\},$$ $$\{(0,0,1,0), (0,0,-1,0)\}, \{(0,0,0,1), (0,0,0,-1)\}$$ and one of the form: $$\{(\pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2)\}.$$ In order to compute the parabolic isometries associated to these five cusps we need to look at horospherical neighbourhoods about each ideal vertex. Recall that given any ideal vertex of the form $(\pm 1,0,0,0)$, $(0,\pm 1,0,0)$, $(0,0,\pm 1, 0)$, $(0,0,0,\pm 1)$, such a vertex lies on a given side $S$ if the centre vector of the sphere defining the side has an equal non-zero entry in the same position as the vertex. For example the ideal vertex $(-1,0,0,0)$ lies on the side $S_{(-1,0,0,1)}$ but not on the side $S_{(1,0,0,1)}$ or on the side $S_{(0,1,1,0)}$. Any ideal vertex of the form $(\pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2)$ lies on a side $S$ if the non-zero entries of the centre of the sphere defining the side have the same sign as the non-zero entries in the same position of the ideal vertex. For example the ideal vertex $(1/2, 1/2, -1/2, 1/2)$ lies on the side $S_{(+1,0,-1,0)}$ but not on the side $S_{(-1,+1,0,0)}$.
The above recollection shows us that each ideal vertex lies on precisely six sides. Since each side of the 24-cell $P$ intersects precisely four other sides, and does so at right angles, we find that a horospherical neighbourhood of any of the ideal vertices is a cube. From this fact it follows that for each of the equivalence classes $$\{(1,0,0,0), (-1,0,0,0)\}, \{(0,1,0,0), (0,-1,0,0)\},$$ $$\{(0,0,1,0), (0,0,-1,0)\}, \{(0,0,0,1), (0,0,0,-1)\}$$ the associated boundary cusp will have fundamental domain consisting of two cubes. For example, if we take the vertex class $\{(1,0,0,0), (-1,0,0,0)\}$ then we can think of a fundamental domain for the associated cusp cross section as a horospherical neighbourhood about the ideal vertex $(1,0,0,0)$ together with a horospherical neighbourhood about the ideal vertex $(-1,0,0,0)$. Similarly for the class $\{(\pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2)\}$, we can think of the boundary cusp cross section as coming from sixteen cubes, each one corresponding to a horospherical neighbourhood about an ideal vertex making up the equivalence class. In order to understand the structure of these cusp cross sections we need to understand the parabolic isometries associated to them. Through the general theory of non-compact hyperbolic manifolds of finite volume we know that the stabiliser subgroup of an ideal vertex induces the Euclidean structure on the cusp cross section. The way one generally sees this is to observe that if one takes a horospherical neighbourhood about an ideal vertex, then the stabiliser subgroup acts as a group of Euclidean affine transformations on this neighbourhood. Viewing this neighbourhood as a copy of ${\ensuremath{\mathbb R}}^3$ one observes that the quotient via the action of the stabiliser subgroup is a Euclidean 3-manifold, the one corresponding to the cusp cross section (the reader who is not familiar with this material can consult [@benedetti] Thm.D.3.3, p.145). In order to compute the stabiliser subgroup one can take a horospherical neighbourhood about an ideal vertex and see how it transforms under the side pairing transformations associated to the sides making up the horospherical neighbourhood. We are going to give explicit examples of how to do this shortly. Before we do this let us remind the reader that the Euclidean structure associated to each of the five cusps is given in the Ratcliffe-Tschantz census (see [@ratcliffe]) under the column headed **LT**. A quick glance at the census shows that many of the manifolds in the census have very different cusp structures, this means that in general one needs to carry out such computations for the particular manifolds they are interested in. That being said, the general principle of how to compute the cusp structures works for all the Ratcliffe-Tschantz manifolds, and is in fact a standard technique from the theory of non-compact hyperbolic manifolds of finite volume.
As promised we are going to give a calculation of the cusp structure for a particular manifold using horospherical neighbourhoods as described above. Since we have already dealt with aspects of manifold no. 3 we choose to use it again. For the convenience of the reader we recall its basic construction. The side pairing code for manifold no. 3 is **1477B8**, decoding this gives the following side pairing transformations: $$\xymatrixcolsep{5pc}\xymatrix{ S_{(+1,+1,0,0)} \ar[r]^a_{k_{_{(-1,+1,+1,+1)}}} & S_{(-1,+1,0,0)} } \hspace{2cm} \xymatrix{S_{(+1,-1,0,0)} \ar[r]^b_{k_{_{(-1,+1,+1,+1)}}} & S_{(-1,-1,0,0)} }$$
$$\xymatrixcolsep{5pc}\xymatrix{ S_{(+1,0,+1,0)} \ar[r]^c_{k_{_{(+1,+1,-1,+1)}}} & S_{(+1,0,-1,0)} } \hspace{2cm} \xymatrix{S_{(-1,0,+1,0)} \ar[r]^d_{k_{_{(+1,+1,-1,+1)}}} & S_{(-1,0,-1,0)} }$$
$$\xymatrixcolsep{5pc} \xymatrix{ S_{(0,+1,+1,0)} \ar[r]^e_{k_{_{(-1,-1,-1,+1)}}} & S_{(0,-1,-1,0)} } \hspace{2cm} \xymatrix{S_{(0,+1,-1,0)} \ar[r]^f_{k_{_{(-1,-1,-1,+1)}}} & S_{(0,-1,+1,0)} }$$
$$\xymatrixcolsep{5pc} \xymatrix{ S_{(+1,0,0,+1)} \ar[r]^g_{k_{_{(-1,-1,-1,+1)}}} & S_{(-1,0,0,+1)} } \hspace{2cm} \xymatrix{S_{(+1,0,0,-1)} \ar[r]^h_{k_{_{(-1,-1,-1,+1)}}} & S_{(-1,0,0,-1)} }$$
$$\xymatrixcolsep{5pc} \xymatrix{ S_{(0,+1,0,+1)} \ar[r]^i_{k_{_{(-1,-1,+1,-1)}}} & S_{(0,-1,0,-1)} } \hspace{2cm} \xymatrix{S_{(0,+1,0,-1)} \ar[r]^j_{k_{_{(-1,-1,+1,-1)}}} & S_{(0,-1,0,+1)} }$$
$$\xymatrixcolsep{5pc} \xymatrix{ S_{(0,0,+1,+1)} \ar[r]^k_{k_{_{(+1,+1,+1,-1)}}} & S_{(0,0,+1,-1)} } \hspace{2cm} \xymatrix{S_{(0,0,-1,+1)} \ar[r]^l_{k_{_{(+1,+1,+1,-1)}}} & S_{(0,0,-1,-1)} }.$$
The labelling of the sides is given in the following table:\
----- ------------------- ------------------------------------------------ ------ ------------------- -------------------------------------------------
$A$ $S_{(+1,+1,0,0)}$ $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$ $A'$ $S_{(-1,+1,0,0)}$ $(\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$
$B$ $S_{(+1,-1,0,0)}$ $(\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 0)$ $B'$ $S_{(-1,-1,0,0)}$ $(\frac{-1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 0)$
$C$ $S_{(+1,0,+1,0)}$ $(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$ $C'$ $S_{(+1,0,-1,0)}$ $(\frac{1}{\sqrt{2}}, 0, \frac{-1}{\sqrt{2}})$
$D$ $S_{(-1,0,+1,0)}$ $(\frac{-1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$ $D'$ $S_{(-1,0,-1,0)}$ $(\frac{-1}{\sqrt{2}}, 0, \frac{-1}{\sqrt{2}})$
$E$ $S_{(0,+1,+1,0)}$ $(0, \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}})$ $E'$ $S_{(0,-1,-1,0)}$ $(0, \frac{-1}{\sqrt{2}} ,\frac{-1}{\sqrt{2}})$
$F$ $S_{(0,+1,-1,0)}$ $(0, \frac{1}{\sqrt{2}} ,\frac{-1}{\sqrt{2}})$ $F'$ $S_{(0,-1,+1,0)}$ $(0, \frac{-1}{\sqrt{2}} ,\frac{1}{\sqrt{2}})$
$G$ $S_{(+1,0,0,+1)}$ $(1 + \sqrt{2}, 0, 0)$ $G'$ $S_{(-1,0,0,-1)}$ $(1 - \sqrt{2}, 0, 0)$
$H$ $S_{(+1,0,0,-1)}$ $(-1 + \sqrt{2}, 0, 0)$ $H'$ $S_{(-1,0,0,+1)}$ $(-1 - \sqrt{2}, 0, 0)$
$I$ $S_{(0,+1,0,+1)}$ $(0, 1 + \sqrt{2}, 0)$ $I'$ $S_{(0,-1,0,+1)}$ $(0, -1 - \sqrt{2}, 0)$
$J$ $S_{(0,+1,0,-1)}$ $(0, -1 + \sqrt{2}, 0)$ $J'$ $S_{(0,-1,0,-1)}$ $(0, 1 - \sqrt{2}, 0)$
$K$ $S_{(0,0,+1,+1)}$ $(0, 0, 1 + \sqrt{2})$ $K'$ $S_{(0,0,+1,-1)}$ $(0, 0, -1 + \sqrt{2})$
$L$ $S_{(0,0,-1,+1)}$ $(0, 0, -1 - \sqrt{2})$ $L'$ $S_{(0,0,-1,-1)}$ $(0, 0, 1 - \sqrt{2})$
----- ------------------- ------------------------------------------------ ------ ------------------- -------------------------------------------------
\
\
Consider the equivalence class of ideal vertices $\{(1,0,0,0), (-1,0,0,0)\}$, if we start with the ideal vertex $(1,0,0,0)$ then by the above discussion it is easy to see that it lies on the sides $A$, $B$, $C$, $C'$, $G$ and $H$. The vertex $(1,0,0,0)$ can be thought of as the unit basis vector $e_1$ in ${\ensuremath{\mathbb R}}^4$, a horospherical neighbourhood represented by pieces of the sides $A$, $B$, $C$, $C'$, $G$ and $H$ can then be thought of as a cube in ${\ensuremath{\mathbb R}}^3$ (viewed as an orthogonal hyperplane to $e_1$). The orientation of the orthogonal copy of ${\ensuremath{\mathbb R}}^3$ will always be taken to be:
{width="3cm" height="3cm"}
The horospherical neighbourhood will then look like:
{width="6cm" height="5cm"}
As the side $A$ is defined via the sphere $S_{(+1,+1,0,0)}$ with centre vector $(+1,+1,0,0)$ we see that its intersection with a horospherical neighbourhood about $(1,0,0,0)$ can be thought of as a plane centred at $(x,0,0)$ (here the three co-ordinates are $(e_2, e_3, e_4)$), where $x \in {\ensuremath{\mathbb R}}$, and parallel to the plane defined by the equation $e_2 = 0$. The actual value of $x$ is unimportant, it depends on where exactly we form our horospherical neighbourhood about $(1,0,0,0)$, which in turn comes down to how high up into the horoball about $(1,0,0,0)$ we are. The side $B$ is defined via the sphere $S_{(+1,-1,0,0)}$ with centre vector $(+1,-1,0,0)$, its intersection with a horospherical neighbourhood about $(1,0,0,0)$ can be thought of as a plane centred at $(-x,0,0)$ and parallel to the plane defined by the equation $e_2 = 0$. The side $C$ is defined by the sphere $S_{(+1,0,+1,0)}$ with centre vector $(+1,+1,0,0)$, similar reasoning to the above shows that its intersection with a horospherical neighbourhood about the ideal vertex $(1,0,0,0)$ can be viewed as a plane centred at $(0,y,0)$, where $y \in {\ensuremath{\mathbb R}}$, parallel to the plane $e_3 = 0$. The side $C'$ can be viewed as a plane centred at $(0,-y,0)$ parallel to the plane $e_3 = 0$. The side $G$ is defined by the sphere $S_{(+1,0,0,+1)}$ with centre vector $(+1,0,0,+1)$, intersecting it with a horospherical neighbourhood about the ideal vertex $(1,0,0,0)$ we obtain a plane centred at $(0,0,z)$, where $z \in {\ensuremath{\mathbb R}}$ and parallel to the plane $e_4 = 0$. The side $H$ can be viewed as a plane centred at $(0,0,-z)$ parallel to the plane $e_4 = 0$. Using this description we can label the above cube, which represents a horospherical neighbourhood based at $(1,0,0,0)$, as follows:
{width="6cm" height="4cm"}
The equivalence class corresponding to the ideal vertex $(+1,0,0,0)$ also contains the ideal vertex $(-1,0,0,0)$, therefore in order to describe the parabolic isometries giving rise to the cusp corresponding to this class we need to describe a horospherical neighbourhood about $(-1,0,0,0)$. The procedure is exactly analogous to what we did above. However, for this vertex we are going to orient the orthogonal copy of ${\ensuremath{\mathbb R}}^3$ to the vertex $(-1,0,0,0)$ as follows:
{width="5cm" height="4cm"}
The reason for this choice will become apparent shortly, for now let us observe that it leads to the following labelling of a horospherical neighbourhood:
{width="5cm" height="4cm"}
A fundamental domain for a cross section of the cusp corresponding to the above ideal vertex equivalence class consists of these two cubes.
{width="6cm" height="5cm"}
We can see that when we apply the transformation $h$ the bottom face of the top box gets joined to the top face of the bottom box. In principle the bottom face of the top box has four different ways it can be joined to the top face of the bottom box. However, observe that when applying the transformation $h$, the side $B$ maps to $A'$, side $A$ maps to $B'$, side $C$ maps to $D'$, and side $C'$ maps to $D$. This means we can visualise a fundamental domain as a rectangular box with sides labelled as follows:
{width="6cm" height="5cm"}
This is why we chose to orient the orthogonal copy of ${\ensuremath{\mathbb R}}^3$ to the ideal vertex $(-1,0,0,0)$ the way we did. We then see that the top face of the rectangular box will be identified to the bottom face of the rectangular box by the parabolic transformation $g^{-1}h$ (note that $g^{-1}h$ fixes the ideal vertex $(1,0,0,0)$, hence is clearly an element of the stabiliser subgroup of the ideal vertex $(1,0,0,0)$). Similarly the front face will be identified to the back face via the transformation $c$ (note one could also take the transformation $d$, they both act in the same way). Finally, the identification of the face on the right side with the left side is done by the transformation $a^{-1}h$, this is because the top right $A$ get identified to the bottom left $A'$, and the bottom right $B'$ gets identified to the top left $B$. Thus what we see that is happening is that the top face is being identified to the bottom face via a translation, the front face is being identified to the back face via a translation, but the side faces are being identified by a “twist” (also called a screw parabolic transformation). This tells us that the stabiliser subgroup associated to the vertex equivalence class $\{(1,0,0,0), (-1,0,0,0)\}$ is generated by the three transformations $\langle c, g^{-1}h, a^{-1}h \rangle$. Put another way the parabolic subgroup associated to the vertex class $\{(1,0,0,0), (-1,0,0,0)\}$ is the subgroup $\langle c, g^{-1}h, a^{-1}h \rangle$.
Once one has understood the parabolic subgroup associated to a cusp, one can try to understand how the Euclidean structure on the cusp cross section comes about. For a general non-compact hyperbolic manifold of finite volume this can be a difficult task, the reason being that in order to work out the Euclidean structure on the cross section one needs a solid understanding of how elements of the associated parabolic subgroup act as Euclidean transformations on a horospherical neighbourhood. For the Ratcliffe-Tschantz manifolds the action of the parabolic subgroup on a horospherical neighbourhood can be completely understood due to some nice symmetry of the 24-cell, and the fact that the side pairing transformations are very easy to describe.
Continuing with the above example, we outline how to describe the Euclidean structure on a cusp cross section associated to the ideal vertex class $\{(1,0,0,0), (-1,0,0,0)\}$. We already know that the associated parabolic subgroup is given by $\langle c, g^{-1}h, a^{-1}h \rangle$, we want to understand how these transformations act on a horospherical neighbourhood centred about $(1,0,0,0)$. A Euclidean transformation on ${\ensuremath{\mathbb R}}^3$ is just an affine transformation of the form: $$x \mapsto \Lambda\cdot x + v$$ where $\Lambda \in O(3)$ and $v \in {\ensuremath{\mathbb R}}^3$. For each of the transformations $c, g^{-1}h, a^{-1}h$ we want to understand what the matrix $\Lambda$ looks like, and what the translation vector $v$ looks like. Each side pairing transformation consists of two components, a k-part and an r-part, the k-part is given by a diagonal matrix, hence its action on the copy of ${\ensuremath{\mathbb R}}^3$ is easy to understand. The r-part consists of reflection in the image side of the k-part, in this case its action on ${\ensuremath{\mathbb R}}^3$ is also easy to understand, this is because, as was mentioned before, the intersection of the sides that the ideal vertex $(1,0,0,0)$ lies on with a horospherical neighbourhood can be thought as the box (remember our orientation convention):
{width="5cm" height="4cm"}
Since the sides intersect the horospherical neighbourhood in planes parallel to the $e_2-e_3$, $e_2-e_4$ and $e_3-e_4$ planes we see that the r-part is an affine transformation with $\Lambda$-matrix corresponding to one of reflections in the $e_2-e_3$, $e_2-e_4$ or $e_3-e_4$ planes. For example suppose we take the side pairing transformation $c$. From the picture above we see that the part of the side $C'$ that intersects the horospherical neighbourhood can be thought of as a plane parallel to the plane $e_3 = 0$ and centred at the point $(0,-y,0)$.
{width="8cm" height="8cm"}
This tells us that the r-part of the transformation $c$, which remember is reflection in the side $C'$, is a reflection in this plane. But it is easy to see that such a reflection is the affine transformation: $$w \mapsto
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\cdot w +
\begin{pmatrix}
0 \\
-2y \\
0
\end{pmatrix}$$ The k-part of the transformation $c$ is given by the matrix: $$\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}$$ whose action on the horospherical neighbourhood (which we are viewing as a copy of ${\ensuremath{\mathbb R}}^3$) is given by the matrix: $$\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{pmatrix}$$ Therefore the action of the transformation $c$ on this horospherical neighbourhood being the composition of the k-part and the r-part, is given by the transformation: $$w \mapsto
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{pmatrix} \cdot w +
\begin{pmatrix}
0 \\
-2y \\
0
\end{pmatrix} = w +
\begin{pmatrix}
0 \\
-2y \\
0
\end{pmatrix}$$ This tells us that the Euclidean transformation that corresponds to the action of $c$ on a horospherical neighbourhood about $(1,0,0,0)$ is nothing more than a translation by the vector $(0,-2y,0)$.
We can apply the same techniques to understand the action of the parabolic transformation $g^{-1}h$ on the horospherical neighbourhood. In this case we need to understand the k and r-parts of two transformations, $g^{-1}$ and $h$.
The k-part of the transformation $h$ is given by the matrix: $$\begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}$$ The side $H'$ is parallel to the plane $e_4 = 0$ and centred at the vector $(0,0,-z)$. Therefore the r-part of the transformation $h$ is given by the affine transformation: $$w \mapsto
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{pmatrix} \cdot w +
\begin{pmatrix}
0 \\
0 \\
-2z
\end{pmatrix}$$ The k-part of the transformation $g^{-1}$ is the same is that of $h$, the r-part is given by reflection in the plane defined by the side $G$. The associated plane is parallel to the plane $e_4 = 0$ and centred at the point $(0,0,z)$. Therefore the r-part for $g^{-1}$ is given by the affine transformation: $$w \mapsto
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{pmatrix} \cdot w +
\begin{pmatrix}
0 \\
0 \\
2z
\end{pmatrix}$$ The composition $g^{-1}h$ consists of the two k-parts one from $g^{-1}$ and one from $h$, as these are both equal they simply give the identity. Therefore to understand $g^{-1}h$ as an affine transformation we need only understand the composition of the r-parts of $h$ and $g^{-1}$. This is easy to calculate: $$w \mapsto
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{pmatrix} \left(
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{pmatrix} \cdot w +
\begin{pmatrix}
0 \\
0 \\
-2z
\end{pmatrix} \right) +
\begin{pmatrix}
0 \\
0 \\
2z
\end{pmatrix}$$ which is the affine transformation: $$w \mapsto x +
\begin{pmatrix}
0 \\
0 \\
4z
\end{pmatrix}$$ Finally, let us work out how the parabolic transformation $a^{-1}h$ behaves as a Euclidean transformation on a horospherical neighbourhood. The k-parts of $h$ and $a^{-1}$ are given by the following matrices: $$\begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix},
\begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}$$ We have already computed the r-part of $h$, as for the r-part of $a^{-1}$ observe that the side $A$ is parallel to the $e_2 = 0$ plane, and centred at the vector $(x,0,0)$. Reflection in this plane is given by the affine transformation: $$w \mapsto
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix} \cdot w +
\begin{pmatrix}
2x \\
0 \\
0 \\
\end{pmatrix}$$ The composition $a^{-1}h$ is then given by the affine transformation: $$w \mapsto
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix} \left(
\begin{pmatrix}
-1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1 \\
\end{pmatrix} \cdot w +
\begin{pmatrix}
0 \\
0 \\
-2z \\
\end{pmatrix} \right) +
\begin{pmatrix}
2x \\
0 \\
0 \\
\end{pmatrix}$$ which when we expand out we obtain: $$\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1 \\
\end{pmatrix} \cdot w +
\begin{pmatrix}
2x \\
0 \\
-2z \\
\end{pmatrix}$$ The reader will recall that before we started showing how to compute the action of these parabolic isometries on a horospherical neighbourhood, we showed how a cusp cross section arises through a fundamental domain consisting of a rectangular box. We went on to say that the right and left sides of this fundamental domain (which were being identified by the parabolic transformation $a^{-1}h$) were identified via a “twist”. The reader who was not content with the use of the word “twist” at that time should now be at ease, for the “twisting” we speak of is given by the matrix $$\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1 \\
\end{pmatrix}$$ in the above formula for the affine transformation associated to the parabolic isometry $a^{-1}h$.
In summary, we have shown that the parabolic subgroup corresponding to the ideal vertex equivalence class $\{(1,0,0,0), (-1,0,0,0)\}$ is given by the group $\langle c, g^{-1}h, a^{-1}h \rangle$. Furthermore, for each of these parabolic isometries we have shown how they act on a horospherical neighbourhood as a Euclidean transformation. The following table summarises this information
-------------------------------------------------------
Parabolic isometry Affine transformation
-------------------- ----------------------------------
$c$ $\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}$, $\begin{pmatrix}
0 \\
-2y \\
0 \\
\end{pmatrix}$
$g^{-1}h$ $\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}$, $\begin{pmatrix}
0 \\
0 \\
4z \\
\end{pmatrix}$
$a^{-1}h$ $\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1 \\
\end{pmatrix}$, $\begin{pmatrix}
2x \\
0 \\
-2z \\
\end{pmatrix}$
-------------------------------------------------------
where in the second column the matrix is in $O(3)$ and the vector corresponds to the translation vector.
The information presented above is enough for one to obtain the Euclidean structure of the cusp corresponding to the class $\{(1,0,0,0), (-1,0,0,0)\}$. Recall that there are ten classes of closed flat 3-manifolds denoted by $\textbf{A}$, $\textbf{B}$, $\textbf{C}$, $\textbf{D}$, $\textbf{E}$, $\textbf{F}$, $\textbf{G}$, $\textbf{H}$, $\textbf{I}$, and $\textbf{J}$ in the Hantzsche-Wendt notation see [@hantzsche]. These are denoted by $\mathcal{G}_1$, $\mathcal{G}_2$, $\mathcal{G}_3$, $\mathcal{G}_4$, $\mathcal{G}_5$, $\mathcal{G}_6$, $\mathcal{B}_1$, $\mathcal{B}_2$, $\mathcal{B}_3$ and $\mathcal{B}_4$ respectively using the notation of Wolf see [@wolf] Thm.3.5.5, p.117. The first six are the orientable ones, with $\textbf{A}$ being the 3-torus, and $\textbf{B}$ being the orientable $S^1$-fibre bundle over the Klein bottle. The last four are all non-orientable. Using the classification of orientable compact flat 3-manifolds (see [@wolf] Thm.3.5.5, p.117) we see that this cusp has type $\textbf{B}$ using the Hantzsche-Wendt notation or type $\mathcal{G}_2$ using Wolf’s notation.
One can carry out an analogous procedure to work out generators for the parabolic subgroups corresponding to the other ideal vertex equivalence classes and their corresponding affine transformations. The only slight difference is that when dealing with the class $\{(\pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2)\}$ one has to take sixteen boxes to form a fundamental domain. Also, computing the associated affine transformations in this case is slightly harder as one does not have such a nice parameterisation of horospherical neighbourhoods as in the case for the other equivalence classes. We will not need explicit formulas for the corresponding affine transformations, hence will not bother the reader with an explanation of how to find them in this situation. For the sake of completeness we have included the following table showing the end results of the above computations for all other ideal vertex equivalence classes.\
We point out that if we order the ideal vertices according to our table above then we see that the Euclidean structures on the associated cusps are given by **BAAFB**. However, if we go to manifolds no.3 in the Ratcliffe-Tschantz census we see that under the column headed **LT** the Euclidean structure on the cusps is given by **AABBF**. This is because they chose to write out the Euclidean structures in alphabetical order, as opposed to any ordering on the ideal vertices.
The Euclidean structure on each cusp is found by an appeal to the classification theorem of compact connected orientable flat 3-dimensional Riemannian manifolds (see [@wolf] Thm.3.5.5, p.117). One simply needs to identify the linear holonomy group, and one can do this from the computations of the associated affine transformations. The matrices of each generator (when viewed as an affine transformation on a horospherical neighbourhood) generate the linear holonomy group. For example, consider the first entry in the above table, we see that two of the generators have the identity matrix as their $O(3)$ component, when viewed as an affine transformation, and one of the generators has $\biggl(\begin{smallmatrix}
1&0&0\\ 0&-1&0\\ 0&0&-1
\end{smallmatrix} \biggr)$ as its $O(3)$ component. Therefore the linear holonomy group is generated by $\biggl(\begin{smallmatrix}
1&0&0\\ 0&-1&0\\ 0&0&-1
\end{smallmatrix} \biggr)$, which has order 2. This implies the linear holonomy group associated to this cusp is ${\ensuremath{\mathbb Z}}_2$. Using the classification theorem (see [@wolf] Thm.3.5.5, p.117) we see that it corresponds to a Euclidean structure of type **B** (or **$\mathcal{G}_2$** in Wolf’s notation).
Observe that if we look at the cusp corresponding to the ideal vertex class $\{(0,1,0,0), (0,-1,0,0)\}$ we see that the Euclidean structure is given by $\textbf{A}$, which is a 3-torus. Viewing the 3-torus as an $S^1$-fibre bundle over the 2-torus we see that there is a natural 4-manifold that it bounds. Namely, the associated disk bundle, which we denote by $\widetilde{\textbf{A}}$. If we then take a horoball neighbourhood about the cusp corresponding to $\{(0,1,0,0), (0,-1,0,0)\}$ and chop it off, we obtain a boundary 3-manifold given by $\textbf{A}$. We can then glue in the 4-manifold $\widetilde{\textbf{A}}$ by identifying boundary components, thus killing the cusp corresponding to $\{(0,1,0,0), (0,-1,0,0)\}$. At this point there are a few remarks that we need to make. The first one is that since we are gluing in a disc bundle over a flat surface the Euler characteristic does not change. The second remark is that the way we glue in the solid 3-torus $\widetilde{\textbf{A}}$ is not unique, it depends on a choice. Namely, in choosing how the 3-torus fibres over the 2-torus we made a choice of an $S^1$-fibre, you can think of this as fixing a meridian. When we carry out the gluing process we are filling in this meridian with a copy of $D^2$. Finally, the 4-manifold we chose to glue in was the solid torus and this choice is not unique. What we mean by this is that since there are other 4-manifolds bounded by the 3-torus we could have just as well chosen one of them to glue in. However as our interest is in constructing codimension two link complements we will always focus on the disk bundle case.
From the classification theorem of closed flat 3-manifolds (see [@wolf] Thm.3.5.5, p.117) one knows that the only orientable closed flat 3-manifolds that are $S^1$-fibre bundles are given by types $\textbf{A}$ and $\textbf{B}$ (or $\mathcal{G}_1$ and $\mathcal{G}_2$ using Wolf’s notation). Furthermore, all the non-orientable ones $\textbf{G}$, $\textbf{H}$, $\textbf{I}$, $\textbf{J}$ ($\mathcal{B}_1$, $\mathcal{B}_2$, $\mathcal{B}_3$, $\mathcal{B}_4$ in Wolf’s notation) are $S^1$ fibre bundles. There are many hyperbolic 4-manifolds in the Ratcliffe-Tschantz census whose cusp structure has type $\textbf{A}$, $\textbf{B}$, $\textbf{G}$, $\textbf{H}$, $\textbf{I}$, $\textbf{J}$. For such manifolds we can obtain a closed 4-manifold by gluing in associated disk bundles as described above, such a gluing procedure does not change the Euler characteristic. Let $E \rightarrow S$ denote a closed flat 3-manifold written as an $S^1$ fibre bundle over a compact surface $S$ (note that from the classification theorem we know that $S$ must be either a 2-torus or a Klein bottle). The associated disk bundle can then be written as $$\pi : \widetilde{E} \rightarrow S$$ The 4-manifold $\widetilde{E}$ has a nice handle decomposition coming from the handle decomposition of the surface $S$. $S$ is a compact surface, hence can be given a handle decomposition consisting of one 0-handle, $n$ 1-handles ($n = 2 - \chi(S)$), and one 2-handle (see [@gompf] p.131 last paragraph). The $k$-handles of $\widetilde{E}$ are given by preimages under $\pi$ of $k$-handles of $B$ (see [@gompf] p.131 last paragraph). This means that $\widetilde{E}$ has a handle decomposition consisting of one 0-handle, two 1-handles and one 2-handle, since $\chi(S) = 0$.
On the level of a Kirby diagram the gluing in of the disk bundle $\widetilde{E}$ is done by adding one 2-handle, two 3-handles and one 4-handle (see [@akbulut] chap.3, p.35). If we are given a hyperbolic 4-manifold with cusps all of which are $S^1$-fibre bundles then we can glue in a disk bundle to each cusp to produce a closed 4-manifold. In our paper [@sarat] we showed how to construct a Kirby diagram for any one of the Ratcliffe-Tschantz manifolds. Provided we can understand where to add the 2-handle corresponding to the gluing of the disk bundle we can then produce a Kirby diagram for the “filled in” manifold. We can then try and apply certain handle slides/cancellations to try and reduce the Kirby diagram of this “filled in” manifold. The hope is that we can reduce it to the Kirby diagram of a familiar closed 4-manifold that we can explicitly identify. Provided we are successful in this reduction process, we would have then found an explicit four dimensional hyperbolic link complement.
Our general method of constructing four dimensional hyperbolic link complements whose diffeomorphism type can be identified can thus be broken down in to three general steps. The first step involves finding finite volume non-compact hyperbolic 4-manifolds with cusp structure given by one of the closed flat 3-manifolds that is an $S^1$-fibre bundle over a flat surface. The second step involves being able to construct a Kirby diagram for the filling of the hyperbolic 4-manifold. Our approach in this step is to use a construction of a Kirby diagram for the hyperbolic 4-manifold, then explicitly work out how to add 2-handles to the diagram to obtain a Kirby diagram for the filling. Finally the third step involves carrying out various handle slides/cancellations. We are going to give an explicit example of a filling using a hyperbolic 4-manifold obtained from the Ratcliffe-Tschantz census soon. The handle slides/cancellations we use are based on three very simply moves from the theory of Kirby calculus. In the next section we will take the time to explain exactly what these moves are and how they will be used.
The main manifold we are going to be concerned with in terms of the filling process is numbered 1011 in the census. This is a non-orientable five cusped hyperbolic 4-manifold with each cusp structure given by the non-orientable $S^1$ fibre bundle with flat structure given by type $\textbf{G}$ (or $\mathcal{B}_1$ in Wolf’s notation). It has orientable double cover the flat manifold given by type $\textbf{A}$ (or $\mathcal{G}_1$ in Wolf’s notation), the 3-torus. The structure of each cusp can be found using methods exactly analogous to what we did for manifold no. 3. The following table summarises this information.
Elementary Moves {#elementary}
================
In this section we are going to go through the three main reduction moves we will be using to simplify the Kirby diagram of a filling of one of the Ratcliffe-Tschantz manifolds. We call these three main reductions moves the “three elementary moves”. In [@sarat] we showed how to construct the Kirby diagram of manifold no. 3 in the Ratcliffe-Tschantz census. The reader might want to refer to that paper for the following discussion.
We showed that the Kirby diagram of manifold no. 3 can be viewed via the four following diagrams:
{width="11cm" height="11cm"}
The top two diagrams represent that part of the Kirby diagram that lies in the $x-y$ and $x-z$ planes, reading left to right. The bottom left diagram represents that part of the diagram contained in the $y-z$ plane. Finally, the bottom right diagram represents the six 2-handles that did not all lie in a single 2-plane. The attaching circles of the 2-handles are colour coded, each colour representing the attaching circle of one 2-handle. The reader can refer to [@sarat] for an explanation on how we obtain the 2-handle structure.
Consider the boundary component given by the code $\textbf{A}$ (or $\mathcal{G}_1$ in Wolf’s notation), this is the 3-torus and it bounds the solid 4-manifold $S^1 \times S^1 \times D^2$. When we glue in the solid 3-torus we need to add one 4-handle, two 3-handles (as the Euler characteristic of $T^2 = 0$) and one 2-handle. If we go back to the table outlining the generators of each parabolic subgroup corresponding to each boundary component, we see that the first boundary component labelled $\textbf{A}$ has a generator given by the translation $a$. This means that algebraically the translation $a$ represents an $S^1$-fibre of the boundary component. Therefore when we glue in a solid torus to this boundary component we can do so along the $S^1$-fibre corresponding to the translation $a$. Algebraically this means we are killing the transformation $a$, hence the 2-handle we are attaching must be a straight line segment running between $A-A'$ once. As the 1-handle $A-A'$ lies in the $x-y$ plane we draw this 2-handle as a dashed line segment lying in the $x-y$ plane, it must also be added to the diagram showing the six 2-handles that do not all lie in a single 2-plane.
{width="10cm" height="10cm"}
There are two subtle points we need to address with this gluing procedure. First of all, we did not explain how the normal bundle to the added 2-handle looks in the Kirby diagram. In order to understand this one must carry out a similar analysis as was done when trying to understand how the normal bundles of the 2-handles in the Kirby diagram look like (see Sect.5 in [@sarat]). The point is that these added 2-handles will, most of the time, lie in a single plane, hence the trivialisation of their normal bundle is easy to understand. When the added 2-handle does not lie in a single plane we will find that it has a planar framing (see Sect.5 in [@sarat]), in other words a parallel curve to the 2-handle behaves as if the added 2-handle was lying in a plane. The second point to address has to do with how exactly we know where to put the added 2-handle in our Kirby diagram. In the above diagram the added 2-handle lies in the $x-y$ plane, just before the diagram we said that we can draw this 2-handle as a straight line running between $A,A'$ in the $x-y$ plane because both $A$ and $A'$ lie in the $x-y$ plane. The question is, why is this the right place for the added 2-handle? The basic idea of why this is the right place has to do with how the fundamental domain of this boundary component looks. It consists of two cubes, one coming from the ideal vertex $(0,1,0,0)$, and another coming from $(0,-1,0,0)$. The cube centred at $(0,1,0,0)$ has two of its sides being $A$ and $A'$ and the $S^1$-fibre that we are filling along corresponds to a straight line joining $A$ and $A'$. When we formed the Kirby diagram we did so by taking the dual polyhedron to the 24-cell $P$, when we do this the cubes will look like octahedrons in the Kirby diagram, as the dual of a cube is an octahedron. The point is that we can then go to our Kirby diagram find the associated dual octahedron, and then identify the corresponding $S^1$-fibre as a straight line running between some 1-handles. If we do this for the above mentioned boundary component we find that the added 2-handle does indeed lie where we have drawn it in the above diagram. This explanation may seem convoluted, but for now we insist the reader to not pay too much attention to it as the primary aim of this section is to explain the “elementary moves” we will be using to reduce a Kirby diagram, and we don’t want the reader to get bogged down with some minor details. In the next section we will deal with an explicit example, and give full details of exactly how the dual octahedra look like in the Kirby diagram, which in turn will tell us exactly how the added 2-handles look like, and how they should be added.
Coming back to the above diagram we can see that the added 2-handle only passes over $A-A'$ once, in other words the attaching sphere of the added 2-handle transversely intersects the belt sphere of the 1-handle $A-A'$ once. This means this 1-handle and 2-handle pair form a cancelling pair and can be erased from the diagram. Any other 2-handles that pass over the 1-handle $A-A'$ must first be slid over the added 2-handle, and then we can erase the pair of handles from the diagram. Observe that because all the added 2-handles, coming from boundary fillings, are unknotted and have parallel curves that do not twist around the 2-handle in any way (i.e. they are planar framed), whenever we slide 2-handles over them nothing non-trivial will happen, making the sliding process very straightforward. This standard handle cancellation move is the **first elementary move** we will be using to try and reduce our Kirby diagram. Let us show how the diagram in the $x-y$ plane changes when we carry out such a cancellation.
{width="6cm" height="6cm"}
After this cancellation has taken place we can see that a few 2-handles have slid into new positions. In particular notice how the blue 2-handle has a component that now loops back into $I$, similarly the red 2-handle has a component that loops back into $J$.
The **second elementary move** we will be making use of is to push such 2-handles through the 1-handle piece they loop back into. For example if we consider the component of the blue 2-handle that loops back into $I$, we see that we can push it through $I$ to come out as a component that loops back into $I'$.
{width="6cm" height="6cm"}
We can then slide the blue 2-handle off $I'$, giving a blue 2-handle that runs between $B-B'$ once.
{width="6.5cm" height="5.8cm"}
We can also carry out the same move for the component of the red 2-handle that loops back into $J$:
{width="6cm" height="6cm"}
In the above move it is very important that we have that the 2-handle we are pushing through being unknotted. This is because some of our 1-handles are being identified via orientation preserving diffeomorphisms, hence pushing 2-handles through will cause bits of knots to get mirrored, however if the 2-handles are all unknotted then one does not have to worry about such minor technicalities.
We now have a blue 2-handle and a red 2-handle passing over the 1-handle $B-B'$ once. The **third elementary move** we will be making use of can be described as follows. We can cancel the 1-handle $B-B'$ using the blue 2-handle that now runs over it once, in so doing the red 2-handle will form an unknotted circle:
{width="6cm" height="6cm"}
Note that this unknotted circle now has a well-defined notion of a framing number. You can see that because the 2-handles in the $x-y$ plane all have parallel curves that do not twist around them in any way the unknotted circle must have framing zero. This unknotted circle then cancels a 3-handle and can be deleted from the diagram, a proof of this fact can be found in [@gompf] prop.5.1.9, p.148.
The three moves described above will be heavily used in various situations in the sections to come, this is why we took the time to explain each one carefully. On top of this we will carry out various handle slides, as all our 2-handle are unknotted and have a planar framing the handle slides we carry out will always be straightforward. If at any point we exploit the use of a non-trivial move we will take the time to carefully explain how one proceeds.
Filling in the boundary components of any one of the Ratcliffe-Tschantz manifolds produces a compact 4-manifold (possibly non-orientable) for which we know how to build a Kirby diagram for. Using the elementary moves outlined above one can try and reduce the Kirby diagram of this smooth 4-manifold, the hope is that after sufficiently many reductions the end Kirby diagram is that of a compact smooth 4-manifold that we can identify. This gives a way of trying to identify which compact smooth 4-manifolds have smooth complements that are given by the Ratcliffe-Tschantz manifolds, and in turn allows one to obtain explicit examples of smooth hyperbolic link complements. In general this proves to be a very difficult task, the reason being is that we have at least twenty four 2-handles to deal with and we view many of them in certain 2-planes. The added 2-handles (coming from filling boundary components) will in general pass through some of these 2-planes giving rise to intersection points that have to be carefully tracked when carrying out various elementary moves. In the next section we will give an explicit example of a situation in which we can identify the filled in 4-manifold up to diffeomorphism.
An explicit example
===================
In this section we will show that the double cover of the Ratcliffe-Tschantz manifold numbered 1011 is a smooth complement in the standard smooth 4-sphere. In other words if we perform a boundary filling on the orientable double cover of manifold 1011 we get a closed smooth 4-manifold that is diffeomorphic to $S^4$.
For the remainder of this section we are going to denote the manifold numbered 1011 in the Ratcliffe-Tschantz census by $M$. The structure of the Kirby diagram for $M$ was computed in [@sarat]. We remind the reader of the structure of the Kirby diagram for $M$:
The following picture shows those 2-handles lying in the $x-y$ plane, with the table following explaining the colour coding.
{width="10cm" height="10cm"}
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colour equivalence class
-------- ------------------------------------------------------------------------------------------------------------------
green $\xymatrix{
A \cap H \ar[r]^a & A'\cap G'\ar[r]^{g^{-1}} & B\cap G \ar[r]^{b} & B'\cap H' \ar[r]^{h^{-1}} & A\cap H }$
red $\xymatrix{
A \cap J \ar[r]^a & A'\cap J\ar[r]^j & B'\cap J' \ar[r]^{b^{-1}} & B\cap J' \ar[r]^{j^{-1}} & A\cap J }$
brown $\xymatrix{
A \cap G \ar[r]^a & A'\cap H' \ar[r]^{h^{-1}} & B\cap H \ar[r]^{b} & B'\cap G' \ar[r]^{g^{-1}} & A\cap G }$
blue $\xymatrix{
A \cap I \ar[r]^a & A'\cap I\ar[r]^i & B'\cap i' \ar[r]^{b^{-1}} & B\cap I' \ar[r]^{i^{-1}} & A\cap I }$
pink $\xymatrix{
G \cap I \ar[r]^g & G'\cap J' \ar[r]^{j^{-1}} & G'\cap J \ar[r]^{g^{-1}} & G\cap I' \ar[r]^{i^{-1}} & G\cap I }$
black $\xymatrix{
H \cap J \ar[r]^h & H'\cap I' \ar[r]^{i^{-1}} & H'\cap I \ar[r]^{h^{-1}} & H\cap J' \ar[r]^{j^{-1}} & H\cap J }$
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\
\
The 2-handles that lie in the $x-z$ plane can be seen in the following picture.
{width="10cm" height="10cm"}
---------------------------------------------------------------------------------------------------------------------------
colour equivalence class
-------- ------------------------------------------------------------------------------------------------------------------
green $\xymatrix{
D \cap K \ar[r]^d & D'\cap L \ar[r]^{l} & D'\cap L' \ar[r]^{d^{-1}} & D\cap K' \ar[r]^{k^{-1}} & D\cap K }$
red $\xymatrix{
G \cap K \ar[r]^g & G'\cap L' \ar[r]^{l^{-1}} & H'\cap L \ar[r]^{h^{-1}} & H\cap K' \ar[r]^{k^{-1}} & G\cap K }$
brown $\xymatrix{
C \cap G \ar[r]^c & C'\cap G\ar[r]^g & D\cap G' \ar[r]^{d} & D'\cap G' \ar[r]^{g^{-1}} & C\cap G }$
blue $\xymatrix{
C \cap H \ar[r]^c & C'\cap H \ar[r]^h & D\cap H' \ar[r]^{d} & D'\cap H' \ar[r]^{h^{-1}} & C\cap H }$
pink $\xymatrix{
C \cap K \ar[r]^c & C'\cap L \ar[r]^{l} & C'\cap L' \ar[r]^{c^{-1}} & C\cap K' \ar[r]^{k^{-1}} & C\cap K }$
black $\xymatrix{
G \cap L \ar[r]^g & G'\cap K' \ar[r]^{k^{-1}} & H'\cap K \ar[r]^{h^{-1}} & H\cap L' \ar[r]^{l^{-1}} & G\cap L }$
---------------------------------------------------------------------------------------------------------------------------
\
\
The 2-handles that lie in the $y-z$ plane can be seen in the following picture.
{width="10cm" height="10cm"}
---------------------------------------------------------------------------------------------------------------------------
colour equivalence class
-------- ------------------------------------------------------------------------------------------------------------------
green $\xymatrix{
I \cap L \ar[r]^i & I'\cap L \ar[r]^{l} & J'\cap L' \ar[r]^{j^{-1}} & J\cap L' \ar[r]^{l^{-1}} & I\cap L }$
red $\xymatrix{
I \cap K \ar[r]^i & I'\cap K \ar[r]^{k} & J'\cap K' \ar[r]^{j^{-1}} & J\cap K' \ar[r]^{k^{-1}} & I\cap K }$
brown $\xymatrix{
F \cap L \ar[r]^f & F'\cap K' \ar[r]^{k^{-1}} & F'\cap K \ar[r]^{f^{-1}} & F\cap L' \ar[r]^{l^{-1}} & F\cap L }$
blue $\xymatrix{
E \cap K \ar[r]^e & E'\cap L' \ar[r]^{l^{-1}} & E'\cap L \ar[r]^{e^{-1}} & E\cap K' \ar[r]^{k^{-1}} & E\cap K }$
pink $\xymatrix{
E \cap J \ar[r]^e & E'\cap I' \ar[r]^{i^{-1}} & F\cap I \ar[r]^{f} & F'\cap J' \ar[r]^{j^{-1}} & E\cap J }$
black $\xymatrix{
E \cap I \ar[r]^e & E'\cap J' \ar[r]^{j^{-1}} & F\cap J \ar[r]^{f} & F'\cap I' \ar[r]^{i^{-1}} & E\cap I }$
---------------------------------------------------------------------------------------------------------------------------
\
\
Finally, the 2-handles that do not lie in any one of the above three planes can be seen in the following picture.
{width="10cm" height="10cm"}
-------------------------------------------------------------------------------------------------------------------
colour equivalence class
-------- ----------------------------------------------------------------------------------------------------------
green $\xymatrix{
A \cap E \ar[r]^a & A'\cap E\ar[r]^e & B\cap E' \ar[r]^{b} & B'\cap E' \ar[r]^{e^{-1}} & A\cap E }$
red $\xymatrix{
B \cap C \ar[r]^b & B'\cap D\ar[r]^d & B'\cap D' \ar[r]^{b^{-1}} & B\cap C' \ar[r]^{c^{-1}} & B\cap C }$
brown $\xymatrix{
A \cap C \ar[r]^a & A'\cap D\ar[r]^d & A'\cap D' \ar[r]^{a^{-1}} & A\cap C' \ar[r]^{c^{-1}} & A\cap C }$
blue $\xymatrix{
A \cap F \ar[r]^a & A'\cap F\ar[r]^f & B\cap F' \ar[r]^{b} & B'\cap F' \ar[r]^{f^{-1}} & A\cap F }$
pink $\xymatrix{
C \cap E \ar[r]^c & C'\cap F \ar[r]^f & D\cap F' \ar[r]^{d} & D'\cap E' \ar[r]^{e^{-1}} & C\cap E }$
black $\xymatrix{
C \cap F' \ar[r]^c & C'\cap E' \ar[r]^{e^{-1}} & D\cap E \ar[r]^{d} & D'\cap F \ar[r]^{f} & C\cap F' }$
-------------------------------------------------------------------------------------------------------------------
\
\
Recall the manifold $M$ is non-orientable and has five cusps each of which has the type **G** (or in Wolf’s notation type $\mathcal{B}_1$). We denote the 4-manifold that bounds this 3-manifold by $\widetilde{\textbf{G}}$, remember this is the associated disc bundle to **G**. We need to choose a translation in each parabolic subgroup that is going to correspond to an $S^1$-fibre which we will fill in. We have already shown the parabolic information corresponding to each cusp in the table at the end of section \[parabolics\]. From that table the reader can see that we can take the first generator associated to each stabiliser subgroup as the translation corresponding to an $S^1$-fibre. That is, we take the transformations $c$, $a$, $k$, $j$ and $e^{-1}g$. We are going to fill in each of these five $S^1$-fibres by gluing in the manifold $\widetilde{\textbf{G}}$, on the level of the above Kirby diagrams we need to show where the added 2-handles go.
In order to understand where these added 2-handles will go in our Kirby diagram let us go back and see how we obtained these parabolic translations. The idea was to take each ideal vertex then take a horospherical neighbourhood about each of these vertices, and then by applying various isometries we could work out the parabolic translations. For example when we took the ideal vertex $\{(1,0,0,0)\}$ we found that its equivalence class consisted of two points $\{(1,0,0,0), (-1,0,0,0)\}$, and hence a fundamental domain for the parabolic subgroup corresponding to this class were two cubes, one centred at the ideal vertex $(1,0,0,0)$ and one centred at $(-1,0,0,0)$. The cube around the ideal vertex $(1,0,0,0)$ took the form:
{width="5cm" height="4cm"}
We then found that the isometry $c$ was a parabolic translation in the parabolic subgroup corresponding to the class $\{(1,0,0,0), (-1,0,0,0)\}$. When we formed the handle decomposition of the 24-cell what we were really doing was taking a dual cell structure. This means that if we take the dual of the above box we will be getting that part of the fundamental domain in the handle decomposition of the 24-cell. As the dual of a cube is an octahedron, taking the dual of the above cube gives:
{width="6cm" height="6cm"}
Filling in the $S^1$-fibre of the associated boundary component involves attaching a 2-handle from $C$ to $C'$. In our Kirby diagram this involves drawing an attaching circle between $C$ and $C'$ in our octahedron. Note that the $S^1$-fibre is identified in the fundamental domain by a straight line joining side $C$ to side $C'$ and running inside the rectangular box making up the fundamental domain. Therefore, in our Kirby diagram the added 2-handle, representing a filling of the $S^1$-fibre corresponding to the translation $c$, will be a straight line running from $C$ to $C'$ and contained in the dual octahedron.
{width="6cm" height="6cm"}
We can also see how this looks in part of our ambient handle decomposition diagram. Recall, the coordinates of $C$, $C'$, $A$, $B$, $G$ and $H$ are:
----- ------------------- ----------------------------------------------- ------ ------------------- ------------------------------------------------
$A$ $S_{(+1,+1,0,0)}$ $(\frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}}, 0)$ $B$ $S_{(+1,-1,0,0)}$ $(\frac{1}{\sqrt{2}} ,\frac{-1}{\sqrt{2}}, 0)$
$C$ $S_{(+1,0,+1,0)}$ $(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$ $C'$ $S_{(+1,0,-1,0)}$ $(\frac{1}{\sqrt{2}}, 0, \frac{-1}{\sqrt{2}})$
$G$ $S_{(+1,0,0,+1)}$ $(1 + \sqrt{2}, 0, 0)$ $H$ $S_{(+1,0,0,-1)}$ $(-1 + \sqrt{2}, 0, 0)$
----- ------------------- ----------------------------------------------- ------ ------------------- ------------------------------------------------
The following shows a picture of this octahedron (in purple) in part of the ambient handle decomposition diagram.
{width="9cm" height="6cm"}
The filling of $C-C'$ can be seen in the following picture, with the attaching circle of the 2-handle we are using to do the filling being shown in green. You can clearly see that it lies in the $x-z$ plane.
{width="9cm" height="6cm"}
We can do the same for all the other fillings we are going to carry out. The reader should keep in mind that each attaching circle of an added 2-handle, corresponding to a filling, comes from a straight line joining two sides of the rectangular box making up the fundamental domain of the boundary component we are filling.
If we consider the ideal vertex $\{(0,1,0,0)\}$ we saw that its equivalence class consisted of the two vertices $\{(0,1,0,0), (0,-1,0,0)\}$. The horosphere about the vertex $\{(0,1,0,0)\}$ looks like:
{width="5cm" height="4cm"}
Taking the dual gives the following octahedron:
{width="6cm" height="6cm"}
Filling in the $S^1$ fibre of the associated boundary component involves attaching a 2-handle from $A$ to $A'$ in the following way.
{width="6cm" height="6cm"}
A picture of this dual octahedron in part of the three dimensional handle decomposition diagram is shown in the following picture, with the octahedron given by the purple edges.
{width="6cm" height="6cm"}
The filling of $A-A'$ can be seen in the following picture, with the added 2-handle in green. One can clearly see that it lies in the $x-y$ plane, and does not interfere with the other 2-handles.
{width="6cm" height="6cm"}
For the ideal vertex $(0,0,1,0)$ we found that the equivalence class was $\{(0,0,1,0), (0,0,-1,0)\}$, the horosphere about the vertex $(0,0,1,0)$ looks like:
{width="5cm" height="4cm"}
The dual octahedron then takes the form:
{width="6cm" height="6cm"}
Filling in the $S^1$ fibre of the associated boundary component involves attaching a 2-handle from $K$ to $K'$, and in this case we take the following attaching circle.
{width="6cm" height="6cm"}
A picture of this dual octahedron in part of the handle decomposition is shown in the following picture, with the octahedron given by the purple edges
{width="9cm" height="7cm"}
The attaching circle of the 2-handle we are attaching is shown in the following picture, it is clear that this 2-handle lies in the $x-z$ and $y-z$ planes, and does not interfere with any of the other 2-handles.
{width="9cm" height="8cm"}
For the vertex class $\{(0,0,0,1), (0,0,0,-1)\}$ we found that the isometry $j$ was a parabolic translation. In this case we need to look at the horosphere about the ideal vertex $(0,0,0,-1)$.
{width="5cm" height="3.5cm"}
The dual octahedron is then given by
{width="6cm" height="6cm"}
and the added 2-handle can be seen in the following picture.
{width="6cm" height="6cm"}
A picture of this dual octahedron in part of the handle decomposition is shown in the following picture, with the octahedron given by the purple edges.
{width="7cm" height="6cm"}
The attaching circle of the 2-handle we are attaching is shown in the following picture, it is clear that this 2-handle lies in the $x-y$ and $y-z$ planes, and does not interfere with any of the other 2-handles.
{width="8cm" height="7cm"}
The final vertex class to consider consists of the ideal vertices $\{(\pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2)\}$. In this case we found that the isometry $e^{-1}g$ corresponded to a parabolic translation. Therefore, when we fill the fibre corresponding to this isometry we need to add a 2-handle component from $E$ to $G$ and one from $E'$ to $G'$. In this case we need to consider two horospheres, the one about the vertex $(1/2, 1/2, 1/2, 1/2)$ and the one about the vertex $(-1/2, -1/2, -1/2, -1/2)$. Both these horosphere are shown in the following picture, the one on the left corresponding to $(1/2, 1/2, 1/2, 1/2)$.
{width="5cm" height="3.5cm"}
The dual octahedrons are then given by:
{width="6cm" height="5cm"}
The following shows a three dimensional picture of the first octahedron above.
{width="7cm" height="10cm"}
The 2-handle that we are going to add has one component running from $E$ to $G$, and can be seen in the following diagram. The reader should note that we have drawn the added component as a curved arc when it really should be a straight line running from $E$ to $G$. The reason for drawing it as a curved arc is simply because it is easier to view in the diagram, the reader should really picture this as a straight line.
{width="8cm" height="10cm"}
Observe that the added component falls outside of the diagrams showing the $x-y$, $x-z$, $y-z$ planes and the diagrams showing the six 2-handles that do not all lie in a single 2-plane. Due to this, any handle cancellation/slides we do which take place in the $x-y$, $x-z$ and $y-z$ planes or the diagram corresponding to the six 2-handles that do not all lie in a single 2-plane, will not interfere with this added 2-handle component running from $E$ to $G$.
The following shows a three dimensional picture of the second octahedron above
{width="6cm" height="7cm"}
The 2-handle component that we are adding runs from $E'$ to $G'$ and can be seen in the following diagram:
{width="6cm" height="7cm"}
In this case the added 2-handle runs inside the diagram corresponding to the six 2-handles that do not all lie in a single 2-plane. However, it does not lie in any of the planes $x-y$, $x-z$ or $y-z$. Furthermore, all the handle cancellations/slides we do to begin with will not interfere with this 2-handle.
The following picture shows the Kirby diagrams with the added attaching circle of the added $2$-handles that pass over $A-A'$, $J-J'$, $K-K'$ and $C-C'$, they are the black dashed lines. We note that these 2-handles pass over the 1-handles in question once. Also, the added 2-handle running between $C-C'$ in the $x-z$ plane is drawn as a curved arc, as opposed to a straight line, purely for ease of viewing.
{width="11cm" height="10cm"}
The added 2-handle that runs from $E$ to $G$ and $E'$ to $G'$ are shown in the following picture.
{width="8cm" height="9cm"}
The above shows the Kirby diagram associated to the smooth closed 4-manifold that is obtained from filling in the boundary components of $M$. In sect.5 of [@sarat] we explained how the 2-handles associated to the Kirby diagram of $M$ had what we called a planar framing. This had the feature that whenever we took a parallel curve to such a 2-handle it would never cross over the 2-handle. We want to briefly show that the same feature is possessed by the added 2-handles making up the closed boundary filling of $M$.
We need to start by outlining exactly how the attaching regions of the 1-handles are being identified. We gave explicit computations of the identifying diffeomorphisms in sect.5 of [@sarat]. The following table shows what the identifying diffeomorphism for the attaching regions of the 1-handles are.
------------------ ----------------------------------------------------------------------------------------------------
1-handle Identifying diffeomorphism
$A,A'$ & $B, B'$ $(x,y,z) \mapsto (-x,y,z)$
$C,C'$ & $D, D'$ $(x,y,z) \mapsto (x,y,-z)$
$E,E'$ & $F, F'$ $(x,y,z) \mapsto \bigg(-\frac{x}{x^2+y^2+z^2},-\frac{y}{x^2+y^2+z^2},-\frac{z}{x^2+y^2+z^2}\bigg)$
$G,G'$ & $H, H'$ $(x,y,z) \mapsto \bigg(-\frac{x}{x^2+y^2+z^2},-\frac{y}{x^2+y^2+z^2},-\frac{z}{x^2+y^2+z^2}\bigg)$
$I,I'$ & $J, J'$ $(x,y,z) \mapsto (x,-y,z)$
$K,K'$ & $L, L'$ $(x,y,z) \mapsto \bigg(\frac{x}{x^2+y^2+z^2},\frac{y}{x^2+y^2+z^2},\frac{z}{x^2+y^2+z^2}\bigg)$
------------------ ----------------------------------------------------------------------------------------------------
\
The first four added 2-handles all lie in at least one single 2-plane, it is easy to check that a parallel curve to such a 2-handle cannot cross over the 2-handle in any way, and lies in the same plane. We will show this for the added 2-handle that runs between $A, A'$. This added 2-handle lies in the $x-y$ plane, we take a parallel curve just above it that is going into $A$, and that also lies in the $x-y$ plane. It is shown as the orange curve going into $A$ in the following diagram.
{width="7cm" height="7cm"}
The attaching map for $A, A'$ is given by the reflection $(x,y,z) \mapsto (-x,y,z)$, it is therefore clear that when the parallel curve goes into $A$ it comes out of $A'$ also above the added 2-handle. The following diagram shows what the parallel curve looks like.
{width="7cm" height="7cm"}
We see that a parallel curve to the added 2-handle running between $A,A'$ does not at any point cross over the added 2-handle. A similar argument for the added 2-handles running between $C,C'$, $J,J'$, $K,K'$ shows that these added 2-handles have parallel curves behaving in a similar way.
Let us show that the added 2-handle running from $E$ to $G$ and from $E'$ to $G'$ has parallel curve exhibiting a similar behaviour. We start by taking a parallel curve component to the piece going into $E$, we choose the parallel curve component to lie just above this added 2-handle piece. The following picture shows the added 2-handle (corresponding to the boundary filling) in green, and the parallel curve component in orange. Remember that we are drawing the added 2-handle component, in green, as a curved arc purely for ease of viewing. The reader should really think of this as a straight line running from $E$ to $G$. Due to this we have had to draw the parallel curve as a curved arc, again the reader should think of this as a straight line running into $E$ and that is parallel to what should be a straight line running between $E$ and $G$.
{width="8cm" height="7cm"}
The attaching map for $E, E'$ was given by the composition of the inversion in $S^2$ followed by the antipodal map. A rough computation shows that the orange parallel curve comes out of $E'$ slightly above the added green 2-handle (we suggest the reader to look at sect.5 in [@sarat] for an example of this type of computation).
{width="7cm" height="5cm"}
The attaching map for $G, G'$ is the same as the one for $E, E'$, and a little rough analysis shows that when the parallel curve comes out of $G'$ it does so just above the added green 2-handle.
{width="10cm" height="8cm"}
We thus see that the parallel curve never crosses over the added 2-handle, and in this regard has a planar framing. The importance of this observation is that the closed filled in manifold, obtained from filling in the boundary of $M$, has a Kirby diagram with every 2-handle having a planar framing. In turn this allows us to carry out handle slides in a very straight forward manner. In particular, for those 2-handles that reside in the diagram corresponding to the six 2-handles that do not all lie in a single plane the handle slides can be carried out as if these 2-handles were all lying in a single 2-plane.
There is one technicality to do with the **second elementary move** we described in section \[elementary\]. This was the handle slide that involved pushing a component of 2-handle through one attaching sphere of a 1-handle so that it came out of the second attaching sphere. When we described this handle slide, we showed how it took place in the case that the 2-handles were all residing in a single 2-plane. In this situation it was straightforward to understand how the 2-handle component behaved as it went through one attaching sphere and then out the other. We also want to carry out such handle slides in the diagram corresponding to the six 2-handles that do not all lie in a single 2-plane. In this case one has to be a bit careful, the reason being that if the attaching map identifying the attaching spheres is some “wild” diffeomorphism then it may well be the case that when we push a component of 2-handle through one attaching sphere it could come out the other in a very non-trivial way. The point is that for all the Kirby diagrams we consider this will never be a problem as the attaching maps between attaching spheres are very straightforward. Let us explain this in a bit more detail.
Suppose we have a 1-handle $S, S'$ and we have a component of a 2-handle that goes in to $S$, as shown in the following diagram.
{width="4cm" height="2cm"}
We can then push the blue 2-handle component through $S$ so that it comes out of $S'$. What we are really doing when we carry out such a move is we are isotoping the blue curve onto the attaching sphere $S$, producing a curve on $S$. For example, the following diagram shows one such isotopy in which the blue 2-handle component has been moved to give a curve on the attaching sphere $S$.
{width="4cm" height="2cm"}
Now, we have an attaching map from $S$ to $S'$, which is some diffeomorphism identifying the spheres. Therefore, the blue curve on $S$ will map to some image curve on $S'$, once we understand how this image curve on $S'$ looks we can then understand what the blue 2-handle component looks like after we have pushed it through $S$. It is at this point that one must be very careful, for the attaching map identifying $S$ to $S'$ could be some “wild” diffeomorphism, which in turn may map the blue curve on $S$ to some curve on $S'$ that winds around $S'$ several times. If there are other bits of 2-handle coming out of $S'$, then this image curve could wrap around some of these other bits of 2-handle, and this is precisely why it is important to know what the attaching map between the attaching spheres is. In our case all attaching maps are reflections or compositions of reflections with inversion in $S^2$, therefore it is a straightforward process to work out what a 2-handle component looks like when we push it through a 1-handle.
We give an explicit example of this sort of computation. Consider the diagram consisting of the six 2-handles that did not all lie in a single plane together with the added 2-handle running from $A$ to $A'$, corresponding to a filling of a boundary component.
{width="6cm" height="6cm"}
When we use the added 2-handle running between $A$ and $A'$ (black dashed line) to cancel $A,A'$ the green 2-handle component moves in to the following position.
{width="6cm" height="5cm"}
We can then slide this green 2-handle component through $E$ so that it comes out of $E'$. To start with this involves isotoping the green curve looping back into $E$ to the following curve on $E$.
{width="6cm" height="6cm"}
We have shown a close up of the curve on $E$ and how it looks in the whole diagram. The curve sits in a plane parallel to the x-z plane
If we want to push it through so that it comes out of $E'$ we need to understand what the attaching map for $E, E'$ is. Observe that there are other 2-handles that are hitting $E'$, we have a pink and black 2-handle component in the above diagram, but $E'$ also sits in the y-z plane, and there are blue, black and pink 2-handle components in that plane meeting $E'$ (see diagram showing 2-handles in the y-z plane). Therefore it is possible that when we send the green curve on $E$, using the attaching map of $E, E'$, we could end up with an image curve on $E'$ that winds around some of these other 2-handle components meeting $E'$.
In this situation we know exactly how $E$ is identified to $E'$, we showed that the identifying map was given by $$(x,y,z) \mapsto \bigg(-\frac{x}{x^2+y^2+z^2},-\frac{y}{x^2+y^2+z^2},-\frac{z}{x^2+y^2+z^2}\bigg)$$ which is the composition of the inversion in $S^2$ followed by the antipodal map. Recall that the attaching sphere corresponding to $E$ is a sphere (of some small arbitrary radius) centred at the point $(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$. The green curve on $E$ lies inside $S^2$:
{width="3cm" height="3cm"}
When we apply the inversion map it will map to a curve outside of $S^2$. The following diagram shows what it looks like when we apply the inversion map.
{width="4cm" height="4cm"}
We then apply the antipodal map to get the following curve on $E'$.
{width="2cm" height="2cm"}
In the whole diagram we then see that the image of the green curve, which was residing on $E$, under the attaching map from $E$ to $E'$ looks like:
{width="6cm" height="6cm"}
Thus, what we are able to conclude is that when we push the original green curve through $E$ it comes out of $E'$ in straightforward manner, and does not “wind around” any of the other 2-handle components meeting $E'$. We can then proceed to push the green 2-handle component off $E'$ to obtain a green 2-handle component passing over $B, B'$ once and that does not interfere with any of the other 2-handles
{width="6cm" height="6cm"}
In general we will carry out several moves where we push a 2-handle component through the attaching sphere of a 1-handle. As all our attaching maps are reflections or reflections composed with inversion in $S^2$, we will find that we will never be in a situation where the curve on one attaching sphere is pushed to a curve in the image attaching sphere in such a way that it winds around other bits of 2-handle.
Before we proceed to doing some handle cancellations we remark that some of these added 2-handles will intersect some of the other planes. For example consider the added 2-handle that runs between $C-C'$, as the co-ordinate of $C = (\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$ and $C' = (\frac{1}{\sqrt{2}}, 0, \frac{-1}{\sqrt{2}})$, we can see that in the bottom right picture above, this 2-handle must intersect the $x-y$ plane. We have drawn this intersection point as a black dot in the top left picture below. Similarly the added 2-handle that runs over $A-A'$ intersects the $y-z$ plane in a point, and this has been drawn as a black dot in the bottom left picture below. Finally, the 2-handle passing over $J-J'$ intersects the $x-z$ plane in a point, and this can be seen as a black dot in the top right picture below.
{width="12cm" height="13cm"}
When we start cancelling handles and doing handle slides we have to be careful that we do not cut through any of the other 2-handles for then we would be doing an “illegal” move, and that if one of our 2-handles goes around another, then a handle slide on one may cause the other to change position and we have to keep track of this. It turns out that none of this really creates any problems as all our moves are primarily done in the planes in question, and it can be clearly seen that they do not tangle around any of the added 2-handles that intersect these planes.
The above discussion may seem confusing so we go through a few handle cancellations in detail to make our point.
The added 2-handles corresponding to boundary fillings pass over a 1-handle once and hence form a handle cancellation pair, therefore we may cancel them from our diagram. We recall that if we have a handle cancelling pair that does not meet any other 2-handles in a Kirby diagram then we can simply delete them from the diagram. However, if there are some 2-handles that pass over the 1-handle that we are cancelling, we must push those 2-handles through the 2-handle that is cancelling the 1-handle.
We start by cancelling $A-A'$ (the reader should compare this with the first elementary move we showed in the previous section) using the added 2-handle that passes over it once. This only affects the 2-handles in the $x-y$ plane and the six extra 2-handles that do not lie in any one plane. The following pictures show the Kirby diagrams of these handles once we have done the cancellation.
{width="10cm" height="10cm"}
Recall that this 2-handle that we just used to cancel $A-A'$ intersected the $y-z$ plane, hence when we use it to cancel $A-A'$ we will introduce new intersection points in the $y-z$ plane, and it is important to keep track of these new intersection points.
We start by giving an analysis of what happens when we cancel $A-A'$ from the $x-y$ plane. First of all, the cancellation from the $x-y$ plane introduces some new 2-handles in the $x-y$ plane. We have the blue 2-handle that starts at $I$ and loops back into $I$. This creates an intersection point in the $y-z$ plane close to $I$, you can see it as the black dot in the following diagram:
{width="6cm" height="5cm"}
We also have the new brown 2-handle in the $x-y$ plane that goes between $G$ and $H'$, this creates a second intersection point with the $y-z$ plane:
{width="6cm" height="6cm"}
Remaining in the $x-y$ plane we also have the new green 2-handle that passes between $G'$ and $H$, this introduces a third intersection point in the $y-z$ plane.
{width="6cm" height="6cm"}
Finally, we have the red 2-handle that starts at $J$ and loops back into it, this gives a fourth intersection point.
{width="7cm" height="6.5cm"}
In total we obtain four new intersection points from cancelling $A-A'$ from the $x-y$ plane. However we are not done yet, there will also be some intersection points arising from the cancellation of $A-A'$ from the diagram containing the six 2-handles that do not all lie in any 2-plane.
The analysis of this situation is exactly analogous to what we did above. We start with the green 2-handle that starts at $E$ and loops back into it. This gives an intersection point in the $y-z$ plane near $E$ as shown in the following diagram.
{width="7cm" height="7cm"}
The brown 2-handle that runs between $C$ and $D$ gives an intersection point that looks like:
{width="7cm" height="7cm"}
There is also the brown 2-handle that runs between $C'$ and $D'$, this gives a point of intersection that looks like:
{width="7cm" height="7cm"}
Finally, we have the 2-handle that starts at $F$ and loops back into it, it gives an intersection point near $F$:
{width="7cm" height="7cm"}
In total we get 8 new intersection points, to make things easier we want to keep track of how each intersection came about. Namely, we want to keep track of which 2-handle in which diagram gave us a particular intersection point. To do this we will introduce a simple coding system that will allow us to keep track of where these points of intersection are coming from. Each intersection point will be labelled with either four letters or two letters. In the case of four letters, the first two will tell us that the intersection point is coming from the $x-y$, $x-z$ or $y-z$ planes, and the next two letters will tell us the 1-handles that the 2-handle causing the intersection point is passing between, the actual 2-handle will be clear from context. In the case of a two letter code, we are to interpret that the 2-handle causing the intersection point is coming from the diagram representing the six 2-handles that do not all lie in a single 2-plane. The two letters denote the two 1-handles that this 2-handle passes over, again the actual 2-handle in question will be clear from the context.
At times it will be difficult for us to show the labels of all intersection points in the whole diagram simply because there may be too many intersection points and not enough space. In these instances we will always show two diagrams, a whole diagram with unlabelled points of intersection and a close up diagram of the labelled intersection points. It will be easy to tell which point corresponds to which point in the two diagrams.
It is time to show an explicit example of what exactly we mean by this code, and how exactly it looks. The following picture shows the $y-z$ plane with a black dot near $I$ labelled **XYII**, the first two letters being XY tell us that the 2-handle giving this point of intersection is coming from the $x-y$ plane, the second two letters are $II$ and this tells us that the 2-handle in the $x-y$ plane is starting at $I$ and looping back into $I$.
{width="6cm" height="5.5cm"}
This is precisely the point of intersection we talked about above which arose through cancelling $A-A'$ in the $x-y$ plane, and we showed a picture of it previously. There were also many other intersection points (remember there were 8 in total!), as another example the following picture shows a labelled intersection point coming from cancelling $A-A'$ in the diagram corresponding to the six 2-handles that do not all lie in a single 2-plane. The labelling consists of just two letters, which immediately tells us that the 2-handle contributing this point of intersection is coming from the diagram corresponding to the six 2-handles that do not all lie in a single 2-plane. The two letters are CD, which tell us that the 2-handle in question is running between the 1-handle $C$ and the 1-handle $D$.
{width="6cm" height="5.5cm"}
When we cancelled $A-A'$ we saw that there were a total of 8 new points of intersection. The following shows a close up of all these points of intersection with their corresponding labels.
{width="7.5cm" height="6cm"}
So far we have done one handle cancellation, and in some detail explained how we get extra points of intersection. The following picture shows what we have done.
{width="9cm" height="10cm"}
In the top left diagram we have a 2-handle, corresponding to (the one in blue)
$\xymatrix{
A \cap I \ar[r]^a & A'\cap I\ar[r]^i & B'\cap i' \ar[r]^{b^{-1}} & B\cap I' \ar[r]^{i^{-1}} & A\cap I }$
which has a component that loops back into $I$. In the above picture this is the blue arc at the top meeting $I$ twice. We can then slide it over the 1-handle $I-I'$ to give a 2-handle that meets $B-B'$ once.
There is also a second handle slide we can do in the $x-y$ plane. Namely, we have the 2-handle in red that starts at $J$ and loops back into it. We can then slide it over the 1-handle $J-J'$ to give a 2-handle in red that also meets $B-B'$ once.
{width="6cm" height="5.5cm"}
In carrying out these two handle slides the intersection points labelled **XYII** and **XYJJ** in the $y-z$ plane will disappear. However, in the $x-y$ plane the 2-handle (in blue) that meets $B-B'$ once will give a point of intersection in the $y-z$ plane in the region bounded by the 1-handles $I'$, $F'$, $J'$ and $E'$ and the 2-handles that run between them, and the red 2-handle that meets $B-B'$ will also give an intersection point in the $y-z$ plane in the region bounded by the 1-handles $I'$, $F'$, $J'$, $E'$ and the 2-handles that run between them.
The picture below shows these new intersection points along with the other intersection points that we have encountered so far. The two diagrams at the bottom of the picture show close ups of the regions that these intersection points lie in and the labels of the intersection points.
{width="8cm" height="9cm"}
We can also carry out two handle slides in the diagram consisting of the 2-handles that do not all lie in a single 2-plane. We can slide the green 2-handle that starts and ends at $E$ through $E$ and then off $E'$ to obtain a 2-handle that meets $B-B'$ once. Similarly we can slide the blue 2-handle that starts and ends at $F$ through $F$ and then off $F'$ to give another 2-handle that meets $B-B'$ once.
{width="6cm" height="5.5cm"}
These handle slides that we have carried out have not interfered with the added 2-handles running from $E$ to $G$ and $E'$ to $G'$. In the case of the added 2-handle that runs between $E$ and $G$ this is easy to see as it lies outside of the diagrams for which these handle cancellations and slides are being done. In the case of the 2-handle that runs between $E'$ and $G'$ it is also easy to see, using some three dimensional insight, that none of the cancellations and slides we have done so far affect the handle. The following picture shows how this 2-handle sits after we have carried out the above cancellations and slides, it is drawn in yellow. The picture is supposed to give the reader some insight in to why it is the case that carrying out the above cancellations and slides does not affect the added 2-handle that runs between $E'$ and $G'$.
{width="6cm" height="5cm"}
The two handle slides just carried out above will cause the intersection points in the $y-z$ plane labelled **EE** and **FF** to disappear, but the two new 2-handles between $B-B'$, that arose through these handle slides, will give two new points of intersection in the $y-z$ plane in the region bounded by $I'$, $F'$, $J'$, $E'$ and the 2-handles that run between them.
{width="8cm" height="9cm"}
The following picture shows the end result of cancelling $A-A'$, along with the various handle slides we under took after this cancellation
{width="8cm" height="10cm"}
and the picture below shows how the added 2-handles running between $E$ and $G$, and $E'$ and $G'$ sit in our Kirby diagram so far.
{width="09cm" height="10cm"}
We now cancel $K-K'$, in this case only the 2-handles in the $x-z$ and $y-z$ planes change.
The following picture shows how the Kirby diagrams change after we have carried out this cancellation.
{width="8cm" height="8cm"}
At this point one can ask if cancelling $K-K'$ has introduced any new intersection points in any of the other diagrams ? The only candidate is the $x-y$ plane. In this case it is easy to see that no intersection points are created. Let us give a brief explanation of why this is the case. Start with the $x-z$ plane, this meets the $x-y$ plane along a straight line that passes through the one handles $H'$, $G'$, $H$ and $G$. In the following picture you can see this line as the horizontal yellow line through $H'$, $G'$, $H$ and $G$, where the top diagram is that of the $x-y$ plane, and the bottom diagram is that of the $x-z$.
{width="8cm" height="9cm"}
When we cancel $K-K'$ from the $x-z$ plane it is easy to see that none of the cancellations interfere with the yellow line, meaning that in cancelling $K-K'$ from the $x-z$ plane none of the 2-handles that emerge from this cancellation cross the yellow line. This means none of these 2-handles contribute to any intersection points in the $x-y$ plane.
In the case of the $y-z$ plane we have that it intersects the $x-y$ plane along a vertical line through $I'$, $J'$, $J$ and $I$, as shown in the following picture.
{width="8cm" height="8cm"}
In this case you can also clearly see that when we cancel $K-K'$ from the $y-z$ plane none of the new 2-handles that emerge from this cancellation cross the horizontal yellow line in the $y-z$ plane. This means none of these 2-handles intersect the $x-y$ plane, and hence we do not get any points of intersection from cancelling $K-K'$ from the $y-z$ plane.
Coming back to the pictures of the $x-z$ and $y-z$ planes, after cancelling $K-K'$ we see that we can do some handle slides. First of all, in the $x-z$ plane we have a green 2-handle that starts at $D$ and loops back into it, and a pink 2-handle that starts at $C$ and loops back into it. We can slide the green 2-handle through $D$ to $D'$ and then off $D'$ to give a 2-handle meeting $L-L'$ once. We can do the same with the pink 2-handle to get another 2-handle meeting $L-L'$ once.
In the case of the $y-z$ plane we have the red 2-handle that starts at $F'$ and loops back into it, and we have the blue 2-handle that starts at $E$ and loops back into it. We can then slide these (just as we did above) to give two 2-handles that meet $L-L'$ once.
{width="9cm" height="8cm"}
Observe that in carrying out these handle slides we have not added any new points of intersection to the $x-y$ plane. For example the two handle slides we did in the $x-z$ plane gave us two 2-handles that met $L-L'$ once. These two 2-handles do not cross the intersection line of the $x-y$ plane with the $x-z$ plane, hence these new 2-handles cannot intersect the $x-y$ plane. A similar analysis for the handle slides in the $y-z$ plane shows that they do not add any points of intersection as well.
It is clear that in cancelling the 1-handle $K-K'$ we have introduced no new intersection points in the other diagrams. For example when we cancelled $K-K'$ in the $x-z$ plane, we introduced a 2-handle that loops back in to $C$, it is the pink 2-handle in the first cancellation diagram. This 2-handle does not introduce any intersection points in the $x-y$ plane because $C$ lies above the $x-y$ plane. Similarly we also introduced a 2-handle that goes from $H$ to $G$, it is drawn as the red 2-handle in the picture above. This 2-handle also does not create any points of intersection with the $x-y$ plane. A similar analysis shows that when we first cancel $K-K'$ none of the new 2-handles formed create any new intersection points.
We then proceeded to doing some handle slides, and we can ask whether these handle slides create any new intersection points. It turns out that they do not, and one can easily see this by simply thinking of what happens in a 3-dimensional picture. Let us give an explicit analysis of this, we will work with the $y-z$ plane and the 2-handle that loops back in to $E$, it is drawn in blue in the first picture showing what happens when we cancel $K-K'$ (see the picture before the three above pictures). We push this 2-handle through $E-E'$ to get a 2-handle meeting $L-L'$, note that $E'$, $L'$ and $L$ all lie below the $x-y$ plane. Hence when we push this 2-handle through $E-E'$ to give a 2-handle between $L-L'$ we create no intersection points with the $x-y$ plane. Similarly, this new 2-handle does not create any intersection points with the $x-z$ plane.
So far we have filled in two boundary components of the manifold $M$ corresponding to the fibres given by the isometries $a$ and $k$. We then carried out various handle cancellations and handle slides. The following picture puts all that we have done so far together.
{width="9cm" height="10cm"}
The labelling of the intersection points are shown in the following diagram. The top two diagrams show close ups of the regions around the intersection points in the $x-y$ and $x-z$ planes respectively (viewing from left to right). The bottom two diagrams show close ups of the intersection points in the $y-z$ plane.
{width="9cm" height="8cm"}
The next step is to cancel $J-J'$, this cancellation will only affect the $x-y$ and $y-z$ planes. The diagrams on the right correspond to sliding the obvious handles in the left picture over 1-handles and then off them to obtain 2-handles that only meet certain 1-handles once. For example in cancelling $J-J'$ from the $x-y$ plane we obtain a pink 2-handle that starts at $G'$ and loops back into it (see the top left diagram in the picture below), we can then slide this through $G'$ and off $G$ to obtain a pink 2-handle that meets $I-I'$ once (see the top right diagram in the picture below).
{width="11cm" height="10cm"}
When we first cancel $J-J'$ in the $x-y$ plane we introduce some new 2-handles, namely the one in pink that loops back into $G'$ and the one in black that does the same with $H$. These two 2-handles create two new points of intersection with the $x-z$ plane:
{width="7cm" height="5cm"}
When we cancel $J-J'$ in the $y-z$ plane we introduce a 2-handle between $I-I'$, $E-F'$, $E'-F$, and one that loops back into $L'$. These four 2-handles each create intersection points with $x-z$ plane as well, you can see them as the four vertical black dots:
{width="7cm" height="5cm"}
The labelling of these 6 new intersection points is given in the following picture.
{width="5.5cm" height="5.5cm"}
After cancelling $J-J'$ we carried out some some handles slides. In the $x-y$ plane we slid the new 2-handle that starts at $G'$ and loops back into it (the one in pink) through $G'$ and then off $G$ to give a 2-handle that passes over $I-I'$ once. This will cause the intersection point in the $x-z$ plane, labelled **XYG’G’**, to disappear. However, a new intersection point will arise corresponding to the new 2-handle that passes over $I-I'$ once. This new intersection point is shown in the following diagram as the black dot to the right of $G$, you can also see that the intersection point labelled **XYG’G’** is no longer in the picture.
{width="5.5cm" height="5.5cm"}
The other handle slide we did in the $x-y$ plane was to take the black 2-handle that starts at $H$ and loops back into it and slide it through $H$ and then off $H'$ to give a new 2-handle that passes over $I-I'$ once. This handle slide causes the intersection point labelled **XYHH** to disappear with a new intersection point appearing to the left of $H'$ (viewed in the $x-z$ plane).
{width="6cm" height="4cm"}
We also carried out one handle slide in the $y-z$ plane. This handle slide corresponded to taking the green 2-handle that starts at $L'$ and loops back into it, pushing it through $L'$ so it comes out at $L$, and then sliding it off to give a new 2-handle that passes over $I-I'$ once. This causes the intersection point labelled **YZL’L’** to disappear, with a new intersection point just below $L$ to appear. The following picture shows this new intersection point and all the others with their corresponding label.
{width="6cm" height="4.5cm"}
If we put everything we have done so far we get the following:
{width="8cm" height="6.5cm"}
The pictures showing how the added 2-handle running between $E$ and $G$, and $E'$ and $G'$ is the same as before.
We move on to cancelling $C-C'$, this will only affect the $x-z$ plane and the six 2-handles not lying in any one plane.
{width="5cm" height="6cm"}
Recall that the added 2-handle that passed between $C-C'$ (the one we just used above to cancel $C-C'$) intersected the $x-y$ plane. Therefore the above handle cancellation will give rise to some points of intersection in the $x-y$ plane. We take the time to explain how these points of intersection look like.
Start with the $x-z$ plane, when we cancel $C-C'$ we got two new 2-handles one in blue which starts at $H$ and loops back into it, and one in brown that starts at $G$ and loops back into it. These will give two new intersection points in the $x-y$ plane, as shown below:
{width="6cm" height="6cm"}
Moving to the case of the six 2-handles that did not all lie in a single 2-plane, we see that we get four new points of intersection. First of all, we have the red 2-handle that starts and ends at $B$. Secondly, we have the black 2-handle that runs from $E'$ to $F'$ and the pink 2-handle that runs from $F$ to $E$. Finally, we have the brown 2-handle that runs from $D$ to $D'$. All four of these 2-handles give four points of intersection in the $x-y$ plane. The following picture show these points of intersections in the $x-y$ plane, with the bottom diagram being a close up showing the labelling of these intersection points.
{width="7cm" height="8cm"}
We also have to deal with the points of intersection in the $y-z$ plane. We have two points of intersection labelled **C’D’** and **CD**, when we cancel $C-C'$ from the diagram that consisted of the six 2-handles that did not all lie in a single 2-plane the part of the brown 2-handle component running from $C'$ to $D'$ and the part running from $C$ to $D$ join together to give a brown 2-handle running from $D$ to $D'$. Thus in the $y-z$ plane we will still see two points of intersection but their labelling will be **DD’** because this new brown 2-handle running from $D$ to $D'$ intersects the $y-z$ plane in two distinct points.
{width="6cm" height="7cm"}
We can then carry out three handle slides. In the $x-z$ plane we can slide the blue 2-handle that starts and ends at $H$ through $H$ and then off $H'$ to give a blue 2-handle between $D-D'$. Similarly, we can slide the red 2-handle that starts and ends at $G$ off $G'$ to give a red 2-handle that meets $D-D'$ once. Finally, in the diagram corresponding to the six 2-handles that did not all lie in a single 2-plane, we have the red 2-handle that starts and ends at $B$. We can slide it off $B'$ to give a red 2-handle meeting $D-D'$ once.
{width="9cm" height="9cm"}
The first two handle slides we undertook will cause the points of intersection labelled **XZHH** and **XZGG** to disappear from the $x-y$ plane, however we will get two new points of intersection coming from the new 2-handles that run between $D-D'$, these are shown in the picture below with labelling **XZDD’**. The second handle slide we undertook will cause the point of intersection labelled **BB** to disappear, but the new 2-handle that we obtained running from $D$ to $D'$ will give a new intersection point labelled **DD’**. These new points of intersection can be seen in the picture below.
{width="6cm" height="6cm"}
These handle slides do not affect the $y-z$ plane.
So far we have we have cancelled the 1-handles $A-A'$, $K-K'$, $J-J'$ and $C-C'$, the result of all these cancellations and various handle slides are shown in the picture below along with points of intersections that arise when we cancel/slide handles.
{width="11cm" height="9cm"}
It is clear that none of the handle cancellations and slides we have done so far have interfered with the added 2-handles running from $E$ to $G$ and $E'$ to $G'$. This will be the case for many of the handle cancellations and slides we do in the following, and because of this we will often omit drawing the 2-handles between $E$ to $G$ and $E'$ to $G'$. However, it is recommended that the reader keep a mental image of these two 2-handles so as to help convince themselves that none of the handle cancellations/slides we carry out do indeed affect these two 2-handles.
In the above picture you can see several 2-handles that pass over certain 1-handles only once. For example, if we look at the $x-y$ plane we can see that there is a red 2-handle and a blue 2-handle that passes between the 1-handle $B-B'$ once, hence we may use either of these to cancel $B-B'$.
Cancelling $B-B'$ only affects the handle diagram in the $x-y$ plane and the diagram corresponding to the six $2$-handles not lying in any one plane.
{width="10cm" height="7.2cm"}
The unknotted circles in the above picture denote 2-handles not meeting any 1-handles and have zero framing. Hence they can be used to cancel a 3-handle, and so we may simply erase them from the diagram (recall elementary move number 3, which we outlined in the previous section).
Recall that the 2-handles running between $B$ and $B'$ gave points of intersection in the $y-z$ plane. To remind the reader the picture below shows all the points of intersection in the $y-z$ plane.
{width="8cm" height="6cm"}
When we cancel $B-B'$ from the $x-y$ plane using the red 2-handle, the intersection point corresponding to this 2-handle in the $y-z$ plane will disappear. However, a new one corresponding to the green 2-handle running from $G$ to $H'$ will appear, and one corresponding to the brown 2-handle running from $G'$ to $H$ will appear. The blue 2-handle running between $B$ and $B'$ (still staying in the $x-y$ plane) also corresponds to an intersection point in the $y-z$ plane, when we cancel $B-B'$ this 2-handle forms a closed loop with framing 0. It then gives a point of intersection in the $y-z$ plane, however since the framing of this 2-handle is 0 it cancels a 3-handle, and so we can simply delete it from our diagram. This means that we can simply delete the corresponding point of intersection from the $y-z$ plane. Similarly, when we cancelled $B-B'$ from the diagram that consisted of the 2-handles that did not all lie in a single 2-plane we obtained two 2-handles, one in blue and the other in green which are loops with zero framing. Hence they each cancel a 3-handle respectively and can be deleted from the diagram. This means that the points of intersection they give in the $y-z$ plane can be deleted.
{width="6cm" height="6cm"}
Observe that there are two points of intersection labelled **XYGH’**, and another two labelled **XYG’H**. It is easy to work out which one corresponds to which 2-handle in the $x-y$ plane. For example, the point of intersection between $E'$, $I'$ and $F'$ labelled **XYGH’** corresponds to the green 2-handle running between $G$ and $H'$ in the $x-y$ plane, and the point of intersection labelled **XYGH’** between $E$, $I$ and $F$ corresponds to the brown 2-handle running between $G$ and $H'$ in the $x-y$ plane.
Before we show how all diagrams look like after the cancellation of $B-B'$, we want to go back to the diagram that corresponded to the six 2-handles that did not all lie in a single 2-plane.
{width="5cm" height="4.5cm"}
Look at the brown 2-handle that runs between $D$ and $D'$. We were going to do an isotopy that moves this 2-handle into a different position. When carrying out such an isotopy we have to be careful that we do not pass through any other 2-handles. Therefore in order to do this in a correct manner we need to keep track of the intersection points with the other planes.
We are going to push the brown 2-handle in the following direction:
{width="5cm" height="4.5cm"}
This will cause the brown 2-handle (over time) to move in the following way:
{width="8cm" height="7cm"}
During this moving of the brown 2-handle the intersection point corresponding to this 2-handle in the $x-y$ plane also moves, and the trajectory it takes looks like:
{width="6cm" height="5.5cm"}
Observe that so far, in moving the brown 2-handle we have not passed through any of the other 2-handles, this is because in the $x-y$ plane we have kept track of how the intersection point has moved. In the $x-z$ plane there is nothing to check as the brown 2-handle that we are moving does not intersect this plane. Finally, in the case of the $y-z$ plane we have that this 2-handle intersects it in two points, thus in carrying out this isotopy of the 2-handle in question there will definitely be some change in the $y-z$ plane. In order to understand what goes on observe that when we continue to move the 2-handle in the direction shown above we will come to a stage where a vertical arc of this 2-handle will lie in the $y-z$ plane. Thus when we look at the $y-z$ plane we will see an arc between the two points of intersection corresponding to this brown 2-handle. The following picture shows this arc in the $y-z$ plane.
{width="6cm" height="6cm"}
As we continue to move this 2-handle in the direction indicated we will see this arc in the $y-z$ plane disappear along with the original two points of intersection that this 2-handle gave.
The final position of this 2-handle is shown in the following picture, with the bottom diagram showing the final position of the intersection point in the $x-y$ plane.
{width="7cm" height="7cm"}
The following picture shows all diagrams so far, notice that two points of intersection in the $y-z$ plane have disappeared.
{width="10cm" height="10cm"}
We are going to move on to cancelling $D-D'$. In this case we have a few options in regard to which 2-handle we use to carry out the cancellation. In the $x-z$ plane we have the choice of the brown and blue 2-handles that pass between $D-D'$ once, and in the diagram that corresponded to the six 2-handles that did not all lie in a single 2-plane, we have the choice of the red and brown 2-handle that passes between $D-D'$ once. We will choose the red 2-handle in the diagram that corresponds to the six 2-handles that did not all lie in a single 2-plane.
{width="9cm" height="8.5cm"}
Observe that after cancelling $D-D'$ all the 2-handles that passed between $D-D'$ become loops with framing 0. We can see two in the $x-z$ plane, one in blue and the other in brown, and we can see one brown one in the other diagram. As all these two handles have framing 0 we know that they form a cancellation pair with a 3-handle, and hence we can erase them from our diagram. In doing so we will see the intersection points in the $x-y$ plane labelled **XZDD’** and **DD’** disappear, but we will get two new ones corresponding to the pink 2-handle running from $E'$ to $F'$ and the black 2-handle running from $E$ to $F$, both in the diagram that corresponds to the six 2-handles that did not all lie in a single 2-plane.
{width="6cm" height="5cm"}
All diagrams look like:
{width="8cm" height="8cm"}
Let us remark that the cancellations and slides we have done so far have not interfered in any way with the two added 2-handles that run from $E$ to $G$ and $E'$ to $G'$.
We move on to cancelling $L-L'$ using the green 2-handle in the $x-z$ plane. This cancellation only affects the diagrams in the $x-z$ and $y-z$ planes.
{width="6cm" height="7cm"}
In undertaking this cancellation we obtain three 2-handles that become loops which carry framing number 0, these will then cancel a 3-handle and hence we can simply erase them from our diagrams.
Note that as none of the 2-handles that ran between $L-L'$ intersected any of the other planes, the points of intersection in the various planes do not change. Furthermore it is easy to see that cancelling $L-L'$ does not cause the added 2-handles that run from $E$ to $G$ and $E'$ to $G'$ to change.
We can then cancel $I-I'$ using the red 2-handle that lies in the $y-z$ plane.
{width="8cm" height="6cm"}
We obtain three 2-handles that are loops with framing 0, which we can simply erase from the diagrams. As for the intersection points, the only plane we have to worry about is the $x-z$ plane. The intersection points labelled **XYII’** disappear and nothing new takes their place. The points of intersection labelled **YZII’** also disappear, but we get two new points of intersection at the top of the $y-z$ plane corresponding to the black 2-handle in the $y-z$ plane that runs between $E$ and $F'$ and the pink 2-handle in the $y-z$ plane that runs between $E'$ and $F$. The labelling of these points of intersection are **YZEF’** and **YZE’F** respectively.
The following picture shows the diagrams so far with labelled points of intersection. The first diagram represents the $x-y$ plane, the one below it is the $x-z$ plane, and the next two diagrams are the $y-z$ plane and the diagram that corresponded to the six 2-handles that did not all lie in a single plane.
{width="9cm" height="8cm"}
We can isotope the pink 2-handle in the $y-z$ plane joining $E'$ to $F$, so that it does not enclose the 1-handles $E$ and $F'$. The isotopy moves the 2-handle in the direction of the $x$-axis. The following picture shows the direction in which we move this 2-handle.
{width="6cm" height="4cm"}
It is easy to see that in moving this 2-handle in this direction, we do not pass through any of the other 2-handles, hence it is a well defined isotopy. The final position of this 2-handle can be seen in the following picture.
{width="5cm" height="4.5cm"}
We can also isotope the 2-handles in the diagram that corresponded to the six 2-handles that did not all lie in a single plane to get:
{width="5cm" height="4cm"}
The 2-handles now lie in planes parallel to the $x-z$ plane.
We observe that this will cause the points of intersection in the $x-y$ plane to move into the following position:
{width="4cm" height="4cm"}
We can now put everything together to give a picture in 3-space. Remember the co-ordinates of the 1-handles $H$, $H'$, $G$, $G'$ and $E$, $E'$, $F$, $F'$ are given as:\
----- ------------------- ------------------------------------------------ ------ ------------------- -------------------------------------------------
$E$ $S_{(0,+1,+1,0)}$ $(0, \frac{1}{\sqrt{2}} ,\frac{1}{\sqrt{2}})$ $E'$ $S_{(0,-1,-1,0)}$ $(0, \frac{-1}{\sqrt{2}} ,\frac{-1}{\sqrt{2}})$
$F$ $S_{(0,+1,-1,0)}$ $(0, \frac{1}{\sqrt{2}} ,\frac{-1}{\sqrt{2}})$ $F'$ $S_{(0,-1,+1,0)}$ $(0, \frac{-1}{\sqrt{2}} ,\frac{1}{\sqrt{2}})$
$G$ $S_{(+1,0,0,+1)}$ $(1 + \sqrt{2}, 0, 0)$ $G'$ $S_{(-1,0,0,-1)}$ $(1 - \sqrt{2}, 0, 0)$
$H$ $S_{(+1,0,0,-1)}$ $(-1 + \sqrt{2}, 0, 0)$ $H'$ $S_{(-1,0,0,+1)}$ $(-1 - \sqrt{2}, 0, 0)$
----- ------------------- ------------------------------------------------ ------ ------------------- -------------------------------------------------
\
The following picture shows a diagram of these 1-handles in 3-space along with the various 2-handles that run between them.
{width="6cm" height="6cm"}
Observe that in the top diagram the 2-handle denoted by $e$ passes over the 1-handle $H-H'$ once, hence they form a cancelling pair, and the cancellation can be done in the $x-y$ plane without affecting any of the other 2-handles.
{width="7cm" height="6cm"}
The 2-handles denoted by $f$ then cancels a 3-handle and can be erased from the diagram.
In our 3-space picture we then have:
{width="7cm" height="5cm"}
The 2-handle denoted $k$ runs over $F-F'$ once, and hence we can use it to cancel $F-F'$. This cancellation can be done in the $y-z$ plane without affecting the other 2-handles.
{width="7cm" height="5cm"}
The 2-handle $l$ slides off $E-E'$ and cancels a 3-handle, we are then left with:
{width="7cm" height="6cm"}
We can then slide the 2-handle denoted $j$ along $i$ to obtain a 2-handle that has one component looping back into $E$ and another into $E'$. We can push either component through the attaching sphere it loops back into obtaining an unknot with framing zero. This then cancels a 3-handle and can be deleted from the diagram.
Recall we still have not dealt with the 2-handle that corresponds to filling in the last boundary component. This is represented by $e^{-1}g$, which corresponds to attaching a 2-handle component from $E$ to $G$, and then one from $E'$ to $G'$. So far the handle cancellations and slides we have done have not affected these two components, and hence it suffices to put them in now.
{width="7cm" height="5cm"}
We can then use them to cancel $G-G'$:
{width="6cm" height="5cm"}
At this point one can perform a handle slide, one can slide the 2-handle labelled $b$ along $a$ the end result is that $b$ will have two components, one that loops back into $E$ and one that loops back into $E'$. We can then push one component through the attaching sphere it loops back into obtaining an unknot with zero framing. This cancels with a 3-handle and can be erased from the diagram. We are then left with the same diagram as above with only the 2-handles labelled $a$ and $i$ remaining.
The fundamental group of this manifold is clearly ${\ensuremath{\mathbb Z}}_2$. We want to take the double cover of the above 4-manifold to get a simply connected 4-manifold. Let us denote the above 4-manifold by $Y$, then we want to form the double cover $X \rightarrow Y$. More precisely using the Kirby diagram of $Y$ obtained above, we want to obtain a Kirby diagram of $X$. In order to do this we proceed as follows. First of all since $Y$ has only one 1-handle we can identity the 1-skeleton of $Y$ with $S^1 \times D^3$. As $X$ is a cyclic double cover we have that the 1-skeleton of $Y$ is 2-fold covered in the obvious way. I.e. the 1-skeleton of $X$ is $S^1 \times D^3$ and this double covers $S^1 \times D^3$ (corresponding to the 1-skeleton of $Y$) in the usual way. Each of the remaining handles of $Y$ lift to two copies of that handle in $X$. There is a subtle issue in what we have said so far, namely we did not say how the data determining the trivialisation of the normal bundle to each 2-handle lift to the double cover. In our case this is not a problem as all our 2-handles have a planar framing, and so when we lift them we still get 2-handles with a planar framing (i.e. parallel curves do not twist around). In the general case one only has to observe that a choice of trivialisation on each normal bundle to each knot (representing a 2-handle) lifts to that trivialisation on each lifted knot, and changing the trivialisation to the normal bundle of a 2-handle in $Y$ by one twist causes the trivialisation on each lifted knot to change by one twist, hence we know how to lift any choice of trivialisation. Focusing on our case we make an important observation, the 2-handles in our manifold $Y$ do not twist around each other in any way, hence when we lift them to $X$ we can focus on their lifts separately. In our situation we have two 2-handles denoted $a$ and $i$. When we lift each one, we will get two 2-handles each passing the lifted 1-handle once. As none of them twist around each other we use any lifted one to cancel the lifted 1-handle, leaving us with three unknot’s with framing zero. Each of these then cancels a 3-handle, and we see that the manifold we are left with is $S^4$. The boundary components of manifold 1011 where given by the non-orientable closed flat 3-manifold labelled $\textbf{G}$ ($\mathcal{B}_1$ in Wolf’s notation). Appealing to the classification of closed flat 3-manifolds we know that the orientable double cover of $\textbf{G}$ is the flat 3-manifold given by $\textbf{A}$ ($\mathcal{G}_1$ in Wolf’s notation), which is the 3-torus.
Thus we have proved the following theorem:
\[mainthm\_1\] There exists a collection $L$ of five linked 2-tori embedded in a smooth 4-manifold $X$ such that $X$ is diffeomorphic to a standard $S^4$, and $X-L$ admits a finite volume hyperbolic geometry.
[10]{}
Akbulut, S. *4-manifolds (2014)*, lecture notes available at http://www.math.msu.edu/ akbulut/.
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Coxeter, H.S.M *Regular Complex Polytopes. Second Edition.*, Cambridge University Press, Cambridge, 1991.
Gompf, R.E. and Stipsicz, A.I *4-manifolds and Kirby Calculus*, Graduate Studies in Mathematics, Providence, Rhode Island, 1999.
Hantzsche, W. and Wendt, H. *Dreidimensionale euklidische Raumformen*, Math. Ann. 110 (1935), 593-611.
Ivan$\check{s}$i$\acute{c}$, D. *Hyperbolic structure on a complement of tori in the 4-sphere*, Adv. Geom. 4, no. 1 (2004), 119–139.
Kerckhoff, S.P. and Storm, P.A. *From the hyperbolic 24-cell to the cuboctahedron*, Geometry & Topology 14 (2010) 1383–1477.
Kirby, R. *A Calculus for Framed Links in $S^3$*, Inventiones math., 45 (1978), 35-56.
Ratcliffe, J.G. and Tschantz, S.T. *The Volume Spectrum of Hyperbolic 4-manifolds*, Experiment. Math. Volume 9, Issue 1 (2000), 101-125.
Saratchandran, H. *Kirby diagrams and the Ratcliffe-Tschantz hyperbolic 4-manifolds*, arXiv:math:GT/1503.06722.
Thurston, W. *The geometry and topology of three-manifolds*, Princeton Univ. Math. Dept. Notes (1979), available at http://www.msri.org/communications/books/gt3m
Thurston, W. *Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)6, no. 3 (1982), 357–381.
Ivan$\check{s}$i$\acute{c}$, D. and Ratcliffe, J.G. and Tschantz, S.T. *Complements of tori and Klein bottles in the 4-sphere that have hyperbolic structure*, Algebr. Geom. Topol. 5 (2005), 999–1026.
Wolf, J.A. *Spaces of constant curvature*, McGraw-Hill, United States of America, 1967.
|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'Let $X$ be a symplectic variety equipped with an action of a torus ${\mathsf{A}}$. Let $\mc \subset {\mathsf{A}}$ be a finite cyclic subgroup. We show that K-theoretic stable envelope of subvarieties $X^{\mc}\subset X$ can be obtained via various limits of the elliptic stable envelopes of $X$. An example of $X$ given by the Hilbert scheme of points in the complex plane is considered in detail.'
author:
- 'Yakov Kononov, Andrey Smirnov'
bibliography:
- 'bib.bib'
title: 'Pursuing quantum difference equations I: stable envelopes of subvarieties'
---
Introduction
============
The development of the theory of elliptic stable envelope was initiated by M. Aganagic and A. Okounkov in [@AOElliptic]. Since then the theory has found remarkable applications to various areas of mathematics. To list just a few, stable envelopes can be related to the so-called Bethe vectors in integrable models [@OkBethe], they provide a new description of canonical bases for quantum groups [@Hik], they manifest themselves as weight functions for solutions of the qKZ equations [@varch; @RTV; @konno1; @konno2], they provide explicit formulas for the R-matrices of various algebras (Yangians, quantum loop algebras and elliptic quantum groups) [@MO; @Neg; @ZinByk; @InstR]. Stable envelopes also find important applications in Donaldson-Thomas theory of threefolds [@KOO], quantum field theory and string theory [@KorGa; @Dim], and combinatorics of symmetric polynomials [@NegGor; @Wen; @NegPier; @GenJacks; @MS].
Initially, the theory was built as a tool to describe the monodromy of qKZ-like equations and quantum difference equations associated with the quiver varieties [@OS]. These ideas were outlined in [@OkBethe; @AOF] as a generalization of earlier developments in [@TV1; @TV2; @TV3; @EV; @TolMon].
In geometric approach, qKZ equations and quantum difference equations describe $q$-holonomic modules generated by [*vertex functions*]{} of symplectic varieties [@pcmilect]. These developments revealed a deep interaction between Gromov-Witten type enumerative theories and representation theory. We refer to [@Pushk1; @Pushk2; @KorZe] for recent progress in this direction, see also [@Dink1; @Dink2] for the description of vertex functions in more specific situations.
The elliptic stable envelope relates the enumerative invariants of symplectic varieties to enumerative invariants of the [*symplectic dual*]{} varieties [@AOElliptic]. This suggests that stable envelopes provide a natural tool to work with symplectic duality (or 3d-mirror symmetry). This idea was first emphasised by A. Okounkov in his talk “[*Enumerative symplectic duality*]{}” during the 2018 MSRI workshop “Structures in Enumerative Geometry” and further examined in several special cases in [@RSVZ1; @MirSym2; @SZ].
An interesting problem in enumerative geometry of symplectic varieties is to find a better description of the corresponding $q$-difference equations. Even though this problem has been partly addressed in [@pcmilect; @OS], the treatment developed there is not entirely geometric and relies on the techniques of the Hopf algebras invented earlier in [@EV].
The analysis of the [*monodromy*]{} of these equations leads to a new geometric approach, which describes the building blocks of the $q$-difference equations (for instance the dynamical wall-crossing operators,see Section 5.3.1 in [@OS]) by special limits of the elliptic stable envelopes. This paper was mainly motivated by this idea and we consider it as a first natural step in this research direction. Here we study special limits of the elliptic stable envelopes which arise in the following way: let ${\mathsf{A}}$ be a torus acting on a symplectic variety $X$ by automorphisms, let $\mc \subset {\mathsf{A}}$ be a cyclic subgroup of finite order. The inclusion $X^{\mc} \subset X$ induces a morphism of the elliptic cohomology schemes $i: \textrm{Ell}_{{\mathsf{T}}}(X^{\mc})\to \textrm{Ell}_{{\mathsf{T}}}(X)$. In this setup, the elliptic cohomology scheme of the $\mc$-fixed subset admits certain transformations $\omega_{\wall}: \textrm{Ell}_{{\mathsf{T}}}(X^{\mc})\to \textrm{Ell}_{{\mathsf{T}}}(X^{\mc})$ which preserve its structure. These transformations act by shifting the equivariant parameters $\omega_{\wall}: a \to a q^{\wall}$ by special elements $\wall \in {\mathrm{Lie}}_{\matQ}({\mathsf{A}})$ ($q$ denotes the modular parameter of the underlying elliptic curve $E$).
In Theorem \[manth\] we prove that the elliptic stable envelope of $X$ twisted by $\omega_{\wall}$ in the limit $q=0$ converges to the K-theoretic stable envelope of the $\mc$-fixed subvarieties. Schematically,
\[stblim\]
where $z\to 0_{\mathfrak{D}}$ denotes certain vanishing of Kähler parameters which controls the slope of the K-theoretic stable envelopes.
In Section \[hssec\] we apply this result to $X$ given by the Hilbert scheme of points in $\matC^2$. In this case the components of the fixed set $X^{\mc}$ are isomorphic to the Nakajima quiver varieties associated with cyclic quivers. Theorem \[hsthm\] then establishes exact correspondence between stable envelopes for these varieties. Our results here are related to the conjectures proposed in [@NegGor], and we expect that their conceptual proofs will be obtained along these lines.
In the last section we consider a special case of $\mc$ given by a subgroup of framing torus of a Nakajima quiver variety $X$. In this situation the twists $\omega_{\wall}$ and fixed sets $X^{\mc}$ are labeled by certain arrangement of hyperplanes in ${\mathrm{Lie}}_{\matR}({\mathsf{A}})$. The K-theoretic stable envelopes of $X^{\mc}$ arising in the limit (\[stblim\]) for all choices of $\omega_{\wall}$ are described by Theorem \[manth2\].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank our teacher Andrei Okounkov for drawing our attention to the problem. We also thank Boris Feigin, Henry Liu and Zijun Zhou for useful discussions. This work was initiated during the AMS Mathematics Research Community meeting on Geometric Representation Theory and Equivariant Elliptic Cohomology at Rhode Island in June 2019 and the workshop “Elliptic cohomology days” at the University of Illinois, Urbana-Champaign. The authors are indebted to the organizers and all participants for very fruitful discussions and creative scientific atmosphere.
This work is supported by the Russian Science Foundation under grant 19-11-00062.
Elliptic stable envelopes
=========================
Let $X$ be a symplectic variety with an action of algebraic torus ${\mathsf{T}}$. As usual, we assume that the action of ${\mathsf{T}}$ scales the symplectic form with a character which we denote $\hbar^{-1}$. We denote by ${\mathsf{A}}:=\ker(\hbar^{-1})\subset {\mathsf{T}}$ the codimension one subtorus preserving the symplectic form.
We assume that $X^{{\mathsf{T}}}$ is finite. We assume also that elliptic stable envelope exists for $X$. It is well known that the class of symplectic varieties satisfying this condition is quite large. For example, it includes all Nakajima quiver varieties, see Theorem 3 in [@AOElliptic].
For the definition of the elliptic stable envelope and basics of elliptic cohomology we refer to Sections 2-3 in [@AOElliptic] and Sections 2 in [@EllipticHilbert], in particular, the Subsection 2.13 in [@EllipticHilbert] deals with the case of finite $X^{{\mathsf{T}}}$.
Let $\lambda \in X^{{\mathsf{T}}}$ be a fixed point. By our assumption, for any choice of a chamber ${\mathfrak{C}}\subset {\mathrm{Lie}}_{\matR}({\mathsf{A}})$ and a polarization $P\in K_{{\mathsf{T}}}(X)$ (for definitions see, for instance, Section 2 of [@EllipticHilbert]) we have the well defined elliptic stable envelope ${\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)$. By definition, ${\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)$ is a section of a certain bundle (Section 2.13 in [@EllipticHilbert]) over the extended elliptic cohomology scheme $$\mathsf{E}_{{\mathsf{T}}}(X)=\textrm{Ell}_{{\mathsf{T}}}(X)\times {\mathscr{E}}_{{\mathrm{Pic}}(X)}$$ where $\textrm{Ell}_{{\mathsf{T}}}(X)$ denotes the ${\mathsf{T}}$-equivariant elliptic cohomology scheme of $X$ and ${\mathscr{E}}_{{\mathrm{Pic}}({X})}=E\otimes_{\matZ} {\mathrm{Pic}}(X)$ for a family of elliptic curves $E=\matC^{\times}/q^{\matZ}$ over the punctured disc $0<|q|<1$.
Recall that $\mathsf{E}_{{\mathsf{T}}}(X)$ is a scheme over the extended elliptic cohomology scheme of a point: $$\mathsf{E}_{{\mathsf{T}}}(X)\stackrel{\pi}{\longrightarrow}{\mathscr{B}}_{{\mathsf{T}},X}:=\textrm{Ell}_{{\mathsf{T}}}(pt)\times {\mathscr{E}}_{{\mathrm{Pic}}(X)}\cong E^{\dim({\mathsf{T}})+\textrm{rk}({\mathrm{Pic}}(X))}.$$ The coordinates on the abelian variety $\textrm{Ell}_{{\mathsf{T}}}(pt)$ are usually called the equivariant parameters. We denote them by $a$ (for those corresponding to ${\mathsf{A}}$) and $\hbar$ (cosponsoring to ${\mathsf{T}}/{\mathsf{A}}$). The coordinates in ${\mathscr{E}}_{{\mathrm{Pic}}(X)}$ are referred to as the Kähler parameters and are denoted by $z$.
We recall that the elliptic cohomology scheme has the following structure: $$\mathsf{E}_{{\mathsf{T}}}(X)=\left(\coprod\limits_{\lambda\in X^{{\mathsf{T}}}} \widehat{{\textsf{O}}}_{\lambda}\right)/ \Delta$$ where $\widehat{{\textsf{O}}}_{\lambda}\cong {\mathscr{B}}_{{\mathsf{T}},X}$ and $\Delta$ denotes the data describing how the fixed point components $\widehat{{\textsf{O}}}_{\lambda}$ glue to form $\mathsf{E}_{{\mathsf{T}}}(X)$, see Section 2 in [@RSVZ1]. We denote the restriction of the elliptic stable envelope to the fixed point components by $$T_{\lambda,\mu}(a,z)=\left.{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)\right|_{\widehat{{\textsf{O}}}_{\mu}}.$$ The components $T_{\lambda,\mu}(a,b)$ represent sections of certain line bundles over the abelian varieties $\widehat{{\textsf{O}}}_{\mu}$ and thus can be expressed in terms of the odd Jacobi theta function associated with $E$: \[prodform\] (x)=(x\^[1/2]{}-x\^[-1/2]{})\_[i=1]{}\^ (1-x q\^[i]{})(1-q\^[i]{}/x). Note that in the multiplicative notations odd means \[oddthe\] (1/x)=-(x). The quasiperiods of these sections are governed by \[thetatransf\] (x q)=- (x). Given a K-theory class $P$ we will denote by $\Theta(P)$ the corresponding elliptic Euler class. For example, if $$P=\sum_i a_i - \sum_j b_j \in K_{{\mathsf{A}}}(pt)$$ where $a_i,b_i$ are some characters of ${\mathsf{A}}$ then, explicitly \[thdef\] (P)=. We denote the $q=0$ limit of the theta function by: $$\ss(x):=\left.\vartheta(x)\right|_{q=0}=x^{1/2}-x^{-1/2}$$ and extend it by the linearity to sums of characters, so that $\ss(P)$ is given by the right side of (\[thdef\]) with symbol $\vartheta$ replaced by $\ss$.
By definition of the elliptic stable envelope, the sections $T_{\lambda,\mu}(a,z)$ are [*holomorphic*]{} in equivariant parameters $a$. The important feature of sections $T_{\lambda,\mu}(a,z)$ is that they are also [*balanced*]{}.
Let ${\mathscr{E}}=E^{n}\times E^{m}$ be an abelian variety. We denote the coordinates on the factors by $a=(a_1,\dots, a_n)$ and $z=(z_1,\dots,z_m)$. Let $s(a,z)$ be a meromorphic section of degree zero line bundle over ${\mathscr{E}}$.
[*We say that $s(a,z)$ is balanced in the variables $a$ if in coordinates it can be represented in the following form: \[balform\] s(a,z)= \_[l]{} where $a^l=a_1^{l_1}\dots a_n^{l_n}$ denote monomials in the variables $a$ and $\dots$ stands for monomials in the rest of variables $z$.* ]{}
For example, the following section over $E\times E$: $$s(a,z)={\frac {\vartheta \left( a z \right) }{\vartheta \left( a \right) \vartheta
\left( z \right) }}+{\frac {\vartheta \left( {a}^{2} z \right) \vartheta
\left( a \right) }{\vartheta \left( {a}^{2} \right) \vartheta \left( a z
\right) }}$$ is balanced in variable $a$. It is also balanced in variable $z$. But it is not balanced in variables $(a,z)$.
As $q\to 0$ the elliptic curve converges to nodal elliptic curve with smooth locus isomorphic to $\matC^{\times}$. The balanced sections are characterized by good behavior in this limit.
\[wlim\] [*For any $\wall=(\wall_1,\dots,\wall_n) \in \matR^{n}$ and a section $s(a,z)$ balanced in variables $a$ the following limit exists \[exlims\] \_[q0]{} s(a q\^ ,z) (a,z) where $a q^{\wall}=(a_1 q^{\wall_1},\dots,a_n q ^{\wall_n})$ and $\sqrt{z}$ denotes the square root of some monomial in variables $z_1,\dots,z_m$.*]{}
Let $\wall\in \matR$. The Lemma follows immediately from the following identity \[thetlim\] \_[q0]{} = {
[ll]{} z\^[-- 1/2 ]{}, &\
z\^[--1/2]{}, &
. where $\lfloor \wall \rfloor$ stands for the integral part of $\wall$. This identity, in turn, can be derived from (\[prodform\]).
Natural examples of balanced sections are provided by restrictions of the elliptic stable envelopes to the components of the fixed points. For $\lambda,\mu \in X^{{\mathsf{T}}}$ let us consider the following section \[balsec\] s(a,z)==. Here $a$ and $z$ denote the equivariant and Kähler parameters, which are the coordinates on abelian variety $\widehat{{\textsf{O}}}_{\mu}$.
*If $X$ is a hypertoric variety then (\[balsec\])*
1\) is balanced in the equivariant parameters $a$,
2\) is balanced in the Kähler parameters $z$.
3\) has poles separately in $a$ and $z$
The property 3) means that for $s(a,z)$ it is allowed to have factors $\vartheta(a)\vartheta(z)$ but not $\vartheta(az)$ in denominators of (\[balform\]).
For the hypertoric varieties, the formulas for the elliptic stable envelopes of fixed points can be described very explicitly as certain products of theta functions, see Section 4.1.3 of [@AOElliptic] or Section 3.2 in [@SZ]. These hypertoric formulas are explicitly balanced separately in equivariant and Kähler parameters, and have separated poles.
\[balcor\] [*If $X$ is a quiver variety with finite $X^{{\mathsf{T}}}$ then (\[balsec\]) has properties 1), 2), 3).*]{}
For quiver varieties, the elliptic stable envelope of a fixed point $\lambda \in X^{{\mathsf{T}}}$ can be expressed in terms of the elliptic stable envelopes of fixed points in the hypertoric variety given by the [*abelianization*]{} of $X$. We refer to Section 4 of [@AOElliptic] (in particular Section 4.3) where the details of the abelianization procedure are explained.
We expect that these properties of the elliptic stable envelope hold in general.
[*Let $X$ be a smooth symplectic variety with finite $X^{{\mathsf{T}}}$ for which the elliptic stable envelope exists. Then 1),2),3) hold for (\[balsec\]).* ]{}
The K-theoretic stable envelope (we refer to Section 9 of [@pcmilect] for its definition) can be obtained from the elliptic as the following limit:
[*For generic $s\in {\mathrm{Pic}}(X)\otimes \matR$ we have: \[kthlim\] (P)\^[-1/2]{}\_[q0]{} (.\^[Ell]{}\_[X,,P]{}()|\_[z=q\^s]{}) (P\_[,0]{})\^[1/2]{} = \^[Kth,\[s\]]{}\_[X,,P]{}() where ${\mathrm{Stab}}^{Kth,[s]}_{X,{\mathfrak{C}},P}(\lambda)$ is the K-theoretic stable envelope of $\lambda$ with slope $s$. ($P_{\lambda,0}$ denotes the component of $P_{\lambda}$ which has zero degree in $a$, as in (\[polparts\])).*]{}
The $K$-theoretic stable envelopes for the slopes which are close to $0\in {\mathrm{Pic}}(X) \otimes \matR$ play a special role in representation theory, see Theorem 10.2.11 in [@pcmilect] for an example. If $\mathscr{U}_{0} \subset {\mathrm{Pic}}(X) \otimes \matR$ is a small analytic neighborhood of $0$ then the K-theoretic stable envelope changes only when the slope crosses certain hyperplanes passing through $0\in {\mathrm{Pic}}(X) \otimes \matR$. These hyperplanes divide the neighborhood into a set of chambers: \[dcham\] \_[0]{} { } = \_i \_i. We will denote K-theoretic stable envelopes with the slope from these chambers by: $${\mathrm{Stab}}^{\mathfrak{D}}_{X,{\mathfrak{C}},P}(\lambda):={\mathrm{Stab}}^{Kth,[s]}_{X,{\mathfrak{C}},P}(\lambda), \ \ \ s\in \mathfrak{D}.$$ If we denote $$\lim\limits_{z\to 0_{\mathfrak{D}}} f(z):= \lim\limits_{q\to 0} f(q^{s}), \ \ s\in \mathfrak{D},$$ then for small slopes (i.e., from $\mathscr{U}_{0}$) the above proposition gives:
\[kthcorol\] [*If $$\label{zlim}
S(\lambda):= \det(P)^{-1/2}\otimes \left( \lim\limits_{q\to 0} {\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)\right) \otimes \det(P_{\lambda,0})^{1/2} \in K_{{\mathsf{T}}}(X)\otimes \matC(z)$$ then \[zlim2\] \_[z0\_]{} S() = \^\_[X,,P]{}()* ]{}
From the definition of the elliptic stable envelope we know that the section (\[balsec\]) has the following quasiperiods: $$s(a q^{\sigma},z)=z^{\chi_{\lambda}(\sigma,\cdot)-\chi_{\mu}(\sigma,\cdot)} s(a,z) \ \ \ s(a ,zq^{\delta})= a^{\chi_{\lambda}(\cdot,\delta)-\chi_{\mu}(\cdot,\delta)} s(a,z)$$ where $\sigma \in \textrm{cochar}({\mathsf{A}})$ and $\delta \in \textrm{cochar}({\mathsf{K}})$, and $\chi_{\lambda}$ is a pairing: $$\chi_{\lambda}: \textrm{cochar}({\mathsf{A}})\times \textrm{cochar}({\mathsf{K}}) \to \matZ.$$ (here we denote the Kähler torus of $X$ by ${\mathsf{K}}={\mathrm{Pic}}(X)\otimes_{\matZ}\matC^{\times}$). By Lemma \[wlim\] this section has a well defined limit when $q\to 0$, moreover:
\[limexlem\] [ *If $\wall \in \mathrm{cochar}({\mathsf{A}})\otimes \matR$ and $\textsf{v}\in \mathrm{cochar}({\mathsf{K}}) \otimes \matR$ then the limits $$\lim_{z\to0_{\mathfrak{D}}}\Big(z^{-\chi_{\lambda}(\wall,\cdot)+\chi_{\mu}(\wall,\cdot)} \lim\limits_{q\to 0} s(a q^{\wall},z)\Big)
\ \ \
\text{and} \ \ \
\lim_{a\to0_{\mathfrak{C}}}\Big(a^{-\chi_{\lambda}(\cdot,\textsf{v})+\chi_{\mu}(\cdot,\textsf{v})} \lim\limits_{q\to 0} s(a ,z q^{\textsf{v}})\Big)$$ exist for all chambers $\mathfrak{C}$ and $\mathfrak{D}$.*]{}
Assume that both ${\mathsf{A}}$ and ${\mathsf{K}}$ are one-dimensional. The general case then follows from choosing arbitrary one-dimensional subtori in ${\mathsf{A}}$ and ${\mathsf{K}}$. We prove the Lemma for the first limit. For the second the argument is the same after switching the roles of $a$ and $z$.
We need to show that the limits $z\to 0$ and $z\to \infty$ of $\lim\limits_{q\to 0} s(a q^{\wall},z)$ exist. As $s(a,z)$ is balanced, it must have the form: \[terms\] s(a,z) = f(a) g(z)\_[i]{} where $f(a)$ and $g(z)$ are some balanced sections of depending only on $a$ and $z$ respectively. In the one-dimensional case $\wall \in \matR$ and $\chi_{\lambda}(\cdot,\cdot) \in \matZ$. Moreover, from (\[thetatransf\]) we compute that \[lsum\] \_(,)-\_(,) =- \_[i]{} n\_i m\_i, and thus $
-\chi_{\lambda}(\wall,\cdot)+\chi_{\mu}(\wall,\cdot)=\wall \sum\limits_{i} n_i m_i.
$ The terms in the sum (\[terms\]) are sections of the same line bundle, and thus (\[lsum\]) must be the same for each term. From (\[thetlim\]) we see that $$\lim\limits_{q\to 0} \Big( f(a) g(z)\prod\limits_{i} \dfrac{\vartheta(a^{n_i} z^{m_i})}{\vartheta(a^{n_i}) \vartheta(z^{m_i})} \Big) = r(a,z) \prod\limits_{i} \dfrac{z^{-\lfloor \wall n_i\rfloor m_i }}{z^{m_i}-1}$$ with $r(a,z)$ such that the limits $$\lim\limits_{z\to 0} r(a,z) \in \matQ(a), \ \ \lim\limits_{z\to \infty} r(a,z) \in \matQ(a)$$ exist and are nontrivial. We note that $
0 \leq \wall n_i - \lfloor \wall n_i \rfloor < 1,
$ which means that $$\lim\limits_{z\to 0} \dfrac{z^{m_i(\wall n_i - \lfloor \wall n_i \rfloor )}}{z^{m_i}-1}<\infty , \ \ \lim\limits_{z\to \infty} \dfrac{z^{m_i(\wall n_i - \lfloor \wall n_i \rfloor )}}{z^{m_i}-1}<\infty$$ and the lemma follows.
Subvarieties invariant under finite subgroups
=============================================
\[wshift\]
----------
Let $\mc \subset {\mathsf{A}}$ be a cyclic subgroup of order $b$ and let $X^{\mc}$ be its fixed set. The action of ${\mathsf{A}}$ on $X^{\mc}$ factors through the map $
\psi: {\mathsf{A}}\to {\mathsf{A}}^{'} = {\mathsf{A}}/\mc.
$ We denote $\mathsf{Z}= \psi^{-1} (q^{\textrm{cochar}({\mathsf{A}}')})$. The group $\mathsf{Z}$ acts on $\textsf{E}_{{\mathsf{T}}}(X^{\mc})$ by translations in the equivariant parameters $a\to a q^{\wall}$.
We fix an element $q^{\wall} \in \mathsf{Z}$ such that $(q^{\wall})^{b}\in q^{\textrm{cochar}({\mathsf{A}}')\setminus \{0\}}$ but $(q^{\wall})^{m}\not \in q^{\textrm{cochar}({\mathsf{A}})}$ for $0<m<b$. We denote the corresponding translation of the elliptic cohomology scheme by $\omega_{\wall}$: $$\xymatrix{
E_{{\mathsf{T}}}(X^{\mc})\ar[r]^{\omega_{\wall} }\ar[d]^{\pi^{*}}&
E_{{\mathsf{T}}}(X^{\mc})\ar[d]^{\pi^{*}}\\
{\mathscr{B}}_{{\mathsf{T}},X^{\mc}}\ar[r]^{a\to a q^{\wall}}&{\mathscr{B}}_{{\mathsf{T}},X^{\mc}}. }$$
The restriction of the polarization to a fixed point has the following decomposition: \[polparts\] P\_=P\_[,>0]{}P\_[,<0]{}P\_[,0]{} K\_(pt) where the three terms denote the parts whose ${\mathsf{A}}$-characters take positive, negative or zero values at the chamber ${\mathfrak{C}}$. The positive part is called [*index*]{} of the fixed point $\lambda$: $$\ind_{\lambda} = P_{\lambda,>0} \in K_{{\mathsf{A}}}(pt).$$ Similarly we, have a decomposition of the tangent spaces at the fixed points: $$T_{\lambda} X=N^{+}_{\lambda} \oplus N^{-}_{\lambda}.$$ For a Laurent polynomial $$\ind_{\lambda}=\sum\limits_{\sigma \in \textrm{char}({\mathsf{A}})} \, a^{\sigma}$$ and $\wall \in \textrm{cochar}({\mathsf{A}})\otimes \matR$ we denote $$\lfloor \ind_{\lambda} \cdot \wall \rfloor = \sum_{\sigma\in \textrm{char}({\mathsf{A}})} \, \lfloor \langle\sigma, \wall\rangle \rfloor .$$ We also define $\sigma_{\lambda}\in \textrm{char}({\mathsf{A}})$ by \[sigpol\] (P\_) = a\^[\_]{} K\_(pt).
\[lllem\] [ *If $P^{\mc}_{\lambda},N^{-,\mc}_{\lambda},{\ind^{\mc}_{\lambda}}$ denote $\mc$-invariant parts of $P,N^{-}_{\lambda}$ and ${\ind_{\lambda}}$, then for $\wall$ as in Section \[wshift\] we have:*]{} $$\begin{aligned}
\label{lemlim}
\lim\limits_{q\to 0} \left(\left[\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(P_{\lambda})}\right]_{a=a q^{\wall}} \right) = (-1)^{\mathrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda})} \hbar^{\lfloor \mathrm{ind}_{\lambda} \cdot \wall \rfloor} \dfrac{\ss(N^{-,\mc}_{\lambda})}{\ss(P^{\mc}_{\lambda})} \nonumber \\
=(-1)^{\mathrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda})}
\dfrac{\hbar^{\lfloor\ind_{\lambda} \cdot \wall \rfloor+\mathrm{rk}(\ind_{\lambda})/2 } }{\det(\ind_{\lambda}) \det(P_{\lambda,0})^{1/2}}
\dfrac{\Lambda^{\!\bullet}(\bar{N}^{-,\mc}_{\lambda}) }{\Lambda^{\!\bullet}(\bar{P}^{\mc}_{\lambda})}\end{aligned}$$
Follows directly from (\[thetlim\]).
We recall that the K-theoretic stable envelope can be obtained as a $q\to 0$ limit of $$\det(P)^{-1/2} \circ {\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P} \circ \det(P_{0})^{1/2}$$ see Section 4.5.2 in [@AOElliptic]. In particular, the K-theoretic stable envelope in this approach is normalized by its diagonal components: \[ksdia\] (P)\^[-1/2]{} \^[Ell]{}\_[X,,P]{} (P\_[0]{})\^[1/2]{} \^ (|[N]{}\^[-]{}\_)
The chamber ${\mathfrak{C}}$ and the $\mc$-invariant part of the polarization $P^{\mc} \in K_{{\mathsf{T}}}(X^{\mc})$ define the elliptic and K-theoretic stable envelopes for $X^{\mc}$. The inclusion $X^{\mc} \to X$ induces a map of extended elliptic cohomology schemes: $$i: \ \ \textsf{E}_{{\mathsf{T}}}(X^{\mc})\to \textsf{E}_{{\mathsf{T}}}(X).$$ If $\mathfrak{D}$ is a chamber from (\[dcham\]) then we denote by $\mathfrak{D}'$ the corresponding chamber for $X^{\mu_b}$ defined by the property: \[chamdp\] () ’ where $\kappa:{\mathrm{Pic}}(X)\otimes \matR\to {\mathrm{Pic}}(X^{\mc})\otimes \matR$ is the induced map.
For $\wall \in \textrm{cochar}({\mathsf{A}})\otimes \matR$ let us define a $\textrm{char}({\mathsf{K}})$-valued function on $X^{{\mathsf{T}}}$ by $$\lambda \to \chi_{\lambda}(\wall,\cdot) \in \textrm{char}({\mathsf{K}}).$$ Here is our main result.
\[manth\] [*Let $\omega_{\wall}$ be the translation in equivariant parameters as in Section \[wshift\]. Define $$S:=\lambda \to \Lambda^{\!\bullet}(\bar{P}^{\mc}) \circ \omega^{*}_{\wall} \circ i^{*} \left( \dfrac{{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P} (\lambda) }{\Theta(P)} \right) \circ \det(P_{0,\lambda})^{1/2},$$ then $$\begin{aligned}
\label{mainlim}
\lim\limits_{z\to 0_{\mathfrak{D}}} \left( z^{\chi(\wall,\cdot)-\chi_{\lambda}(\wall,\cdot)} \lim\limits_{q\to 0} S(\lambda) \right) \\ \nonumber
& =(-1)^{\mathrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda})} \hbar^{\lfloor \ind_{\lambda}\cdot \wall \rfloor}\, {\mathrm{Stab}}^{\mathfrak{D}'}_{X^{\mc},{\mathfrak{C}},P^{\mc}}(\lambda). \end{aligned}$$* ]{}
Let us assume that ${\mathsf{A}}\cong \matC^{\times}$. If not, we can choose a cocharacter $\matC^{\times} \to {\mathsf{A}}$ whose image contains $\mc$, then the shift $\omega^{*}_{\wall}$ does not affect the equivariant parameters in ${\mathsf{A}}/\matC^{\times}$ and thus they do not change the limit $q\to 0$.
Let us denote $$E_{\lambda,\mu}(a,z):= i^{*}\left.\left(\dfrac{{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P} (\lambda) }{\Theta(P)}\right)\right|_{\mu}.$$ These are the fixed point components of certain meromorphic section over $\textsf{E}_{{\mathsf{T}}}(X^{\mc})$. We have \[ecomp\] E\_[,]{}(a q\^,z)=.\^[\*]{}\_ i\^[\*]{} ()|\_ By Corollary \[balcor\] the sections $E_{\lambda,\mu}(a,z)$ are balanced, i.e., are of the form (\[balform\]). By Lemma \[wlim\] the limit $$\lim\limits_{q\to 0} E_{\lambda,\mu}(a q^{\wall},z)$$ exists. By (\[thetlim\]), only the factors $\vartheta(a^n\dots)$ ( $\dots$ stands for monomials in $z$ and $\hbar$) in the numerator and denominator of $E_{\lambda,\mu}(a q^{\wall},z)$ with $n\mid b$ can contribute a nontrivial function of $a$ in this limit. The factors with $n\nmid b$ in the limit $q\to 0$ can only produce monomials in $z$ and $\hbar$. The factors $\Theta(P)$ with $n\mid b$ are exactly those from the invariant part $\Theta(P^{\mc})$. Thus, one can cancel all poles in the limit of (\[ecomp\]) by tensoring it with $\Lambda^{\!\bullet}(\bar{P}^{\mc})$. We conclude that $$K_{\lambda,\mu}(a,z)=\Lambda^{\!\bullet}(\bar{P}^{\mc}_{\mu}) \otimes \lim\limits_{q\to 0} E_{\lambda,\mu}(a q^{\wall},z) \otimes \det(P_{\lambda,0})^{1/2}$$ are holomorphic in equivariant parameters $a$. These are the fixed point components of a holomorphic (in $a$) function on $\textrm{Spec}(K_{{\mathsf{T}}}(X^{\mc}))\otimes {\mathsf{K}}$, which we denote by \[kclass\] K():=\^(|[P]{}\^) \_[q0]{} \^[\*]{}\_ i\^[\*]{} ( ) (P\_[,0]{})\^[1/2]{} From the support condition for ${\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)$ (see Section 3.3.5 [@AOElliptic]) we find that $K(\lambda)$ is supported at: \[support\] (K()) X\^ \^[f]{}\_X() =\^[f]{}\_[X\^]{}(). By definition of the elliptic stable envelope $\left.{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P} (\lambda)\right|_{\lambda}=\Theta(N^{-}_{X,\lambda})$. The factors in $\Theta(N^{-}_{X,\lambda})$ with $n\mid b$ are exactly those in $\Theta(N^{-}_{X^{\mc},\lambda})$. From Lemma \[lllem\] we find that the diagonal components of $K(\lambda)$ have the form: \[nrm\] .K()|\_= (-1)\^[(\_-\^\_)]{} \^[ ]{}(|[N]{}\^[- ]{}\_[X\^,]{}). The K-theoretic stable envelope is characterized by $a$-degree bound on its fixed point components, see Section 9.1.9 in [@pcmilect]. In particular, Corollary \[kthcorol\] implies that we have the following bounds: $$\begin{aligned}
\nonumber
& \deg_{\mathsf{A}}\left( \lim\limits_{z\to 0_{\mathfrak{D}}}\Big( \Lambda^{\!\bullet}(\bar{P}_{\mu})\otimes \lim\limits_{q\to0}\left.\dfrac{{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)}{\Theta(P)}\right|_{\mu} \Big)\right) \\ \label{bound}
& \subset \deg_{\mathsf{A}}\left( \lim\limits_{z\to 0_{\mathfrak{D}}}\Big( \Lambda^{\!\bullet}(
\bar{P}_{\mu})\otimes \lim\limits_{q\to0}\left.\dfrac{{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\mu)}{\Theta(P)}\right|_{\mu} \Big)\otimes s_{\lambda} \otimes s_{\mu}^{-1}\right)\end{aligned}$$ where $s_{\lambda}$ denotes the restriction of a line bundle $s\in \mathfrak{D}$ from (\[dcham\]).
If we consider the same limits with additional shift $\omega^{*}_{\wall}$ as in (\[kclass\]) the only the terms $\vartheta(a^n\dots)$ with $n\mid b$ contribute. Thus, taking the $\mc$-invariant part of (\[bound\]) we obtain: $$\begin{aligned}
\label{window}
& \deg_{{\mathsf{A}}}( \lim\limits_{z\to 0_{\mathfrak{D}}} z^{\chi_{\lambda}(\wall,\cdot)-\chi_{\mu}(\wall,\cdot)} \left.K(\lambda)\right|_{\mu})\subset
\deg_{{\mathsf{A}}}(\lim\limits_{z\to 0_{\mathfrak{D}}}\left.K(\mu)\right|_{\mu} \otimes s_{\lambda} \otimes s_{\mu}^{-1})
\end{aligned}$$ (note that the limits exist by Lemma \[limexlem\]). Now, (\[support\]), (\[nrm\]) and (\[window\]) say that the K-theory class $$\lim\limits_{z\to 0_{\mathfrak{D}}} z^{\chi_{\lambda}(\wall,\cdot)-\chi(\wall,\cdot)} K(\lambda)$$ satisfies all three defining properties of the K-theoretic stable envelope with slope $s$, see Section 9 in [@pcmilect].
Comparing (\[nrm\]) with (\[ksdia\]) we find that the normalization of this K-theoretic stable envelope differs from the one accepted in [@pcmilect; @AOElliptic] by a factor $$(-1)^{\mathrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda})} \hbar^{\lfloor\ind_{\lambda}\cdot \wall\rfloor}.$$ The theorem follows from the uniqueness of the stable envelope in K-theory see Proposition 9.2.2 in [@pcmilect].
For practical computations, it might be more convenient to formulate the above theorem as follows. Let us consider the normalized matrix of restrictions: \[ttild\] \_[,]{}(a,z):=. This is a triangular matrix with trivial diagonal $\tilde{T}_{\lambda,\lambda}(a,z)=1$ and other coefficients given by certain elliptic functions. Similarly we denote $$\tilde{K}_{\lambda,\mu}(a,\hbar):=\dfrac{\left.{\mathrm{Stab}}^{\mathfrak{D}'}_{X^{\mc},{\mathfrak{C}},P^{\mc}}(\lambda)\right|_{\mu} }{\left.{\mathrm{Stab}}^{\mathfrak{D}'}_{X^{\mc},{\mathfrak{C}},P^{\mc}}(\lambda)\right|_{\lambda}}$$ the matrix of K-theoretic stable envelopes of $X^{\mc}$ with a slope from $\mathfrak{D}'$ normalized in the same fashion.
\[thm2\] [*The matrix $\tilde{K}(a,\hbar)$ can be obtained from the matrix $\tilde{T}_{\lambda,\mu}(a,z)$ in the following limit: \[seclim\] \_[z0\_]{} Z (\_[q0]{} (a q\^,z)) Z\^[-1]{} =H (a,) H\^[-1]{} where $Z$ denotes the diagonal matrix $$Z:=\left.\mathrm{diag}(z^{\chi_{\lambda}(\wall,\cdot)})\right|_{\lambda \in X^{{\mathsf{T}}}}$$ and $H$ denotes the diagonal matrix $$H:=\left.\mathrm{diag}\Big((-1)^{\mathrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda})} \hbar^{m_{\lambda}(\wall)/2}\Big)\right|_{\lambda \in X^{{\mathsf{T}}}} ,$$ for $$m_{\lambda}(\wall)=\langle \sigma_{\lambda},\wall \rangle-\sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(N^{-}_{\lambda})} \lfloor \langle \sigma,\wall \rangle \rfloor . \ \ $$ with $\sigma_{\lambda}$ from (\[sigpol\])*]{}.
By definition $\left.{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)\right|_{\lambda}=\Theta(N^{-}_{\lambda})$, therefore the elliptic functions in (\[mainlim\]) (before taking the limits) differ from those in (\[ttild\]) by the ratio \[balratio\] . We note that \[npol\] N\_\^[-]{}=P\_[,<0]{}+|[P]{}\_[,>0]{} and thus by (\[polparts\]) we have $$\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(P_{\lambda})}=\dfrac{(-1)^{\textrm{rk}(P_{\lambda,>0}) }}{\Theta(P_{\lambda,0})} \dfrac{\Theta(P_{\lambda,>0} \hbar^{-1})}{\Theta(P_{\lambda,>0})}.$$ We conclude that this ratio is a balanced function in the equivariant parameters $a$. Dividing any balanced function by this ratio is clearly a balanced function again and thus all elliptic functions (\[ttild\]) are balanced in $a$. By Lemma \[wlim\] we conclude that the limits $q\to 0$ in (\[seclim\]) are well defined for all $\wall$.
Conjugation by the diagonal matrix $Z$ gives: $$\tilde{T}_{\lambda,\mu}(a,z) \to z^{\chi_{\mu}(\wall,\cdot)-\chi_{\lambda}(\wall,\cdot)} \tilde{T}_{\lambda,\mu}(a,z),$$ which is the same monomial $z$-prefactor as in (\[mainlim\]). Thus, the existence of the limit $z\to 0_{\mathfrak{D}}$ follows from Theorem \[manth\] (note that the ratio (\[balratio\]) does not depend on the Kähler parameters and thus can not affect asymptotic behavior at $z\to 0_{\mathfrak{D}}$).
We conclude that the left-hand side of (\[seclim\]) is equal to the restriction matrix for the class (\[mainlim\]) if we normalize it as in (\[ttild\]) namely: $$\begin{aligned}
& \left(\lim\limits_{z\to 0_{\mathfrak{D}}} Z \Big(\lim\limits_{q\to 0} \tilde{T}(a q^{\wall},z,\hbar,q)\Big) Z^{-1}\right)_{\lambda,\mu} = & \nonumber \\ \nonumber
& \dfrac{(-1)^{\textrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda})} \hbar^{\lfloor \ind_{\lambda}\cdot \wall \rfloor}\, \left.{\mathrm{Stab}}^{\mathfrak{D}'}_{X^{\mc},{\mathfrak{C}},P^{\mc}}(\lambda)\right|_{\mu}}{(-1)^{\textrm{rk}(\ind_{\mu}-\ind^{\mc}_{\mu})} \hbar^{\lfloor \ind_{\mu}\cdot \wall \rfloor}\, \left.{\mathrm{Stab}}^{\mathfrak{D}'}_{X^{\mc},{\mathfrak{C}},P^{\mc}}(\mu)\right|_{\mu}}= \tilde{K}_{\lambda,\mu}(a,\hbar) H_{\lambda}/H_{\mu}&\end{aligned}$$ The theorem follows from the identity $$\lfloor \ind_{\lambda}\cdot \wall \rfloor-\lfloor \ind_{\mu}\cdot \wall \rfloor=m_{\lambda}(\wall) - m_{\mu}(\wall)$$ provided by the Lemma \[diflem\].
\[diflem\] [ *For $\lambda,\mu \in (X^{\mc})^{{\mathsf{T}}}$ we have $$\lfloor \ind_{\lambda}\cdot \wall \rfloor-\lfloor \ind_{\mu}\cdot \wall \rfloor=m_{\lambda}(\wall) - m_{\mu}(\wall)$$ where $$m_{\lambda}(\wall)=\langle \sigma_{\lambda},\wall \rangle-\sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(N^{-}_{\lambda})} \lfloor \langle \wall, \sigma\rangle \rfloor . \ \ $$ with $\sigma_{\lambda}$ from (\[sigpol\])*]{}
We start from (\[lemlim\]) and write $$\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(P_{\lambda})}=\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(N^{-}_{\lambda} \hbar^{-1/2})}\dfrac{\Theta(N^{-}_{\lambda} \hbar^{-1/2})}{\Theta(P_{\lambda})}.$$ From (\[npol\]) we find: $$\Theta(N^{-}_{\lambda} \hbar^{-1/2})=\Theta((P_{\lambda,<0}+\hbar \bar{P}_{\lambda,>0})\hbar^{-1/2})=(-1)^{\textrm{rk}(\ind_{\lambda})}\Theta((P_{\lambda,<0}+P_{\lambda,>0}) \hbar^{-1/2})$$ where the last equality follows from (\[oddthe\]). Using (\[polparts\]) we find: \[thetid\] (N\^[-]{}\_ \^[-1/2]{})= (P\_ \^[-1/2]{}) and overall we obtain: $$\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(P_{\lambda})}=
\Big(\dfrac{(-1)^{\textrm{rk}(\ind_{\lambda})}}{\Theta(P_{\lambda,0} \hbar^{-1/2})}\Big)
\Big(\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(N^{-}_{\lambda} \hbar^{-1/2})}\Big)
\Big(\dfrac{\Theta(P_{\lambda} \hbar^{-1/2})}{\Theta(P_{\lambda})}\Big).$$ Note that each multiple in this expression is balanced in $a$, (the first one does not depend on $a$). Applying (\[thetlim\]) we find: $$\lim\limits_{q\to 0}\Big(\dfrac{(-1)^{\textrm{rk}(\ind_{\lambda})}}{\Theta(P_{\lambda,0} \hbar^{-1/2})}\Big) = \dfrac{(-1)^{\textrm{rk}(\ind_{\lambda})}}{\ss(P_{\lambda,0} \hbar^{-1/2})}$$ $$\lim\limits_{q\to 0} \Big(\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(N^{-}_{\lambda} \hbar^{-1/2})}\Big)_{a=a q^{\wall}}=\dfrac{\ss(N^{-,\mc}_{\lambda})}{\ss(N^{-,\mc}_{\lambda} \hbar^{-1/2})} \prod\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(N^{-}_{\lambda})} \hbar^{-\lfloor \langle \sigma,\wall\rangle \rfloor/2-1/4}$$ $$\lim\limits_{q\to 0} \Big(\dfrac{\Theta(P_{\lambda} \hbar^{-1/2})}{\Theta(P_{\lambda})}\Big)_{a=a q^{\wall}} = \dfrac{\ss(P^{\mc}_{\lambda} \hbar^{-1/2})}{\ss(P^{\mc}_{\lambda})} \prod\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(P_{\lambda})} \hbar^{\lfloor \langle \sigma,\wall\rangle \rfloor/2+1/4}$$ Note that (\[thetid\]) also holds if we substitute $\Theta$ by $\ss$ , and thus, for $\mc$-invariant part we have $$\dfrac{\ss(N^{-,\mc}_{\lambda})}{\ss(P^{\mc}_{\lambda})}=
\Big(\dfrac{(-1)^{\textrm{rk}(\ind^{\mc}_{\lambda})}}{\ss(P^{\mc}_{\lambda,0} \hbar^{-1/2})}\Big)
\Big(\dfrac{\ss(N^{-,\mc}_{\lambda})}{\ss(N^{-,\mc}_{\lambda} \hbar^{-1/2})}\Big)
\Big(\dfrac{\ss(P^{\mc}_{\lambda} \hbar^{-1/2})}{\ss(P^{\mc}_{\lambda})}\Big).$$ Combining all these limits we find: $$\begin{aligned}
& \lim_{q\to 0} \left( \left[\dfrac{\Theta(N^{-}_{\lambda})}{\Theta(P_{\lambda})}\right]_{a=a q^{\wall}}\right) = \nonumber \\ \nonumber
& (-1)^{\textrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda}) }\dfrac{\ss(N^{-,\mc}_{\lambda})}{\ss(P^{\mc}_{\lambda})}
\prod\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(N^{-}_{\lambda})} \hbar^{-\lfloor \langle \sigma,\wall\rangle \rfloor/2-1/4}
\prod\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(P_{\lambda})} \hbar^{\lfloor \langle \sigma,\wall\rangle \rfloor/2+1/4}.\end{aligned}$$ Comparing this with (\[lemlim\]) we obtain: $$\lfloor \mathrm{ind}_{\lambda} \cdot \wall \rfloor = \sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(N^{-}_{\lambda})} (-\lfloor \langle \sigma,\wall\rangle \rfloor/2-1/4)
+\sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(P_{\lambda})} \,(\lfloor \langle \sigma,\wall\rangle \rfloor/2+1/4).$$ The last step is to note that the characters appearing in $P_{\lambda}$ and $P_{\mu}$ for $\lambda,\mu \in (X^{\mc})^{{\mathsf{T}}}$ can only differ by an integral part of $\wall$ which means that $$\begin{aligned}
\nonumber
& \sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(P_{\lambda})} \,(\lfloor \langle \sigma,\wall\rangle \rfloor/2+1/4)-\sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(P_{\mu})} \,(\lfloor \langle \sigma,\wall\rangle \rfloor/2+1/4)=\\ \nonumber
& 1/2 \Big(\sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(P_{\lambda})} \, \langle \sigma,\wall\rangle- \sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(P_{\mu})} \, \langle \sigma,\wall\rangle\Big)= (\langle \sigma_\lambda,\wall \rangle-\langle \sigma_\mu,\wall \rangle)/2,\end{aligned}$$ and thus $$\begin{aligned}
\nonumber
& \lfloor \mathrm{ind}_{\lambda} \cdot \wall \rfloor-\lfloor \mathrm{ind}_{\mu} \cdot \wall \rfloor=\\ \nonumber
& = (\langle \sigma_\lambda,\wall \rangle-\sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(N^{-}_{\lambda})} \lfloor \langle \sigma,\wall\rangle \rfloor )/2-
(\langle \sigma_\mu,\wall \rangle-\sum\limits_{\sigma \in \mathrm{char}_{{\mathsf{A}}}(N^{-}_{\mu})} \lfloor \langle \sigma,\wall\rangle \rfloor )/2=\\ \nonumber
& = m_{\lambda}(\wall)-m_{\mu}(\wall).\end{aligned}$$
Application to the case of the Hilbert Scheme \[hssec\]
=======================================================
In this section we consider an application of Theorem \[manth\] to the case of $X$ given by the Hilbert scheme of $n$ points on $\matC^{2}$. In this section we follow notations of [@EllipticHilbert], where the explicit formula for the elliptic stable envelope for this variety was obtained. In particular, the tori ${\mathsf{A}}\subset{\mathsf{T}}$ acting on $X$, the set of fixed points $X^{{\mathsf{T}}}$, the choice of the polarization $P$ and the chambers ${\mathfrak{C}}$ were described in Section 3 of [@EllipticHilbert].
Recall that the Hilbert scheme $X$ is a Nakajima variety associated to the quiver in Fig.\[jord\], with dimension $n$, framing dimension $1$ and stability conditions: $$\theta_{\pm }: g\to \det(g)^{\pm 1}$$ see [@NakajimaLectures1] or Section 3.3 in [@EllipticHilbert].
![\[jord\] The quiver defining the Hilbert scheme $X$.](jordan){width="4cm"}
We consider the cyclic subgroup: $$\mc= \{ w^{k}, k=0,\dots,n-1 \}\subset {\mathsf{A}}\cong \matC^{\times}.$$ of $b$-th roots of $1$.
[*The fixed set of $\mc$ has the following form $$X^{\mc}=\coprod_{{n_0,n_1,\cdots, n_{b-1}} \atop {n_0+\dots+n_{b-1}=n} } X(n_0,\dots,n_{b-1})$$ where $X(n_0,\dots,n_{b-1})$ is the Nakajima quiver variety associated with the cyclic quiver of length $b$ (see Fig.\[cyclic\]) with dimensions $n_0,\dots,n_{b-1}$, framing dimensions $r=(1,0,\dots,0)$ and stability conditions $$\theta^{b}_{\pm}: (g_0,\dots, g_{b-1})\to \prod\limits_{i=0}^{b-1} \det(g_i)^{\pm 1}.$$.*]{}
![\[cyclic\] The quiver defining $X(n_0,n_1,\dots,n_{b-1})$.](cyclic){width="4.5cm"}
We note that it is possible that $X(n_0,\dots,n_{b-1})=\varnothing$ for some choices $(n_0,\dots,n_{b-1})$.
Here is the sketch of a proof. As a Nakajima variety associated to Fig.\[jord\], $X$ is given by the symplectic reduction of $$T^* R=T^* \textrm{Hom}(\matC,\matC^n)\oplus T^* \textrm{Hom}(\matC^n,\matC^n).$$ by the natural action of $GL(n)$. Recall that the torus ${\mathsf{A}}$, and thus $\mc$, act by scaling the loop in the quiver. It means that, if $(J,Y)$ is an element from $R$ then $\mc$ acts by $(J,Y)\to (J,Y \omega)$.
We have a decomposition \[cdecom\] \^[n]{}=\_[i=0]{}\^[b-1]{} \^[n\_i]{}, .|\_[\^[n\_i]{}]{}=e\^. The $\mc$-invariant part of $R$ then has the form $$R^{\mc} = \textrm{Hom}(\matC,\matC^{n_0}) \oplus \bigoplus_{i=0}^{b-1} \textrm{Hom}(\matC^{n_i},\matC^{n_i+1}) \ \ \textrm{with} \ \ \matC^{n_b}:=\matC^{n_0}.$$ and the symplectic reduction of $T^* R^{\mc}$ is exactly the quiver variety associated to Fig.\[cyclic\].
To complete the proof we also need to show that $\theta^{b}_{\pm}$-stable points in $T^{*} R^{\mc}$ satisfying the moment map condition for $\prod_i GL(n_i)$ are also $\theta_{\pm}$-semistable in $T^{*} R$ and satisfy the moment map condition for $GL(n)$. This is straightforward and we leave it to the reader.
Recall that the fixed points $X^{{\mathsf{T}}}=X^{{\mathsf{A}}}$ are labeled by the Young diagrams with $n$ boxes. It is also clear from the previous proposition that $$X^{{\mathsf{T}}}=\coprod\limits_{{n_0,n_1,\cdots, n_{b-1}} \atop {n_0+\dots+n_{b-1}=n} } X(n_0,\dots,n_{b-1})^{{\mathsf{T}}}.$$
[*For a fixed point $\lambda \in X^{{\mathsf{T}}}$ we have $$\lambda\in X(n_0,\dots,n_{b-1}) \ \ \Leftrightarrow \ \ |\{\Box\in \lambda: c(\Box)\!\!\! \mod b=i\}|=n_i, \ \ i=1,\dots,b-1.$$ where $c(\Box)$ is the content of a box $\Box$ in the Young diagram $\lambda$.*]{}
It is convenient to use the description of $X$ as a space of ideals in $\matC[x,y]$, see Section 3.1-3.2 in [@EllipticHilbert]. A box in the Young diagram $\lambda$ with coordinates $(i,j)$ then corresponds to the monomial $x^{j-1} y^{i-1}$. These monomials form a basis of $\matC^{n}$ in (\[cdecom\]) above. The ${\mathsf{A}}$-character of this monomial equals $i-j~=~c_{\Box}$. It means that $\omega$ acts on it by $e^{\frac{2 \pi \sqrt{-1} c_{\Box} }{b}}$ and thus it is from $\matC^{n_{c_{\Box}\!\!\mod b} }$. Since these monomials form a basis, we have $$n_{i} =\dim \matC^{n_{i} }=|\{\Box\in \lambda: c(\Box)\!\!\! \mod b=i\}|.$$
We have ${\mathrm{Pic}}(X)\cong \matZ$ with a generator given by the line bundle $\mathscr{O}(1)$. We denote $$d_{\lambda}=\deg_{{\mathsf{A}}}(\left.\mathscr{O}(1)\right|_{\lambda} )=\sum\limits_{\Box\in \Lambda} c_{\Box}.$$
\[sgilem\] [*The character (\[sigpol\]) for the Hilbert scheme $X$ has the following form: $$\sigma_{\lambda}=d_{\lambda} +|\lambda|^2$$*]{}
The Lemma is obvious from the following expression for the polarization: \[hpol\] P\_=\_[i,j]{} a\^[c\^\_i-c\^\_j+1]{}-\_[i,j]{} a\^[c\^\_i-c\^\_j]{}+\_[i ]{} a\^[c\^\_i]{} K\_(pt). see Section 3.7 in [@EllipticHilbert].
\[hoklem\] [*The ${\mathsf{A}}$-characters of $N^{\pm}_{\lambda}$ have the following form $$N^{\pm}_{\lambda}=\sum\limits_{\Box\in \lambda} a^{\pm \mathrm{hook}_{\lambda}(\Box)}$$ where $\mathrm{hook}_{\lambda}(\Box)$ denotes the hook length of a box $\Box$ in the Young diagram $\lambda$.*]{}
For the tangent space we can write: $$T_{\lambda} X = N^{+}_{\lambda}\oplus N^{-}_{\lambda}=P_{\lambda}\oplus \bar{P}_{\lambda}$$ and the result follows from (\[hpol\]) after some algebra.
[*Assume that $\lambda,\mu \in X(n_0,\dots,n_{b-1})^{{\mathsf{T}}}$ and $\wall\in \matQ$ then $$\lfloor \ind_{\lambda} \cdot \wall \rfloor - \lfloor \ind_{\mu} \cdot \wall \rfloor=m_{\lambda}(\textsf{w})-m_{\mu}(\textsf{w})$$ where $m_{\lambda}(\textsf{w}) \in \matQ$ is defined by: $$m_{\lambda}(\textsf{w})=\textsf{w} \, d_{\lambda} -\sum\limits_{\Box\in\lambda} \lfloor \mathrm{hook}(\Box) \textsf{w} \rfloor$$* ]{}
Follows from Lemmas \[diflem\],\[sgilem\] and \[hoklem\].
For the Hilbert scheme $X$ we have ${\mathsf{K}}={\mathrm{Pic}}(X)\otimes \matC^{\times}=\matC^{\times}$ and there are two chambers in (\[dcham\]) corresponding to $$z\to 0 \ \ \textrm{or} \ \ z\to \infty.$$ We will denote by $\mathfrak{D}^{\pm}$ the corresponding chambers (\[chamdp\]) for a $\mc$-fixed point component $X(n_0,\dots,n_b)$. These chambers correspond to the slopes from [*canonical*]{} and [*anticanonical*]{} alcoves of $X(n_0,\dots,n_b)$.
If $s(a,z)$ is as in (\[balsec\]), then for the Hilbert scheme $X$ it has the following transformation laws: $$s(a q,z)=z^{d_{\lambda}-d_{\mu}} s(a,z), \ \ s(a,z q)=a^{d_{\lambda}-d_{\mu}} s(a,z)$$ and thus $\chi_{\lambda}(\wall,\cdot)=\wall d_{\lambda}$.
Let us choose a $\mc$-fixed component $X(n_{0},\dots,n_{b-1})\subset X$, and consider the matrix: $$\tilde{T}_{\lambda,\mu}(a,z)=\dfrac{\left.{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)\right|_{\mu}}{\left.{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\mu)\right|_{\mu}}, \ \ \ \lambda,\mu \in X(n_0,\dots,n_{b-1})^{{\mathsf{T}}}.$$ and let $$\tilde{K}^{\pm}_{\lambda,\mu}(a,\hbar)=\dfrac{\left.{\mathrm{Stab}}^{\mathfrak{D}^{\pm}}_{X(n_0,\dots,n_{b-1}),{\mathfrak{C}},P^{\mc}}(\lambda)\right|_{\mu}}{\left.{\mathrm{Stab}}^{\mathfrak{D}^{\pm}}_{X(n_0,\dots,n_{b-1}),{\mathfrak{C}},P^{\mc}}(\mu)\right|_{\mu}}$$ be the normalized matrix of restrictions of K-theoretic stable envelopes for the cyclic quiver variety $X(n_0,\dots,n_{b-1})$, with slopes corresponding to the canonical and anticanonical alcoves, then the Theorem \[thm2\] gives:
\[hsthm\] [*Let $\wall=\frac{a}{b}\in \matQ$ such that $\mathrm{gcd}(a,b)=1$, then $$\lim\limits_{z\to 0} Z \Big(\lim_{q\to 0} \tilde{T}(a q^{\wall},z)\Big) Z^{-1}= H \tilde{K}^{+}(a,\hbar) H^{-1},$$ $$\lim\limits_{z\to \infty} Z \Big(\lim_{q\to 0} \tilde{T}(a q^{\wall},z)\Big) Z^{-1}= H \tilde{K}^{-}(a,\hbar) H^{-1},$$ where $$Z=\left.\mathrm{diag}(z^{\wall \,d_\lambda})\right|_{\lambda \in X(n_0,\dots,n_{b-1})^{{\mathsf{T}}}}, \ \ H=\left.\mathrm{diag}( (-1)^{\mathrm{rk}(\ind_{\lambda}-\ind^{\mc}_{\lambda})} \hbar^{m_{\lambda}(\wall)})\right|_{\lambda \in X(n_0,\dots,n_{b-1})^{{\mathsf{T}}}}.$$*]{}
Let us also denote $$\tilde{T}_{\lambda,\mu}(a,z)=\dfrac{\left.{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\lambda)\right|_{\mu}}{\left.{\mathrm{Stab}}^{Ell}_{X,{\mathfrak{C}},P}(\mu)\right|_{\mu}}, \ \ \ \lambda,\mu \in X^{{\mathsf{T}}}.$$ We note that $X^{\mc}$ may have nontrivial fixed components (i.e. not just $X^{{\mathsf{T}}}$) only if $b\leq n$. The above theorem then gives:
[*The limits are non-trivial: $$\lim\limits_{z\to 0} Z \Big(\lim_{q\to 0} \tilde{T}(a q^{\wall},z)\Big) Z^{-1}\neq \mathrm{Id}$$ ($\mathrm{Id}$ denotes the identity matrix of size $|X^{{\mathsf{T}}}|$) only for the following points: $$\wall \in \Big\{ \dfrac{a}{b}\in \matQ: \mathrm{gcd}(a,b)=1, \ 1 \leq b\leq n \Big\}.$$*]{}
Finite subgroups of framing torus
=================================
For this section $X(\nn,\rr)$ denotes a Nakajima quiver variety with the dimension vector $\nn=(n_1,\dots, n_l)$ and the framing dimensions $\rr=(r_1,\dots, r_l)$ where $l$ is the number of vertices in the quiver (see [@GinzburgLectures; @NakALE] for introductions to quiver varieties). The framing torus acting on $X(\nn,\rr)$ has the form $${\mathsf{A}}=(\matC^{\times})^{r_1}\times \dots \times (\matC^{\times})^{r_l}.$$ We denote by $a_1,\dots,a_{|\rr|}$ with $|\rr|=r_1+\dots+r_l$ the coordinates on ${\mathsf{A}}$. We fix the hyperplane arrangement in ${\mathrm{Lie}}_{\matQ}({\mathsf{A}})$ defined by the equations: $$H^{(n)}_{i,j}=\{ \tilde{a}_i-\tilde{a}_j=n \} \subset {\mathrm{Lie}}_{\matQ}({\mathsf{A}}), \ \ i,j \in I, \ \ n\in \matZ,$$ where $\tilde{a}_i$, $i \in I=\{1,\dots,|\rr|\}$ denote the corresponding coordinates on ${\mathrm{Lie}}_{\matQ}({\mathsf{A}})$.
For a point $\wall \in {\mathrm{Lie}}_{\matQ}({\mathsf{A}})$ we denote $$B_{\wall}=\{ H^{(n)}_{i,j}: \wall\in H^{(n)}_{i,j} \}.$$ Let ${\mathsf{Z}}(\wall) \subset {\mathsf{A}}$ be the subtorus defined by the condition $${\mathrm{Lie}}({\mathsf{Z}}(\wall) )=\bigcap_{H^{(n)}_{i,j}\in B_{\wall}} \, H^{(0)}_{i,j}.$$ A choice of $\wall$ also provides the decompositions
\[isubs\] I=I\_[1]{}…I\_[m]{} so that $i,j$ belong to the same subset if there is $H^{(n)}_{i,j} \in B_{\wall}$ and \[rdec\] =\_1+…+\_m with $|\rr_k|=|I_k|$. We recall the following well known property of quiver varieties known as tensor product structure:
\[framlem\] [*The fixed point set of the torus ${\mathsf{Z}}(\wall)$ has the following form: $$X(\nn,\rr)^{{\mathsf{Z}}(\wall)}=\coprod_{\nn_1+\dots+\nn_m=\nn} \, X(\nn_1,\rr_1)\times \dots \times X(\nn_m,\rr_m).$$ The characters which appear in the normal bundle $N_{X(\nn,\rr)^{{\mathsf{Z}}(\wall)}}$ are of the form $a_i/a_j$ with $i$ and $j$ from different subsets of (\[isubs\]).* ]{}
See Section 2.4 in [@MO].
For example, if $\wall=0$ then ${\mathsf{Z}}(\wall)=1$ is trivial and $X(\nn,\rr)^{{\mathsf{Z}}(\wall)}=X(\nn,\rr)$. In the “opposite” case, if $\wall$ is such that $B_{\wall}=\varnothing$ then ${\mathsf{Z}}(\wall)={\mathsf{A}}$ and thus $X(\nn,\rr)^{{\mathsf{Z}}(\wall)}=X(\nn,\rr)^{{\mathsf{A}}}$.
Informally speaking, we have the following picture. For each point $\wall\in {\mathrm{Lie}}_{\matQ}({\mathsf{A}})$ we associate a subvariety $X(\nn,\rr)^{{\mathsf{Z}}(\wall)}$ in $X(\nn,\rr)$. For a point $\wall$ which is in the complement of all hyperplanes, this subvariety is simply $X(\nn,\rr)^{{\mathsf{A}}}$. If $\wall$ arrives at a hyperplane then the subvariety gets larger. Further, if $\wall$ is at an intersection of two hyperplanes the fixed point set gets even larger and so on. Finally, when we arrive at the intersection of maximal number of hyperplanes the corresponding variety gets maximally large, i.e., $X(\nn,\rr)$.
Let ${\mathfrak{C}}$ and $P$ be a choice of a chamber and a polarization for a quiver variety $X(\nn,\rr)$. We denote the ${\mathsf{Z}}(\wall)$-invariant part of $P$ by $P(\wall)$. Clearly, $$P(\wall)=\bigoplus_{i=0}^{m} P_{i}$$ where $P_i$ is a polarization for $X(\nn_i,\rr_i)$. We denote by $\ind^{\wall}_{\lambda}$ the index of $\lambda$ associated with $P(\wall)$ and the chamber ${\mathfrak{C}}$.
As the varieties $X(\nn,\rr)$ and $X(\nn_i,\rr_i)$ are all associated to the same quiver, the map $\kappa$ is an isomorphism and we write $\mathfrak{D}'=\mathfrak{D}$.
For a point $\wall\in {\mathrm{Lie}}_{\matQ}({\mathsf{A}})$, as in the previous section, we denote by $\omega_{\wall}$ the translation acting on sections of line bundles over ${\mathscr{E}}_{{\mathsf{A}}}$ by: \[wsh\] \^[\*]{}\_f(a\_1,…, a\_[||]{})=f(a\_1 q\^[\_1]{},…, a \_[||]{} q\^[\_[||]{}]{}). The Theorem \[manth\] then gives:
\[manth2\] [*For any $\wall\in {\mathrm{Lie}}_{\matQ}({\mathsf{A}})$ we have: $$\begin{aligned}
\label{mainlim2}
&\Lambda^{\!\bullet}(\bar{P}(\wall)) \otimes \lim\limits_{z\to 0_{\mathfrak{D}}} \left( z^{\chi(\wall,\cdot)-\chi_{\lambda}(\wall,\cdot)} \lim\limits_{q\to 0} \omega^{*}_{\wall} \circ i^{*} \left( \dfrac{{\mathrm{Stab}}^{Ell}_{X(\nn,\rr),{\mathfrak{C}},P} (\lambda) }{\Theta(P)} \right) \right) \\ \nonumber
& =(-1)^{\mathrm{rk}(\ind_{\lambda}-\ind^{\wall}_{\lambda})} \hbar^{\lfloor \ind_{\lambda}\cdot \wall \rfloor}\, {\mathrm{Stab}}^{\mathfrak{D}}_{X(\nn,\rr)^{{\mathsf{Z}}(\wall)},{\mathfrak{C}},P(\wall)}(\lambda).
\end{aligned}$$*]{}
We have $\wall=(\wall_{1},\dots,\wall_{|r|})$ with $\wall_i\in \matQ$. We consider a cyclic subgroup of ${\mathsf{A}}$ with generator: \[nudef\] (a)=(a\_1,…, a\_[||]{}) (a e\^[2i ]{}) = (a\_1 e\^[2i \_1]{},…, a\_[||]{} e\^[2i \_[||]{}]{}). We note that $X(\nn,\rr)^{\mc}=X(\nn,\rr)^{{\mathsf{Z}}(\wall)}$. Indeed, all ${\mathsf{A}}$-weights appearing in the normal bundle to $X(\nn,\rr)^{{\mathsf{Z}}(\wall)}$ is of the form $a_i/a_j$ with $i$ and $j$ from different subsets of (\[isubs\]). From the definition of the hyperplane arrangement $$\wall_i-\wall_j\in \matZ \iff i,j \ \ \textrm{are in same subset of} \ \ (\ref{isubs}),$$ which gives $X(\nn,\rr)^{\mc}\subset X(\nn,\rr)^{{\mathsf{Z}}(\wall)}$. Is clear from (\[nudef\]) that $\mc \subset {\mathsf{Z}}(\wall)$ and so $X(\nn,\rr)^{{\mathsf{Z}}(\wall)}\subset X(\nn,\rr)^{\mc}$.
We see that the shift (\[wsh\]) satisfies the conditions described in Section \[wshift\] for $\mc$, and the result follows from Theorem \[manth\].
Yakov Kononov\
Department of Mathematics,\
Columbia University,\
New York, NY 10027, USA\
[email protected]
Andrey Smirnov\
Department of Mathematics,\
University of North Carolina at Chapel Hill,\
Chapel Hill, NC 27599-3250, USA\
[email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present an analysis of deep [*HST*]{}/WFC3 near-IR (NIR) imaging data of the globular cluster M4. The best-photometry NIR colour-magnitude diagram (CMD) clearly shows the main sequence extending towards the expected end of the Hydrogen-burning limit and going beyond this point towards fainter sources. The white dwarf sequence can be identified. As such, this is the deepest NIR CMD of a globular cluster to date. Archival [*HST*]{} optical data were used for proper-motion cleaning of the CMD and for distinguishing the white dwarfs (WDs) from brown dwarf (BD) candidates. Detection limits in the NIR are around $F110W\approx26.5$ mag and $F160W\approx27$ mag, and in the optical around $F775W\approx28$ mag. Comparing our observed CMDs with theoretical models, we conclude that we have reached beyond the H-burning limit in our NIR CMD and are probably just above or around this limit in our optical-NIR CMDs. Thus, any faint NIR sources that have no optical counterpart are potential BD candidates, since the optical data are not deep enough to detect them. We visually inspected the positions of NIR sources which are fainter than the H-burning limit in $F110W$ and for which the optical photometry did not return a counterpart. We found in total five sources for which we did not get an optical measurement. For four of these five sources, a faint optical counterpart could be visually identified, and an upper optical magnitude was estimated. Based on these upper optical magnitude limits, we conclude that one source is likely a WD, one source could either be a WD or BD candidate, and the remaining two sources agree with being BD candidates. For only one source no optical counterpart could be detected, which makes this source a good BD candidate. We conclude that we found in total four good BD candidates.'
author:
- 'A. Dieball'
- 'L. R. Bedin'
- 'C. Knigge'
- 'R. M. Rich'
- 'F. Allard'
- 'A. Dotter'
- 'H. Richer'
- 'D. Zurek'
title: 'Deep near-IR observations of the Globular Cluster M4: Hunting for Brown Dwarfs.'
---
Introduction {#intro}
============
Globular clusters (GCs) are the oldest and most massive stellar aggregates in our Galaxy. As such, they are the best natural laboratories to study large, co-eval populations of stars at known distance and metallicity. Indeed, much of our understanding of star formation and evolution has been derived from observational studies of GCs. Nonetheless, we still lack an understanding of the very low mass stars (VLMSs) around the faint end of the Hydrogen-burning main sequence (MS) and of objects beyond that limit, i.e., sub-stellar sources, so called “Brown Dwarfs” (BDs). This is especially true for objects at the low-metallicities typical for the GCs in our Milky Way (see below).
BDs present a link between stars and planets, and thus are important for our understanding of both star and planet formation and evolution. BDs are sub-stellar objects that are not massive enough to ignite and sustain Hydrogen burning. Thus, where low-mass stars will retain their luminosity for a Hubble time or longer, a BD will continue to cool and become fainter with age. Like giant gas planets, BDs have complex atmospheres [@burrows1997; @burrows2001]. The distinction between stars, BDs and planets is either based on mass or on formation. In general, stars have masses $> 80 M_J$ and can sustain Hydrogen burning, BDs have masses between 80 and 13 $M_J$ and cannot sustain Hydrogen burning but a short period of Deuterium burning, and giant planets have masses below 13 $M_J$ and cannot sustain Deuterium burning (Burrows et al. 1997, Stamatellos 2014, but see also Sect. \[Hmass\]). Because low-mass stars and BDs have life-times much longer than the age of the Galaxy, the GC VLMSs and BDs are also important tracers of Galactic formation and chemical evolution.
The formation of BDs is a matter of considerable dispute. They might have formed in the same way as (low-mass) stars from turbulent cloud fragmentation [@elmegreen1999; @wg2005; @andre2012], which would imply a continuous extension of the IMF into the sub-stellar regime. On the other hand, BDs might form e.g. from the ejection of stellar embryos or sub-stellar clumps which did not have the chance to accumulate enough mass [@rc2001; @bv2012]. Kroupa & Bouvier (2003) suggested that BDs form via photo-evaporation of protostars through nearby massive stars. This might also suggest an increase in the number of BDs with cluster mass, as more massive clusters have more O stars which can produce more BDs. BDs might also form from the fragmentation of circumstellar disks (Stamatellos et al. 2011, Thies et al. 2010, Kaplan et al. 2012). In this case, the number of BDs in clusters could be enhanced in dense clusters as dynamical interactions between cluster stars lead to more disk fragmentation. Thies et al. (2015) concluded that BDs likely form not just via one formation scenario, but from a combination of various channels.
Large surveys undertaken in the past decade have detected large numbers of BDs. For example, the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006), the Sloan Digital Sky Survey (SDSS; York et al. 2000), the United Kingdom Infrared telescope Deep Sky Survey (UKIDSS; Lawrence et al. 2007), the Wide-field Infrared Survey Explorer (WISE, Wright et al. 2010) all sky survey have been very successful in finding such cool, low-mass objects.[^1] However, for most of these BDs, key physical properties like metallicity and age are unconstrained (see e.g.). In fact, determining the physical parameters is extremely difficult and a major hurdle in BD research. Observations of open star clusters and star forming regions, where all sources are at the same distance and metallicity, can mitigate this problem (e.g. Steele et al. 1995, Rebolo et al. 1996, Martín et al.2001, Pinfield et al. 2003, Boudreault & Lodieu 2013, Casewell et al. 2014), but only for young and metal-rich objects. Thus the need for benchmark sources is especially evident for the metal-poor regime, and indeed we still do not know much about [*old, metal-poor*]{} BDs. So far, only very few low-metallicity, old VLMSs near the H-burning limit, and even fewer sub-stellar (candidate) halo objects have been identified (e.g. Burgasser et al. 2003, 2009, Lépine et al. 2004, Burgasser & Kirkpatrick 2006, Cushing et al. 2009, Sivarani et al. 2009, Murray et al. 2011, Mace et al. 2013, Pinfield et al. 2014, Luhman & Sheppard 2014, Burningham et al.2014, Kirkpatrick et al. 2014).
This is where GCs come in: they are massive, and thus might have produced BDs in large numbers, and they are also the oldest and most metal-poor stellar aggregates in our Galaxy. Potentially, GCs are the ideal hunting ground for old, metal-poor benchmark VLMSs and BDs which are much needed if we are to test stellar and sub-stellar formation and evolution theories and models of metal-poor (sub-)stellar atmospheres.
However, identifying substellar objects in GC is challenging due to their intrinsic faintness. Therefore, the closest GCs make the best targets for this kind of research. Out of the GCs in our Galaxy, M4 (NGC6121) is the closest GC to us; distance estimates range from 1.7 kpc (Hansen et al. 2004) to $\approx$ 2 kpc (e.g. Bedin et al.2009). Braga et al. (2015) estimated a true distance modulus of 11.28 mag based on RR Lyrae period-luminosity and period-Wesenheit relations, resulting in a distance of 1.8 kpc, which agrees well with previous estimates. Malavolta et al. (2014) analyzed 7250 spectra for 2771 cluster stars and found a metallicity of $\rm[Fe/H]=-1.07$ dex (RGB) to $\rm[Fe/H]=-1.16$ dex (sub giant branch and MS stars), which is well in line with previous metallicity estimates (e.g. $\rm[Fe/H]=-1.15$ dex according to the 2010 update of the Harris, 1996, catalogue of globular clusters in the Milky Way).
As such, M4 is a prime target for ultra-deep observational studies. Indeed, deep optical studies with the Hubble Space Telescope ([*HST*]{}) have been undertaken by Richer et al. (1997, 2004) and Bedin et al. (2009), yielding impressive results. Richer et al. (1997, 2004) estimated the fraction of similar-mass photometric binaries to be small (just 2% in their outer field, falling to just 1% towards the cluster core). Milone et al. (2012) suggested a much higher total binary fraction, raising from 10% at the halfmass radius to 15% towards the cluster core. The present day mass function of the lower-mass MS stars is flat (Bedin et al. 2001), with a slope of $\alpha = 0.1$ and a further flattening towards the cluster centre (Richer et al. 2004). The cluster WDs suggest that the initial mass function (IMF) above 0.8 $M_\odot$ was much steeper than the present day mass function. Bedin et al. (2009) presented the deepest optical colour-magnitude diagram (CMD) to date of this cluster and located the faint end of the WD cooling sequence in M4 at $F606W
= 28.5$ mag, suggesting an age of 11.6$\pm$0.6 Gyr. This agrees with the finding of Hansen et al. (2004), who found a WD based age of 12.1 Gyr. The ongoing [*HST*]{} M4 core project (PI L. Bedin, GO-12911) searches for binary dark companions to MS stars. The high-accuracy astrometry and photometry of this data-set, together with archival material and a part of the deep near-IR (NIR) data presented in this paper, have already been used to identify two distinct sequences along the lower-mass MS [@milone2014].
M4 is also so far the only GC known to host a planetary system, PSR B1620-26 (Sigurdsson et al. 2003), which challenged the planet-metallicity relation of the standard planet formation model at that time [@fischer]. Recently, Hasegawa & Hirashita (2014) suggested that the critical metallicity for gas giant formation is \[Fe/H\]$\approx −1.2$ dex, which agrees with M4’s metallicity. Beer et al. (2004) suggested a metallicity-independent formation scenario, in which the planet in M4 formed through dynamically induced instability in a circumbinary disc. If true, then we can expect many planets to form especially in the dense GCs in which dynamical interactions between cluster stars are ubiquitous. Since BDs might form in a similar way, we then might also expect that many more BDs form in dense GCs compared to open clusters.
This paper is structured as follows. In Section \[obs\] we describe the observations and reduction of our NIR and archival optical data. In Section \[cmds\] we present and discuss the NIR and optical-NIR CMDs, and our results and conclusions are summarized in Section \[summary\].
Observations {#obs}
============
NIR Data {#IRobs}
--------
The NIR observations of the globular cluster M4 were carried out in April 2012 with the Wide Field Camera 3 (WFC3) on board the [*HST*]{}, using the $F110W$ and $F160W$ filters (program GO–12602, PI: Dieball). All observations were made at a single pointing position on a field centred at about one core radius North East from the cluster centre. This region in the cluster had been the focus of the programs GO-5461 (Richer et al. 1997, 2004) and GO–10146 (Bedin et al. 2009) and thus is fully covered by deep optical observations. A standard 4-point WFC3-IRDITHER-BOX-MIN dither pattern and a sampling of NSAMP14 SPARS50 was applied during the observations to get a well-sampled point spread function (PSF). The WFC3 field of view is $136\arcsec
\times 123\arcsec$ in the IR channel, with a resolution of $0.13\arcsec \times 0.121\arcsec$ per pixel. All NIR observations were carried out during two consecutive orbits on April 16th ($F110W$) and four orbits on April 20th to 21st ($F160W$), comprising a total of 8 individual images in $F110W$ and 16 images in $F160W$, each 653 seconds, resulting in total exposure times of 5223 seconds ($F110W$) and 10447 seconds ($F160W$).
Using the pipeline produced flat-fielded (FLT) images, we first created a master image for each of our IR filters, using [ multidrizzle]{} running under [PyRAF]{} (the Python-based interface to IRAF[^2]). [multidrizzle]{} corrects geometric distortions that are present in the input images and combines them into a master image. Shifts between the individual images are expected, as we have applied a dither pattern. On top of this, telescope breathing can affect guide star tracking and as a result can cause small shifts, typically on a sub-pixel scale (see e.g. the [multidrizzle]{} handbook available on the STScI webpages). In order to ascertain that all shifts are taken into account, we created geometric-distortion corrected individual images. Based on the coordinates of the same 10 stars in each of these images (selected to be well distributed over the field of view), accurate shifts between the individual and the master images were then determined using [tweakshift]{}.
The master images created in this way are displayed in Figures \[f110\] and \[f160\]. Note that these master images serve as reference images for the positions of the stars detected by [ DOLPHOT]{}, but the photometry was actually performed on the individual images (see \[dolphot\] below).
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Photometry of the NIR data {#dolphot}
--------------------------
Photometry was performed on the individual FLT images using the [ DOLPHOT]{} software[^3] developed by A. Dolphin as a generalized version of [HSTphot]{} [@dolphin2000]. [DOLPHOT]{} runs on the FLT images downloaded from the STScI archive, i.e., no further processing of the images is necessary nor recommended. Photometry is performed on each of the input images, using the WFC3 module which replaces the analytic PSF model with a look-up table computed using Tiny Tim PSFs (Krist & Hook 2011). For our data, we used Jay Anderson’s PSF libraries for the WFC3/IR filters and the pixel area maps available from the [ DOLPHOT]{} webpage. The WFC3 module also includes a photometric calibration to the VEGAmag system. As a reference frame for a common physical (image) coordinate system we use our deepest image, i.e., the master image created with [multidrizzle]{}. The [DOLPHOT]{} package also provides routines to mask all pixels that are flagged as bad in the data quality arrays, to multiply with the pixel area maps, to calculate sky images, and to align all input images (or rather the source coordinates found in each input image) to the reference frame (our drizzled master images) using user defined lists of stellar coordinates in each input and the reference image. For these steps, we used the recommended WFC3 IR parameter settings.
Note that performing photometry on the drizzled image is expected to provide sub-optimal photometry, because the drizzling process affects the PSF and the noise characteristics of the drizzled images. Instead, [DOLPHOT]{} runs on all input images simultaneously and thus is capable of providing deep photometry. We started with the recommended parameters, and then refined some parameters to push our photometry as deep as possible.[^4]
[DOLPHOT]{} can run on data from multiple filters and cameras, thus we were able to do the photometry on both $F110W$ and $F160W$ data sets simultaneously. The output file includes x and y positions, photometric parameters like sharpness, crowding (the magnitude difference to the measured magnitude if no neighbouring sources would be fit simultaneously), the object type, and magnitudes in the VEGAmag system for all sources detected in the $F110W$ and $F160W$ data. A calibrated NIR CMD is plotted in Fig. \[cmdIRall\]. As can be seen, the CMD is exceptionally deep. The total catalogue contained 51311 sources, but a large number of those will be spurious detections.
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Optical Data and photometry {#optobs}
---------------------------
Among the archival [*HST*]{} material the only images which offer any hope of detecting BDs in the optical are those deep and in the reddest filters available, i.e., $F775W$ and $F814W$ (indeed, technically $>$2/3 of these pass-bands are already in the NIR wavelength range).
We extensively searched in the [*HST*]{} archive, and identified four $\sim$1200 seconds deep images taken with ACS/WFC in $F775W$ under program GO-10146 (PI: Bedin) as the optimal for our purposes, as those images were taken in low-sky mode, are well dithered, and collected in a well defined epoch in 2005.48 (i.e., about 6.8 years before the GO-12602 data).
Images were reduced with the software described in great detail in Anderson et al. (2008). Briefly, the method is essentially a PSF-fitting where all pixels from all images are simultaneously fitted using the appropriate PSFs which account for the spatial and temporal shape-variation on each individual image. The key to the method is an optimal knowledge on how to transform those pixels into a common reference frame and an exquisite empirical PSF modeling. This software is well tested and used in all the twelve works of the series “The ACS Survey of Galactic Globular Clusters” (see, Sarajedini et al.2007).
The first photometric run returned 19990 optical detections. As the detection of BDs might be only marginal in these optical images, we relaxed the finding criteria in a second photometric run, imposing to save all the local maxima, and not only the significant ones, resulting in 1.5 million detections. Although most of these local maxima likely are just fluctuations of the background noise, we can still use them to set an optical upper limit to the NIR-detected BD candidates.
For visual inspection of the WDs and BD candidates, stacked images were created from the CTE-corrected FLT images. The stacked images have pixels values resulting from median clipping of the CTE-corrected FLT individual images, and are supersampled by a factor of 2 in each direction (see Anderson et al. 2008 and Anderson & Bedin 2010 for further details).
The Colour-Magnitude Diagrams {#cmds}
=============================
We used the data in the two NIR filters $F110W$ and $F160W$ to create a NIR CMD. As the full NIR catalogue contains a large number of spurious detections, we created a best-photometry NIR CMD (described below in Sect. \[nircmds\]). However, since WDs and BD candidates have similar magnitudes and colours in the NIR, BD candidates cannot be distinguished based on the NIR data alone. We then used additional optical F775W data to create proper motion cleaned optical-NIR CMDs (see Sect. \[nircmds\]) in which the WD and MS sequence are clearly separate. Thus, the optical data are used to distinguish the WDs from the BD candidates.
Best-photometry NIR CMDs {#nircmds}
------------------------
In order to produce a clean CMD of only “good” stellar sources, we selected the output catalogue in such a way that the faint sources will be dominated by true stellar detections while keeping the number of spurious detections at bay.[**[^5]**]{} The resulting best-photometry catalogue includes 2526 sources, i.e, $\approx 5\%$ of all detections. The best-photometry CMD is remarkably clean, as can be seen in Fig. \[cmdIRsel\]. We can clearly see the MS delineating down towards the expected H-burning limit and going beyond that to fainter sources. Note that this CMD is not proper motion cleaned, as we show all NIR sources that satisfy our selection criterion on the photometry. All sources with magnitudes fainter than 24 mag in $F110W$ were visually inspected on the $F110W$ master image, resulting in 177 visually confirmed NIR sources around or fainter than the H-burning limit in $F110W$ (see Sect \[Hmass\] below).
The MS is narrow between the two MS “knees” ($F110W \approx 15$ mag and $F110W \approx 18$ mag). The first “knee” occurs around $T_{eff}\approx4500$ K and a mass of $\approx0.55 M_\odot$ and is due to the formation of molecules in the cool stellar atmospheres, and the second knee is a result of increasing electron degeneracy in the (sub)-stellar interior close to the H-burning limit (Baraffe et al. 1997). Below the second knee, the MS broadens towards fainter and lower-mass MS stars. Indeed, the low-mass MS splits into two branches, as shown in Milone et al. (2014), who had first pointed out multiple stellar generations among VLMSs in M4, based on high-precision deep optical [*HST*]{} data from the [*HST*]{} M4 core project and our NIR data. The split can be clearly seen in the NIR CMD (see Milone et al. 2014, their Fig. 2), as opposed to the optical data, demonstrating that the NIR is also an ideal waveband to search for multiple sequences along the lowest-mass and hence faintest MS. Note that the goal in Milone et al. (2014) was to search for multiple generations along the low mass MS based on very precise photometry. In contrast, in this paper, our goal is to go deep and well beyond the H-burning limit.
In our Fig. \[cmdIRsel\], we can see the MS delineating further down towards the expected end of the H-burning sequence, marked with red slashed lines and light-red shaded area. See Sect. \[Hmass\] below for a discussion on the mass and NIR magnitude at the H-burning limit. Around $F110W \approx 24.5$ mag the number of sources on the MS decreases and the MS peters out. On the blue side of the faint MS, the WD sequence can be seen, starting around $F110W < 22$ mag and $F110W - F160W \approx 0.2$ mag and going fainter and redder.
An increase in source number seems to be apparent below $F110W > 25$ mag, i.e., below the expected end of the H-burning sequence. This is the area in the CMD where we expect the BDs to appear. This happens to coincide with the WD sequence as well, i.e., this is the region where WD and BD cooling sequences would be expected to cross. Unfortunately, this also means that we cannot disentangle WDs and BD candidates based on our NIR data alone. In order to help with both cluster membership determination and distinguishing WDs and BD candidates, the deep optical data from GO–10146 were used (see Sect. \[optobs\] and Sect, \[optnircmds\]).
The WD as well as the BD regions have been indicated in Fig. \[cmdIRsel\]. For orientation purposes, we have plotted a WD sequence (blue line). The WD cooling sequence was constructed by interpolating on the Wood (1995) grid of theoretical WD cooling curves, adopting a mean WD mass of $0.55 M_\odot$. Using a grid of synthetic DA WD spectra kindly provided by B. Gänsicke (see Gänsicke et al. 1995) we carried out synthetic photometry with [ PySynphot]{}. Note that we have shifted the WD cooling sequence to get a reasonable match to the underlying CMD. This required a rather large distance of 2.2 kpc and a reddening of $E(B-V)=0.55$ mag (a standard reddening law (Seaton 1979) is built into [PySynphot]{}). Our cooling sequence starts at $T_{eff} = 50\,000$ K and terminates at $T_{eff} = 8\,000$, but note that the coolest WDs in M4 have temperatures as low as $T_{eff} = 4\,000$ (Bedin et al. 2009).
In addition, we have marked the location of the known field BD SDSS-J125637.13-022452.4 [@burgasser2009], which has a metallicity similar to M4 but is likely several Gyr younger. As a consequence, its cooling time is shorter and thus it is expected to be brighter than the M4 BDs. The observed $J$- and $H$-band magnitudes agree with a 5 Gyr old, $0.078 M_\odot$ source at a metallicity of \[M/H\]=-0.5 dex. We scaled the observed $J$- and $H$-band magnitudes to the WFC3 NIR filters and applied M4’s distance and reddening (following Hendricks et al. 2012, we adopted a reddening of $E(B-V) =
0.37$ mag, a true distance modulus of 11.28 mag, $R = 3.67$, $A_J =
0.302 * A_V$ and $A_H = 0.191 * A_V$) and over-plotted the field BD on our CMD. Its location supports that our data are indeed deep enough to reach well into the BD zone.
Two 12 Gyr isochrones have been overplotted on the best-photometry CMD, a BT-Settl model based on the Asplund et al. (2009) solar abundances and the Barber & Tennyson (2006) line list (Allard et al. 2012), for a metallicity of $\rm{[M/H]}=-1$ dex; and a Dartmouth model for $\rm{[Fe/H]}=-1.2$ and $\rm{[\alpha/Fe]}=+0.4$ (Dotter et al. 2008). Note that we do not attempt to derive cluster parameters from the isochrone fitting, instead, we have fit the isochrones by eye so that they best overlap with the underlying CMD[^6]. Note also that both sets terminate at a stellar mass of 0.083 $M_\odot$ (BT-Settl) or 0.1 $M_\odot$ (Dartmouth), i.e., they do not reach to the H-burning limit.
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Optical-NIR CMDs {#optnircmds}
----------------
Our NIR and optical catalogues have been matched using a six-parameter linear transformations between the star positions in the different epochs. As reference stars for the transformations we used only well-measured, isolated, non-saturated cluster stars with a high signal-to-noise and low residuals. The [*predicted positions*]{} of the first epoch sources in the second epoch are compared with the [ *observed positions*]{} and the displacements between first and second epoch are calculated. The top panel in Figures \[cmdj\], \[cmdh\], \[pmcmdIR110\] and \[pmcmdIR160\] shows the displacements in WFC3/IR, based on the total [DOLPHOT]{} NIR (51311 detections) and optical (19990 sources) catalogue. Since cluster stars have been used for the reference list, we expect cluster members to agglomerate around zero in the displacement vector point diagram. Indeed, two populations can be distinguished: a dense and tight agglomeration of data points around $\Delta Y = 0$ and $\Delta X = 0$ which denotes the cluster members, and a more widely spread data region centering around $\Delta Y \approx 0.75$ and $\Delta X \approx -0.5$ which denotes field sources. The latter are mostly Bulge sources, reflecting the low tangential motion of M4 around the Galactic center (see also Bedin et al. 2003).
The corresponding CMDs are plotted in the second row in Figures \[cmdj\], \[cmdh\], \[pmcmdIR110\] and \[pmcmdIR160\]. The left CMDs show all sources within up to two WFC3 pixels displacement, the middle CMDs show only sources with a displacement of no more than 0.1 pixels which suggests that they are cluster members, and all non-cluster sources are shown in the right diagrams. A displacement of 0.1 pixels corresponds to a proper motion of 1.9 mas/year, based on our timeline of 6.8 years and a pixel scale of 0.13$\arcsec$. This agrees well with e.g. Bedin et al. (2003, 2009) and Zloczewski et al. (2012). As can be seen, the proper-motion cleaned “cluster” CMDs (middle diagrams) show a well defined MS, terminating around $F775W\approx26$ mag in both CMDs in Figures \[cmdj\] and \[cmdh\], as well as a well defined WD sequence (light-blue data points) going down to the bottom of the WD sequence around $F775W\approx28$ mag.
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On the other hand, the proper-motion cleaned optical-NIR CMDs do not show any BD candidates, i.e., sources fainter than $F775W > 28$ mag and redder than $F775W - F110W > 3.5$ or $F775W - F160W > 4.5$ mag (see also Fig. \[models\]). This was expected, as the deep optical CMD presented in Bedin et al. (2009) did not show any potential MS sources fainter than the H-burning limit. However, and most importantly in the context of this paper, the BDs are expected to be much redder than the WDs in the optical-NIR CMDs. Indeed, the WD sequence and the MS are clearly separated in our optical-NIR CMDs. Therefore we can use the deep optical data set to identify the WDs in our NIR CMDs and disentangle WDs from potential BDs.
The minimum mass at the Hydrogen-burning limit {#Hmass}
----------------------------------------------
What is the minimum mass at the Hydrogen-burning limit in a low-metallicity cluster like M4? And, as a consequence, at which NIR magnitude and colour do we expect the H-burning limit in our NIR CMD? Early theoretical work (Kumar 1963) suggested a lower limit to the stellar MS at $\approx 0.07 M_\odot$ for population I and $\approx
0.09 M_\odot$ for the metal-poor population II stars, to which also GCs belong. Hayashi & Nakano (1963) suggested that stars less massive than $0.08 M_\odot$ cannot undergo Hydrogen burning. They further suggested that the limiting mass is not very different for Population I and Population II stars. Burrows et al. (1993) presented a zero metallicity theoretical model (i.e., Population III) which suggests a limiting H-burning mass as high as $0.094 M_\odot$. Treatment of the atmosphere has a considerable impact on the predicted limiting mass, as can be seen in Chabrier et al. (2000) who suggested a limiting mass of $0.065 M_\odot$ for models that include dust formation, Saumon & Marley (2008) who suggested a limiting mass of $0.075 M_\odot$ for cloudless models, and $0.070 M_\odot$ for cloudy models; all for solar metallicity and an age of 10 Gyr.
Previous theoretical works suggest that the H-burning limit is at higher masses for more metal-poor stars. The models roughly agree on a limiting mass of $0.075 M_\odot$ for solar metallicities. Following Hayashi & Nakano (1963), we conservatively assume that the H-burning mass for a population as metal-poor as M4 is between $0.075 M_\odot$ and $0.08 M_\odot$.
Unfortunately, detailed sub-stellar models for sub-solar metallicities are presently not available. Updated models that extend well into the BD regime are currently being computed (Allard, private communication).
However, the most recent set of BT-Settl models (Allard et al. 2012; 2013) suggest a strong metallicity dependence of the shape and luminosity of the low-mass MS. These models only go down to $0.083
M_\odot$ and are close to the H-burning limit (Allard private communication), but do not go beyond the stellar sequence into the sub-stellar regime. Thus, we linearly extrapolated the BT-Settl models (based on the Asplund et al. 2009 solar abundances) for $\rm[M/H]=-1$ dex and an age of 12 Gyr (closest to M4’s parameters) down to sub-stellar masses of $0.068 M_\odot$, and applied distance and reddening as in Fig. \[cmdIRsel\] for the BT-Settl model. The extrapolated models are plotted in Fig. \[models\], and the magnitudes around the H-burning limit are listed in Table \[mlim\]. As mentioned above, no sub-stellar models for low metallicities are currently available. Different models exist for a metallicity of $\rm{[M/H]} = 0.0$ dex. Thus, in Fig. \[compare\] we compare our extrapolated models with these more metal-rich models, all for an age of 12 Gyr and scaled to M4’s distance and reddening. The effect of the different atmospheric physics can be clearly seen in the shape and colour of the models. However, the expected end of the H-burning sequence between a $0.075 M_\odot$ and a $0.08 M_\odot$ is at comparable magnitudes in all sub-stellar models. This gives us some confidence that we can use the extrapolated metal-poor BT-Settl models to get an estimate of the magnitude and colour range of the H-burning limit. For more exact values we will have to wait for the updated metal-poor models, but we remind the reader that the main purpose of this project is to provide the metal-poor benchmark sources and thereby fill the observational plane with data points that are needed to constrain theoretical models.
Unlike stars, BDs cannot retain their luminosities via nuclear fusion. As a consequence, they cool with time, and a BD will be at fainter luminosities in an old GC compared to a young BD of the same mass and metallicity. To get an estimate of this effect, we used NextGen models (Baraffe et al. 1997, Baraffe et al. 1998; see Fig. \[BDcool\]). In the left diagram, we plot mass against $T_{eff}$ for various ages. The right diagram shows 1 Gyr and 10 Gyr NextGen models for different metallicities. Since substellar isochrones at M4’s low metallicity of \[M/H\]=-1 dex and ages of $\geq
10$ Gyr currently do not exist, we also plot isochrones for a metallicity of \[M/H\]=-0.5 and 0.0 dex which go down to lower masses. As can be seen, metallicity has a considerable impact on the shape of the isochrones as well as on the cooling time scale and hence fading of low-mass, substellar objects. The \[M/H\]=0 dex isochrones continue to extend with time to fainter magnitudes and redder colours, suggesting that a 0.075 $\rm{M}_\odot$ source fades by $\approx 0.7$ mag in $F110W$ from 1 to 10 Gyr, and a 0.08 $\rm{M}_\odot$ source becomes fainter by $\approx 0.2$ mag. The \[M/H\]=-0.5 dex isochrones suggest that a source of 0.079 $\rm{M}_\odot$ becomes fainter by $\approx 0.7$ mag, and a 0.08 $\rm{M}_\odot$ source becomes fainter by $\approx 0.5$ mag, but the sources also become bluer rather than redder. The \[M/H\]=-1 dex 10 Gyr isochrones terminates at 0.083 $\rm{M}_\odot$. According to the models, such a metal-poor, low-mass source already fades by 0.3 mag in $F110W$ from 1 to 10 Gyr. The models suggest that the blue-turn is more pronounced for lower-metallicities, and also metal-poor sources become fainter compared to metal-richer sources at the same mass, i.e. they cool faster.
Our optical-NIR CMDs presented in Figures \[cmdj\] and \[cmdh\] show that the observed MS peters out around $F775W\approx26$ mag, suggesting that we are approaching the end of the H-burning sequence. Our NIR CMD in Fig. \[cmdIRsel\] suggests that the MS peters out just below $F110W\approx24$ mag where the density of stars decreases. At fainter magnitudes, the WD sequence crosses the MS and the star density increases again. Based on the CMDs, we estimate our detection limits around $F775W\approx28$ mag, $F110W\approx26.5$ mag and $F160W\approx27$ mag. The detection limits are also indicated in Fig. \[models\] with a dotted line. Comparing the theoretical models in Fig. \[models\] with our observed CMDs and taking the detection limits and the predicted H-burning limit into account, we conclude that we have reached beyond the H-burning limit in our NIR CMD and are probably just above or around this limit in our optical-NIR CMDs.
[cccc]{} mass \[$M_\odot$\] & $F110W$ \[mag\] & $F160W$ \[mag\] & $F775W$ \[mag\]\
0.075 & 25.170 & 24.736 & 30.105\
0.077 & 24.771 & 24.281 & 29.327\
0.080 & 24.172 & 23.598 & 28.160\
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BD candidates {#bdcands}
-------------
As with the best-photometry faint NIR sources, we visually inspected all cluster WD candidates, i.e., sources whose proper motion (or rather displacements) suggest that they are cluster members and whose position in the optical-NIR CMDs suggest that they are WDs. Our first photometric run on the optical data did not return an optical counterpart for 59 of the best-photometry faint NIR sources. We then over-plotted the positions of those 59 faint NIR sources (i.e., [ *without*]{} an initial optical counterpart) on the optical $F775W$ master image and inspected each position by eye. Visual inspection of the $F775W$ master image showed a faint optical source at or close to the location of the NIR source in most cases. Thus, for the second optical photometric run (see Sect. \[optobs\]), the parameter settings were relaxed so that all local maxima were retained, resulting in 1.5 million detections. Nearly all of those are just spurious detections (i.e., background noise or spikes in PSF wings that are not real stellar sources), however, a further 47 optical counterparts to the faint best-photometry NIR sources were detected and thus were added to the initial list of optical-NIR matches.
For the remaining twelve faint NIR sources, still no optical counterpart was returned. However, out of those, five are located on PSF streaks and two in the ACS WFC chip gap, so that nothing can be said about an optical counterpart. For four of the remaining five sources, visual inspection of the $F775W$ master image seem to indicate an optical source on the position of the NIR source, but no photometric measurement was possible. However, we used nearby stars of similar brightness (based on pixel counts) to estimate magnitude limits. One of these sources appears to have an optical counterpart in the centre of our search circle, probably at $F775W\approx26$ mag. This is too bright for a BD candidate and thus makes this source a WD candidate. Thus, we do not consider it further. The remaining four sources, id 1 to 4, with no optical photometry are listed in Table \[bdcand\], and images are shown in Fig. \[bdimas\]. For comparison, we also show images of WDs selected from the proper motion cleaned CMD and which have similar NIR magnitudes to the four NIR sources without optical photometry (see Fig. \[wdimas\] and Table \[wdcand\]).
At the rim of our search circle for source 1, an optical source can be seen. This, however, is too far away ($0\farcs1$ away from the position of the NIR source) and would not agree with being a cluster member. In the centre of our search circle, an optical source just might be visible. If true, this source would have $F775W\gtrsim28$ mag, i.e., the local detection limit. For source 2, an optical counterpart is visible in the centre of the search circle, again this source would be close to the detection limit at $F775W\gtrsim28$ mag. Thus, both Source 1 and 2 are likely (massive) BD candidates. Unfortunately, Source 2 is close to a saturation streak from a nearby bright stars in the optical image. The potential counterpart, however, can be clearly distinguished. Source 3 shows a “bright” optical source at the rim of our search circle, probably at $F775W\approx25.8$ mag, again based on the magnitudes of nearby stars that appear to be of similar brightness. However, this optical source is $0\farcs15$ away from the position of the NIR source and thus too far away to agree with being a cluster member. A very faint optical source might just be in the centre of the search circle, if true, then this optical source would be at $F775W>28.6$ mag, the local detection limit, making source 3 a good BD candidate. Just one source, source 4, does not show an optical counterpart at the location of the NIR source (i.e., within our search circle), and is thus our best BD candidate. Since the optical photometry did not return magnitudes fainter than $F775W = 32.9$ mag, an optical counterpart to BD candidate 4 must be fainter than this absolute optical limit. Note that Source 4 appears somewhat extended and could probably consist of two or three faint sources, or possibly an extended object (although we expect galaxies to be much bluer, see Bedin et al. 2009).
The positions of the four BD candidates are indicated in the CMDs in Fig. \[pmcmds\]. Since we do not have an optical measurement, we provide the upper optical magnitude limits given in Table \[bdcand\]. We also overplot the extrapolated BT-Settl 12 Gyr isochrone, using the distance and reddening parameters derived from the best fit in the NIR CMD. As can be seen, all four BD candidates are very close to the extension of the MS into the BD regime, which supports our classification of these sources as good BD candidates. Source 2 is blueward of the isochrone. Its position in the optical-NIR CMDs agrees with this source being a faint WD at the very bottom of the (optical) WD cooling sequence in M4, but it also agrees with this source being a VLMS star or a massive BD. In the NIR CMD this source is [*not*]{} at the very bottom of the WD sequence. We suggest that this source is a good BD candidate. Sources 1, 3 and 4 all are very close to the MS, and their position in the optical-NIR CMDs does not suggest that these sources are WDs, but rather BDs. WDs are much bluer and brighter than our BD candidates.
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[lllllll]{} id$_{BD}$ & RA & DEC & $F110W$ & $F160W$ & $F775W$ & comment\
& \[h:m:s\] & \[$^\circ:\arcmin:\arcsec$\] & \[mag\] & \[mag\] & \[mag\] &\
1 & 16:23:41.701 & -26:29:17.82 & $24.27\pm0.02$ & $23.39\pm0.11$ & $\gtrsim$28 & BD candidate\
2 & 16:23:45.443 & -26:30:05.58 & $25.41\pm0.05$ & $24.75\pm0.50$ & $\gtrsim$28 & WD/BD candidate\
3 & 16:23:44.711 & -26:29:30.90 & $24.36\pm0.02$ & $23.60\pm0.11$ & $>28.6$ & BD candidate\
4 & 16:23:45.371 & -26:29:37.40 & $26.75\pm0.16$ & $26.13\pm0.18$ & $>$32.9 & BD candidate\
[llllll]{} id$_{WD}$ & RA & DEC & $F110W$ & $F160W$ & $F775W$\
& \[h:m:s\] & \[$^\circ:\arcmin:\arcsec$\] & \[mag\] & \[mag\] & \[mag\]\
1 & 16:23:44.354 & -26:30:26.09 & $25.70\pm0.06$ & $25.10\pm0.24$ & $27.184\pm0.44$\
2 & 16:23:36.848 & -26:29:33.58 & $26.30\pm0.11$ & $25.62\pm0.40$ & $27.618\pm0.44$\
3 & 16:23:36.953 & -26:29:33.81 & $25.67\pm0.06$ & $25.14\pm0.48$ & $26.607\pm0.23$\
4 & 16:23:47.062 & -26:30:37.80 & $25.68\pm0.07$ & $25.04\pm0.37$ & $26.368\pm0.90$\
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Figures \[pmcmdIR110\] and \[pmcmdIR160\] again show the NIR CMDs, but only for those NIR sources for which an optical counterpart had been found (black data points). The middle CMDs show only NIR sources whose optical counterparts agree with being cluster members, based on the displacement vector point diagram. We also show best-photometry NIR sources [*without*]{} optical counterparts, plotted in red. As mentioned in Sect. \[nircmds\], all 177 best-photometry NIR sources fainter than $F110W = 24$ mag have been inspected on the $F110W$ and the optical $F775W$ master images. Out of these faint 177 NIR sources, 165 have an optical counterpart. Out of these 165 faint NIR/optical sources, 48 are cluster members based on their proper motion (displacements), and are located on the WD sequence. The remaining five faint best-photometry NIR sources which are not on PSF streaks or on the chip gap, and [*without an optical counterpart*]{} (and hence without a proper motion estimate), all make good BD candidates to start with. Visual inspection of these five sources and estimates on the optical magnitude limit (see above) suggests that one is probably a WD candidate, and the remaining four sources, listed in Table \[bdcand\], are good BD candidates.
Expected number of BDs
----------------------
How many BDs can we expect? This number is highly uncertain and depends on the assumed BD formation scenario (see Sect. \[intro\]). Furthermore, given our detection limits we can only expect to find the most massive BDs with masses larger than 0.068$M_\odot$ (based on the extrapolated BT-Settl models). Richer et al. (2004) derived a rather flat present-day mass function for M4. Extrapolating towards fainter and lower-mass stars, they estimate that between 15 to 50 VLMSs with masses between 0.085 and 0.095 $M_\odot$ should be in their field of view (GO-8679, WFPC2 data). Using the theoretical models (Fig. \[models\]) we can count the number of VLMSs in our field. If we only consider NIR sources that have an optical counterpart and a proper motion that suggests that they are cluster members, and best-photometry NIR sources without an optical counterpart, we find 23 VLMSs in a mass interval between 0.08 and 0.09 $M_\odot$. Assuming that the mass function is flat and that the slope of the mass function does not change considerably across the stellar/sub-stellar border, we can expect a similar number of BDs with masses between 0.070 and 0.08 $M_\odot$. However, the number of BDs formed per star is probably more around $\frac{1}{5}$ (see e.g. Thies et al. 2007). In this case, we can expect $\approx$ 5 BDs. This is of course a very rough estimate, but it does agree with our finding of four BD candidates.
Field contamination
-------------------
How many foreground or background sources can we expect in our cluster CMD? We can simply count the number of sources that are well outside the area covered by the cluster and the clump of field stars around $\Delta Y \approx 0.75$ pixels and $\Delta X \approx -0.5$ pixels in the vector point diagram. We find a field star density of 165 field stars per $\Delta \rm{pixels}^2$. Scaling this number to the area covered by our cluster stars, i.e., a displacement of 0.1 WFC3 pixels, we find that we can expect 5.2 field stars in our cluster CMDs.
How many foreground stars can we then expect to have NIR magnitudes and colours similar to our BD candidates? Selecting only field stars with $24 < F110W < 27$ mag and $0 < F110W-F160W < 1$ mag, the field star density is reduced to 50 stars/$\Delta \rm{pixels}^2$, and scaled to the area covered by the cluster in the vector point diagram , we then find that 1.6 such sources can be expected among our pm selected faint cluster members. Thus, about half of our suggested BD candidates might actually be foreground or background sources that happen to move with the cluster velocity across the plane of the sky.
Summary and Conclusion {#summary}
======================
We have presented the deepest NIR [*HST*]{}/WFC3 study of the GC M4 to date. The NIR data were proper-motion cleaned using archival deep optical [*HST*]{}/ACS ($F775W$) data. Our best-photometry NIR CMD reveals a narrow MS delineating down towards the expected end of the H-burning sequence. 177 best-photometry NIR sources fainter than the H-burning limit in $F110W$ ($F110W > 24$ mag) could be identified in our $F110W$ master image. For 165 of these faint NIR sources, an optical counterpart was found, 48 of these are cluster members according to their proper motion. All of these 48 faint cluster sources are on the WD sequence.
We found in total five faint NIR sources for which the optical photometry did not return a measurement (and which are not on PSF streaks or on the chip gap). We then visually inspected the positions of these faint NIR sources on the optical images and estimated, where possible, upper optical magnitude limits of potential optical counterparts that just might be visible. One source is likely another WD and rejected as a BD candidate. Based on the upper optical magnitude limits, we indicate the position of the remaining four sources in the optical-NIR CMDs. One of the sources (source 4 in Table \[bdcand\]), does not show an optical counterpart at all, which implies that its optical counterpart must be fainter than the absolute optical detection limit of $F775W > 32.9$ mag. This source appears to be somewhat extended in the NIR image, which might indicate multiple faint sources, i.e. multiple BDs, or possibly a galaxy. However, its position in the CMDs does agree with this source being a BD. One source (source 2 in Table \[bdcand\]) might be another WD candidate, but its position in the optical-NIR CMD also agrees with this source being a massive BD or a VLMS star at the bottom of the MS. The remaining two sources also have positions that indicate that these sources are massive BDs. We conclude that we have found four good BD candidates, but we caution that further studies and deeper optical data are necessary to confirm their status and cluster membership.
We are grateful to an anonymous referee for her/his valuable comments which helped to improve this paper. A.D. thanks Andrew Dolphin for helpful discussions about [DOLPHOT]{}. This work was supported by NASA through grant GO-12602 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555. A.D. acknowledges support from the People Program (Marie Curie Actions) of the European Union’s Seventh Framework Program FP7-PEOPLE-2013-IEF under REA grant agreement number 629579. L.R.B.acknowledges PRIN-INAF 2012 funding under the project entitled: “The M4 Core Project with Hubble Space Telescope”.
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[^1]: Compilations of BDs and low-mass stars can be found on DwarfArchives.org, the “List of Brown Dwarfs” maintained by W. R. Johnston on http://www.johnstonsarchive.net/astro/browndwarflist.html, and the “List of all ultracool dwarfs” maintained by J. Gagne on https://jgagneastro.wordpress.com/list-of-ultracool-dwarfs/
[^2]: IRAF (Image Reduction and Analysis Facility) is distributed by the National Astronomy and Optical Observatory, which is operated by AURA, Inc., under cooperative agreement with the National Science Foundation.
[^3]: http://americano.dolphinsim.com/dolphot/
[^4]: We used [Force1 = 1]{}, [FlagMask = 4]{} to eliminate saturated stars, [WFC3IRpsfType = 1]{} for the Anderson PSF cores, [FitSky = 2]{}, [SigFind = 1.5]{}, [ SigFindMult = 0.8]{}, [SigFinal = 1.5]{}, and [RPSF = 15]{}.
[^5]: For this purpose, we settled on a selection that only includes source with object type 2 or less (i.e., only “stars”). We selected a sharpness between -0.05 and 0.05 (a sharpness of zero denotes a perfectly-fit star, a negative value indicates a broader source, a positive value indicates a source too “sharp”, for example a cosmic ray - for an uncrowded field, sharpness values between -0.3 and 0.3 are recommended in the [DOLPHOT]{} manual, so we apply a stricter selection criterion here). The crowding parameter indicates how much brighter a source would be if nearby stars would not have been measured simultaneously. A crowding of zero indicates an isolated star. We allowed for a crowding of no more than 0.3.
[^6]: The best fit was achieved with the reddening law from Hendricks et al. 2012 but a larger $A_V = 1.9$ for the Dartmouth isochrone. For the BT-Settl isochrone, we chose a smaller distance modulus of 11 mag and $A_V = 1.8$. The difference in shape of the isochrones, as well as the difference in the best-fit parameters, reflect the differences in the underlying physics, i.e. treatment of the stellar atmospheres including molecules. For a more in-depth discussion of the input physics to the models we refer the reader to the BT-Settl and Dartmouth webpages and the references given there (https://phoenix.ens-lyon.fr/Grids/BT-Settl/ and http://stellar.dartmouth.edu/models/).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report precision Doppler measurements of intermediate–mass subgiants obtained at Lick and Keck Observatories. All stars show variability in their radial velocities consistent with planet–mass companions in Keplerian orbits. We find a planet with a minimum mass $M_P\sin{i} = \msiniC$ in a day orbit around , a planet with a minimum mass of in a day orbit around , and a planet with a minimum mass of in a day orbit around . Mass estimates from stellar interior models indicate that all three stars were formerly A–type, main–sequence dwarfs with masses ranging from to . These long–period planets would not have been detectable during their stars’ main–sequence phases due to the large rotational velocities and stellar jitter exhibited by early–type dwarfs. There are now 9 “retired” (evolved) A–type stars ($M_* > 1.6$ ) with known planets. All 9 planets orbit at distances $a \geq 0.78$ AU, which is significantly different than the semimajor axis distribution of planets around lower–mass stars. We examine the possibility that the observed lack of short–period planets is due to engulfment by their expanding host stars, but we find that this explanation is inadequate given the relatively small stellar radii of K giants ($R_* < 32$ $=0.15$ AU) and subgiants ($R_* < 7$ $ = 0.03$ AU). Instead, we conclude that planets around intermediate–mass stars reside preferentially beyond $\sim$0.8 AU, which may be a reflection of different formation and migration histories of planets around A–type stars.'
author:
- 'John Asher Johnson, Debra A. Fischer, Geoffrey W. Marcy, Jason T. Wright, Peter Driscoll, R. Paul Butler, Saskia Hekker, Sabine Reffert, Steven S. Vogt'
title: 'Retired A Stars and Their Companions: Exoplanets Orbiting Intermediate–Mass Subgiants$^1$'
---
Introduction
============
Very little is known about the occurrence rate and orbital properties of planets around A–type stars, corresponding to stellar masses ranging from 1.6 to 3.0 . Inspection of the Catalog of Nearby Exoplanets (CNE)[^1] reveals that only 6 of the 173 stars with securely detected planetary companions have masses in excess of 1.6 [@butler06]. This small number of detections is not a true reflection of the occurrence of planets around A–type stars, but rather the result of a strong selection bias against early–type, main–sequence stars in precision Doppler surveys.
Measuring precise Doppler shifts of early–type dwarfs is complicated by their rotationally broadened spectral features, high surface temperatures, and high levels of excess radial velocity noise, or “jitter” [@saar98; @wright05]. @galland05 find that Doppler precision for early–type dwarfs is limited to $\sim 40$ at spectral type F5V, and 90–200 for A stars, rendering Doppler measurements of these stars sensitive only to planets with large masses and short orbital periods. The lowest mass companion so far detected around an A star is the brown dwarf orbiting HD180777 [@galland06]. Even though the 28 day orbital solution has a large velocity semiamplitude, $K
=1200$ , the signal is only a 3$\sigma$ detection above the stellar jitter and measurement uncertainties.
Most of what is known about planet formation around intermediate–mass stars comes from two primary sources: direct imaging of disks around young stars and Doppler detections of planets around evolved stars. While A–type dwarfs are poor Doppler targets, their high intrinsic luminosities facilitate the detection and direct imaging of material in their circumstellar environments. More than a decade before the discoveries of the first extrasolar planets, evidence of planet formation outside of our Solar System came from the infrared detection of collision–generated dust around the A–type, main sequence stars Vega [@aumann84] and $\beta$ Pic [@smith84]. Since then, advances in high–contrast imaging have resulted in the detection of an optically thick disk around a pre–main–sequence Herbig Ae star [@perrin06], as well as scattered light images of optically thin “debris disks” around 11 main–sequence stars—the majority of which have spectral types F5V or earlier [Table 2 of @kalas06; @schneider06; @wahhaj07 and references therein]. Recent observations of the debris disk around the young A star Fomalhaut have revealed a perturbation in the disk structure that may be due to the influence of an orbiting Jovian planet [@kalas05]. Studying the relationships between the architectures of disks around young A stars and the distribution of planet properties around their older counterparts will provide key tests of planet formation models.
A key to finding planets around A stars using Doppler methods is provided by the effects of stellar evolution. As stars evolve away from the main sequence, they become cooler and rotate slower, which increases the number of narrow absorption lines in their spectra . Several Doppler surveys have focused on evolved, intermediate–mass stars on the red giant branch [@frink02; @mitchell04; @hatzes05; @lovis05] and clump giant branch [@sato03; @setiawan03]. These surveys have resulted in the discovery of 6 substellar companions orbiting former A–type stars (Table \[massive\_table\]). That none of these planets would have been detectable during their host stars’ main–sequence phases highlights the important role evolved stars play in the study of planets around intermediate–mass stars.
Here we present new planet candidates around stars with $M_*
> 1.6$ . These detections come from our precision Doppler survey of evolved stars on the subgiant branch of the H–R diagram. We discussed the selection criteria of our target stars in @johnson06b, along with the discovery of an eccentric hot Jupiter orbiting the 1.28 subgiant HD185269. We discuss our spectroscopic observations and Doppler measurement technique in § \[observations\]. In § \[stars\], we present the characteristics of the host stars along with the orbital solutions for their planet candidates. We conclude with a comparison of the semimajor axis distributions of planets around A–type stars and lower–mass stars in § \[summary\].
Observations
============
We are monitoring a sample of 159 evolved stars at Lick and Keck Observatories [@johnson06b]. At Lick Observatory, the Shane 3m and 0.6m Coude Auxiliary Telescopes (CAT) feed the Hamilton spectrometer [@vogt87], which has a resolution of $R \approx 50,000$ at $\lambda = 5500$ Å. Spectroscopic observations at Keck Observatory were obtained using the HIRES spectrometer with a resolution of $R \approx
80,000$ at $\lambda = 5500$ Å [@vogt94]. Doppler shifts are measured from each spectrum using the iodine cell method described by @butler96 [see also @marcy92b]. A temperature–controlled Pyrex cell containing gaseous iodine is placed at the entrance slit of the spectrometer. The dense set of narrow molecular lines imprinted on each stellar spectrum from 5000 to 6000 Å provides a robust wavelength scale for each observation, as well as information about the shape of the spectrometer’s instrumental response.
Traditionally, the Doppler shift of each stellar observation is made with respect to an observed, iodine–free stellar template spectrum. These template observations require higher signal and resolution than normal radial velocity observations, which leads to increased exposure times. Given our large target list and the small aperture of the CAT, obtaining an observed template for each star would represent a prohibitive cost in observing time. We therefore perform a preliminary analysis of each star’s observations using a synthetic, “morphed” template spectrum following the method described by @johnson06. Stars showing conspicuous Doppler variations are reanalyzed using a traditional, observed template to verify the signal and search for a full orbital solution.
Doppler measurements from Keck and Lick Observatories for four stable subgiants are shown in Figure \[std\_stars\]. The error bars represent the internal uncertainties of each measurement, which are approximated by the weighted standard deviation of the mean velocity measured from each of the 700 individual 2 Å wide chunks in each spectrum [@butler96]. We typically achieve internal measurement uncertainties of 1-2 for Keck observations and 3-5 at Lick. Subgiants have an additional 4-6 of “jitter”—velocity scatter in excess of internal errors due to astrophysical sources such as pulsation and rotational modulation of surface features [@saar98; @wright05]. We therefore adopt a jitter value of 5 for our subgiants, which is added in quadrature to the internal uncertainties of the measurements before searching for a best–fit orbital solution.
After determining the best–fit Keplerian solution using a Levenberg–Marquardt, least–squares minimization, we estimate the orbital parameter uncertainties using a bootstrap Monte Carlo method. We first subtract the best–fit Keplerian from the measured velocities. The residuals are then scrambled and added back to the original measurements, and a new set of orbital parameters is obtained. This process is repeated for 1000 trials, and the standard deviations of the parameters from all trials are adopted as the formal, 1$\sigma$ uncertainties.
Stellar Properties and Orbit Solutions {#stars}
======================================
Estimates of Stellar Properties {#stellar}
-------------------------------
We estimated the stellar properties of our target stars using two primary methods: the LTE spectral synthesis method (SME) described by @valenti05, and the Padova[^2] stellar interior models. The spectral synthesis method uses a non–linear least–squares algorithm to vary the parameters of a synthetic spectrum to search for a fit to an iodine–free stellar template spectrum. The free parameters in the fit are the abundances of heavy elements; effective surface temperature, $T_{\rm eff}$; surface gravity, $\log{g}$; and broadening effects due to the star’s projected rotation velocity, . @valenti05 estimate a precision of 0.04 dex in metallicity, 44 K in effective temperature, 0.3 dex in $\log{g}$, and 0.5 in rotational velocity.
To estimate stellar masses, radii, luminosities and ages, we used the Padova theoretical stellar models, which have been transformed into several photometric systems by @girardi02. Stellar properties can be inferred by interpolating a star’s color, absolute magnitude and metallicity onto these model grids. However, the @girardi02 model grids are defined at widely–spaced metallicity intervals, with \[Fe/H\] $ = $ -0.4, 0.0, +0.18 and +0.30. Since the uncertainties in our spectroscopically derived metallicity estimates are much less than the model grid intervals, and because the derived stellar properties do not vary linearly with \[Fe/H\], we could not simply perform a linear, 3-dimensional interpolation of $M_V$, $B-V$ and \[Fe/H\]. Instead, we first linearly interpolate the stars’ colors and absolute magnitudes onto each of the four metallicity grids. We then use a cubic spline interpolation between the grid points to measure the desired stellar property (e.g. mass) at the star’s measured \[Fe/H\]. Our procedure is illustrated in Figure \[massint\], which shows stellar mass as a function of \[Fe/H\] for each star’s absolute magnitude and color. The same procedure was used for stellar radii, luminosities and ages.
We compared our interpolated stellar properties to the @takeda06 theoretical interior models of the stars in the Spectroscopic Properties of Cool Stars catalog [SPOCS @valenti05]. We found a subset of 11 evolved stars in the catalog with $2.0 < M_V < 3.0$ and $0.7 < B-V < 1.1$. Differences between our inferred values and those from @takeda06 had an rms scatter of 7% in mass, 12% in radius, with a median offset of -2% and -4% in each parameter, respectively. Ages of this subset of evolved stars estimated by the two methods have a difference of -0.4 Gyr with and rms scatter of 1.1 Gyr. We therefore adopt fractional uncertainties of 7% for our derived masses, 12% for radii and 1 Gyr for ages. We list the full set of derived stellar properties of the three candidate planet host stars in Table \[stellartable\]. We summarize each star’s properties and orbital solution in the following subsections.
() is listed with a G5 spectral type in the *Hipparcos* Catalog, with $V = \vmagC$, $B - V = \bvC$ and a parallax–based distance of pc [@hipp]. However, no luminosity class is given. Based on its distance, we calculate $M_V = \mvC$, which at its $B-V$ color places the star 3.7 mag above the mean main–sequence of stars in the Solar neighborhood, as defined by @wright04. Based on its color and absolute magnitude, we find that is likely a G8IV subgiant near the base of the red giant branch. Commensurate with its evolved status, is chromospherically inactive, with $=$ and $=$ as measured from the CaII H&K line core and averaged over all observations [@wright04b].
Based on our LTE spectral analysis, we find that is metal–poor, with $\rm [Fe/H] = -0.15$, and slowly rotating, with $ = \vsiniC$ . The other stellar parameters derived from our spectral analysis are listed in Table \[stellartable\]. We interpolated the star’s color, absolute magnitude and metallicity onto the @girardi02 theoretical stellar model grids using the method described in § \[stellar\]. Our interpolation yields a stellar mass $M_* =
\mstarC$ , radius $R_* = \rstarC$ , and an age of Gyr.
We began observing in 2004 May at Lick Observatory using the 3 m Shane Telescope and 0.6 m CAT. Table \[vel192699\] lists our velocity measurements, along with their times of observation and internal measurement uncertainties (without jitter). Our first 7 observations, initially analyzed using a synthetic stellar template spectrum [@johnson06], showed correlated variations spanning two observing seasons. We obtained a high–quality observed template using the Shane 3m telescope and initiated intensive follow–up observations during the Fall 2006 observing season. The Keplerian signal is visible to the eye (Figure \[orbitC\]), obviating a periodogram analysis.
The best–fit Keplerian orbit has a period of $P = \pC$ d, velocity amplitude $K =
\kC$ , and eccentricity $e = \eC \pm \eeC$. With an assumed stellar mass of , we estimate a minimum planet mass $ =
\msiniC$ and orbital separation $a = \arelC$ AU. The fit has $\rm rms = \rmsC$ and a reduced $= \chiC$, consistent with the measurement errors and jitter. The full set of orbital parameters and uncertainties is listed in Table \[orbittable\].
{#starA}
(, HR8461) is listed in the Hipparcos catalog as a K1 star (no luminosity class given) with $V = \vmagA$, $B-V = \bvA$, and a parallax–based distance of pc. Given its distance and apparent magnitude, we calculate an absolute magnitude $M_V = \mvA$, which places it 4.2 mag above the average main sequence of stars in the Solar neighborhood [@wright04]. We therefore estimate that is a class K1IV subgiant near the base of the red giant branch.
Based on our LTE spectral analysis, we find that is somewhat metal–rich, with $=\feA \pm 0.04$, and slowly rotating, with $ = \vsiniC$ . Our interpolation of the star’s color, absolute magnitude and metallicity onto the @girardi02 stellar model grids yields a stellar mass $M_* =
\mstarA$ , stellar radius $R_* = \rstarA$ , and an age of Gyr. Consistent with its post–main–sequence evolutionary status, is chromospherically inactive with $=$ and $=$ , as measured from its CaII H&K emission [@wright04b]. The other stellar parameters derived from our spectral analysis and stellar model interpolation are listed in Table \[stellartable\].
We began monitoring in 2004 August at Lick Observatory. The first 9 observations were Doppler–analyzed using a synthetic template, and showed excessive variability with $\rm
rms=19$ . We then obtained a traditional, observed template to confirm the variations with higher Doppler precision. The full set of velocities is listed in Table \[vel210702\] (without jitter) and plotted in Figure \[orbitA\]. The error bars in Figure \[orbitA\] have been augmented by adding 5 of jitter in quadrature to the internal measurement uncertainties.
The best–fit Keplerian orbital solution is shown in Figure \[orbitA\] overplotted on the velocities. The solution has a day period, an eccentricity $e = \eA \pm \eeA$, and a semiamplitude $K = \kA$ . The fit residuals have $\rm rms =
\rmsA$ and reduced $=$ , consistent with the internal measurement uncertainties and jitter. Assuming a stellar mass $M_* = \mstarA$ , the best–fit solution yields a relative separation $a
= \arelA$ AU.
We find that the inclusion of a linear trend in the orbital solution yields a slight improvement in the quality of fit, decreasing the rms scatter of the residuals from to 6.7 , and the reduced from to 1.00 after accounting for the extra free parameter in the Keplerian–plus–trend model. We tested the validity of the trend using the prescription of @wright07, and found a false–alarm probability of 49%. The large FAP indicates that the apparent linear trend is likely due to noise rather than and additional orbital companion. Indeed, the trend appears to be driven primarily by the three outliers near JD $=$ 100, 400 and 800 (Figure \[orbitA\]). We therefore favor the single–planet Keplerian model summarized in Table \[orbittable\].
{#sectionB}
() is listed in the *Hipparcos* Catalog as a G8V star with $V = \vmagB$, $B-V = \bvB$ and a parallax–based distance of pc [@hipp]. Given its distance, the star has $M_V = \mvB$, placing it 3.5 mag above the mean main–sequence of stars in the Solar neighborhood [@wright04]. Like most evolved stars, is chromospherically quiet with $=$ and $=$ [@wright04b]. Its low chromospheric activity and location in the H–R diagram indicate that is most likely a luminosity class IV star on the subgiant branch, rather than a class V dwarf.
is listed in the SPOCS Catalog [@valenti05] with a metal abundance slightly below Solar ( $ = \feB \pm 0.04$) and projected rotational velocity $= \vsiniB$ . Interpolation of the star’s $B-V$ color, absolute magnitude and metallicity onto the @girardi02 stellar model grids yields a stellar mass $M_* =
\mstarB$ , radius $R_* = \rstarB$ , and an age of Gyr. The interior models of @takeda06 yield $M_* = 1.52$ and $R_* = 3.72$ . The SPOCS Catalog lists $M_* =
1.74$ , and $R_* = 4.11$ [@valenti05]. The variances of these different mass and radius estimates are 0.1 and 0.2 , respectively, which are consistent with our estimate of uncertainties in §\[stellar\]. The other stellar properties are listed in Table \[stellartable\].
was one of the original stars added to the CCPS Keck program in 1996, and was subsequently added to our list of intermediate–mass stars in 2004. Table \[vel175541\] lists our Doppler measurements along with their observation dates and internal uncertainties (without jitter). Figure \[velplotB\] shows that the rms scatter of the velocity measurements is a factor of 6 greater than the mean internal uncertainty ($\bar{\sigma_v} \approx 2$ ), and 2–3 times larger than the rms scatter of stable Keck subgiants (Figure \[std\_stars\]). A Lombe–Scargle periodogram analysis of the velocities reveals a pronounced peak near $P=300$ d, with an analytical false–alarm probability $\rm < 0.1\%$ (Figure \[pgramB\]).
To search for the best–fit orbital solution, we added 5 of jitter in quadrature to the internal measurement uncertainties. We find that a Keplerian with $P=\pB$ d, $K = \kB$ and $e = \eB$ provides the best fit to the data, resulting in $\rm rms = \rmsB$ and $ = $ . Figure \[orbitB\] shows the the radial velocities phased at $P = \pB$ day, along with the best–fit orbital solution (the gray points show the measurements at phases outside of phases 0.0 and 1.0, in order to guide the eye). Assuming a stellar mass of , we estimate a minimum planet mass $=
\msiniB$ and orbital separation $a = \arelB$ AU.
While the strong periodogram peak and low are indicative of a correlated signal resulting from an orbiting planet, it is still possible that random variability could conspire to produce a false periodicity in our sparse series of measurements. To test the null hypothesis, we used the “scrambled” velocity false–alarm test described by @marcy05b. For $10^4$ separate trials, we held the observation times constant and scrambled the order of the measurements using a pseudo random number generator. This has the effect of keeping the sampling constant while removing any true temporal coherence, if such a signal exists. For each of the scrambled trials, we perform a full search for the best–fit Keplerian orbital solution—with jitter—and record the from the fit.
The distribution of generated from the scrambled–velocity trials is then compared to the fit obtained from the original time series, as shown in the lower panel of Figure \[pgramB\]. None of the $10^4$ scrambled trials produced a equal to or lower than the best–fit solution to the original time series, resulting in a false–alarm probability of $< 0.01$%. From this test, we conclude that the temporally correlated signal seen in the velocity time series is likely real, rather than an artifact of random noise. We find that the best explanation of the periodic signal is the presence of an unseen planetary companion orbiting .
Summary and Discussion {#summary}
======================
We present precision Doppler measurements of intermediate–mass subgiants that show periodic variations in their radial velocities consistent with planet–mass orbital companions. Interpolation of the stars’ absolute magnitudes, colors and metallicities onto the @girardi02 stellar interior models shows that all stars have masses ranging from 1.65 to 1.85 . Figure \[sg\_hr\] shows these massive host stars on an H–R diagram, along with their theoretical evolution tracks. Following the tracks back to the zero–age main sequence reveals that these present–day subgiants were originally early–type dwarfs with $B-V \lesssim 0.2$ and spectral types ranging from A2V to A5V. The long–period planets presented here would not have been detectable during their stars’ main–sequence phases due to the jitter and rotational line broadening typical for intermediate–mass dwarfs. These planets orbiting “retired” A stars illustrate how evolved stars provide a unique window into stellar mass and planetary domains otherwise inaccessible to Doppler–based planet searches.
There are now 9 former A–type stars ($1.6 \lesssim M_* < 3.0$ ) with planetary companions. We list some of the properties of these massive host stars and their planets in Table \[massive\_table\]. All 9 planets orbit beyond $\sim$0.78 AU from their stars. This paucity of planets with semimajor axes $a < 0.78$ AU is unlikely to be due to a detection bias. For a given planet mass and stellar mass, the velocity semiamplitude of a star scales as $K \sim a^{-1/2}$, making planets in smaller orbits easier to detect. The detectability of close–in planets is also facilitated by the increased number of orbital cycles that are observable over a given time span.
We consider two possible explanations for the observed lack of close–in planets around intermediate–mass stars. The first possibility is that planets around A–type stars have the same semimajor axis distribution as planets orbiting lower–mass stars, but the close–in planets were destroyed by the expanding atmospheres of their giant host stars. Alternatively, planets orbiting A–type stars may have a different semimajor axis distribution than lower–mass stars, with planets residing preferentially in long–period orbits beyond $\sim0.8$ AU.
These possibilities can be explored by comparing the properties of planets in Table \[massive\_table\] to planets orbiting lower–mass stars listed in the CNE. We exclude extremely low–mass planets with $K < 15$ that would not be easily detectible around higher–mass subgiants and giants. We use a one–sided Kolmogorov–Smirnov (K–S) test to compare the semimajor axis distributions of planets around intermediate–mass and lower–mass stars [@press]. We find the probability that the two distributions are identical is only 0.06%. Under the assumption that the semimajor axis distribution of planets is independent of stellar mass, short–period planets orbiting evolved A–type stars must be efficiently destroyed by the expanding atmospheres of their giant host stars. The validity of this hypothesis depends on whether the radii of $\sim2$ giants are large enough to engulf planets out to $\sim0.8$ AU.
Figure \[rstar\_hj\] shows the evolution of the radius of a 2.0 star according to the @girardi02 stellar evolution models. As the star crosses the Hertzsprung Gap during its subgiant phase, its radius remains nearly constant at $a = 5$ $ =
0.023$ AU, which is within the orbit of a $P = 3$ day hot Jupiter. Not until the star begins to ascend the RGB does its outer atmosphere begin to encroach on the orbits of short–period planets. But even at the tip of the RGB (near the helium flash), the radius of a 2 star is only at the distance of a 10 day hot Jupiter at $a
\approx 26$ $=0.12$ AU (the radius of a 2.5 red giant is not much larger at $a \approx 32$ $=0.15$ AU). Thus, engulfment cannot be solely responsible for the lack of close–in planets around subgiants and K giants. Indeed, engulfment can only be important for 4 of the stars in Table \[massive\_table\]: the post–helium–flash clump giants HD104985, HD11977 and $\epsilon$ Tau, and HD13189 which has a poorly constrained radius due to its highly uncertain parallax.
The evolution of planetary orbits from 0.05–0.15 AU in the presense of an expanding stellar atmosphere has not been examined in detail. The effects of planet engulfment on its host star have been studied by @siess99, but a key assumption in their model is that the substellar companion is destroyed. Since it is unclear what happens to a planet when it interacts with the atmosphere of its expanding host star, we simply assume that planets orbiting within the radius of a giant star are destroyed[^3]. Under this assumption, we would expect a deficiency of hot Jupiters around clump giants out to $\sim$0.15 AU, but no corresponding deficiency around subgiants and K giants.
We now analyze the lack of close–in planets around the sample in Table \[massive\_table\] accounting the possible destruction of hot Jupiters around clump giants. For subgiants and giants we can use the K–S test as before, which yields a probability of 0.7% that the semimajor axis distribution is the same as lower–mass stars in the CNE. For clump giants we exclude planets from the CNE with $a <
0.15$ AU, and the corresponding probability from the K–S test is 1.7%. Thus, the distribution of close–in planets around former A–type stars remains inconsistent with the distribution of planets in the CNE. Since engulfment does not provide an adequate explanation for the lack of close–in planets in Table \[massive\_table\], we are left with the possibility that the semimajor axis distribution of planet around A–type stars is significantly different than the distribution around lower–mass stars ($M_* < 1.6$ ).
Differences between the semimajor axes of planets around stars of various masses has previously been investigated by @burkert06. From their study of the orbital properties of known exoplanets, they find evidence of a gap in the semimajor axis distribution around stars with masses $M_* \geq 1.2$ , with fewer planets between 0.08 AU and 0.6 AU compared to lower–mass stars. They were able to reproduce this gap in their Monte Carlo simulations of planet migration, and they attribute the gap to the shorter depletion timescales of disks around intermediate–mass stars.
The semimajor axis distribution of planets as a function of stellar mass can be investigated further with the inclusion of a larger sample of intermediate–mass subgiants in Doppler-based planet searches. As Figure \[rstar\_hj\] shows, Doppler surveys of subgiants can probe occurrence of Jovian planets at orbital distances ranging from many AU down to as close as 0.05 AU, the realm of hot Jupiters. The smaller radii of subgiants also result in higher surface gravities compared to giants, which leads to lower levels of pulsation–induced jitter. @hekker06 show that giants with $B-V > 1.2$ typically have jitter values greater than 20 , ostensibly due to radial and non–radial pulsation modes. Only giants blueward of this limit are stable to within 20 , compared to the 4–6 of jitter seen in subgiants ($B-V < 1.0, M_V \lesssim 2.0$). This increased velocity stability, coupled with their relatively small radii, therefore make subgiants ideal proxies for A–type dwarfs in Doppler–based planet searches.
The primary limitation of subgiants is their relative scarcity, which restricts the number of bright targets suitable for high–resolution spectroscopic observations. The time it takes stars to cross the Hertzsprung Gap is small compared to the star’s lifetime—of order 100 Myr—rendering Hertzsprung Gap stars within 200 pc rare compared to main–sequence stars and giants. Additional targets can be found further from the Sun, with fainter apparent magnitudes ($V
\gtrsim 7.5$). In the near future, we plan to expand our sample of subgiants using the Keck telescope and HIRES spectrometer in order to further investigate the orbital properties, planet masses and occurrence rate of planets orbiting intermediate–mass stars. As the number of subgiants included in Doppler surveys increases, it will become apparent whether the lack of short–period planets around intermediate–mass stars is a result of different formation and migration mechanisms in the disks of A–type stars, or simply a consequence of the small number of massive subgiants currently surveyed.
We extend our gratitude to the many CAT observers who have helped with this project, including Howard Isaacson, Julia Kregenow, Karin Sandstrom, Bernie Walp, Peter Williams, Katie Peek and Shannon Patel. Special thanks to Hervé Buoy and Francisco Ramos-Stierle for lending a portion of their 3m observing time to observe HD192699 before it set in 2006. We thank Michael Fitzgerald and Marshall Perrin for their useful discussions, and Tim Robishaw for sharing his data display expertise and IDL plotting routines. We also gratefully acknowledge the efforts and dedication of the Lick Observatory and Keck Observatory staff, and the time assignment committees of NASA, NOAO and University of California for their generous allocations of observing time. We appreciate funding from NASA grant NNG05GK92G (to GWM), and the NSF for its grant AST-0307493 (to SSV) for supporting this research. DAF is a Cottrell Science Scholar of Research Corporation and acknowledges support from NASA Grant NNG05G164G that made this work possible. This research has made use of the Simbad database operated at CDS, Strasbourg France, and the NASA ADS database. The authors wish to extend special thanks to those of Hawaiian ancestry on whose sacred mountain of Mauna Kea we are privileged to be guests. Without their generous hospitality, the Keck observations presented herein would not have been possible.
[lllll]{} V & & &\
$M_V$ & & &\
B-V & & &\
Distance (pc) & & &\
${\rm [Fe/H]}$ & (0.04) & (0.04) & (0.04)\
$T_{eff}$ (K) & (44) & (44) & (44)\
() & (0.5) & (0.5) & (0.5)\
$\log{g}$ & (0.3) & (0.3) & (0.3)\
$M_{*}$ () & (0.12) & (0.13) & (0.12)\
$R_{*}$ () & (0.51) & (0.57) & (0.46)\
$L_{*}$ () & (0.3) & (0.3) & (0.3)\
Age (Gyr) & (1.0) & (1.0) & (1.0)\
$S_{HK}$ & & &\
$\log R'_{HK}$ & & &\
[lllll]{} P (d) & () & () & ()\
T$_p$ (JD) & () & () & ()\
e & () & () & ()\
K$_1$ () & () & () & ()\
$\omega$ (deg) & () & () & ()\
() & & &\
$a$ (AU) & & &\
Fit RMS () & & &\
& & &\
N$_{\rm obs}$ & & &\
[^1]: For the updated catalog of extrasolar planet and their parameters see http://exoplanets.org.
[^2]: See also http://pleiadi.pd.astro.it/
[^3]: @maxted06 discovered a short–period substellar companion that apparently survived engulfment as its parent star evolved into a white dwarf. However, no Jovian planet has yet been detected around a white dwarf.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Holonomy $R$-matrices parametrized by finite-dimensional representations are constructed for quantized universal enveloping algebras of simple Lie algebras at roots of 1.'
author:
- 'R. Kashaev, N.Reshetikhin'
title: 'Invariants of links with flat connections in their complements.II. Holonomy $R$-matrices related to quantized universal enveloping algebras at roots of 1.'
---
Introduction
============
In the previous paper [@KR] we gave the construction of invariants of tangles with flat $G$-connections in the complement. The construction was based on the notion of holonomy $R$-matrices. Such $R$-matrices are operator-valued functions on $G\times G$ which satisfy the holonomy Yang-Baxter equation [@KR].
In this paper we show that quantized universal enveloping algebras at roots of 1 provide examples of such $R$-matrices.
In section 2 we present basic facts about quantized universal enveloping algebra when the quantization parameter is generic. In section 3 we show that the regular part of the universal $R$- matrix specialized at roots of 1 give examples of holonomy $R$- matrices. In section 4 we analyze the invariants of tangles with a flat connection in the complement derived from such holonomy $R$-matrices. This work was supported by the NSF grant DMS-0070931 and by the DMS grant RM1-2244.
Quantized Universal Enveloping Algebras
=======================================
Let ${{\mathfrak g}}$ be a simple Lie [algebra]{} of rank $r$ with the root system ${\Delta}$. Denote by $P$ its weight lattice and by $Q$ its root lattice. Fix simple roots ${\alpha}_1,\dots ,{\alpha}_r\in{\Delta}$ and denote by $({a_{ij}})^r_{i,j=1}$ the corresponding Cartan matrix. Denote by $\omega_1,\dots ,\omega_r\in P$ the basis of fundamental weights (dual to the basis of simple roots $\{{\alpha_i}\}$) and let $d_i$ be the the length of the $i$-th simple root.
Quantized universal enveloping algebras
---------------------------------------
Let $Q\leq M\leq P$ be a lattice. The quantized universal enveloping [algebra]{} $U^M_q({{\mathfrak g}})$ is the associative [algebra]{} with 1 over ${{\mathbb C}}(q)$ generated by $L_\mu$, $\mu\in M$, and $E_i,F_i$, $i=1,\dots ,r$ with defining relations: $$\begin{aligned}
L_\mu L_\nu &=& L_\nu L_\mu \ , \quad
L_0=1 \ , \\ L_\mu E_i &= & q^{{\alpha}_i(\mu)}E_iL_\mu \\
L_\mu F_i &=& q^{-{\alpha_i}(\mu)}F_iL_\mu \ , \qquad
E_iF_j -F_jE_i = \delta_{ij}
\frac{L_{{\alpha_i}}-L_{{\alpha_i}}^{-1}}{q_i-q_i^{-1}} \ , \\
&&\sum^{1-{a_{ij}}}_{k=0} (-1)^k
\left[ \begin{array}{c} 1-{a_{ij}}\\ k\end{array}\right]_{q_i}\,
E_i^{1-k-{a_{ij}}} E_j\, E_k^k =0 \ , \quad i\neq j\\
&&\sum^{1-{a_{ij}}}_{k=0} (-1)^k
\left[ \begin{array}{c} 1-{a_{ij}}\\ k\end{array}\right]_{q_i}\,
F_i^{1-k-{a_{ij}}} F_j\, F_k^k =0 \ , \quad i\neq j\end{aligned}$$ Here $q_i=q^{d_i}$, $$\left[ \begin{array}{c} m\\n\end{array}\right]_q\,
= \frac{[m]_q!}{[m-n]_q![n]_q!} \ ,\qquad
[n]_q!=[n]_q\dots [2]_q[1]_q \ , \qquad
\left[n\right]_q = \frac{q^n-q^{-n}}{q-q^{-1}} \ .$$ The map ${\Delta}$ acting on generators as $$\begin{aligned}
\label{comult}
{\Delta}L_\mu &=& L_\mu{\otimes}L_\mu \ , \\
{\Delta}E_i &=& E_i{\otimes}1 + L_{{\alpha_i}}{\otimes}E_i \ ,\\
{\Delta}F_i &=& F_i{\otimes}L_{-{\alpha_i}} + 1{\otimes}F_i\end{aligned}$$ extends to the homomorphism of [algebra]{}s ${\Delta}:U^M_q({{\mathfrak g}})\to
U^M_q({{\mathfrak g}}){\otimes}U^M_q({{\mathfrak g}})$. The pair $(U^M_q({{\mathfrak g}}),{\Delta})$ is a Hopf [algebra]{} with the counit ${\varepsilon}(L_\mu)=1$, ${\varepsilon}(E_i)={\varepsilon}(F_i)=0$.
It is clear that if $P\leq M\leq M'\leq Q$ we have the embedding $U^M_q({{\mathfrak g}})\hookrightarrow U^{M'}_q({{\mathfrak g}})$ of Hopf [algebra]{}s.
Quantum Weyl group
------------------
Let $B_W$ be the braid group associated with the Weyl group $W$, $$B_W=\{{\mbox}{generated by} \ T_i\mid {\mbox}{with defining relations} \
\underbrace{T_iT_jT_iT_j\dots}_{m_{ij}} =
\underbrace{T_jT_iT_jT_i\dots}_{m_{ij}}\}$$ Here $m_{ij}=2$ if $i$ and $j$ are not connected in the Dynkin diagram, $m_{ij}=3$ if $a_{ij}a_{ji}=1$, $m_{ij}=4$ if $a_{ij}a_{ji}=2$ and $m_{ij}=6$ if $a_{ij}a_{ji}=3$.
This group acts on $U^M_q({{\mathfrak g}})$ by auto[morphism]{}s $[L]$: $$\begin{aligned}
T_i(L_\mu) &=& L_{s_i(\mu)} \ , \\
T_i(E_i) &=& -F_iL_{{\alpha_i}} \ ,\\
T_i(E_j) &=&\sum_{r+s=-{a_{ij}}}(-1)^r q^s_i E^{(r)}_i E_j E_i^{(s)}\\
T_i(F_i) &=& -L_{{\alpha_i}}^{-1} E_i \\
T_i(F_j) &=& \sum_{r+s=-{a_{ij}}} (-1)^r q^s_i F_i^{(s)} F_jF_i^{(r)}\end{aligned}$$ where $X_i^{(r)}=\frac{X_i^r}{[r]_{q_i}!}$.
Fix a reduced decomposition of the longest element $w_0\in W$. If $w_0=s_{j_1}\dots s_{j_N}$, $N=|{\Delta}_+|$, then $$\label{order}
{\beta}_a = s_{j_1}\dots s_{j_{a-1}}{\alpha}_{j_a} \ , \ a=1,\dots, N \ ,$$ were ${\alpha}_1\dots {\alpha}_r$ are simple roots. This gives a total convex ordering $\beta_1<\dots<\beta_N$ on the set of roots $\Delta$ of $\mathfrak g$. Such construction give all convex orderings and vice versa.
According to [@Lu] define root elements of $U_q({\mathfrak g})$ as $$E_{\beta_a}=T_{j_1}\dots T_{j_{a-1}} (E_{j_a}),$$ $$F_{\beta_a}=T_{j_1}\dots T_{j_{a-1}} (F_{j_a})$$
The elements $${E}^{k_1}_{{\beta}_1}\dots { E}^{k_N}_{{\beta}_N} \ L_\mu \, {
F}^{\ell_N}_{{\beta}_N}\dots {F}_{{\beta}_1}^{\ell_1}$$ form a linear basis in the algebra $U_q({\mathfrak g})$.
Integral form of quantized universal eneveloping algebras
---------------------------------------------------------
Define the ${{\mathbb C}}[q,q^{-1}]$-sub[algebra]{} ${\mathcal
U}^M_q({{\mathfrak g}})\subset U^M_q({{\mathfrak g}})$ as the smallest $B_W$-stable ${{\mathbb C}}[q,q^{-1}]$-sub[algebra]{} of $U^M_q({{\mathfrak g}})$ containing the elements $$\overline{E_i} = (q_i-q_i^{-1})E_i \ , \qquad
\overline{F_i} = (q_i-q_i^{-1})F_i \ .$$ Set $\overline{E_{\alpha}}=(q_{\alpha}-q_{\alpha}^{-1})E_{\alpha}$, $\overline{F_{\alpha}}=(q_{\alpha}-q_{\alpha}^{-1})F_{\alpha}$, then monomials $$\label{pbw}
{\bar E}^{k_1}_{{\beta}_1}\dots {\bar E}^{k_N}_{{\beta}_N} \
L_\mu \, {\bar F}^{\ell_N}_{{\beta}_N}\dots {\bar F}_{{\beta}_1}^{\ell_1}$$ form a linear basis in ${\mathcal U}^M_q({{\mathfrak g}})$. Here we used the enumeration of positive roots corresponding to a reduced decomposition of the longest element of the Weyl group (see above).
Poisson Lie groups $G$ and $G^*$ {#PL}
--------------------------------
It is well known that the algebra ${\mathcal
U}^Q_q({{\mathfrak g}})$ can be regarded as a Hopf algebra deformation of the algebra of polynomial functions on the Poisson Lie group $G^*=\{(b_+,b_-)\in B_+\times B_-| [b_+]_0=[b_-]_0^{-1} \}$. Notice that as a Lie group $G^*$ is naturally isomorpic to the semidirect product $H\ltimes (N_+{\times}N_-)$ where $H$ act naturally on $N_\pm$. The tangent Lie bialgebra for this Poisson Lie group is dual to the standard Lie bialgebra structure on $\mathfrak g$ [@CP]. In this sense the Poisson Lie group $G^*$ is dual to the Poisson Lie group $G$.
Similarly for any lattice $M, \ Q\leq M\leq P$ the covering group $G_M^*$ of $G^*$ is also a Poisson Lie group which is dual to the standard Poisson Lie group structure on $G$ in a sence that their tagent Lie bialgebras are dual.
The algebra $C(G_M^*)$ of algebraic functions on the Poisson Lie group $G_M^*$ is a Poisson Hopf algebra. As a Poisson algebra it is generated by elements $k_\mu, e_i, f_i$, $\mu\in M$, $i=1,\dots, r$ with defining relations $$\{k_\mu,k_\nu\}=0, \{k_\mu, e_j\}=\mu(\alpha_i)k_\mu e_j$$ $$\{e_i,f_j\}=\delta_{ij} (k_{\alpha_i}-k_{\alpha_i}^{-1})$$ $$\underbrace{\{e_i,\dots,\{e_i }_{-a_{ij}+1} ,
e_j\}^{(d_ia_{ij})}\dots \}^{(-d_ia_{ij})} = 0$$ $$\underbrace{\{f_i,\dots,\{f_i }_{-a_{ij}+1} ,
f_j\}^{(d_ia_{ij})}\dots \}^{(-d_ia_{ij})} = 0$$ where $\{ X,Y \}^{(n)} = \{X,Y\}-nXY$.
The comultiplication acts on generators as in (\[comult\]). The elements $k_\mu$ are coordinate functions on the Cartan subgroup of $G_M^*$ which is a finte cover of the Catran subgroup of $G$. The elements $e_i$ and $f_i$ are coordinate functions on the nilpotent subgroups $N^\pm\subset
G^*$ corresponding to the simple roots.
The braid group $B_W$ acts on $C(G_M^*)$ by Poisson automorphisms.
$$\begin{aligned}
\tau_i(k_\mu) &=& k_{s_i(\mu)} , \\
\tau_i(e_i) &=&-f_i k_{\alpha_i}^{-1}, \\
\tau_i(f_i) &=& -e_i k_{\alpha_i} , \\
\tau_i(e_j) &=& \frac{(-1)^{a_{ij}}}{(-a_{ij})!}
\{e_i,\dots\{e_i,e_j\}^{(a_{ij}d_i)}\}^{(d_i(a_{ij}+2))}\dots\}^
{d_i(-a_{ij}-2)}, \\
\tau_i(f_j) &=& \frac{1}{(-a_{ij})!} \{f_i,\dots \{f_i,f_j\}^{(a_{ij}d_i)}\}^
{(d_i(a_{ij}+2))}\dots \}^{(d_i(-a_{ij}-2))}\end{aligned}$$
One can define coordinates corresponding to all positive and negative roots on $G_M^*$ similarly to how it was done for $U^M_q({\mathfrak g})$.
Fix a linear isomorphism between ${\mathcal U}_q^M({\mathfrak g})$ and $C(G_M^*)$ by identifying monomials (\[pbw\]) with corresponding monomials in $k_\mu$, $e_{\alpha}$, $f_{\alpha}$. Then it is clear that the Hopf algebra structure on ${\mathcal U}_q^M({\mathfrak g})$ is a Hopf algebra deformation of the Poisson Hopf algbera structure on $C(G_M^*)$ described above.
Symplectic leaves of $G_M^*$
----------------------------
According to the general structural facts about Poison Lie groups symplectic leaves of $G^*$ are orbits of the (local) dressing action of the dual Poisson Lie group $G$ [@STS]. This action can be describe as follows.
Let $I: G^*\to G$ be the natural map $(x_+,x_-)\mapsto
x_+(x_-)^{-1}$. This map intertwines the dressing action with the adjoint action of $G$ on $G$.
The map $I$ brings the dressing action of $G$ on $G^*$ to the action of $G$ on itself by conjugations, i.e. if we will write $g: x\mapsto g(x)$ for the dressing action of $g\in G$ on $x\in G^*$ we have: $$I(g(x))=gI(x)g^{-1}$$ Thus, orbits of dressing action in $G^*$ are connected components of orbits of adjoint action of $G$ on itself.
The image of the map $I$ is open dense in $G$. Over generic point in $G$ it is a branched cover map with $2^r$-fibers and it gives an isomorphism between a neighborhood of 1 in $G$ and neighborhoods of points $(\sigma, \sigma^{-1}) \in G^*$ where $\sigma\in H$, $H$ is a Cartan subgroup in $G$ and $\sigma^2=1$. Using this ismorphisms we can identify these neighborhoods of $G^*$ and $G$. We will call it a realization of $G^*$ on $G$.
The natural projection $G_M^*\to G^*$ is Poisson and is a finite cover. Therefore symplectic leaves of $G_M^*$ are connected components of preimages of symplectic leaves in $G^*$.
Formal Poisson Lie group $G_M^*$ {#epsilon}
--------------------------------
Let $\Gamma_M$ be the finite subgroup in $G_M^*$ which is the pre-image of $1\in G^*$ with respect to the natural projection $G_M^*\to G^*$. Let $\varepsilon_\mu\in \Gamma_M$ be the element corresponding to the weight $\mu\in M/Q$.
Denote by $F(G_M^*)$ the completion of $C(G_M^*)$ by formal power series in $k_\mu\epsilon_\mu^{-1}-1$, $e_\alpha$ and $f_\alpha$. This Poisson Hopf algebra is the formal Poisson Lie group $G_M^*$. Instead of formal variables $k_\mu$ we can work with $z_\mu$ such that $k_\mu=\varepsilon_\mu\exp(z_\mu)$.
Quantized universal enveloping algebras at roots of 1
=====================================================
The algebra ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$ and its center
---------------------------------------------------------------------------
Let $\ell$ be an odd integer such that $\ell >
\max_i(d_i)$ and ${\varepsilon}\in{{\mathbb C}}$ be a primitive $\ell$-th root of 1. Denote by ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$ be the quotient [algebra]{}$${\mathcal U}^M_{\varepsilon}({{\mathfrak g}}) = \frac{{\mathcal U}^M_q({{\mathfrak g}})}{(q-{\varepsilon}){\mathcal U}^M_q({{\mathfrak g}})} \ .$$ The center $Z^M_{\varepsilon}=Z({\mathcal U}^M_{\varepsilon}({{\mathfrak g}}))$ has natural structure of Poisson [algebra]{} and, as a Poisson [algebra]{}, it acts on ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$ by derivations [@DKP].
Denote by $Z^M_0$ the sub[algebra]{} in $Z^M_{\varepsilon}$ generated by ${\bar E}^\ell_{\alpha}$, ${\bar F}^\ell_{\alpha}$, $L^\ell_\mu$, ${\alpha}\in{\Delta}_+$, $\mu\in P$.
The following is known (see \[DP\] and references therein):
\[zroot\]
- The subalgebra $Z^M_0$ is a Hopf subalgebra in ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}}) $ .
- It is also a Poisson subalgebra in $Z^M_{\varepsilon}$ and together with the Hopf algbera structure is a Poisson-Hopf algebra.
- $Z^M_{\varepsilon}$ is integrally closed
- $Z^M_{\varepsilon}$ is a free $Z^M_0$ module of the rank $\ell^r$.
- ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}}) $ is finite-dimensional over $Z^M_0$ with $\dim_{Z^M_0}({\mathcal U}^M_{\varepsilon}({{\mathfrak g}}))=\ell^{\dim{{\mathfrak g}}}$.
- There is an isomorphism of Poisson Hopf algebras $Z^M_0\simeq C(G^*_M)$.
Here $G_M^*$ is the finite covering of the Poisson Lie group dual to the Poisson Lie group $G$ (see section \[PL\]). The isomorphism $Z^M_0\simeq C(G_M^*)$ is given by the map $\phi$: $$\phi(L_\mu^\ell)=k_\mu, \ \phi(\bar{E}_i^\ell)=e_i, \ \phi(\bar{F}_i^\ell)=f_i.$$
Geometrically, the algebra ${\mathcal U}^M_{\varepsilon}({\mathfrak g})$ can be regarded as a sheaf of algebras over $G_M^*$ such that it is a bundle of algberas over each symplectic leaf of $G_M^*$ with a flat connection over each simplectic leaf.
The completion $\overline{{\mathcal U}}^M_{{\varepsilon}}({{\mathfrak g}})$ and its center
-------------------------------------------------------------------------------------------
Let $\ell$ be a positive integer.
There exists a unique algebra structure $\overline{{\mathcal U}}^M_{\!q}({{\mathfrak g}})$ over ${{\mathbb C}}[q^{\pm 1}][[
q^{\ell}-1]]$ on the space formal power series $$\sum_{k_1\dots k_N\geq 0\atop {{{m_1\dots m_N\geq 0}\atop
{n_1\dots n_N\geq 0}}\atop {s\geq 0}}} C^S_{\{k\}\{m\}\{n\}}
{\bar E}^{k_1}_{{\beta}_1}\dots {\bar E}^{k_N}_{{\beta}_N} \ \prod^N_{i=1}
\big(L^\ell_{\mu_i}-1\big)^{n_i}\, {\bar F}^{m_N}_{{\beta}_N}\dots
{\bar F}^{m_1}_{{\beta}_1}$$ such that restricted to polynomials in ${\bar E}_\beta,{\bar
F}_\beta$, $L_{\mu_i}$, $q^\ell-1$,$q^{\pm 1}$ it coincides with ${\mathcal U}^M_q({{\mathfrak g}})$. Here $C^S_{\{K\}\{m\}\{n\}}\in {{\mathbb C}}[L^{\pm
1}_{\mu_i}, q^{\pm 1}][[q^{\ell}-1]]$, $N\!=\!|{\Delta}_+|$, ${\beta}_1,\dots ,{\beta}_N$ is a convex ordering on ${\Delta}_+$, $\mu_1,\dots
,\mu_r$ are generators of $M$ and ${\bar E}_{{\beta}_i}$ and ${\bar
F}_{{\beta}_i}$ are as in (\[order\]) .
Specializing $q$ to ${\varepsilon}$ as in the previous section we obtain the completion $\overline{{\mathcal U}}^M_{{\varepsilon}}({{\mathfrak g}})$ of ${\mathcal U}^M_{{\varepsilon}}({{\mathfrak g}})$.
The center $\overline{Z}^M_{{\varepsilon}}=Z(\overline{{\mathcal U}}^M_{{\varepsilon}}({{\mathfrak g}}))$ has a natural Poisson algebra structure. The following proposition is the formal version of the proposition \[zroot\].
\[FQUA\]
- The subalgebra $\overline{Z}^M_0\subset \overline{Z}^M_{{\varepsilon}}$ generated by formal power series in\
${\bar E}^{\ell}_{{\alpha}}, {\bar
F}^{\ell}_{{\alpha}},L^{\ell}_{\mu}\!-\!1$ is a Hopf-Poisson subalgebra in $\overline{{\mathcal U}}^M_{{\varepsilon}}({{\mathfrak g}})$.
- $\overline{Z}^M_0$ is isomorphic to the formal group $F(G_M^*)$.
- $\overline{Z}^M_{{\varepsilon}}$ is a free $Z^M_0$-module of rank $\ell^r$.
- $\overline{{\mathcal U}}^M_{{\varepsilon}}({{\mathfrak g}})$ is f.d. over $\overline{Z}^M_0$ with [[dim]{}]{}${}_{\overline{Z}^M_0}(
\overline{{\mathcal U}}^M_{{\varepsilon}}({{\mathfrak g}}))= \ell^{\rm{dim}\,{{\mathfrak g}}}$
Let us introduce formal variables $z_{\mu}$ as $L^\ell_\mu ={\varepsilon}_\mu\exp(z_\mu)$ where ${\varepsilon}_\mu$ are elements of the finite order which generate the group of automorphisms of the covering map $G_M^*\to G^*$ in a neighborhood of 1 ( the same ${\varepsilon}_\mu$ that were used in section \[epsilon\]. Then ${\bar L}_\mu = L_\mu\exp\left(- \
\frac{z_{\mu}}{\ell}\right)$ are elements of finite order.
The universal $R$-matrix for $\overline{{\mathcal U}}^P_{{\varepsilon}}({{\mathfrak g}})$
------------------------------------------------------------------------------------------
For each positive root ${\beta}$ let $z_{{\beta}},{\bar
E}_{{\beta}},{\bar F}_{{\beta}}$ be corresponding elements of $\overline{{\mathcal U}}^P_{{\varepsilon}}({{\mathfrak g}})$. Let $\overline{{\mathcal U}}^P_{{\varepsilon}}({{\mathfrak g}})^{{\hat{\otimes}} 2}$ be the completion of the tensor product with respect to the gradation given by the degree function $deg(z_{\beta})=deg(\bar{E}_{\beta})= deg(\bar{F}_{\beta})=1$. Define the outer automorphism ${\mathcal R}^{(n)}_{{\beta}}$ of $\overline{{\mathcal U}}^P_{{\varepsilon}}({{\mathfrak g}})^{{\hat{\otimes}} 2}$ as $${{\mathcal R}}^{(n)}_{\beta}(x) = \exp \left(\frac{1}{\ell^2} Li_2\big( {\bar
E}_{{\beta}}^{\ell}{\otimes}{\bar F}_{{\beta}}^{\ell}\big)\right)(x)$$ where $\exp(y)\circ x={\displaystyle}{\sum^{\infty}_{n=0} \frac{1}{n!}}\,
\{y\{y\dots \{y,x\}\dots\}\}$ and $$Li_2(x)=-\int_0^x \frac{log(1-s)}{s}ds =\sum_{n=1}^\infty \frac{x^n}{n^2}$$
Define the outer automorphism ${\mathcal R}^{(n)} =
{\mathcal R}^{(n)}_{{\beta}_N}\circ\dots\circ {\mathcal R}^{(n)}_{{\beta}_1}$.
Define the element ${\tilde R}^{(n)}_{\beta}\in
\overline{{\mathcal U}}^M_{{\varepsilon}}({{\mathfrak g}})^{{\hat{\otimes}} 2}$ as $${\tilde R}^{(n)}_{\beta}=
\prod^{\ell -1}_{m=0} \big( 1-{\varepsilon}^m {\bar E}_{\beta}{\otimes}{\bar
F}_{\beta}\big)^{-\frac{m}{\ell}} \ .$$
Define the element $R^{(n)}\in \overline{{\mathcal U}}^P_{{\varepsilon}}({{\mathfrak g}})^{{\hat{\otimes}} 2}$ as $$R^{(n)}=R^{(n)}_{{\beta}_N}\dots R^{(n)}_{{\beta}_1}$$ where $$R^{(n)}_{{\beta}_i} = {\mathcal R}^{(n)}_{{\beta}_1}\circ\dots\circ
{\mathcal R}^{(n)}_{{\beta}_i-1}\big({\tilde R}^{(n)}_{{\beta}_i}\big) \ .$$
Define the outer automorphism ${\mathcal R}^{(c)}$ of $\overline{{\mathcal U}}^P_{{\varepsilon}}({{\mathfrak g}})^{{\hat{\otimes}} 2}$ as $${\mathcal R}^{(c)}(x)=\exp(\frac{1}{2\ell^2}\sum_{i,j=1}^r (b^{-1})_{ij}
z_{\alpha_i}\otimes z_{\alpha_j})(x)$$ where $b_{ij}=a_{ij}d_j$ is the symmetrized Cartan matrix.
Define the element $R^{(c)}\in \overline{{\mathcal U}}^P_{{\varepsilon}}({{\mathfrak g}})^{{\hat{\otimes}} 2}$ as $$R^{(c)}=\sum_{\lambda, \mu \in P/dlP} {\varepsilon}^{(\lambda, \mu)}
P_\lambda\otimes P_\mu$$ Here $d$ is the degree of the covering map $G^*_P\to G^*$ and $P_\lambda$ are idepompotents in the subalgebra generated by $\bar{L}_\mu$ such that $$\bar{L}_\mu P_\lambda=P_\lambda\bar{L}_\mu ={\varepsilon}^{(\lambda,\mu)}
P_\lambda$$
\[m-YB\]
1. The outer automorphism $\mathcal R$ restricted to $\bar{Z}_0\hat{\otimes} \bar{Z}_0$ is a Poisson automorphism.
2. The automorphism ${\mathcal R}={\mathcal R}^{(c)}\circ
{\mathcal R}^{(n)}$ restricted to $\bar{Z}_0\hat{\otimes} \bar{Z}_0$ satisfies the Yang-Baxter equation $$\label{class-ybe}
{\mathcal R}_{12}\circ {\mathcal R}_{13}\circ {\mathcal R}_{23}
={\mathcal R}_{23}\circ {\mathcal R}_{13}\circ {\mathcal R}_{12}$$ Here ${\mathcal R}_{ij}$ act on $\bar{Z}^{\hat{\otimes}3}$, ${\mathcal R}_{12}={\mathcal R}\otimes id$, ${\mathcal R}_{23}=
id\otimes {\mathcal R}$, ${\mathcal R}_{13}=id\otimes \sigma_{23}\circ
{\mathcal R}_{12}\circ id\otimes \sigma_{23}$ and $\sigma_{23}
(x\otimes y\otimes z)=x\otimes z\otimes y$.
3. The element $R=R^{(c)}R^{(n)}$ satisfies the twisted Yang-Baxter relation $$\label{tw-YBE}
({\mathcal R}_{12}^{-1}\circ {\mathcal R}_{13})(R_{23})\cdot
{\mathcal R}_{12}^{-1}(R_{13})\cdot R_{12} =
({\mathcal R}_{23}^{-1}\circ {\mathcal R}_{13}) (R_{12})
{\mathcal R}_{23}^{-1}(R_{13})R_{23} \ ,$$
4. ${\Delta}'(a)={\mathcal R}(R{\Delta}(a)R^{-1})$ for all $a\in \bar{{\mathcal U}}_{\varepsilon}({\mathfrak g})$.
This theorem follows from the asymptotical behavior of the universal $R$-matrix for $sl_2$ [@R-2], from the multiplicative formula for the universal $R$-matrix and from teh Campbell-Hausdorf formula. For arbitrary simple Lie algebra it was proven in [@Ga]. The Poisson automorphism $\mathcal R$ was studied in [@WX] and [@R-1].
Representations of ${\mathcal U}^P_{\varepsilon}({{\mathfrak g}})$ and holonomy $R$-matrices
============================================================================================
Representations of ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$
------------------------------------------------------------------
We will denote by $(\pi_x^V,V)$ a representation $\pi_x^V: {\mathcal U}^M_{\varepsilon}({{\mathfrak g}})\to End(V)$ of ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$ in the vector space $V$ with $Z^M_0$-central character $x\in G_M^*$ . Here we used a natural identification of Poisson-Hopf algebras $Z^M_0\simeq C(G^*_M)$ (see proposition \[FQUA\]).
The group $G$ acts on $G_M^*$ locally by dressing transformations. This $G$-action on $G^*_M\simeq Spec(Z_0)$ lifts to the action of $G$ on ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$ by outer auto[morphism]{}s. We will write $g:a\mapsto \tau_g(a)$ for this action.
We will say that the representation $(\pi_x^V,V)$ is [*$G$-equivalent*]{} to the representation $(\pi_y^W,W)$ if
- $x,y\in G_M^*$ belong to the same $G$-orbit, i.e. if there exists $g\in G$ with $y=g(x)$
- if there exists a linear map $\varphi_{V,W}(x,g):V\to W$ such that $$\pi^W_{g(x)}(a)=\varphi_{V,W}(x,g)\pi^V_x(\tau_g(a))\varphi_{V,W}(x,g)^{-1}.$$
The representation of ${\mathcal U}^M_{\varepsilon}({\mathfrak g})$ dual to $(\pi_x^V, V)$ is the representation in the dual vector space $V^*$ with the algebra acting as $a\mapsto \pi_x^V(S(a))^*$. Here $S$ is the antipode and $f^*$ is the linear map dual to $f:V\to V$. It is clear that if $(\pi_x^V, V)$ has $Z^M_0$-character $x\in G_M^*$ then the dual representation will have the $Z^M_0$-central character $i(x)$ where $i$ is the operation of taking the inverse in the group $G^*$.
Irreducible representations
---------------------------
Because the [algebra]{} ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$ is finite-dimensional over its center, there exists a non-empty Zariski open subset $S^M_{\varepsilon}\subset{\mbox{Spec }}\!(Z^M_{\varepsilon})$ such that ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})/{\langle}(c-\chi(c)){\mathcal U}^M_{\varepsilon}({{\mathfrak g}})\mid c\in Z^M_{\varepsilon}{\rangle}$ is isomorphic to a matrix algebra for any $\chi \in S^M_{\varepsilon}$ ( for more details see [@DP]. We will call such elements generic. Thus, for each generic $\chi\in {\mbox{Spec }}\!(Z^M_{\varepsilon})$ we have unique iso[morphism]{}class of irreducible representations.
Denote by $S^M_0\subset Spec(Z^M_0)$ the image of $S^M_{\varepsilon}$ with respect to the projection $Spec(Z^M_{\varepsilon})\to Spec(Z^M_0)$. The variety $S^M_{\varepsilon}$ is a finite cover of $S^M_0$. Points of $Spec(Z^M_{\varepsilon})$ are “common level surfaces” of all central elements of ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$. Points of $Spec(Z^M_0)$ are “common level surfaces” of elements of central subalgebra generated by $L_\mu^\ell, \bar{F}_i^\ell,
\bar{E}_i^\ell$ and by their Poisson brackets. The number of branches of the projection $S^M_{\varepsilon}\to S^M_0$ over generic point is $\ell^r$. The number of branches of $S^M_0\to S^Q_0$ is $\ell^{|M/Q|}$. So, over generic point of $G^*$ we expect $\ell^{r+|M/Q|}$ irreducible representations of ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$. All these irreducible representations have dimension $\ell^{|{\Delta}_+|}$ [@DKP].
Central elements of ${\mathcal U}^M_{\varepsilon}({\mathfrak g})$ which also belong to the Poisson center of $Z^M_{\varepsilon}$ are constant on dressing orbits. We will call this central subalgebra the Casimir subalgebra.
Let ${\mathcal O}\subset G^*$ be a dressing orbit and $U\in G^*$ be a neighborhood of $1$ on which the local action of $G$ integrates to an action. Let $\{(\pi_x^V,V)|x\in U\subset {\mathcal O}\}$ be a family of irreducible representations of ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$. Assume that these representations have the same central character with respect to the Casimir subalgebra. Because Poisson Casimirs are constant on $G$-orbits and the specter of primitive ideals is a finite cover over the specter of primitive ideals of the Poisson center of $Z^M_0$, this assumption will hold for sufficiently small $U$. Let $g\in G$ and $x\in
{\mathcal O}\cap U\subset G^*$ be such that $x$ and $g(x)$ are generic. Representations $\pi_x^V\circ \tau_g$ and $\pi_{g(x)}$ have the same central characters and therefore isomorphic. Thus, we have a family $\{T(g|x)\}$ of linear automorphisms of $V$ such that $$\label{connect}
\pi_x^V(\tau_g(a))=T(g|x)\pi_{g(x)}^V(a)T(g|x)^{-1}$$
Considering formal neighborhood of $1$ in $G$ and $G^*$ and representations of ${\mathcal U}^M_{\varepsilon}({\mathfrak g})$ over such neighborhood we get what is called formal representations. Such representations are homomorphisms from $\bar{{\mathcal U}}^M_{\varepsilon}({\mathfrak g})$ to the algebra $End(V)[[x]]$ where $V$ is the space of representations and $x$ are formal coordinates in a neighborhood of $1$.
Holonomy $R$-matrices
---------------------
From now on we will use the map $I$ to identify neighborhoods of identities in $G^*$ and $G$. After this the dressing action of $G$ on $G^*$ is identified with the adjoint action of $G$ on itself.
Let $(\pi_x^VV)$ and $(\pi_y^W,W)$ be two generic formal representations of ${\mathcal U}^P_{\varepsilon}({\mathfrak g})$.
\[eval-R\] Let $a, b\in {\mathcal U}^P_{\varepsilon}({\mathfrak g})$ and $\mathcal R$ be the outer automorphism of ${\mathcal U}^P_{\varepsilon}({\mathfrak g})^{\otimes 2}$ defined in the theorem \[m-YB\]. Then $$(\pi_x^V\otimes \pi_y^W)({\mathcal R}(a\otimes b))=
\pi_x^V(\tau_{x_L(x,y)_+}(a))\otimes \pi_y^W(\tau^{-1}_{x_-}(b))$$ where $x_L(x,y)=x_-yx_-^{-1}$
Proof. This proposition follows from the definition of $\mathcal R$ and from results of [@WX].
For generic formal $x$ and $y$ define the element $R^{V, W}(x,y)\in
End(V\otimes W)[[x,y]]$ as $$\label{hol-R}
R^{V,W}(x,y)=(T(x_L(x,y)_+|x)^{-1}\otimes T(x_-|y)^{-1})(\pi_x^V\otimes
\pi_y^W)(R)$$ Here $x$ and $y$ are formal coordinates on a formal neighborhood of $1$ in $G$.
Linear maps (\[hol-R\]) satisfy the holonomy Yang-Baxter equation: $$\begin{aligned}
\label{hYBE}
R^{V,W}_{12}(x_R(x,x_L(y,z)),x_R(y,z)) R^{V,U}_{13}(x,x_L(y,z))
R^{W,U}_{23}(y,z)=
\\ R^{W,U}_{23}(x_L(x,y),x_L(x_R(x,y),z)) R^{V,U}_{13}(x_R(x,y),z) R^{V,W}_{12}(x,y)\end{aligned}$$
Proof. We can choose linear isomorphisms $T(g|x)$ such that $T(g_1g_2|x)$ is proportional to $T(g_1|x)T(g_2|x)$ and $$\label{T-invert}
T(g^{-1}|x)T(g|gxg^{-1})=1$$
From Proposition \[eval-R\] we can evaluate the both sides of the equation (\[tw-YBE\]) in the tensor product of three $G^*_P$ evaluation representations. For the left side we have: $$\begin{aligned}
(\pi_x^V\otimes\pi_y^W\otimes\pi_z^U)(({\mathcal R}_{12}^{-1}\circ
{\mathcal R}_{13})(R_{23})\cdot
{\mathcal R}_{12}^{-1}(R_{13})\cdot R_{12} )=\\
(\pi_x^V\otimes\pi_y^W\otimes\pi_z^U)(\tau_{x_L(x,x_L(y,z)_-)_+}\otimes\tau_{x_L(y,z)_+})(R)_{12}
(id\otimes\tau_{y_-})(R)_{13}R_{23}
\end{aligned}$$ Similarly one can evaluate the right side: $$\begin{aligned}
(\pi_x^V\otimes\pi_y^W\otimes\pi_z^U)(({\mathcal R}_{23}^{-1}\circ {\mathcal R}_{13}) (R_{12})
{\mathcal R}_{23}^{-1}(R_{13})R_{23} )=\\
(\tau_{x_-}^{-1}\otimes\tau_{x_R(x,y)_-}^{-1})(R)_{23}(\tau_{x_L(x,y)_+}^{-1} \otimes
id)(R)_{13}R_{12}\end{aligned}$$ where $x_R(x,y)=x_L(x,y)_+^{-1}xx_L(x,y)_+$.
The holonomy Yang-Baxter equation for linear maps (\[hol-R\]) follows from the identities (\[connect\]) and the identities for $T(x|y)$.
Thus we constructed solutions to formal holonomy Yang-Baxter equation. Let $S$ and $S'$ be two generic symplectic leaves in $G^*_P$. Linear operators (\[hol-R\]) admit analytical continuation to sections of vector bundles over $S\times S'$.
Invariants of tangles
=====================
d-matrix
--------
In the construction of invariants of tangles with flat connection given in [@KR] an important role played linear operators $d^V(x)\in End(V)$ defined in terms of holonomy $R$-matrices as $$d^X(a)=(tr\otimes id)(P((R^{t_1}(a,i(a)^{-1})^{-1})^{t_1})$$ Since we constructed holonomy $R$-matrices for irreducible ${\mathcal U}^P_{\varepsilon}({\mathfrak g})$-modules, we have such $d$ operators for each generic irreducible representation $(\pi_x^V,V)$.
Let $(\pi^V_x, V)$ be an irreducible representation of ${\mathcal U}^P_{\varepsilon}({\mathfrak g})$ with generic $x\in G^*$, then $$d^V(x)=c_V(x)\pi^V_x(L_\rho)$$ where $c_V(x)$ is a non-zero complex number and $\rho=1/2\sum_{\alpha\in {\Delta}_+}\alpha$.
Proof. For each pair of representations $(\pi^V_x, V)$ and $(\pi^W_x, W)$ of ${\mathcal U}^P_{\varepsilon}({\mathfrak g})$ we have: $$\label{cr-1}
R_{12}^{X,Y}(x,y)=d_2(a)^{-1}(((R_{12}^{X,Y}(x,y)
^{-1})^{t_2})^{-1})^{t_2})d_2(y)$$ $$\label{cr-2}
R_{12}^{X,Y}(a,b)=d_1(y)^{-1}(((R_{12}^{X,Y}(a,b)
^{t_1})^{-1})^{t_1})^{-1})d_1(a)$$ These equations imply that for generic $x\in G^*$ and a representation $(\pi^V, V)$ we have: $$\pi^V_x(S^2(a))=d^V(x)\pi^V_x(a)d^V(x)^{-1}$$ On the other hand from the definition of the antipode it is easy to see that $$S^2(a)=L_\rho aL_\rho^{-1}$$ The lemma now follows from the Schur’s lemma.
This lemma implies that the invariant of a knot defined by the functor $F$ constructed in [@KR] is identically zero. The situation is similar to invariants studied in [@Ro] (see also the references therein).
Invariants knots of string knots
--------------------------------
Recall that a string knot is a tangle with one connected component and with two boundary components. If $D_t$ is a diagram of a string knot a generic $G$-coloring of the whole diagram is determined by the corresponding $G$-coloring of one of its boundary component.
Let $t$ is a string knot and $F_V(t,x)$ is the value of the functor $F$ on it. Here we assume that the lower and upper boundaries of $t$ are $G$-colored by $x$ and decorated by ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$-module $(\pi_x^V, V)$. Then the element $F_V(t,x)\in End(V)$ is invariant with respect to gauge transformations, i.e. $F_V(t^g,g(x))=F_V(t,x)$ where $t^g$ is the result of the gauge action of $g\in G$ (see [@KR]) on the $G$-colored tangle $t$ and $g(x)$ is the result of the dressing action of $g\in G$ on $x$.
Proof. The gauge invariance of the functor $F$ (see [@KR]) implies that for any other representation $(\pi^W_x, W)$ of ${\mathcal U}^M_{\varepsilon}({{\mathfrak g}})$ we have: $$(1\otimes F_V(t^{y_+},y_+(x)))R^{V,W}_{21}(x_R(y,x),x_L(y,x))^{-1}=
R^{V,W}_{21}(x_R(y,x),x_L(y,x))^{-1}(1\otimes F_V(t,x))$$ $$(1\otimes F_V(t^{y_-},y_-(x)))R^{W,V}_{12}(y,y_-(x))^{-1}=
R^{W,V}_{21}(y,y_-(x))^{-1}(1\otimes F_V(t,x))$$ These equations with $y=1$ imply that elements $F_V(t,x)$ are central. Same equations for generic $y\in G$ imply that $F_V(t^y,y(x))=F_V(t,x)$.
For generic $q$ the functor $F$ (see [@RT]) applied to a string knot defines an element of a completion of the center of ${\mathcal U}_q({\mathfrak g})$. This element can be evaluated in a finite-dimensional representation and up to a scalar factor (which is equal to the quantum dimension of the representation) coincides with the corresponding invariant of the knot obtained by closing the string knot. This means that the central element itself is not only an invariant of a string knot but is also an invariant of the knot obtained by closing the string knot.
One can argue that the same happens in our case. The “limit” of this central element when $q\to {\varepsilon}$ according to the asymptotical behavior of the universal $R$-matrix [@R-2] has an essential singularity and and a regular part. The essential singularity will give the invariant related to the Poisson $R$-matrix ( see [@WX] and [@R-1]). The regular part will give the invariant discussed here. As it was explained above for generic $q$ the central element which is the invariant of a string knot is also an invariant of the knot obtained by the closing the string knot. Therefore, we should expect the same for roots of 1. We will return to the detailed discussion of this question in a separate publication.
Conclusion
==========
We constructed invariants of tangles with flat connections over the complement using representation theory of quantized universal enveloping algebras at roots of 1.
We conjecture that these invariants for $SL_2$ coincide with the invariants constructed in [@BB] in case when the $3$-manifold is a complement to a tangle. When $G=SL_2$ and the flat connection is trivial they coincide with the invariant constructed in [@Ka].
Since the invariant for $G=SL_2$ and trivial flat connection in the complement gives the hyperbolic volume of the complement when $l\to
\infty$. It would be very interesting to describe the asymptotic of our new invariants in terms of corresponding geometrical invariants.
Let $\phi$ be a flat connection in the complement to a tangle. We expect that in the limit $\phi\to 1$, where $1$ is the trivial flat connection, our new invariant becomes the invariant constructed in [@RT] for roots of 1 and reducible but indecomposable representations of dimension $\ell^{{\Delta}_+}$. For $SL_2$ this gives the relation between the invariant constructed in [@Ka] and the Jones polynomial at roots of 1, which was observed in [@MM].
What has been done in our two papers is a first step in the larger program. Here we will outline of some further steps.
First question is whether there is a topological quantum field theory which can give a geometric description of these invariants. In case of Jones polynomials such phenomenological quantum field theory (Chern-Simons theory for compact simple Lie groups) was proposed by Witten and allowed to describe the invariants in geometrical terms. One can guess that appropriate version of Chen-Simons theory for complex simple Lie groups will describe the large $l$ asymptotic of our invariants.
There is another description of our invariant in terms of triangulated manifolds which is based on “6j-symbols” for the category of modules over ${\mathcal U}_{\varepsilon}({\mathfrak g}))$. It generalizes the construction from [@TV][@Ka] and [@BB]. This construction also gives invariants of more general 3-manifolds with flat connections. We will do it in a separate publication.
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{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Luigi Del Debbio\
SUPA, School of Physics, University of Edinburgh, Edinburgh, EH9 3JZ, UK\
E-mail:
- |
Haralambos Panagopoulos\
Department of Physics, University of Cyprus, Lefkosia, CY-1678, Cyprus\
E-mail:
- |
Ettore Vicari\
Dipartimento di Fisica, Università di Pisa, and INFN, I-56127 Pisa, Italy\
E-mail:
title: ' SU($N$) gauge theories in the presence of a topological term '
---
$\theta$ dependence of the ground-state energy {#gse}
==============================================
Four-dimensional SU($N$) gauge theories have a nontrivial dependence on the angle $\theta$ that appears in the Euclidean Lagrangian as $${\cal L}_\theta = {1\over 4} F_{\mu\nu}^a(x)F_{\mu\nu}^a(x)
- i \theta {g^2\over 64\pi^2} \epsilon_{\mu\nu\rho\sigma}
F_{\mu\nu}^a(x) F_{\rho\sigma}^a(x)
\label{lagrangian}$$ where $q(x)=\frac{g^2}{64\pi^2} \epsilon_{\mu\nu\rho\sigma} F_{\mu\nu}^a(x)
F_{\rho\sigma}^a(x)$ is the topological charge density. Semiclassical instanton solutions are one example of field configurations which have nontrivial topological properties. Moreover, the most plausible explanation of how the solution of the so-called U(1)$_A$ problem can be compatible with the $1/N$ expansion (performed keeping $g^2N$ fixed [@Hooft-74]) requires a nontrivial $\theta$ dependence of the ground-state energy density $F(\theta)$ [@Witten-79; @Veneziano-79], $$\exp[ - V F(\theta) ] =
\int [dA] \exp \left( - \int d^4 x {\cal L}_\theta \right)
\label{vftheta}$$ where $V$ is the volume. Evidence for such a dependence has been obtained by exploiting the lattice formulation of the theory, using numerical Monte Carlo simulations, as will be described in Section \[fte\]. The complex nature of the $\theta$ term in the Euclidean QCD Lagrangian makes the Monte Carlo studies of the $\theta$ dependence quite hard, since the lattice action corresponding to the Lagrangian (\[lagrangian\]) cannot be directly simulated for $\theta\ne 0$. Nevertheless, important information on the $\theta$ dependence of relevant physical quantities, such as the ground-state energy and the spectrum, can also be inferred from results at $\theta=0$, by expanding them about $\theta=0$ and computing the coefficients of the expansion [@DPV-02; @DMPSV-06]. The $\theta$ dependence is particularly interesting in the large-$N$ limit where the issue may also be addressed by other approaches, such as AdS/CFT correspondence applied to nonsupersymmetric and non conformal theories, see e.g. Ref. [@AGMOO-00].
We introduce a [*scaling*]{} energy density $$f(\theta) = {\Delta F(\theta)\over \sigma^2},
\label{scge}$$ where $\Delta F(\theta) \equiv F(\theta) - F(0)$ and $\sigma$ is the string tension at $\theta=0$. By expanding $f(\theta)$ around $\theta=0$, one can study its $\theta$ dependence in the region of small $\theta$ values. The function $f(\theta)$ is conveniently parametrized as $$\begin{aligned}
f(\theta)={1\over 2} C \theta^2 s(\theta),\label{ftheta}\end{aligned}$$ where $C$ is the ratio $\chi/\sigma^2$ and $\chi$ is the topological susceptibility at $\theta=0$, $$\chi = \int d^4 x \langle q(x)q(0) \rangle = {\langle Q^2 \rangle\over V}
\label{chidef}$$ where $Q=\int d^4 x q(x)$. $s(\theta)$ is a dimensionless function of $\theta$ such that $s(0)=1$.
The function $s(\theta)$ can be expanded around $\theta=0$ as $$\begin{aligned}
s(\theta) = 1 + b_2 \theta^2 + b_4 \theta^4 + \cdots.
\label{stheta}\end{aligned}$$ The coefficients of the expansion of $f(\theta)$ are related to the zero-momentum $n$-point connected correlation functions of the topological charge density, and therefore to the moments of the probability distribution $P(Q)$ of the topological charge $Q$. If $s(\theta)=1$, and therefore $b_{2n}=0$, the corresponding distribution $P(Q)$ is Gaussian, i.e. $$P(Q)={1\over \sqrt{2\pi\langle Q^2\rangle}}\,{\rm exp}\left( -{Q^2\over 2\langle Q^2\rangle}
\right).$$ Therefore the coefficients $b_{2n}$ of the expansion of $s(\theta)$ parametrize the deviations from a simple Gaussian behavior. For example, he first non–trivial correction is given by $$\begin{aligned}
&&b_2 = - {\chi_4 \over 12 \chi},\label{b2chi4} \\
&& \chi_4 = {1\over V} \left[
\langle Q^4 \rangle_{\theta=0} - 3 \left(
\langle Q^2 \rangle_{\theta=0} \right)^2 \right]. \label{chi4}\end{aligned}$$ It has been recently shown [@Luscher-04] (see also [@GRT-04]) that correlation functions involving multiple zero-momentum insertions of the topological charge density can be defined in a nonambiguous, regularization-independent way, and therefore the expansion coefficients $b_{2n}$ are well defined renormalization-group invariant quantities.
Behavior in the large-$N$ limit {#largeN}
===============================
Witten argued [@Witten-80] that in the large-$N$ limit $F(\theta)$ is a multibranched function of the type $$F(\theta) = N^2 {\rm min}_k\, H\left( {\theta+2\pi k\over N}\right)
\label{conj1}$$ which is periodic in $\theta$, but not smooth since at some value of $\theta$ there is a jump between two different branches. This issue was also discussed in Ref. [@Ohta]. More recently, the conjecture was refined [@Witten-98] leading to a rather simple expression for $\Delta F(\theta)$ in the large-$N$ limit, that is $$\Delta F(\theta) = {\cal A} \, {\rm min}_k \, (\theta+2\pi k)^2 + O\left(
1/N\right).
\label{conj2}$$ In particular, for sufficiently small values of $\theta$, i.e. $|\theta|<\pi$, $$\Delta F(\theta) = {\cal A} \, \theta^2 + O\left( 1/N\right).
\label{conj2b}$$ Thus possible $O(\theta^4)$ terms are expected to be depressed by powers of $1/N$. This conjecture has been supported using arguments based on duality between large-$N$ gauge theories and string theory [@Witten-98]. It has also been discussed in a field-theoretical framework in Ref. [@Gabadadze-99].
The large-$N$ behavior of the coefficients $b_{2n}$ of the expansion of $f(\theta)$ around $\theta=0$ can be inferred by using general large-$N$ scaling arguments applied to the Lagrangian (\[lagrangian\]). They indicate the ratio $\bar{\theta}\equiv
\theta/N$ as the relevant quantity in the large-$N$ limit of the ground-state energy, and more generally of the spectrum of the theory. Then we expect $$\begin{aligned}
&&f(\theta) = N^2 \bar{f}(\bar{\theta}\equiv \theta/N),
\label{fthetabar} \\
&&\bar{f}(\bar{\theta}) =
{1\over 2} C_\infty \bar{\theta}^2 ( 1 + \bar{b}_2 \bar{\theta}^2 +
\bar{b}_4 \bar{\theta}^4 + \cdots),
\label{lnexp}\end{aligned}$$ where $C_\infty$ is the large-$N$ limit of the ratio $C=\chi/\sigma^2$. Comparing with Eq. (\[ftheta\]), one derives $$\begin{aligned}
C=C_\infty + c_2/N^2 + ... , \qquad b_{2i}=\bar{b}_{2i}/N^{2i}+...,
\label{largeNco}\end{aligned}$$ We recall that a nonzero value of $C_{\infty}$ is essential to provide an explanation to the U(1)$_A$ problem in the ’t Hooft large-N limit, and can be related to the $\eta'$ mass [@Witten-79; @Veneziano-79] through the relation $$\chi_\infty = {f_\pi^2 m_{\eta'}^2\over 4 N_f} + O(1/N).
\label{wittenformula}$$ The quantity $b_2$ also lends itself to a physical interpretation, being related to the $\eta^\prime - \eta^\prime$ elastic scattering amplitude [@Veneziano-79].
Results for the first few terms of the expansion around $\theta=0$ of the ground-state energy {#fte}
=============================================================================================
The $\theta$ dependence of SU($N$) gauge theories has been investigated by Monte Carlo simulations of their Wilson lattice formulation. The lattice action corresponding to the Lagrangian (\[lagrangian\]) cannot be directly simulated for $\theta\ne 0$, by virtue of the complex nature of the $\theta$ term. On the other hand, the coefficients $b_{2n}$ in the expansion of the ground-state energy $F(\theta)$ around $\theta=0$ can be accessed by determining the moments of the topological charge distribution at $\theta=0$. They are dimensionless renormalization-group invariant quantities, which should approach a constant in the continuum limit, with $O(a^2)$ scaling corrections ($a$ is the lattice spacing).
Computing quantities related to topology using lattice simulation techniques is not a simple task. In the case $N=3$ several methods have been employed to determine the topological susceptibility, see e.g. Refs. [@DFRV-81]-[@HIMT-01],[@DPV-02],[@LTW-05]-[@DFHK-07]. Cooling, geometrical, heating techniques have been used to address the problems caused by power–divergent additive contributions and multiplicative renormalizations in definitions of the topological susceptibility based on discretized versions of the topological charge density operator $q(x)$. These methods have their drawbacks, since their systematic errors are not under robust theoretical control.
A substantial progress has been achieved after the introduction of the Neuberger overlap formulation [@Neuberger-98; @Neuberger-01] of fermions, which represented a breakthrough for the lattice formulation of QCD. Overlap lattice fermions satisfy the Ginsparg-Wilson relation [@GW-82] and therefore preserve an exact chiral symmetry [@Luscher-98]. As a by product, the index of the overlap Dirac operator [@HLN-98] provides a well–defined estimator for the topological charge [@Neuberger-01; @GRTV-01; @GRT-04; @Luscher-04], which can also be used in pure gauge theories. This method circumvents completely the problem of renormalization arising in bosonic approaches, even though at a much higher computational cost. Using these methods, the topological susceptibility of the pure SU(3) gauge theory has been investigated in Refs. [@EHN-98]-[@DGP-05], finally obtaining the accurate estimate [@DGP-05] $\chi r_0^4=0.059(3)$ ($r_0$ is the length scale defined in [@Sommer]). This value corresponds to $C=\chi/\sigma^2=0.029(2)$ (using [@NRW-01] $\sigma^{1/2}
r_0=1.193(10)$).
It is important to note that the results obtained by the (less computer-power demanding) bosonic methods are substantially consistent, see e.g. Refs. [@ADD-97; @Teper-00; @LT-01; @DPV-02; @DP-04; @DFHK-07], showing their effectiveness although they are supported by a weaker theoretical ground. For example, we mention the results: [@ADD-97] $C=0.027(4)$, obtained using the heating method, [@DPV-02] $C=0.0282(12)$, obtained using cooling, and the more recent result [@DFHK-07] $C=0.0259(11)$.
For larger values of $N$, results have been obtained only by the cooling method so far [@LT-01; @DPV-02; @LTW-05], up to $N=8$. They fit well the expected large-$N$ behavior: $C=C_\infty+c_2/N^2$, providing an estimate of $C_\infty$, and therefore of the topological susceptibility in the large-$N$ limit: $C_\infty=0.0200(43)$ [@LT-01], $C_\infty=0.0221(14)$ [@DPV-02], $C_\infty=0.0248(18)$ [@LTW-05] (the latter was obtained using $N\le 8$ and keeping $a$ fixed). These results are in substantial agreement with the large-$N$ relation (\[wittenformula\]). We stress that the good agrement for $N=3$ of the cooling method with the more rigorous overlap result make us quite confident on the reliability of results for higher values of $N$, since there are no arguments to suggest that this agreement could be spoiled with increasing $N$ (actually there are reasons in favor of improved agreement [@CTW-02; @RRV-97]). An independent determination of $C_\infty$ using other methods would be welcome.
Higher moments of the topological charge distribution provide estimates of the coefficients $b_{2n}$ of the expansion of the scaling energy density $f(\theta)$, cf. Eqs. (\[ftheta\]) and (\[stheta\]). In particular $b_2$ can be estimated using formulae (\[b2chi4\], \[chi4\]). There are a number of results at $N=3$, obtained by different approaches: Ref. [@DPV-02] used the cooling method, Ref. [@Delia-03] used the heating technique to estimate additive and multiplicative renormalizations in zero-momentum correlations of lattice discretizations of $q(x)$, and finally Ref. [@GPT-07] used the most rigorous and CPU intensive overlap method. The results reported in Table \[b2\] are in good agreement, suggesting that the systematic errors of the various methods are sufficiently small. We mention that the fourth moment of the topological charge distribution has been numerically investigated also in Ref. [@DFHK-07], without arriving at any definite conclusion.
The results of Table \[b2\] provide robust evidence that $b_2$ is nonzero, and therefore that there are deviations from a Gaussian distribution of the topological charge.[^1] However, $b_2$ turns out to be quite small, indeed $|b_2|\ll 1$. Thus deviations from a simple Gaussian behavior are already small at $N=3$.
There are also estimates for larger values of $N$, see Table \[b2\], but only using the cooling method. Again, given the agreement found at $N=3$, higher $N$ results should be sufficiently reliable. They appear to decrease consistently with the expectation from the large-$N$ scaling arguments, i.e. $b_2\approx
\bar{b}_2/N^2$ with $\bar{b}_2\approx -0.2$.
We also mention that the analytical properties at $\theta=0$ have been recently discussed and numerically checked in Ref. [@ADD-05].
Overall, these results support the scenario obtained by general large-$N$ scaling arguments, which indicate $\bar{\theta}\equiv \theta/N$ as the relevant Lagrangian parameter in the large-$N$ expansion. They also show that $N=3$ is already in the regime of the large-$N$ behavior. For $N\ge 3$ the simple Gaussian form $$F(\theta) \approx {1\over 2}\chi \theta^2
\label{gauform}$$ provides a good approximation of the dependence on $\theta$ for a relatively large range of values of $\theta$ around $\theta=0$.
$\theta$ dependence at finite temperature {#gseft}
=========================================
Another interesting issue concerns the behavior of topological properties at finite temperature, and in particular their change at the finite-temperature deconfining transition, which is first order for $N\ge 3$, see e.g. Ref. [@LTW-04] and references therein. This issue has been investigated in a number of numerical works, see e.g. Refs. [@Teper-88; @ADD-97; @DGHS-98; @GHS-02; @DPV-04; @LTW-05], using different methods. They show that the topological properties, and in particular the topological susceptibility $\chi$, vary very little up to $T\lesssim T_c$. They change across the transition, where $\chi$ shows a significant decrease. Then, at high temperature $T\gg T_c$, where the instanton calculus is reliable, a rather different scenario emerges [@Gross:1980br].
Concerning the large-$N$ behavior (investigated by performing simulations at various values of $N\ge 3$ [@DPV-04; @LTW-05]), the results indicate that $\chi$ has a nonvanishing large-$N$ limit for $T<T_c$, as at $T=0$, and that the topological properties, and therefore $F(\theta)$, remain substantially unchanged in the low-temperature phase, up to $T_c$. On the other hand, above the deconfinement phase transition, for $T>T_c$, $\chi$ shows a large suppression, hinting at a vanishing large-$N$ limit for $T>T_c$. These results support the hypothesis put forward in Ref. [@KPT-98]: At large $N$ the topological properties in the high-temperature phase, for $T>T_c$, are essentially determined by instantons that are exponentially suppressed, i.e. behave as $e^{-N}$, and therefore the topological susceptibility gets rapidly suppressed in the large-$N$ limit.
$\theta$ dependence of the spectrum {#spectrum}
===================================
Another interesting issue concerns the $\theta$ dependence of the spectrum of the theory. The analysis of the $\theta$ dependence of the glueball spectrum using AdS/CFT suggests that the only effect of the $\theta$ term in the leading large-$N$ limit on the lowest spin-zero glueball state is that this state becomes a mixed state of $0^{++}$ and $0^{-+}$ glueballs, as a consequence of the fact that the $\theta$ term breaks parity, but its mass does not change [@GI-04].
Ref. [@DMPSV-06] presented an exploratory numerical study of the $\theta$ dependence in the spectrum of SU($N$) gauge theories. Again numerical simulations of the Wilson lattice formulation were employed to investigate the $\theta$ dependence of the string tension $\sigma(\theta)$ and the lowest glueball mass $M(\theta)$. Around $\theta=0$ one can write $$\begin{aligned}
&&\sigma(\theta) = \sigma \left( 1 + s_2 \theta^2 + ... \right),
\label{sigmaex}\\
&&M(\theta) = M\left( 1 + g_2 \theta^2 + ... \right)
\label{gmex}\end{aligned}$$ where $\sigma$ and $M$ are respectively the string tension and the $0^{++}$ glueball mass at $\theta=0$. Then the coefficients of these expansions can be computed from appropriate correlators at $\theta=0$. In particular, $s_2$ can be determined [@DMPSV-06] from the large-$t$ behavior of connected correlation functions of two Polyakov lines at distance $t$ and the square topological charge, such as $$\begin{aligned}
\langle A_P(t) Q^2 \rangle_{\theta=0} -
\langle A_P(t) \rangle_{\theta=0} \langle Q^2 \rangle_{\theta=0}
\label{g2}\end{aligned}$$ where $$A_P(t) = \sum_{x_1,x_2} {\rm Tr}\,P^\dagger(0;0) \; {\rm Tr}\,P(x_1,x_2;t),
\label{apdef}$$ $P(x_1,x_2;t)$ is the Polyakov line along the $x_3$ direction of size $L$, and $Q$ is the topological charge. Analogously, the $O(\theta^2)$ term of the glueball mass can be obtained from appropriate connected correlation functions of plaquette operators and $Q^2$. The $O(\theta^2)$ coefficients $s_2$ and $g_2$ are dimensionless scaling quantities, which should approach a constant in the continuum limit, with $O(a^2)$ scaling corrections.
Ref. [@DMPSV-06] obtained the first estimates of $s_2$ and $g_2$ using the cooling method to determine the topological charge, and for $N=3,4,6$ to also check their large-$N$ behavior. The $O(\theta^2)$ terms in the expansion around $\theta=0$ of the spectrum of SU($N$) gauge theories are small for all $N\ge 3$, especially when dimensionless ratios are considered, such as $M/\sqrt{\sigma}$ and, for $N>3$, the ratios of independent $k$ strings. For example we mention the estimates $s_2=-0.08(1)$ and $g_2=-0.06(2)$ for $N=3$. One may also consider the $\theta$ dependence of the scaling ratio $${M(\theta)\over \sqrt{\sigma(\theta)}} =
{M\over \sqrt{\sigma}} ( 1 + c_2 \theta^2 + ... ),
\label{ratioex}$$ where $c_2=g_2-s_2/2$, thus $c_2=-0.02(2)$ for $N=3$. Moreover, the $O(\theta^2)$ corrections appear to decrease with increasing $N$, and the coefficients do not show evidence of convergence to a nonzero value. This is suggestive of a scenario in which the $\theta$ dependence of the spectrum disappears in the large-$N$ limit, at least for sufficiently small values of $\theta$ around $\theta=0$. In the case of the spectrum, the general large-$N$ scaling arguments of Sec. \[largeN\], which indicate $\bar{\theta}\equiv \theta/N$ as the relevant Lagrangian parameter in the large-$N$ limit, imply that $O(\theta^2)$ coefficients should decrease as $1/N^{2}$. The results of Ref. [@DMPSV-06] appear substantially consistent: In the case of the string tension they suggest $s_2\approx -0.9/N^2$.
Of course, further investigation is required to put this scenario on a firmer ground, using for example other definitions of topological charge.
The case of the two-dimensional CP$^{N-1}$ model {#cpn}
================================================
Issues concerning the $\theta$ dependence can also be discussed in two-dimensional CP$^{N-1}$ models [@DDL-79; @Witten-79b], $${\cal L} = {N\over 2g} \overline{D_\mu z}\, D_\mu z
\label{lagrangiancpn}$$ where $z$ is a $N$-component complex scalar field subject to the constraint $\bar{z}z=1$, $A_\mu=i\bar{z}\partial_\mu z$ is a composite gauge field, and $D_\mu =\partial_\mu +iA_\mu$ is a covariant derivative. They provide an interesting theoretical laboratory. Indeed they present several features that hold in QCD: Asymptotic freedom, gauge invariance, existence of a confining potential between non gauge invariant states (that is eventually screened by the dynamical constituents), and non-trivial topological structure (instantons, $\theta$ vacua). Moreover, unlike four-dimensional SU($N$) gauge theories, a systematic $1/N$ expansion can be performed around the large-$N$ saddle-point solution [@DDL-79; @Witten-79b; @CR-92].
Analogously to four-dimensional SU($N$) gauge theories, one may add a $\theta$ term to the Lagrangian, writing $${\cal L}_\theta = {N\over 2g} \overline{D_\mu z}\, D_\mu z +
i \theta {1\over2\pi}\,\epsilon_{\mu\nu}\, \partial_\mu A_\nu,
\label{lagrangiancpntheta}$$ where $q(x)={1\over2\pi}\,\epsilon_{\mu\nu}\, \partial_\mu A_\nu$ is the the topological charge density. Then one may study the $\theta$ dependence of the ground state and other observables. In the following we discuss this issue within the $1/N$ expansion, performed keeping $g$ fixed. Simple large-$N$ scaling arguments applied to the Lagrangian (\[lagrangiancpn\]) indicate that the relevant $\theta$ parameter in the large-$N$ limit should be $\bar{\theta}\equiv
\theta/N$.
Analogously to SU($N$) gauge theories, the ground state energy $F(\theta)$ depends on $\theta$. One may define a scaling ground state energy $f(\theta)$ and expand it around $\theta=0$, $$f(\theta) \equiv M^{-2} [F(\theta)-F(0)]
= {1\over 2} C \theta^2 \left( 1 + \sum_{n=1} b_{2n} \theta^{2n} \right)
\label{fthetacpn}$$ where $F(\theta)$ is defined as in Eq. (\[vftheta\]), $M$ is the mass scale at $\theta=0$ defined from the second moment of the two-point function of the operator $P_{ij}(x) \equiv \bar{z}_i(x) z_j(x)$, $C$ is the scaling ratio $\chi/M^2$ at $\theta=0$, where $\chi$ is the topological susceptibility. The correlation function of the topological charge density, and in particular the topological susceptibility, has been computed within the $1/N$ expansion [@luescher-78; @CR-91; @v-99]. We have $$C = \chi/M^2 = {1\over 2\pi N} + O(1/N^2)$$ The coefficients $b_{2n}$ are obtained from appropriate $2n$-point correlation functions of the topological charge density operators at $\theta=0$. The analysis of the $1/N$-expansion Feynman diagrams [@CR-92] of the connected correlations necessary to compute $b_{2n}$ shows that they are suppressed in the large-$N$ limit, as [@DMPSV-06] $$b_{2n} = O(1/N^{2n}).$$ This implies that the ground-state energy can be rewritten as $$\begin{aligned}
&&f(\theta) = N \bar{f}(\bar{\theta}\equiv \theta/N),
\label{fthetabarcpn}\\
&& \bar{f}(\bar{\theta}) =
{1\over 2} \overline{C} \bar{\theta}^2
( 1 + \sum_{n=1} \bar{b}_{2n} \bar{\theta}^{2n} ),
\nonumber\end{aligned}$$ where $\overline{C}\equiv N C$ and $\bar{b}_{2n}=N^{2n}b_{2n}$ are $O(N^0)$. Note the analogy with the expected $\theta$ dependence of the ground-state energy in SU($N$) gauge theories, cf. Eq. (\[fthetabar\]). Rather cumbersome calculations lead to the results [@DMPSV-06] $\bar{b}_2= - {27\over 5}$, and $\bar{b}_4= -{1830\over 7}$.
Within the $1/N$ expansion one may also study the dependence of the mass $M$ on the parameter $\theta$. We write $$M(\theta) = M\left( 1 + m_2 \theta^2 + ... \right)
\label{gmexcpn}$$ The analysis of its diagrams in the corresponding $1/N$ expansion indicates that $m_2$ is suppressed as $$m_2 = O(1/N^2)$$ Once again, the relevant parameter is seen to be $\bar{\theta}\equiv \theta/N$.
Critical slowing down of topological modes {#slowdown}
==========================================
Monte Carlo simulations of critical phenomena in statistical mechanics and of quantum field theories, such as QCD, in the continuum limit are hampered by the problem of critical slowing down (CSD) [@Sokal-92]. The autocorrelation time $\tau$, which is related to the number of iterations needed to generate a new independent configuration, grows with increasing length scale $\xi$. In simulations of lattice QCD where the upgrading methods are essentially local, it has been observed, see e.g. Refs. [@ABDDV-96; @DeFetal2; @DPRV-02; @DPV-02; @LTW-04b] that the topological modes show autocorrelation times that are typically much larger than those of other observables not related to topology, such as Wilson loops and their correlators. Actually, the heating method [@DV-92], used to estimate the topological susceptibility, essentially relies on this phenomenon.
Recent Monte Carlo simulations [@DPV-02; @DPRV-02] of the four-dimensional SU($N$) lattice gauge theories (for $N=3,4,6$) provided evidence of a severe CSD for the topological modes, using a rather standard local overrelaxed upgrading algorithm. Indeed, the autocorrelation time $\tau_{\rm
top}$ of the topological charge grows very rapidly with the length scale $\xi\equiv \sigma^{-1/2}$, where $\sigma$ is the string tension, showing an apparent exponential behavior $\tau_{\rm top}\sim \exp (c\xi)$ in the range of values of $\xi$ where data are available. Such a phenomenon worsens with increasing $N$, indeed the constant $c$ appears to increase as $c\propto N$. Of course, this behaviour does not depend on the particular estimator of the topological charge. This peculiar effect has not been observed in plaquette-plaquette or Polyakov line correlations, suggesting an approximate decoupling between topological modes and nontopological ones, such as those determining the confining properties.
These results suggest that the dynamics of the topological modes in Monte Carlo simulations is rather different from that of quasi-Gaussian modes. CSD of quasi-Gaussian modes for traditional local algorithms, such as standard Metropolis or heat bath, is related to an approximate random-walk spread of information around the lattice. Thus, the corresponding autocorrelation time $\tau$ is expected to behave as $\tau\sim\xi^2$ (an independent configuration is obtained when the information travels a distance of the order of the correlation length $\xi$, and the information is transmitted from a given site/link to the nearest neighbors). This guess is correct for Gaussian (free field) models; in general one expects that $\tau\sim \xi^z$ where $z$ is a dynamical critical exponent, and $z\approx 2$ for quasi-Gaussian modes. On the other hand, in the presence of relevant topological modes, he random-walk picture may fail, and therefore we may have qualitatively different types of CSD. These modes may give rise to sizeable free-energy barriers separating different regions of the configuration space. The evolution in the configuration space may then present a long-time relaxation due to transitions between different topological charge sectors, and the corresponding autocorrelation time should behave as $\tau_{\rm top}\sim \exp F_b$ where $F_b$ is the typical free-energy barrier among different topological sectors. However, this picture remains rather qualitative, because it does not tell us how the typical free-energy barriers scale with the correlation length. For example, we may still have a power-law behavior if $F_b \sim \ln \xi$, or an exponential behavior if $F_b\sim \xi^\theta$. It is worth mentioning that in physical systems, such as random-field Ising systems [@Fisher-86] and glass models [@Parisi-92], the presence of significant free-energy barriers in the configuration space causes a very slow dynamics, and an effective separation of short-time relaxation within the free-energy basins from long-time relaxation related to the transitions between basins. In the case of random-field Ising systems the free-energy barrier picture supplemented with scaling arguments leads to the prediction that $\tau\sim
\exp (c \xi^\theta)$ where $\theta$ is a universal critical exponent [@Fisher-86].
The severe CSD experienced by the topological modes under local updating algorithms should be a general feature of Monte Carlo simulations of lattice models with nontrivial topological properties, since the mechanism behind this phenomenon should be similar. This has been also observed in two-dimensional CP$^{N-1}$ models [@DMV-04; @CRV-92]. The numerical study of Ref. [@DMV-04] for various values of $N$ show that an exponential Ansatz, i.e. $\tau_{\rm top}\sim \exp (c\xi^\theta)$ with $\theta\approx 1/2$, and $c\propto N$, provides a good effective description in the range of the correlation length $\xi$ where data are available (however, the statistical analysis of the data did not allow one to exclude an asymptotic power-law behavior $\tau\sim \xi^z$ with $z\gtapprox N/2$ setting in at relatively large $\xi$).
The issue of CSD of topological modes is particularly important for lattice QCD, because it may pose a serious limitation for numerical studies of physical issues related to topological properties, such as the mass and the matrix elements of the $\eta'$ meson, and in general the physics related to the broken U(1)$_A$ symmetry. Indeed, it may substantially worsen the cost estimates of the dynamical fermion simulations for lattice QCD, see, e.g., Ref. [@Jansen-03].
Finally, we note that although the effects of the topological CSD have not been directly observed in plaquette-plaquette or Polyakov line correlations, such a CSD will eventually affect them. The point is that the results of Ref. [@DMPSV-06], summarized in Sec. \[spectrum\], show that the correlators of plaquette operators and topological charge do not vanish at finite $N$, although they are quite small, and therefore there is not a complete decoupling between topological and nontopological modes. Therefore the strong critical slowing down that is clearly observed in the topological sector will eventually affect also the measurements of nontopological quantities, such as those related to the string and glueball spectrum.
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[^1]: An apparently contradictory result has been reported in Refs. [@BCNW-03; @GLWW-03; @DGP-05] for the expected large-volume probability distribution $P(Q)$, i.e. $P(Q)=(2\pi\langle
Q^2\rangle)^{-1/2}e^{-{Q^2\over 2\langle Q^2\rangle}}\left[1
+O(1/V)\right]$. A purely gaussian behaviour would imply an exact quadratic form for $f(\theta)$, and in particular a vanishing $b_2$, thereby contradicting the assumption of a generic expansion of $f(\theta)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Relativistic, kinematically complete phenomenological expressions for the dilepton decay rates of nucleon resonances with arbitrary spin and parity are derived in terms of the magnetic, electric, and Coulomb transition form factors. The dilepton decay rates of the nucleon resonances with masses below $2$ GeV are estimated using the extended vector meson dominance (VMD) model for the transition form factors. The model provides a unified description of the photo- and electroproduction data, $\gamma(\gamma ^{*})N
\rightarrow N^{*}, $ the vector meson decays, $N^{*}\rightarrow
N\rho(\omega) $, and the dilepton decays, $N^{*}\rightarrow N\ell ^{+}\ell
^{-}$. The constraints on the transition form factors from the quark counting rules are taken into account. The parameters of the model are fixed by fitting the available photo- and electroproduction data and using results of the multichannel partial-wave analysis of the $\pi N$ scattering. Where experimental data are not available, predictions of the non-relativistic quark models are used as an input. The vector meson coupling constants of the magnetic, electric, and Coulomb types are determined. The dilepton widths and the dilepton spectra from decays of nucleon resonances with masses below $2$ GeV are calculated.
: nucleon resonances, electromagnetic transition form factors, dileptons
PACS: [25.75.Dw, 13.30.Ce, 12.40.Yx]{}
author:
- |
M.I. Krivoruchenko$^{a,b)}$, B.V. Martemyanov$^{a,b)}$, Amand Faessler$^{a)}$, C. Fuchs$^{a)}$\
$^{a)}$[Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14]{}\
[D-72076 Tübingen, Germany]{}\
[ ]{}$^{b)}$[Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25]{}\
[117259 Moscow, Russia]{}
title: |
Electromagnetic transition form factors\
and dilepton decay rates of nucleon resonances
---
=-110pt =-3mm =1000
Introduction
============
$\;$
Particle properties in the medium are generally known to be different from particle properties in the vacuum. The change of the nucleon mass in the nuclear matter was studied within the Walecka model [@Walecka:1974qa; @Chin:1977iz] already in the 1970’s. During the last decade, the problem of the description of hadrons in dense and hot nuclear matter received new attention [@Drukarev:1988ib; @Tsushima:1991fe; @Brown:1991kk; @Adami:1993tp]. The goal of the current investigations is to determine mass shifts and broadening of the resonances in nuclear matter. The best probe for measuring the in-medium modifications of the vector mesons masses and widths are dileptons which, being produced, leave the reaction zone essentially undistorted by the final-state interactions.
The data on the total photoabsorption cross section on heavy nuclei [@Bianchi:1993nh] give an evidence for a broadening of nucleon resonances in nuclear medium [@Kondratyuk:1994ah]. The physics behind this effect is the same as in the collision broadening of the atomic spectral lines in hot and dense gases, discussed by Weisskopf [@Wei] in early 1930’s.
Dilepton spectra from heavy-ion collisions have been measured by the CERES and HELIOS-3 Collaborations at SPS [@Agakishiev:1995xb; @Masera:1995ck] (a few hundreds GeV per nucleon) and by the DLS Collaboration at the BEVALAC [@Porter:1997rc] (a few GeV per nucleon). The collision broadening makes the peaks associated with the $V\rightarrow e^{+}e^{-}$ vector meson decays in the CERES and HELIOS-3 experiments unobservable, but the spectra can be described theoretically, although the origin of the enhanced dilepton yields in the low mass region i.e. below the $\rho$-meson peak is still a matter of current debate.
However, concerning the DLS experiment, the measured dilepton spectra do not match with the theoretical estimates, even when possible reduction of the $\rho $-meson mass and the $\rho $-meson broadening are taken into account [@Bratkovskaya:1999pr]. The future HADES experiment at GSI will study the dilepton spectra in the same energy range in greater details [@Friese:1999qm].
The dilepton modes of nucleon resonances are important sources of the dilepton production in proton-proton and heavy-ion collisions. In this paper, we derive kinematically complete phenomenological expressions for the dilepton decays of nucleon resonances with arbitrary spin and parity, parameterized in terms of the magnetic, electric, and Coulomb transition form factors, and give numerical estimates for the dilepton spectra and dilepton widths of the nucleon resonances with masses below $2$ GeV.
In order to calculate the dilepton decays, one needs to know the electromagnetic transition form factors of nucleon resonances in the time-like region. The standard vector meson dominance (VMD) with the ground-state $\rho $-$,$ $\omega $-$,$ and $\phi $-mesons predicts monopole form factors with $1/q^{2}$ asymptotics at $q^{2}\rightarrow \infty $. Such asymptotics are, according to the quark counting rules [@Matveev:1973ra], valid for the electromagnetic pion form factor. However, already in the case of the nucleon form factors, radially excited vector mesons $\rho ^{\prime },$ $\rho ^{\prime \prime }$ $...$ etc. should be added into the VMD model in order to provide a dipole behavior for the Sachs form factors and describe the experimental data [@Hohler:1976ax; @Krivoruchenko:1994qb; @Mergell:1996bf; @Dubnicka:1996sp]. It was pointed out [@Faessler:2000md] that the standard VMD model overestimates the photon branching ratios of the nucleon resonances if decay widths of the nucleon resonances are used as an input. It disagrees also with the quark counting rules for the transition form factors. The higher powers of $1/q^{2} $ in the asymptotics imply a destructive interference between contributions of vector mesons at small $q^{2}$. This effect decreases the photon branching ratios. It can help to describe the experimental data for the vector meson and the photon decays of the nucleon resonances within the VMD model framework.
We use the extended VMD model for the description of the transition form factors of the nucleon resonances. The model provides a unified description of photo- and electroproduction data and the vector meson decays of the nucleon resonances. Its parameters are fixed by fitting the available data. The form factors are finally used for the calculation of the dilepton decays of nucleon resonances.
The same philosophy has been used in Ref. [@Faessler:2000de] in order to predict unknown meson decay channels to dileptons. Recent measurements e.g. for $\phi \rightarrow \eta e^+ e^-$, $\eta \rightarrow \pi^+ \pi^- e^+ e^-$ by the CMD-2 Collaboration [@Akhmetshin:2001bw] are in excellent agreement with predictions [@Faessler:2000de]. This gives a support for the extended VMD model in general, and also concerning the reliability of the present investigations of the dilepton decay channels of the nucleon resonances.
The outline of this paper is as follows: In the next Sect., we describe the general framework for the description of the higher-spin resonances, electromagnetic vertexes in terms of the covariant form factors, $F_{k}(q^{2})$, which are free from kinematical singularities, and calculate helicity $\gamma ^{*}N\rightarrow N^{*}$ amplitudes in terms of the covariant form factors $F_{k}(q^{2})$.
The relations between the magnetic, electric, and Coulomb transition form factors and the covariant form factors $F_{k}(q^{2})$ have been established for the $\Delta
(1232)$-resonance by Jones and Scadron [@JS] and for arbitrary spin resonances by Devenish, Eisenschitz and Körner [@DEK]. We find it worthwhile to rederive for methodical purposes in Sect. 3 these relations in view of the controversy existing in the literature concerning the simple $\Delta (1232)$ radiative and dilepton decays (for a discussion see Ref. [@Krivoruchenko:2001hs]). First, we transform amplitudes with the fixed total angular momentum of the photon and orbital momentum of the $\gamma ^{*}N$ system to the magnetic, electric, and Coulomb amplitudes, and, second, transform amplitudes with fixed total angular momentum and the orbital momentum to the helicity basis.
The kinematically complete phenomenological expressions for the $N^{*}\rightarrow N\gamma ^{*}$ decay rates, where $\gamma ^{*}$ is a virtual massive photon, and the $N^{*}\rightarrow Ne^{+}e^{-}$ decay rates are then obtained in terms of the the magnetic, electric, and Coulomb transition form factors. The experimental data on the photo- and electroproduction of the nucleon resonances are quoted for the helicity amplitudes and/or magnetic, electric, and Coulomb transition form factors. The results of Sects. 2 and 3 are sufficient for the description of these data.
The data for the vector meson decays of the nucleon resonances are quoted usually in the partial-wave basis for a fixed total spin and a fixed orbital momentum of the $NV$ system. In Sect. 4, we establish the connection between the partial-wave basis and the helicity basis of the $VN\rightarrow N^{*}$ amplitudes. The results of Sect. 4 are sufficient to fit the data for the vector meson decays of the nucleon resonances.
In Sect. 5, we establish a general representation for the transition form factors in the no-width vector meson limit within the extended VMD model, consistent with the quark counting rules. The quark counting rules reduce the number of the phenomenological parameters which otherwise cannot be determined. The overall sign of the vector meson decay amplitudes is not fixed experimentally with respect to the photo- and electroproduction amplitudes. We use the non-relativistic quark model to fix this sign.
In Sect. 6, numerical results are presented. We perform a fit to the amplitudes of the photo- and electroproduction of the nucleon resonances and to the vector meson decay amplitudes and determine free parameters of the extended VMD model. The minimal extension is found to be sufficient to describe the available data. We use for the vector meson decay amplitudes the data from PDG [@Groom:2000in]. When these data are not available, the results of the multichannel $\pi N$ partial-wave analysis by the Manley and Saleski [@Manley:1992yb] and Longacre and Dolbeau [@Longacre:1977ja] are used. In other cases, we use the quark model predictions by Koniuk [@Koniuk:1982ej] and Capstick and Roberts [@Capstick:1994kb]. We give the coupling constants of the nucleon resonances with the $\rho $-, and $\omega $-mesons, determined form the fit, the total widths of the dilepton decays of the nucleon resonances, and their dilepton spectra.
The $\gamma ^{*}N\rightarrow N^{*}$ helicity amplitudes
=======================================================
$\;$
The electromagnetic transition current between the nucleon and a spin-$J$ nucleon resonance has the form $$J_{\mu }(p_{*},\lambda _{*},p,\lambda )=e\overline{u}_{\beta _{1}...\beta
_{l}}(p_{*},\lambda _{*}){\sf \Gamma }_{\beta _{1}...\beta _{l}\mu }^{(\pm
)}u(p,\lambda )$$ where $m_{*}$ and $m$ are masses, $p_{*}$ and $p$ are momenta, $\lambda
_{*}\ $and $\lambda $ are helicities of the resonance and the nucleon, $e=-\sqrt{4\pi \alpha }$ is the electron charge, $\alpha =1/137$. In the resonance rest frame, $p_{*}=(m_{*},0,0,0),$ $p=(E,0,0,-k).$
The spinor $u_{\beta _{1}...\beta _{l}}(p_{*},\lambda _{*})$ is the generalized Rarita-Schwinger spinor (see e.g. [@BW; @LAN]) that describes fermions with $J=l+\frac{1}{2}$ $\geq \frac{3}{2}$. It is symmetric with respect to the indices $\beta _{1}...\beta _{l}$ and traceless. The spinors are normalized by $$\begin{aligned}
\overline{u}(p,\lambda)u(p,\lambda) &=&2m, \nonumber \\
(-)^{l}\overline{u}_{\beta _{1}...\beta _{l}}(p_{*},\lambda_{*})u_{\beta
_{1}...\beta _{l}}(p_{*},\lambda_{*}) &=&2m_{*} \label{NORM}\end{aligned}$$ (there is a misprint in Eq.(15.7) of Ref. [@LAN]). The matrices ${\sf \Gamma }_{\beta _{1}...\beta _{l}\mu }^{(\pm )}$ stand for the normal- and abnormal parity resonances, $J^{P}=\frac{1}{2}^{-},\frac{3}{2}^{+},\frac{5}{2}^{-},\;...$ (the upper sign) and $J^{P}=\frac{1}{2}^{+},\frac{3}{2}^{-},\frac{5}{2}^{+},...$ (the lower sign).
The photon polarization vectors have the form $$\begin{aligned}
\epsilon _{\mu }^{(\pm 1)}(q) &=&\frac{1}{\sqrt{2}}(0,\mp 1,-i,0), \nonumber
\\
\epsilon _{\mu }^{(0)}(q) &=&\frac{1}{M}(k,0,0,\omega ), \label{PH}\end{aligned}$$ where $q=p_{*}-p=(\omega ,0,0,k)$, $q^{2}=M^{2}.$ These vectors are transversal, $q_{\mu }\epsilon _{\mu }^{(\lambda )}(q)=0,$ and normalized by $$\epsilon _{\mu }^{(\lambda )}(q)^{*}\epsilon _{\mu }^{(\lambda ^{\prime
})}(q)=-\delta _{\lambda \lambda ^{\prime }}.$$
In the limit $M\rightarrow 0$, $\epsilon _{\mu }^{(0)}(q)=q_{\mu }/M+O(M)$. Due to the current conservation $q_{\mu }J_{\mu }=0,$ the longitudinal component of the vector current equals $\epsilon _{\mu }^{(0)}(q)J_{\mu
}=O(M),$ so it vanishes for physical photons at $M=0$.
The $\gamma ^{*}N\rightarrow N^{*}$ vertexes
--------------------------------------------
$\;$
[*Spin* ]{}$J\geq \frac{3}{2}$[* resonances.*]{} The resonances with arbitrary spin have three independent helicity amplitudes in the $\gamma
^{*}N\rightarrow N^{*}$ transitions. It means that there are three independent scalar functions to fix the vertexes. The most general decomposition of the vertex ${\sf \Gamma }_{\beta _{1}...\beta _{l}\mu
}^{(\pm )}$ over the Lorentz vectors and the Dirac gamma matrices has the form [@JS; @DEK; @Trueman:1969wn] $${\sf \Gamma }_{\beta _{1}...\beta _{l}\mu }^{(\pm )}=q_{\beta
_{1}}...q_{\beta _{l-1}}{\sf \Gamma }_{\beta _{l}\mu }^{(\pm )} \label{dec}$$ where $${\sf \Gamma }_{\beta \mu }^{(\pm )}=\sum_{k}{\sf \Gamma }_{\beta \mu }^{(\pm
)k}F_{k}^{(\pm )}. \label{dec_F}$$ In Eq.(\[dec\]), the symmetrization over the indices $\beta _{1},...,\beta
_{l}$ is assumed. The values $F_{k}^{(\pm)}$ are scalar functions of $q^2$ and are called covariant form factors in the following. In this representation, the Dirac structure of the transition amplitudes is fully separated off and expressed by the ${\sf \Gamma }_{\beta \mu }^{(\pm)k}$ matrices.
For the normal-parity case, the matrices ${\sf \Gamma }_{\beta \mu }^{(+)i}$ ($i=1,2,3$) have the form $$\begin{aligned}
{\sf \Gamma }_{\beta \mu }^{(+)1} &=&m_{*}(q_{\beta }\gamma _{\mu }-\not{q}g_{\beta \mu })\gamma _{5}, \label{K1} \\
{\sf \Gamma }_{\beta \mu }^{(+)2} &=&(q_{\beta }P_{\mu }-q\cdot Pg_{\beta
\mu })\gamma _{5}, \label{K2} \\
{\sf \Gamma }_{\beta \mu }^{(+)3} &=&(q_{\beta }q_{\mu }-q^{2}g_{\beta \mu
})\gamma _{5} \label{K3}\end{aligned}$$ where $\gamma _{5}=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3},$ $P=\frac{1}{2}(p_{*}+p)$. For the abnormal-parity case, the matrices ${\sf \Gamma }_{\beta \mu }^{(-)i}$ ($i=1,2,3$) can be taken to be $${\sf \Gamma }_{\beta \mu }^{(-)k}={\sf \Gamma }_{\beta \mu }^{(+)k}\gamma
_{5}. \label{K4}$$ The sets (\[K1\])-(\[K4\]) are simply related to the sets used in Refs. [@JS; @DEK].
[*Spin* ]{}$J=\frac{1}{2}$[* resonances.* ]{}The vertex ${\sf \Gamma }_{\mu
}^{(\pm )}$ ($l=0$) can also be expanded like in Eq.(\[dec\_F\]). There are two matrices ${\sf \Gamma }_{\mu }^{(+)i}$ ($i=1,2$) for the normal-parity case $J^{P}=\frac{1}{2}^{-}$, $$\begin{aligned}
{\sf \Gamma }_{\mu }^{(+)1} &=&(q^{2}\gamma _{\mu }-\not{q}q_{\mu })\gamma
_{5}, \\
{\sf \Gamma }_{\mu }^{(+)2} &=&(P\cdot q\gamma _{\mu }-P_{\mu }{q})\gamma
_{5}, \label{K5}\end{aligned}$$ and two matrices for the abnormal-parity case $J^{P}=\frac{1}{2}^{+},$$${\sf \Gamma }_{\mu }^{(-)k}={\sf \Gamma }_{\mu }^{(+)k}\gamma _{5}.
\label{K6}$$ The sets (\[K5\])-(\[K6\]) are identical to the sets used in Ref. [@DEK]. The vertex dimensions are ${\sf \Gamma }_{\beta _{1}...\beta _{l}\mu
}^{(\pm )}\backsim 1,$ ${\sf \Gamma }_{\beta \mu }^{(\pm )}\backsim 1/
m_{*}^{l-1}$, and $F_{k}^{(\pm )}\backsim 1/m_{*}^{l+1}.$
The $N^{*}\rightarrow N\gamma ^{*}$ decay width in terms of helicity amplitudes
-------------------------------------------------------------------------------
$\;$
The photo- and electroproduction $T$-matrix elements ($S=1+iT$), $$<JJ_{z}|{\it T}|\lambda \lambda _{\gamma }{\bf n}>,$$ depend on the resonance spin, $J$, its projection on the $z$-axis, $J_{z}$, the nucleon and photon helicities, $\lambda $ and $\lambda _{\gamma }$, and on the unit vector, ${\bf n}$, in the direction of the photon momentum. The $N^{*}\rightarrow N\gamma ^{*}$ width has the form $$\Gamma (N^{*}\rightarrow N\gamma ^{*})=\frac{k}{32\pi ^{2}m^{*2}}\int
d\Omega _{{\bf n}}\sum_{\lambda \lambda _{\gamma }}|<\lambda \lambda
_{\gamma }{\bf n}|{\it T}|JJ_{z}>|^{2}~. \label{gammanng}$$
The angular dependence of the matrix element $<JJ_{z}|{\it T}|\lambda
\lambda _{\gamma }{\bf n}>$ is a universal function (see e.g. [@Rose]) determined by the resonance total spin and its spin projections: $$<JJ_{z}|{\it T}|\lambda \lambda _{\gamma }{\bf n}>=D_{\lambda _{*}J_{z}}^{J}({\bf n})^{*}<J\lambda _{*}{\bf n}|{\it T}|\lambda \lambda _{\gamma }{\bf n}>$$ where $\lambda _{*}=-\lambda +\lambda _{\gamma }$ is the resonance helicity. The rotation matrices $$D_{\lambda _{*}J_{z}}^{J}({\bf n})=<J\lambda _{*}{\bf n}|JJ_{z}> \label{D}$$ are the amplitudes of finding the resonance with the spin projection $\lambda _{*}$ on the unit vector ${\bf n}$ in a state with the spin projection $J_{z}$ on the $z$-axis. The helicity amplitudes $<J\lambda _{*}{\bf n}|{\it T}|\lambda \lambda _{\gamma }{\bf n}>$ do not depend on the vector ${\bf n}$, so the symbol ${\bf n}$ can be suppressed. There exist six helicity amplitudes, three ones with positive $\lambda _{*}$’s and three ones with negative $\lambda _{*}$’s. The $P$-invariance of the electromagnetic interactions gives a symmetry relation for the amplitudes with opposite signs of the helicities (see e.g. [@LAN], Eq.(70.13)): $$<J-\lambda _{*}|{\it T}|-\lambda -\lambda _{\gamma }>=\mp <J\lambda _{*}|{\it T}|\lambda \lambda _{\gamma }>.$$
The functions $D_{\lambda _{*}J_{z}}^{J}({\bf n})$ are the unitary matrices with respect to the indices $\lambda _{*}$ and $J_{z}$. The normalization condition reads $$\int d\Omega _{{\bf n}}D_{\lambda _{*}J_{z}}^{J}({\bf n})^{*}D_{\lambda
_{*}^{\prime }J_{z}^{\prime }}^{J^{\prime }}({\bf n})=\frac{4\pi }{2J+1}\delta ^{JJ^{\prime }}\delta ^{\lambda _{*}\lambda _{*}^{\prime }}\delta
^{J_{z}J_{z}^{\prime }}.~ \label{O}$$ Using the properties of the helicity amplitudes and of the $D_{\lambda
_{*}J_{z}}^{J}({\bf n})$ matrices, one obtains the $N^{*}\rightarrow N\gamma
^{*}$ decay width in terms of the three helicity amplitudes: $$\Gamma (N^{*}\rightarrow N\gamma ^{*})=\frac{k}{32\pi ^{2}m_{*}^{2}}\frac{8\pi }{2J+1}\sum_{\lambda _{*}=-\lambda +\lambda _{\gamma }>0}|<\lambda
\lambda _{\gamma }|{\it T}|J\lambda _{*}>|^{2}~. \label{gammanng1}$$
Helicity amplitudes in terms of the covariant form factors F$_{k}^{(\pm )}$
---------------------------------------------------------------------------
$\;$
Since the amplitudes $<\lambda \lambda _{\gamma }|{\it T}|J\lambda _{*}>$ do not depend on the vector ${\bf n,}$ it is convenient to choose it in the direction of the $z$-axis. The helicity amplitudes can then be calculated in terms of the covariant form factors $F_{k}^{(\pm )}\ $from equation $$\left\langle J\lambda _{*}^{(\pm )}|{\it T}|\lambda \lambda _{\gamma
}\right\rangle =-e\overline{u}_{\beta _{1}...\beta _{l}}(p_{*},\lambda _{*}){\sf \Gamma }_{\beta _{1}...\beta _{l}\mu }^{(\pm )}u(p,\lambda )\epsilon
_{\mu }^{(\lambda _{\gamma })}(q). \label{amplitude}$$ The sign $\pm$ refers to the natural- and abnormal-parity resonances. We use the following notations for these amplitudes: $$\begin{aligned}
{\frak F}_{\frac{3}{2}}^{(\pm )} &=&\left\langle J\frac{3}{2}^{(\pm )}\left|
T\right| -\frac{1}{2}1\right\rangle , \nonumber \\
{\frak F}_{\frac{1}{2}}^{(\pm )} &=&\left\langle J\frac{1}{2}^{(\pm )}\left|
T\right| +\frac{1}{2}1\right\rangle , \nonumber \\
\frac{M}{m_{*}}{\frak C}_{\frac{1}{2}}^{(\pm )} &=&\left\langle J\frac{1}{2}^{(\pm )}\left| T\right| -\frac{1}{2}0\right\rangle . \label{AAS}\end{aligned}$$ These amplitudes describe, respectively, the double-spin-flip, no-spin-flip, and single-spin-flip transitions. For $l=0$, the amplitude ${\frak F}_{\frac{3}{2}}^{(\pm )}$ should be set equal to zero.
The experimental data for the helicity amplitudes (\[AAS\]) are quoted by PDG in the non-relativistic normalization for the fermions, including a factor of $1/\sqrt{2\omega _{0}}$ from the photon wave function, and also a sign of the $\pi N\rightarrow N^{*}$ amplitude and an additional sign $\mp $ for nucleon and $\Delta $-resonances. The value $\omega
_{0}=(m_{*}^{2}-m^{2})/(2m_{*})$ is the real-photon energy.
[*Spin* ]{}$J\geq \frac{3}{2}$[* resonances.*]{} The direct calculations give $$\begin{aligned}
\left(
\begin{array}{l}
\pm {\frak F}_{\frac{3}{2}}^{(\pm )} \\
-{\frak F}_{\frac{1}{2}}^{(\pm )} \\
\pm {\frak C}_{\frac{1}{2}}^{(\pm )}
\end{array}
\right) &=&\lambda _{l}^{(\pm )}\frac{2(\pm m)}{3m_{\pm }}\times \nonumber
\\
&&\left(
\begin{array}{lll}
\sqrt{\frac{l+2}{2l}}2m_{\pm }m_{*} & \sqrt{\frac{l+2}{2l}}m_{+}m_{-} &
\sqrt{\frac{l+2}{2l}}2M^{2} \\
\sqrt{\frac{1}{2}}2(m_{\pm }(\mp m)+M^{2}) & \sqrt{\frac{1}{2}}m_{+}m_{-} &
\sqrt{\frac{1}{2}}2M^{2} \\
2m_{*}^{2} & 2m_{*}^{2}-\frac{1}{2}\Delta _{0}^{2} & \Delta _{0}^{2}
\end{array}
\right) \left(
\begin{array}{l}
F_{1}^{(\pm )} \\
F_{2}^{(\pm )} \\
F_{3}^{(\pm )}
\end{array}
\right) \label{HEL3FFF}\end{aligned}$$ where $$\begin{aligned}
m_{\pm } &=&m_{*}\pm m, \\
\Delta _{0}^{2} &=&m_{+}m_{-}+M^{2}.\end{aligned}$$ The coefficients $\lambda _{l}^{(\pm )}$ are defined as $$\lambda _{l}^{(\pm )}=e\frac{3m_{\pm }}{4(\pm m)}\sqrt{m_{\mp }^{2}-M^{2}}k^{l-1}\sqrt{\frac{2^{l}(l!)^{2}(l+1)}{(2l+1)!}} \label{lamb}$$ with $J = l + \frac{1}{2}$. Notice that $$\begin{aligned}
{\frak F}_{\frac{3}{2}}^{(-)}(m_{*},m) &=&-{\frak F}_{\frac{3}{2}}^{(+)}(m_{*},-m), \nonumber \\
{\frak F}_{\frac{1}{2}}^{(-)}(m_{*},m) &=&+{\frak F}_{\frac{1}{2}}^{(+)}(m_{*},-m), \nonumber \\
{\frak C}_{\frac{1}{2}}^{(-)}(m_{*},m) &=&-{\frak C}_{\frac{1}{2}}^{(+)}(m_{*},-m).\end{aligned}$$
[*Spin* ]{}$J=\frac{1}{2}$[* resonances.* ]{}The direct calculation gives the following expression for the helicity amplitudes$:$
$$\left(
\begin{array}{c}
{\frak F}_{\frac{1}{2}}^{(\pm )} \\
\pm \sqrt{2}{\frak C}_{\frac{1}{2}}^{(\pm )}
\end{array}
\right) =\frac{\lambda _{0}^{(\pm )}}{m_{*}}\left(
\begin{array}{ll}
2M^{2} & m_{+}m_{-} \\
-2m_{*}m_{\mp } & -m_{*}m_{\pm }
\end{array}
\right) \left(
\begin{array}{l}
F_{1}^{(\pm )} \\
F_{2}^{(\pm )}
\end{array}
\right) . \label{HEL2FFF}$$
The parameters $\lambda _{0}^{(\pm )}$ are defined by $$\lambda _{0}^{(\pm )}=e\frac{m_{*}}{\sqrt{2}}\sqrt{m_{\pm }^{2}-M^{2}}.$$
Magnetic, electric, and Coulomb transition form factors
=======================================================
$\;$
The photo- and electroproduction of nucleon resonances$\ $can be described in terms of the amplitudes ${\frak A}_{J_{\gamma },\ell }^{(\pm )}$ with the definite total angular momentum $J_{\gamma }=J+\frac{1}{2}$, $J-\frac{1}{2}$ of the photon and the definite orbital momentum $\ell =J_{\gamma }+1,$ $J_{\gamma },$ $J_{\gamma }-1$ of the $N\gamma ^{*}$ system ($\ell$ should not be mixed with $l$ in Eq.(II.1)). There are three nonvanishing amplitudes: $${\frak A}_{J_{\gamma }^{\prime },J_{\gamma }^{\prime }}^{(\pm )},\;{\frak A}_{J_{\gamma },J_{\gamma }+1}^{(\pm )},\;{\frak A}_{J_{\gamma },J_{\gamma
}-1}^{(\pm )} \label{JLM}$$ where $J_{\gamma }^{\prime }=J\mp \frac{1}{2}$ and $J_{\gamma }=J\pm \frac{1}{2}.\;$In the spin $J=\frac{1}{2}$ case, the photo- and electroproduction processes are described by two amplitudes ${\frak A}_{1,2}^{(+)}$ and$\;{\frak A}_{1,0}^{(+)}$ for negative parity resonances and ${\frak A}_{0,1}^{(-)}$ and ${\frak A}_{1,1}^{(-)}$ for positive parity resonances.
We establish first a connection of the magnetic, electric, and Coulomb amplitudes to the amplitudes ${\frak A}_{J_{\gamma },\ell }^{(\pm )}$ and find then a connection of the amplitudes ${\frak A}_{J_{\gamma },\ell
}^{(\pm )}$ to the helicity amplitudes. In this way, using Eqs. (\[HEL3FFF\]) and (\[HEL2FFF\]), we express the magnetic, electric, and Coulomb form factors in terms of the covariant form factors $F_{k}$.
Relation between $J_{\gamma }\ell $ amplitudes and monopole amplitudes
-----------------------------------------------------------------------
$\;$
In the momentum space, the photon vector potentials with the definite total angular momentum, $J_{\gamma },$ its projection, $m$, on the $z$-axis, and orbital momentum, $\ell ,$ have the form $${\bf A}_{J_{\gamma }m\ell }({\bf n})=\sum_{\lambda _{\gamma }}C_{\ell
m-\lambda _{\gamma }1\lambda _{\gamma }}^{J_{\gamma }m}Y_{\ell m-\lambda
_{\gamma }}({\bf n}){\bf \varepsilon }^{(\lambda _{\gamma })}$$ where ${\bf \varepsilon }^{(\lambda _{\gamma })}$ is the space-like part of the polarization vector (\[PH\]) with the vanishing momentum $k=0$. The spherical coordinates of the vector potential have the form $$({\bf A}_{J_{\gamma }m\ell }({\bf n}))_{\lambda _{\gamma }}\equiv -{\bf \varepsilon }^{(\lambda _{\gamma })}\cdot {\bf A}_{J_{\gamma }m\ell }({\bf n})=(-)^{1+\lambda _{\gamma }}C_{\ell m+\lambda _{\gamma }1-\lambda _{\gamma
}}^{J_{\gamma }m}Y_{\ell m+\lambda _{\gamma }}({\bf n}).$$
In momentum space, the magnetic, electric and Coulomb potentials equal [@LAN], $${\bf A}_{J_{\gamma }m}^{T}({\bf n})={\bf a}^{T}Y_{J_{\gamma }m}({\bf n})
\label{MULT}$$ where $T=M,E,C$ and $$\begin{aligned}
{\bf a}^{M} &=&\frac{{\bf l}}{\sqrt{J_{\gamma }(J_{\gamma }+1)}}, \nonumber \\
{\bf a}^{E} &=&\frac{{\bf \nabla }_{n}}{\sqrt{J_{\gamma }(J_{\gamma }+1)}},
\nonumber \\
{\bf a}^{C} &=&{\bf n}.\end{aligned}$$ Here, ${\bf l}=-i{\bf k}\times $ $\partial /\partial {\bf k}$ is the orbital momentum operator and ${\bf \nabla }_{n}=k\partial /\partial {\bf k.}$ Using the Wigner-Eckart theorem (see e.g. [@Rose]), $$<\ell m^{\prime }|({\bf a}^{T})_{\lambda _{\gamma }}|J_{\gamma
}m>=C_{J_{\gamma }m1\lambda _{\gamma }}^{\ell m^{\prime }}<\ell ||{\bf a}^{T}||J_{\gamma }>,$$ the spherical coordinates of the vector potentials ${\bf A}_{J_{\gamma
}m}^{T}({\bf n})$ can be found to be $$({\bf A}_{J_{\gamma }m}^{T}({\bf n}))_{\lambda _{\gamma }}=\sum_{\ell
}(-)^{\ell +1-J_{\gamma }}\sqrt{\frac{2\ell +1}{2J_{\gamma }+1}}<\ell ||{\bf a}^{T}||J_{\gamma }>({\bf A}_{J_{\gamma }m\ell }({\bf n}))_{\lambda _{\gamma
}}. \label{MUJL}$$ Notice that $\ell =J_{\gamma }\pm 1$ for $T=M,E$ and $\ell =J_{\gamma }$ for $T=C.$ Eq.(\[MUJL\]) gives a connection between the $J_{\gamma }\ell $and $J_{\gamma }T$ representations for the photon wave functions and, respectively, for the $\gamma ^{*}N\rightarrow N^{*}$ transition amplitudes.
The linear combinations, ${\frak R}_{J_{\gamma },T}^{(\pm )}$, of the amplitudes ${\frak A}_{J_{\gamma },\ell }^{(\pm )}$ with the coefficients of Eq.(\[MUJL\]) describe absorption of photons of the magnetic, electric, and Coulomb types. In the matrix form, $$\left(
\begin{array}{l}
{\frak R}_{J_{\gamma }^{\prime },M}^{(\pm )} \\
{\frak R}_{J_{\gamma },E}^{(\pm )} \\
{\frak R}_{J_{\gamma },C}^{(\pm )}
\end{array}
\right) =\left(
\begin{array}{lll}
1 & 0 & 0 \\
0 & +\sqrt{\frac{J_{\gamma }}{2J_{\gamma }+1}} & +\sqrt{\frac{J_{\gamma }+1}{2J_{\gamma }+1}} \\
0 & -\sqrt{\frac{J_{\gamma }+1}{2J_{\gamma }+1}} & +\sqrt{\frac{J_{\gamma }}{2J_{\gamma }+1}}
\end{array}
\right) \left(
\begin{array}{l}
{\frak A}_{J_{\gamma }^{\prime },J_{\gamma }^{\prime }}^{(\pm )} \\
{\frak A}_{J_{\gamma },J_{\gamma }+1}^{(\pm )} \\
{\frak A}_{J_{\gamma },J_{\gamma }-1}^{(\pm )}
\end{array}
\right) . \label{MEC}$$
Relation between $J_{\gamma }\ell$ amplitudes and helicity amplitudes
---------------------------------------------------------------------
$\;$
The amplitudes ${\frak A}_{J_{\gamma },\ell }^{(\pm )}$ can be expressed in terms of the helicity amplitudes ${\frak F}_{\frac{3}{2}}^{(\pm )},$ ${\frak F}_{\frac{1}{2}}^{(\pm )},$ and ${\frak C}_{\frac{1}{2}}^{(\pm )}$. Let us consider $T$-matrix elements $$<JJ_{z}|{\it T}|JJ_{z}J_{\gamma }\ell >, \label{JgL}$$ with $|JJ_{z}J_{\gamma }\ell >$ given by
$$|JJ_{z}J_{\gamma }\ell >=\sum_{{s_{z}}J_{\gamma z}}\sum_{{s_{\gamma z}}m}C_{J_{\gamma }J_{\gamma z}\frac{1}{2}{s_{z}}}^{JJ_{z}}C_{\ell m1{s_{\gamma
z}}}^{J_{\gamma }J_{\gamma z}}|\ell m>|\frac{1}{2}{s_{z}}>|1{s_{\gamma z}}>.
\label{DECO}$$
The states $|JJ_{z}J_{\gamma }\ell >$ are eigenstates of the total angular momentum $J,$ its projection on the $z$- axis $J_{z},\;$the total photon angular momentum $J_{\gamma }$, and of the orbital angular momentum $\ell $ of the $N\gamma ^{*}$ system. The values $C_{J_{\gamma }J_{\gamma z}\frac{1}{2}{s_{z}}}^{JJ_{z}}$ and $C_{\ell m1{s_{\gamma z}}}^{J_{\gamma
}J_{\gamma z}}$ are the usual Clebsh-Gordon coefficients (CGC’s) with the phase conventions of PDG [@Groom:2000in]. In Eq.(\[DECO\]) and below, the standard rules for combining the angular momenta and spins are used (see e.g. [@Manley:1984jz]), according to which baryons in CGC’s appear before mesons, the orbital momentum in CGC’s comes before the intrinsic spin, and the angles in the spherical harmonic $Y_{lm}({\bf n})$ are measured with respect to the first particle in the corresponding isospin CGC’s.
Using the set $|\lambda \lambda _{\gamma }{\bf n}>$ of the helicity states, the $T$-matrix elements $<JJ_{z}|{\it T }$ $|JJ_{z}J_{\gamma }\ell >$ can be transformed as follows: $$<JJ_{z}|{\it T}|JJ_{z}J_{\gamma }\ell >=\int d\Omega _{{\bf n}}\sum_{\lambda
\lambda _{\gamma }}<JJ_{z}|{\it T}|\lambda \lambda _{\gamma }{\bf n}><\lambda \lambda _{\gamma }{\bf n}|JJ_{z}J_{\gamma }\ell >. \label{QQQ}$$ The state $|JJ_{z}J_{\gamma }\ell >$ is a superposition of eigenstates of the helicity $\lambda _{*}$ of the resonance: $$|JJ_{z}J_{\gamma }\ell >=\sum_{\lambda _{*}}D_{\lambda _{*}J_{z}}^{J}({\bf n})|J\lambda _{*}J_{\gamma }\ell >.$$ Using the decomposition of Eq.(\[DECO\]) for the states $|J\lambda
_{*}J_{\gamma }\ell >$, the amplitudes $<\lambda \lambda _{\gamma }{\bf n}|JJ_{z}J_{\gamma }\ell >$ can be found to be $$<\lambda \lambda _{\gamma }{\bf n}|JJ_{z}J_{\gamma }\ell >=\sqrt{\frac{2\ell
+1}{4\pi }}D_{\lambda _{*}J_{z}}^{J}({\bf n})C_{J_{\gamma }\lambda _{\gamma }\frac{1}{2}-{\lambda }}^{J\lambda _{*}}C_{\ell 01\lambda _{\gamma
}}^{J_{\gamma }\lambda _{\gamma }}.$$ The CGC’s entering Eq.(III.2) originate from the decomposition of Eq.(\[DECO\]), where one should set $J_{z}=\lambda _{*}$, $J_{\gamma z}={s_{\gamma z}=}\lambda
_{\gamma }$, and $m=J_{\gamma z}-{s_{\gamma z}=0.}$ The coefficient $\sqrt{\frac{2\ell +1}{4\pi }}$ arises from the spherical harmonic $<{\bf n}|\ell
m>=Y_{lm}({\bf n})$ evaluated at $m=0$ and ${\bf n}=(0,0,1)$.
The integration in Eq.(\[QQQ\]) removes the $D$-matrices. The $T$-matrix elements $<JJ_{z}|{\it T}$ $|JJ_{z}J_{\gamma }\ell >$ are expressed in terms of the $T$-matrix elements $<J\lambda _{*}|{\it T}$ $|\lambda \lambda_{\gamma }>$ as follows $$<JJ_{z}|{\it T}|JJ_{z}J_{\gamma }\ell >=\frac{\sqrt{4\pi (2\ell +1)}}{2J+1}\sum_{\lambda \lambda _{\gamma }}C_{J_{\gamma }\lambda _{\gamma }\frac{1}{2}-{\lambda }}^{J\lambda _{*}}C_{\ell 01\lambda _{\gamma }}^{J_{\gamma }\lambda
_{\gamma }}<J\lambda _{*}|{\it T}|\lambda \lambda _{\gamma }>. \label{AAA}$$ The amplitudes ${\frak A}_{J_{\gamma },\ell }^{(\pm )}$ can be defined by
$${\frak A}_{J_{\gamma },\ell }^{(\pm )}=\sqrt{\frac{2J+1}{8\pi }}<JJ_{z}|{\it T}|JJ_{z}J_{\gamma }\ell >.$$
The symmetry of the helicity amplitudes under the $P$-transformation can be used to reduce the summation in Eq.(\[AAA\]) to positive values of $\lambda _{*}=-\lambda +\lambda _{\gamma }>0$, which yields $${\frak A}_{J_{\gamma },\ell }^{(\pm )}=\frac{1\pm (-)^{-l+\ell }}{\sqrt{2}}\sqrt{\frac{2\ell +1}{2J+1}}\sum_{\lambda _{*}=-\lambda +\lambda _{\gamma
}>0}C_{J_{\gamma }\lambda _{\gamma }\frac{1}{2}-{\lambda }}^{J\lambda
_{*}}C_{\ell 01\lambda _{\gamma }}^{J_{\gamma }\lambda _{\gamma }}<J\lambda
_{*}|{\it T}|\lambda \lambda _{\gamma }>. \label{UJgL}$$
Relation between magnetic, electric, and Coulomb transition form factors and helicity amplitudes
------------------------------------------------------------------------------------------------
$\;$
[*Spin* ]{}$J\geq \frac{3}{2}$[* resonances.* ]{}The electric, magnetic and Coulomb transition form factors are defined by $$\begin{aligned}
{\frak R}_{J-\frac{1}{2},M/E}^{(\pm )} &=&\mp \lambda _{l}^{(\pm )}\sqrt{\frac{l+1}{l}}G_{M/E}^{(\pm )}, \nonumber \\
{\frak R}_{J+\frac{1}{2},E/M}^{(\pm )} &=&\mp \lambda _{l}^{(\pm )}\sqrt{(l+1)(l+2)}G_{E/M}^{(\pm )}, \nonumber \\
{\frak R}_{J\pm \frac{1}{2},C}^{(\pm )}\;\;\; &=&\mp \lambda _{l}^{(\pm )}\frac{M}{m_{*}}G_{C}^{(\pm )}.\end{aligned}$$ Here, the $G_{M/E}^{(\pm )}$ stands for $G_{M}^{(+)}$ or $G_{E}^{(-)}$ and the $G_{E/M}^{(\pm )}$ stands for $G_{E}^{(+)}$ or $G_{M}^{(-)}$ with the coefficients $\lambda _{l}^{(\pm )}$ given by Eq.(2.24). Using Eqs.(\[MEC\]) and (\[AAS\]) and substituting the coefficients $C{}_{\ell
01\lambda _{\gamma }}^{J_{\gamma }\lambda _{\gamma }}$ and $C_{J_{\gamma
}\lambda _{\gamma }\frac{1}{2}\lambda }^{J\lambda _{*}}$ in Eq.(\[UJgL\]), we obtain the following relation between the electric, magnetic, and Coulomb form factors and the helicity amplitudes:
$$\lambda _{l}^{(\pm )}\left(
\begin{array}{l}
\sqrt{\frac{l+1}{l}}G_{M/E}^{(\pm )} \\
\sqrt{(l+1)(l+2)}G_{E/M}^{(\pm )} \\
G_{C}^{(\pm )}
\end{array}
\right) =\left(
\begin{array}{lll}
+\sqrt{\frac{l+2}{2(l+1)}} & +\sqrt{\frac{l}{2(l+1)}} & 0 \\
+\sqrt{\frac{l}{2(l+1)}} & -\sqrt{\frac{l+2}{2(l+1)}} & 0 \\
0 & 0 & +1
\end{array}
\right) \left(
\begin{array}{l}
{\frak F}_{\frac{3}{2}}^{(\pm )} \\
{\frak F}_{\frac{1}{2}}^{(\pm )} \\
{\frak C}_{\frac{1}{2}}^{(\pm )}
\end{array}
\right) . \label{EMC3HEL}$$
The transformation matrix is an orthogonal matrix.
The helicity amplitudes are expressed in terms of the covariant form factors $F_{k}$ in Eq.(\[HEL3FFF\]). Eqs.(\[HEL3FFF\]) and (\[EMC3HEL\]) can be combined to give a linear relation between the magnetic, electric, and Coulomb form factors and the covariant form factors $F_{k}$.
The monopole form factors are expressed in terms of the covariant form factors $F_{k}^{(+)}$ as follows:
$$\begin{aligned}
\left(
\begin{array}{l}
G_{M}^{(+)} \\
G_{E}^{(+)} \\
G_{C}^{(+)}
\end{array}
\right) &=&\frac{2m}{3m_{{\large +}}}\left(
\begin{array}{lll}
\frac{\Delta _{l+1}^{2}}{l+1} & \frac{m_{+}m_{-}}{l+1} & \frac{2M^{2}}{l+1}
\\
\frac{\Delta _{0}^{2}}{l+1} & \frac{m_{+}m_{-}}{l+1} & \frac{2M^{2}}{l+1} \\
2m_{*}^{2} & 2m_{*}^{2}-\frac{1}{2}\Delta _{0}^{2} & \Delta _{0}^{2}
\end{array}
\right) \left(
\begin{array}{l}
F_{1}^{(+)} \\
F_{2}^{(+)} \\
F_{3}^{(+)}
\end{array}
\right) , \label{FG_lP} \\
\left(
\begin{array}{l}
G_{M}^{(-)} \\
G_{E}^{(-)} \\
G_{C}^{(-)}
\end{array}
\right) &=&\frac{2m}{3m_{{\large -}}}\left(
\begin{array}{lll}
\frac{\sigma _{-}^{2}}{l+1} & 0 & 0 \\
\frac{\sigma _{-}^{2}}{l+1}+\Delta _{0}^{2} & m_{+}m_{-} & 2M^{2} \\
2m_{*}^{2} & 2m_{*}^{2}-\frac{1}{2}\Delta _{0}^{2} & \Delta _{0}^{2}
\end{array}
\right) \left(
\begin{array}{l}
F_{1}^{(-)} \\
F_{2}^{(-)} \\
F_{3}^{(-)}
\end{array}
\right) \label{FG_lM}\end{aligned}$$
where $$\begin{aligned}
m_{\pm } &=&m_{*}\pm m, \\
\Delta _{l}^{2} &=&m_{+}m_{-}+M^{2}+l\sigma _{+}^{2}, \\
\sigma _{\pm }^{2} &=&m_{\pm }^{2}-M^{2}.\end{aligned}$$ The photon momentum appearing in Eq.(\[PH\]) can be written as $k=\sqrt{\sigma _{-}^{2}\sigma _{+}^{2}}/(2m_{*}).$ The inverse transformations have the form $$\begin{aligned}
\left(
\begin{array}{l}
F_{1}^{(+)} \\
F_{2}^{(+)} \\
F_{3}^{(+)}
\end{array}
\right) &=&\frac{3m_{+}}{2m\sigma _{+}^{2}\sigma _{{\large -}}^{2}}\times
\nonumber \\
&&\left(
\begin{array}{lll}
\;\sigma _{{\large -}}^{2} & -\sigma _{{\large -}}^{2} & 0 \\
-\sigma _{{\large -}}^{2} & 2m_{*}m_{-}+l\Delta _{0}^{2} & -2M^{2} \\
-\frac{1}{2}\sigma _{{\large -}}^{2} & -m_{*}m_{+}-\frac{1}{2}l(4m_{*}^{2}-\Delta _{0}^{2}) & m_{+}m_{-}
\end{array}
\right) \left(
\begin{array}{l}
G_{M}^{(+)} \\
G_{E}^{(+)} \\
G_{C}^{(+)}
\end{array}
\right) , \label{FGPLUS3} \\
\left(
\begin{array}{l}
F_{1}^{(-)} \\
F_{2}^{(-)} \\
F_{3}^{(-)}
\end{array}
\right) &=&\frac{3m_{-}}{2m\sigma _{+}^{2}\sigma _{{\large -}}^{2}}\times
\nonumber \\
&&\left(
\begin{array}{lll}
\;(l+1)\sigma _{{\large +}}^{2} & 0 & 0 \\
-2m_{*}m_{+}-l\sigma _{{\large +}}^{2} & \Delta _{0}^{2} & -2M^{2} \\
m_{*}m_{-}-\frac{1}{2}l\sigma _{{\large +}}^{2} & -2m_{*}^{2}+\frac{1}{2}\Delta _{0}^{2} & m_{+}m_{-}
\end{array}
\right) \left(
\begin{array}{l}
G_{M}^{(-)} \\
G_{E}^{(-)} \\
G_{C}^{(-)}
\end{array}
\right) . \label{FGMINU3}\end{aligned}$$
In terms of the magnetic, electric, and Coulomb form factors, the resonance decay widths equal $$\begin{aligned}
\Gamma (N_{(\pm )}^{*} &\rightarrow &N\gamma ^{*})=\frac{9\alpha }{16}\frac{(l!)^{2}}{2^{l}(2l+1)!}\frac{m_{\pm }^{2}(m_{\mp }^{2}-M^{2})^{l+1/2}(m_{\pm
}^{2}-M^{2})^{l-1/2}}{m_{*}^{2l+1}m^{2}} \nonumber \\
&&\left( \frac{l+1}{l}\left| G_{M/E}^{(\pm )}\right| ^{2}+(l+1)(l+2)\left|
G_{E/M}^{(\pm )}\right| ^{2}+\frac{M^{2}}{m_{*}^{2}}\left| G_{C}^{(\pm
)}\right| ^{2}\right) . \label{GAMMA_l}\end{aligned}$$
There is a symmetry between expressions for decay widths of the normal- and abnormal-parity resonances: $m_{+}\leftrightarrow m_{-},$ $G_{M}^{(+)}\leftrightarrow G_{E}^{(-)},\;G_{E}^{(+)}\leftrightarrow
G_{M}^{(-)},$ $G_{C}^{(+)}\leftrightarrow G_{C}^{(-)}.$ For $l=1$, we recover the result of Ref.[@Krivoruchenko:2001hs].
[*Spin* ]{}$J=\frac{1}{2}$[* resonances.* ]{}In the lowest spin case, the electric and magnetic form factors are defined by $$\begin{aligned}
{\frak R}_{J-\frac{1}{2},M/E}^{(\pm )} &=&G_{M/E}^{(\pm )}\equiv 0, \\
{\frak R}_{J+\frac{1}{2},E/M}^{(\pm )} &=&\mp \sqrt{2}\lambda _{0}^{(\pm
)}G_{E/M}^{(\pm )}, \\
{\frak R}_{J\pm \frac{1}{2},C\;}^{(\pm )}\; &=&\mp \lambda _{0}^{(\pm )}\frac{M}{m_{*}}G_{C}^{(\pm )}.\end{aligned}$$ The helicity amplitudes are simply connected to the electric (magnetic) and Coulomb amplitudes: $$\lambda _{0}^{(\pm )}\left(
\begin{array}{l}
\sqrt{2}G_{E/M}^{(\pm )} \\
G_{C}^{(\pm )}
\end{array}
\right) =\left(
\begin{array}{ll}
-1 & 0 \\
0 & 1
\end{array}
\right) \left(
\begin{array}{l}
{\frak F}_{\frac{1}{2}}^{(\pm )} \\
{\frak C}_{\frac{1}{2}}^{(\pm )}
\end{array}
\right) \label{EMC2HEL}$$
The Eqs. (\[HEL2FFF\]) and (\[EMC2HEL\]) can be combined to give a relation between the magnetic, electric, and Coulomb form factors and the form factors $F_{k}$. In the spin-$\frac{1}{2}$ case, these form factors have the form
$$\left(
\begin{array}{l}
G_{E/M}^{(\pm )} \\
\pm G_{C}^{(\pm )}
\end{array}
\right) =-\frac{1}{\sqrt{2}m_{*}}\left(
\begin{array}{ll}
\ 2M^{2} & \ m_{+}m_{-} \\
\ 2m_{*}m_{\mp } & \ m_{*}m_{\pm }
\end{array}
\right) \left(
\begin{array}{l}
F_{1}^{(\pm )} \\
F_{2}^{(\pm )}
\end{array}
\right) . \label{FG2}$$
The inverse relations are as follows $$\left(
\begin{array}{l}
F_{1}^{(\pm )} \\
F_{2}^{(\pm )}
\end{array}
\right) =\frac{1}{\sqrt{2}m_{\pm }\sigma _{\mp }^{2}}\left(
\begin{array}{ll}
m_{*}m_{\pm } & \ -m_{+}m_{-} \\
\ -2m_{*}m_{\mp } & \ \ 2M^{2}
\end{array}
\right) \left(
\begin{array}{l}
G_{E/M}^{(\pm )} \\
\pm G_{C}^{(\pm )}
\end{array}
\right) .$$
The resonance decay width can be found to be $$\begin{aligned}
\Gamma (N_{(\pm )}^{*} &\rightarrow &N\gamma ^{*})=\frac{\alpha }{8m_{*}}(m_{\pm }^{2}-M^{2})^{3/2}(m_{\mp }^{2}-M^{2})^{1/2} \nonumber \\
&&\left( 2\left| G_{E/M}^{(\pm )}\right| ^{2}+\frac{M^{2}}{m_{*}^{2}}\left|
G_{C}^{(\pm )}\right| ^{2}\right) . \label{GAMMA_0}\end{aligned}$$
We use the normalization for the monopole form factors identical to Ref. [@DEK]. The $\Delta (1232)$-resonance form factors of Refs. [@JS; @Krivoruchenko:2001hs] contain an additional factor of $\sqrt{\frac{2}{3}}.$ In the limit $M\rightarrow 0,$ we recover Eqs.(2.59) and (2.60) of Ref. [@DEK]. Eqs. (\[GAMMA\_l\]) and (\[GAMMA\_0\]) are the main results of this Sect.
The $N^{*}\rightarrow N\gamma ^{*}$ partial-wave amplitudes
===========================================================
$\;$
The decay width of the $N^{*}\rightarrow N\gamma ^{*}$ transition can be calculated in terms of the amplitudes ${\frak H}_{S,\ell }^{(\pm )}$ with the definite total spin, $S$, of the $N\gamma ^{*}$ system and the definite orbital momentum, $\ell ,$ of the $N\gamma ^{*}$ system ($\ell =J\mp \frac{1}{2},J\pm \frac{3}{2}$ for $J^{P}=\frac{3}{2}^{\pm },$ $\frac{5}{2}^{\mp },$ $...$ ). The nucleon decays are described by three independent amplitudes $${\frak H}_{\frac{1}{2},J\mp \frac{1}{2}}^{(\pm )},\;{\frak H}_{\frac{3}{2},J\mp \frac{1}{2}}^{(\pm )},\;{\frak H}_{\frac{3}{2},J\pm \frac{3}{2}}^{(\pm
)},$$ while in the case $J^{P}=\frac{1}{2}^{\mp },$ the decays are described by two independent amplitudes ${\frak H}_{\frac{1}{2},0}^{(+)},$ ${\frak H}_{\frac{3}{2},2}^{(+)}$ and ${\frak H}_{\frac{1}{2},1}^{(-)},$ ${\frak H}_{\frac{3}{2},1}^{(-)}$. The amplitudes ${\frak H}_{S,\ell }^{(\pm )}$ can be connected to the helicity amplitudes ${\frak F}_{\frac{3}{2}}^{(\pm )},$ ${\frak F}_{\frac{1}{2}}^{(\pm )},$ and ${\frak C}_{\frac{1}{2}}^{(\pm )}$.
Relation between partial-wave and helicity amplitudes
-----------------------------------------------------
$\;$
Let us consider the $T$-matrix elements $$<JJ_{z}S\ell |{\it T}|JJ_{z}>.$$ The states $|JJ_{z}S\ell >$ are defined by $$|JJ_{z}S\ell >=\sum_{S_{z}m}C_{\ell mSS_{z}}^{JJ_{z}}|\ell m>|SS_{z}>.
\label{SL}$$ They are eigenstates of the total spin, $S,$ of the nucleon and a virtual photon, of the orbital momentum, $\ell $, and of the total angular momentum, $J$, and its projection on $z$- axis$,J_{z},$ of the decaying resonance $N^{*}$. Using the set of the helicity states $|\lambda \lambda _{\gamma }{\bf n}>$, one can relate the $T$-matrix elements $<JJ_{z}S\ell |{\it T}|JJ_{z}>$ to the $T$-matrix elements $<\lambda \lambda _{\gamma }{\bf n}|{\it T}|JJ_{z}>$: $$<JJ_{z}S\ell |{\it T}|JJ_{z}>=\int d\Omega _{{\bf n}}\sum_{\lambda \lambda
_{\gamma }}<JJ_{z}S\ell |\lambda \lambda _{\gamma }{\bf n}><\lambda \lambda
_{\gamma }{\bf n}|{\it T}|JJ_{z}>~. \label{SLJJ}$$
The state $<JJ_{z}S\ell |$ can be rotated to give a superposition $$<JJ_{z}S\ell |=\sum_{\lambda _{*}}D_{J_{z}\lambda _{*}}^{J}({\bf n})^{*}<J\lambda _{*}S\ell |.$$ Using the decomposition of Eq.(\[SL\]) with $J_{z}=S_{z}=\lambda _{*}$ and $m=0$ for the rotated state $<J\lambda _{*}S\ell |$, we obtain $$<JJ_{z}S\ell |\lambda \lambda _{\gamma }{\bf n}>=\sqrt{\frac{2\ell +1}{4\pi }}C_{\ell 0S\lambda _{*}}^{J\lambda _{*}}C_{\frac{1}{2}-\lambda 1\lambda
_{\gamma }}^{S\lambda _{*}}D_{J_{z}\lambda _{*}}^{J}({\bf n})^{*}.$$ The factor $\sqrt{\frac{2\ell +1}{4\pi }}$ originates from the product $<\ell m|{\bf n}>=Y_{lm}^{*}({\bf n})$ evaluated at $m=0$ and ${\bf n}=(0,0,1) $. The angular dependence in the amplitudes $<\lambda \lambda
_{\gamma }{\bf n}|{\it T}|JJ_{z}>$ is factorized with the help of Eq.(\[D\]). The $D$-matrices are then removed by the angular integration in Eq.(\[SLJJ\]). The $T$-matrix elements in the partial-wave representation, $<JJ_{z}S\ell |{\it T}|JJ_{z}>$, become [@JW]
$$<JJ_{z}S\ell |{\it T}|JJ_{z}>=\frac{\sqrt{4\pi (2\ell +1)}}{2J+1}\sum_{\lambda \lambda _{\gamma }}C_{\ell 0S\lambda _{*}}^{J\lambda _{*}}C_{\frac{1}{2}-\lambda 1\lambda _{\gamma }}^{S\lambda _{*}}<\lambda \lambda
_{\gamma }|{\it T}|J\lambda _{*}>. \label{SLJJOK}$$
We extract from the amplitude the kinematical factors $\lambda _{l}^{(\pm )}$ and define the amplitude ${\frak H}_{S,\ell }^{(\pm )}$ as follows $$\lambda _{l}^{(\pm )}{\frak H}_{S,\ell }^{(\pm )}=\sqrt{\frac{2J+1}{8\pi }}<S\ell |{\it T}|JJ_{z}>.$$ The symmetry of the helicity amplitudes under the $P$-transformation can be used to remove in Eq.(\[SLJJOK\]) the summation over the negative values of $\lambda _{*}.$ We thus obtain $$\lambda _{l}^{(\pm )}{\frak H}_{S,\ell }^{(\pm )}=\frac{1\pm (-)^{-l+\ell }}{\sqrt{2}}\sqrt{\frac{2\ell +1}{2J+1}}\sum_{\lambda _{*}=-\lambda +\lambda
_{\gamma }>0}C_{\ell 0S\lambda _{*}}^{J\lambda _{*}}C_{\frac{1}{2}-\lambda
1\lambda _{\gamma }}^{S\lambda _{*}}<\lambda \lambda _{\gamma }|{\it T}|J\lambda _{*}>.$$
[*Spin* ]{}$J\geq \frac{3}{2}$[* resonances.* ]{}The partial-wave amplitudes are linear combinations of the helicity amplitudes:
$$\begin{aligned}
\lambda _{l}^{(+)}\left(
\begin{array}{c}
{\frak H}_{\frac{1}{2},J-\frac{1}{2}}^{(+)} \\
{\frak H}_{\frac{3}{2},J-\frac{1}{2}}^{(+)} \\
{\frak H}_{\frac{3}{2},J+\frac{3}{2}}^{(+)}
\end{array}
\right) &=&\left(
\begin{array}{ccc}
0 & -\sqrt{\frac{2}{3}} & +\sqrt{\frac{1}{3}} \\
-\sqrt{\frac{3(l+2)}{2(2l+3)}} & -\sqrt{\frac{l}{6(2l+3)}} & -\sqrt{\frac{l}{3(2l+3)}} \\
-\sqrt{\frac{l}{2(2l+3)}} & +\sqrt{\frac{l+2}{2(2l+3)}} & +\sqrt{\frac{l+2}{2l+3}}
\end{array}
\right) \left(
\begin{array}{r}
{\frak F}_{\frac{3}{2}}^{(+)} \\
{\frak F}_{\frac{1}{2}}^{(+)} \\
\frac{M}{m_{*}}{\frak C}_{\frac{1}{2}}^{(+)}
\end{array}
\right), \\
\lambda _{l}^{(-)}\left(
\begin{array}{c}
{\frak H}_{\frac{1}{2},J+\frac{1}{2}}^{(-)} \\
{\frak H}_{\frac{3}{2},J+\frac{1}{2}}^{(-)} \\
{\frak H}_{\frac{3}{2},J-\frac{3}{2}}^{(-)}
\end{array}
\right) &=&\left(
\begin{array}{ccc}
0 & +\sqrt{\frac{2}{3}} & -\sqrt{\frac{1}{3}} \\
+\sqrt{\frac{3l}{2(2l+1)}} & -\sqrt{\frac{l+2}{6(2l+1)}} & -\sqrt{\frac{l+2}{3(2l+1)}} \\
+\sqrt{\frac{l+2}{2(2l+1)}} & +\sqrt{\frac{l}{2(2l+1)}} & +\sqrt{\frac{l}{2l+1}}
\end{array}
\right) \left(
\begin{array}{r}
{\frak F}_{\frac{3}{2}}^{(-)} \\
{\frak F}_{\frac{1}{2}}^{(-)} \\
\frac{M}{m_{*}}{\frak C}_{\frac{1}{2}}^{(-)}
\end{array}
\right).\end{aligned}$$
The matrices entering these equations are orthogonal ones.
[*Spin* ]{}$J=\frac{1}{2}$[* resonances.* ]{}The transformation from the helicity basis to the partial-wave basis is given by $$\begin{aligned}
\lambda _{0}^{(+)}\left(
\begin{array}{c}
{\frak H}_{\frac{1}{2},J-\frac{1}{2}}^{(+)} \\
{\frak H}_{\frac{3}{2},J+\frac{3}{2}}^{(+)}
\end{array}
\right) &=&\left(
\begin{array}{cc}
-\sqrt{\frac{2}{3}} & +\sqrt{\frac{1}{3}} \\
+\sqrt{\frac{1}{3}} & +\sqrt{\frac{2}{3}}
\end{array}
\right) \left(
\begin{array}{r}
{\frak F}_{\frac{1}{2}}^{(+)} \\
\frac{M}{m_{*}}{\frak C}_{\frac{1}{2}}^{(+)}
\end{array}
\right) , \\
\lambda _{0}^{(-)}\left(
\begin{array}{c}
{\frak H}_{\frac{1}{2},J+\frac{1}{2}}^{(-)} \\
{\frak H}_{\frac{3}{2},J+\frac{1}{2}}^{(-)}
\end{array}
\right) &=&\left(
\begin{array}{cc}
+\sqrt{\frac{2}{3}} & -\sqrt{\frac{1}{3}} \\
-\sqrt{\frac{1}{3}} & -\sqrt{\frac{2}{3}}
\end{array}
\right) \left(
\begin{array}{r}
{\frak F}_{\frac{1}{2}}^{(-)} \\
\frac{M}{m_{*}}{\frak C}_{\frac{1}{2}}^{(-)}
\end{array}
\right) .\end{aligned}$$ The inverse transformations are given by the transposed matrices. Eqs.(IV.10) - (IV.13) are in agreement with Ref. [@Koniuk:1982ej] where the transformation matrices are given for $l=0,1,2,3$.
Dilepton decay widths
----------------------
$\;$
The $N^{*}\rightarrow N\gamma ^{*}$ decay width has many equivalent representations that can be obtained from Eqs.(\[gammanng\]) and (\[gammanng1\]) making use of the completeness of sets of the $N\gamma ^{*}$ states: $$\begin{aligned}
\frac{32\pi ^{2}m^{*2}}{k}\Gamma (N^{*} &\rightarrow &N\gamma ^{*})=
\nonumber \\
&=&\int d\Omega _{{\bf n}}\sum_{\lambda \lambda _{\gamma }}|<\lambda \lambda
_{\gamma }{\bf n}|{\it T}|JJ_{z}>|^{2}~ \nonumber \\
&=&\frac{8\pi }{2J+1}\sum_{\lambda _{*}=-\lambda +\lambda _{\gamma
}>0}|<\lambda \lambda _{\gamma }|{\it T}|J\lambda _{*}>|^{2} \nonumber \\
&=&\sum_{J_{\gamma }\ell }|<JJ_{z}J_{\gamma }\ell |{\it T}|JJ_{z}>|^{2}=\sum_{S\ell }|<JJ_{z}S\ell |{\it T}|JJ_{z}>|^{2} \nonumber \\
&=&\frac{8\pi }{2J+1}\sum_{J_{\gamma }\ell }|{\frak A}_{J_{\gamma },\ell
}^{(\pm )}|^{2}=\frac{8\pi }{2J+1}\sum_{J_{\gamma }T}|{\frak R}_{J_{\gamma
},T}^{(\pm )}|^{2} \nonumber \\
&=&\frac{8\pi }{2J+1}\sum_{S\ell }(\lambda_l^{(\pm)})^2|{\frak H}_{S,\ell }^{(\pm )}|^{2}.
\label{OK}\end{aligned}$$ One can add here also expressions (\[GAMMA\_l\]) and (\[GAMMA\_0\]) which calculate the decay widths using the monopole transition form factors. These form factors are the most frequently used ones both in analyzing experimental data and in theoretical works.
If the width $\Gamma (N^{*}\rightarrow N\gamma ^{*})$ is known, the factorization prescription (see [*e.g.*]{} [@Faessler:2000de]) can be used to find the dilepton decay rate:
$$d\Gamma (N^{*}\rightarrow Ne^{+}e^{-})=\Gamma (N^{*}\rightarrow N\gamma
^{*})M\Gamma (\gamma ^{*}\rightarrow e^{+}e^{-})\frac{dM^{2}}{\pi M^{4}},
\label{OK!}$$
where $$M\Gamma (\gamma ^{*}\rightarrow e^{+}e^{-})=\frac{\alpha }{3}(M^{2}+2m_{e}^{2})\sqrt{1-\frac{4m_{e}^{2}}{M^{2}}} \label{OK!!}$$ is the decay width of a virtual photon $\gamma ^{*}$ into the dilepton pair with invariant mass $M$.
The physical $N^{*}\rightarrow N\gamma $ decay rate is given by Eqs.(\[OK\]) in the limit of $M=0$. Eqs.(\[OK\])-(\[OK!!\]) or (\[GAMMA\_l\]), (\[GAMMA\_0\]), (\[OK!\]), and (\[OK!!\]) being combined give the $N^{*}\rightarrow Ne^{+}e^{-}$ decay rates.
Transition form factors and vector meson dominance
==================================================
$\;$
The non-relativistic quark models are successful in the description of static hadron properties and hadron decays. These models, however, are not well suited for the calculation of the dilepton emission, since the electromagnetic form factors should be interpolated into the time-like region. In the time-like region, the vector meson dominance comes into play, whereas the non-relativistic quark models have direct photon-quark couplings. They predict form factors which behave like $exp(q^{2}/\varkappa ),$ whereas the VMD requires a Breit-Wigner shape of the spectral functions for the form factors, centered around the vector meson masses with the physical vector meson widths. In a finite interval of the space-like region, the experimental data can be fitted with an exponential formula, whereas at high negative $q^{2}$ the non-relativistic quark models contradict to the quark counting rules. They contradict also to the Frazer-Fulco unitarity relations for the nucleon form factors [@Frazer:1959gy; @LLME; @SSM; @Hohler:1975ht; @HBOOK; @FUR; @Krivoruchenko:1995cv] in the time-like region. The unitarity relations, from the other side, justify the VMD model. The distinction between expressions given by the exact solution of the unitarity relations for the isovector nucleon form factors and the VMD expression is not strong. As to the pion form factor, the deviation of the naive VMD expression (see Eq.(\[PIF\])) from the FFGS expression [@Frazer:1959gy; @Gounaris:1968mw], obtained by solving the unitarity relations for the pion form factor (see e.g. [@BD]), is quite small. The same is true for the isovector kaon form factor [@Krivoruchenko:1993vd].
Using the VMD model, we satisfy the quark counting rules, analyticity in the complex $q^{2}$-plane, and take into account (approximately) the unitarity relations. It is important also that the VMD model gives simple expressions for the covariant form factors which can easily be embedded into the heavy-ion codes, if residues of the covariant form factors at the vector meson poles are determined.
Extended VMD model
------------------
$\;$
In terms of the vector meson fields, $V_{\mu }$, the electromagnetic current has the form [@SAK] $$J_{\mu }^{em}=-e\sum_{V}\frac{m_{V}^{2}}{g_{V}}V_{\mu } \label{A2}$$ where $m_{V}$ are the vector meson masses. The $SU(3)$ predictions for the coupling constants, $g_{\rho }:g_{\omega }:g_{\phi }=1:3:\frac{-3}{\sqrt{2}}, $ are in good agreement with the values $g_{\rho }=5.03,$ $g_{\omega
}=17.1$, and $g_{\phi }=-12.9$ extracted from the $V\rightarrow e^{+}e^{-}$ decays of the $\rho $-$,$ $\omega $-$,$ and $\phi $-mesons. The expression (\[A2\]) determines the vector meson couplings with the photon.
The VMD model describes well the electromagnetic pion form factor: $$F_{\pi }(q^{2})=\frac{f_{\rho \pi \pi }}{g_{\rho }}\frac{m_{\rho }^{2}}{m_{\rho }^{2}-q^{2}}, \label{PIF}$$ with $f_{\rho \pi \pi }$ being the coupling constant of the effective Lagrangian $$\begin{aligned}
{\cal L}_{\rho \pi \pi } &=&-\frac{1}{2}f_{\rho \pi \pi }
\epsilon _{\alpha \beta \gamma }\rho _{\mu }^{\alpha }
(\pi ^{\beta }{\partial }_{\mu }\pi ^{\gamma }) \nonumber \\
&=&-f_{\rho \pi \pi }(
\rho _{\mu }^{0}\pi ^{-}i{\partial }_{\mu }\pi ^{+}+
\rho _{\mu }^{+}\pi ^{0}i{\partial }_{\mu }\pi ^{-}+
\rho _{\mu }^{-}\pi ^{+}i{\partial }_{\mu }\pi ^{0}) \label{RHOPIPI}\end{aligned}$$ where ${\partial }_{\mu } = \overrightarrow{\partial }_{\mu } - \overleftarrow{\partial }_{\mu }$. The normalization $F_{\pi }(0)=1$ implies $$f_{\rho \pi \pi }/g_{\rho }=1. \label{SAC}$$
The quark counting rules [@Matveev:1973ra] show that the pion form factor decreases like $F_{\pi }(q^{2})\backsim 1/q^{2}$ as $q^{2} \rightarrow \infty .$ The VMD predicts therefore the correct asymptotics.
The electromagnetic nucleon form factors demonstrate experimentally a dipole behavior. The quark counting rules for the Sachs form factors predict $G_{E}(q^{2})\backsim G_{M}(q^{2})\backsim 1/q^{4}$ at $q^{2}\rightarrow
\infty .$ The VMD model with the ground-state $\rho $-$,$ $\omega $-$,$ and $\phi $-mesons cannot describe the nucleon form factors at low values of $q^{2}$ (the isovector charge radius is underestimated) and gives in contrast to the pion incorrect asymptotic behavior. It was proposed [@Hohler:1976ax; @Krivoruchenko:1994qb; @Mergell:1996bf; @Dubnicka:1996sp] to include in the current (\[A2\]) excited states of the vector mesons $\rho
^{\prime },$ $\rho ^{\prime \prime },$ ... etc. The VMD model extended in this way allows to reproduce the low- and intermediate-energy experimental data and yields for the nucleon form factors the correct asymptotic behavior. The minimal extension of the VMD model improves the description of the $\rho
\pi \gamma$ transition form factor that falls off asymptotically as $1/q^{4}
$ [@Faessler:2000de].
The vector meson couplings with the nucleon resonances are defined by the $T$-matrix element of the $VN\rightarrow N^{*}$ process $$<J\lambda _{*}|{\it T}|\lambda \lambda _{V}>=\sum_{k}f_{VNN^{*},k}^{(\pm )}\overline{u}_{\beta _{1}...\beta _{l}}(p_{*},\lambda _{*})q_{\beta
_{1}}...q_{\beta _{l-1}}{\sf \Gamma }_{\beta _{l}\mu }^{(\pm )k}u(p,\lambda
)\epsilon _{\mu }^{(\lambda _{V})}(q) \label{A1}$$ where the vertexes ${\sf \Gamma }_{\beta \mu }^{(\pm )k}$ are the same as for the photon, and $\epsilon _{\mu }^{(\lambda _{V})}(k)$ is the polarization vector of the vector meson $V$ with momentum $q$ and helicity $\lambda _{V}$.
The combination of Eqs.(\[A1\]) and (\[A2\]) allows to calculate the photo- and electroproduction amplitudes $$\begin{aligned}
&<&J\lambda _{*}|{\it T}|\lambda \lambda _{\gamma
}>=\sum_{k}\sum_{V}f_{VNN^{*},k}^{(\pm )}\frac{em_{V}^{2}}{g_{V}}\frac{1}{q^{2}-m_{V}^{2}}\times \nonumber \\
&&\overline{u}_{\beta _{1}...\beta _{l}}(p_{*},\lambda _{*})q_{\beta
_{1}}...q_{\beta _{l-1}}{\sf \Gamma }_{\beta _{l}\mu }^{(\pm )k}u(p,\lambda
)\epsilon _{\mu }^{(\lambda _{\gamma })}(q) \label{A3}\end{aligned}$$ The comparison with Eq.(\[amplitude\]) shows that the covariant form factors have the form $$F_{k}^{(\pm )}(M^{2})=\sum_{V}\frac{f_{VNN^{*},k}^{(\pm )}}{g_{V}}\frac{1}{1-M^{2}/m_{V}^{2}}. \label{VMDF}$$ The $\Delta $-resonance form factors have only contributions from the $\rho $-meson family. If the covariant form factors $F_{k}^{(\pm )}(M^{2})$ are known, the coupling constants $f_{\rho ^{0}NN^{*},k}^{(\pm )}$ for the $\Delta $-resonances can be found from equation $$f_{\rho ^{0}NN^{*},k}^{(\pm )}=-\frac{g_{\rho }}{m_{\rho }^{2}}{\normalsize res}\left\{ F_{k}^{(\pm )}(M^{2}=m_{\rho }^{2})\right\} . \label{RESD}$$ The nucleon resonances receive contributions from the $\rho $-and $\omega $-mesons. The couplings with the nucleon resonances are calculated as residues of a superposition for isospin projections $I_{3}=+\frac{1}{2}$ and $I_{3}=-\frac{1}{2}:$$$f_{VNN^{*},k}^{(\pm )}=-\frac{g_{V}}{2m_{V}^{2}}{\normalsize res}\left\{
F_{k}^{(\pm )}(M^{2}=m_{V}^{2})^{I_{3}=+\frac{1}{2}}\mp F_{k}^{(\pm
)}(M^{2}=m_{V}^{2})^{I_{3}=-\frac{1}{2}}\right\} \label{RESN}$$ where $V=\rho ^{0}(\omega )$ with the corresponding upper (lower) sign between the two isospin form factors.
The quark counting rules predict the following asymptotics for the helicity amplitudes $$\begin{aligned}
{\frak F}_{\frac{3}{2}}^{(\pm )} &=&O(\frac{1}{(-M^{2})^{5/2}}),
\nonumber \\
{\frak F}_{\frac{1}{2}}^{(\pm )} &=&O(\frac{1}{(-M^{2})^{3/2}}),
\nonumber \\
{\frak C}_{\frac{1}{2}}^{(\pm )} &=&O(\frac{1}{(-M^{2})^{5/2}}).
\label{QCR1}\end{aligned}$$ These constraints can be used to reduce the number of free parameters of the model.
The transition form factors of nucleon resonances with high spins decrease stronger than the diagonal nucleon form factors.
[*Spin* ]{}$J\geq \frac{3}{2}$[* resonances.* ]{}Now, taking into account that $\lambda _{l}^{(\pm )}=O((-M^{2})^{l-1/2})\ $at $M^{2}\rightarrow
-\infty ,$ we get the asymptotics of the covariant form factors $F_{k}^{(\pm )}(M^{2})$ at $M^{2}\rightarrow -\infty $: $$\begin{aligned}
F_{1}^{(\pm )}(M^{2}) &=&O(\frac{1}{(-M^{2})^{l+2}}), \nonumber \\
F_{2}^{(\pm )}(M^{2}) &=&O(\frac{1}{(-M^{2})^{l+3}}), \nonumber \\
F_{3}^{(\pm )}(M^{2}) &=&O(\frac{1}{(-M^{2})^{l+3}}).\end{aligned}$$
These constraints can be resolved to give
$$\begin{aligned}
F_{1}^{(\pm )}(M^{2}) &=&\frac{{\sum }_{j=0}^{n+1}C_{1j}^{(\pm )}{M^{2}}^{j}}{{\prod }_{i=1}^{l+3+n}(1-M^{2}/m_{i}^{2})}, \nonumber \\
F_{2}^{(\pm )}(M^{2}) &=&\frac{{\sum }_{j=0}^{n}C_{2j}^{(\pm )}{M^{2}}^{j}}{{\prod }_{i=1}^{l+3+n}(1-M^{2}/m_{i}^{2})}, \nonumber \\
F_{3}^{(\pm )}(M^{2}) &=&\frac{{\sum }_{j=0}^{n}C_{3j}^{(\pm )}{M^{2}}^{j}}{{\prod }_{i=1}^{l+3+n}(1-M^{2}/m_{i}^{2})}. \label{FF_l}\end{aligned}$$
Here, $C_{kj}^{(\pm )}$ are free parameters of the extended VMD model, ${l+3+n}$ is the total number of the vector mesons. For each form factor, the quark counting rules reduce the number of free parameters from ${l+3+n}$ to ${n+2}$ for $k=1$ and to ${n+1}$[ for]{} $k=2,3$. In the simplest case $n=0,$ the knowledge of the four parameters $C_{10}^{(\pm )}$, $C_{11}^{(\pm )}$, $C_{20}^{(\pm )}$, and $C_{30}^{(\pm )}$ is sufficient to fix $F_{k}^{(\pm
)}(M^{2})$. In the zero-width limit, the multiplicative representation (\[FF\_l\]) is completely equivalent to an additive representation of Eq.(\[VMDF\]).
The similar multiplicative representation motivated by the Regge theory is used in Ref. [@DEK], Eqs.(3.7). The asymptotic dominance of the transverse covariant form factors, used in that work as an assumption, does not agree with the quark counting rules.
[*Spin* ]{}$J=\frac{1}{2}$[* resonances.* ]{}In terms of the amplitudes ${\frak F}_{\frac{1}{2}}^{(\pm )}$ and ${\frak C}_{\frac{1}{2}}^{(\pm )}$, the constraints to the asymptotics have the form of Eqs.(\[QCR1\]). Taking into account that $\lambda _{0}^{(\pm
)}=O((-M^{2})^{1/2})\ $at $M^{2}\rightarrow -\infty ,$ we get $$F_{1,2}^{(\pm )}(M^{2})=O(\frac{1}{(-M^{2})^{3}}).$$ The general representation for the covariant form factors in the spin-$\frac{1}{2}$ case has the form $$F_{k}^{(\pm )}(M^{2})=\frac{{\sum }_{j=0}^{n}C_{kj}^{(\pm )}{M^{2}}^{j}}{{\prod }_{i=1}^{3+n}(1-M^{2}/m_{i}^{2})}. \label{FF_0}$$
Relative sign of the photo- and electroproduction amplitudes and amplitudes for the nucleon resonance decays into the vector mesons
-----------------------------------------------------------------------------------------------------------------------------------
$\;$
The experimental data for the photo- and electroproduction amplitudes are quoted by PDG for the amplitudes $$\begin{aligned}
A_{\frac{3}{2}} &=&\frac{\xi _{I}\xi }{\sqrt{8m_{*}m\omega _{0}}}{\frak F}_{\frac{3}{2}}^{(\pm )}, \label{A32} \\
A_{\frac{1}{2}} &=&\frac{\xi _{I}\xi }{\sqrt{8m_{*}m\omega _{0}}}{\frak F}_{\frac{1}{2}}^{(\pm )}, \label{A12} \\
S_{\frac{1}{2}} &=&\frac{\xi _{I}\xi }{\sqrt{8m_{*}m\omega _{0}}}{\frak C}_{\frac{1}{2}}^{(\pm )}, \label{S12}\end{aligned}$$ which include the phase factor of the $N^{*}\rightarrow N\pi $ decay, $\xi
=A(N^{*}\rightarrow N\pi )/|A(N^{*}\rightarrow N\pi )|$ and an isospin factor $\xi _{I}=-1$ for nucleon resonances and $\xi _{I}=+1$ for $\Delta $-resonances. The phase factor $\xi $ appears, since the amplitude $\gamma
^{*}N\rightarrow N^{*}$ is accompanied in the photoproduction experiments by the subsequent pion decay of the nucleon resonance.
The experimental data for the vector meson decays are quoted for the values $\sigma \sqrt{\Gamma _{N^{*}\rightarrow NV}}$ where $\sigma $ is a sign of the amplitudes $N^{*}\rightarrow N\rho ^{0}$ or $N^{*}\rightarrow N\omega $ multiplied by $\xi _{V}\xi ^{*}$, where $\xi $ is the pion decay phase and $\xi _{V}=A(V\rightarrow $pions$)$/ $|A(V\rightarrow $pions$)|.$ The additional factor $\xi _{V}$ appears, since the vector meson production is accompanied by the vector meson decays into pions. The isospin symmetry implies $\Gamma _{\Delta ^{*}\rightarrow N \rho }=\frac{3}{2}\Gamma
_{\Delta ^{*}\rightarrow N \rho ^{0}}$ and $\Gamma _{N^{*}\rightarrow
N\rho }=3\Gamma _{N^{*}\rightarrow N\rho ^{0}}$.
The quark models are very successful in the description of static properties of hadrons and hadron decays. The models [@Koniuk:1980vy; @Warns:1990ic; @Li:1990qu; @Bijker:1994yr; @Capstick:1992uc] give predictions for the photo- and electroproduction amplitudes (\[A32\]) - (\[S12\]). The $^{3}P_{0}$ quark-pair creation model by Yaouanc et al. [@LeYaouanc:1975mr] gives the vector-meson decay amplitudes inclusive of the phase of $\xi _{V}\xi ^{*}.$ The non-relativistic quark model by Koniuk [@Koniuk:1982ej] gives these amplitudes without the factor $\xi _{V}$, so its predictions are valid up to an overall sign. The quark-pair creation models of Refs. [@Capstick:1994kb; @Stassart:1990zt; @Stancu:1993xz] do not include the factor $\xi _{V}$ either. The multichannel $\pi N$-scattering partial-wave analysis of Manley and Saleski [@Manley:1992yb] has an overall sign ambiguity of the $\pi \pi N$ amplitudes with respect to the $\pi N$ amplitudes. The $\pi \pi N$ amplitudes interfere with the $N\rho $ amplitudes due to the $\rho \rightarrow \pi \pi $ decay, so the overall $N\rho $ phase is not fixed.
The extended VMD model, from the other side, gives a unified description of the photo- and electroproduction data and of the resonance decays into the vector mesons, including the signs of the amplitudes. We wish to use the data [@Groom:2000in; @Manley:1992yb; @Koniuk:1982ej; @Capstick:1994kb] to fix the transition form factors. The overall phase of the $N^{*}\rightarrow NV$ decays with respect to the $\gamma^{*}N\rightarrow N^{*}$ amplitudes is not fixed, however, neither experimentally, nor theoretically.
Yaouanc and co-authors [@LeYaouanc:1975mr] calculate the quantity $\xi _{\rho }.$ It depends on the sign of the coupling constant. The product $\xi _{V}\xi ^{*}$ is negative, being an even function of the coupling. They do not analyze, however, the photo- and electroproduction amplitudes. The photo- and electroproduction and vector meson decays of the nucleon resonances are calculated in Refs. [@Bijker:1994yr; @Bijker:1997tr]. The authors give, however, only relative signs of the $N^{*}\rightarrow NV$ amplitudes. The overall sign correlated with the photo- and electroproduction amplitudes is, in principle, provided by the quark models of Refs. [@Koniuk:1982ej; @Koniuk:1980vy] and [@Capstick:1994kb; @Capstick:1992uc]. The $N^{*}\rightarrow NV$ amplitudes of Ref. [@Koniuk:1982ej] depend on the quark coupling with the vector mesons, $g$, whose sign is a matter of convention. The quantity $\xi _{\rho }
$ is, however, proportional to the same coupling, $g,$ so the quantity $\sigma \sqrt{\Gamma _{N^{*}\rightarrow NV}}\ $is an even function of $g$ and therefore well defined. We thus propose a solution of the “sign ambiguity” problem within the non-relativistic Isgur-Koniuk [@Koniuk:1982ej; @Koniuk:1980vy] quark model framework. (The similar analysis can probably be made also in the quark-pair creation model of Ref. [@Capstick:1994kb; @Capstick:1992uc].)
The value of $\xi $ is not calculated in our model. It enters as a common factor to the experimentally measured photo- and electroproduction amplitudes and the vector meson decay amplitudes and can be absorbed by the covariant form factors $F_{k}^{(\pm )}$. The isospin factor $\xi _{I}$ will also be absorbed by the form factors. The common phase of the form factors does not influence the dilepton decay rates.
The value of $\xi _{\rho }$ can easily be found. Using the effective Lagrangian (\[RHOPIPI\]), the $P$-wave amplitude of the $\rho
^{0}\rightarrow \pi ^{+}\pi ^{-}$ decay can be found to be $<\pi \pi |{\it T}|\rho ^{0}>=-f_{\rho \pi \pi }2k_{\pi }$ where $k_{\pi }$ is the absolute value of the pion momentum. Eq.(\[RHOPIPI\]) shows that the coupling $f_{\rho \pi \pi }$ has a meaning of the $\pi ^{+}$ charge with respect to the massive vector field $\rho _{\mu }^{0}.$ In the quark model by Koniuk [@Koniuk:1982ej], the $\rho ^{0}$ static charge of the up-quarks is assumed to be positive [@note]. The value of $f_{\rho
\pi \pi }$ is therefore positive, and so $\xi _{\rho }=-1$.
The vector meson decay amplitudes of the nucleon resonances are proportional to the product $f_{\rho ^{0}NN^{*},k}^{(\pm )}f_{\rho \pi \pi }$. They are invariant with respect to the sign change of the vector meson coupling, since $f_{\rho ^{0}NN^{*},k}^{(\pm )}\backsim g$ and $f_{\rho \pi \pi
}\backsim g$. The VMD in the electromagnetic pion form factor gives rise to Eq.(\[SAC\]), so the coupling $g_{\rho }$ is an odd function of $g$. The sign convention for the coupling $g$ does not affect the isovector parts of the photo- and electroproduction amplitudes, since they are proportional to the ratios $f_{\rho ^{0}NN^{*},k}^{(\pm )}/g_{\rho }$. The model we discuss thus provides a consistent description of the isovector part of the processes $\gamma ^{*}N\rightarrow N^{*}$ and of the $N^{*}\rightarrow N\rho
^{0}$ decays.
Similar arguments could be applied to the effective Lagrangian ${\cal L}_{\omega \rho \pi }$ to determine the $\omega \rightarrow 3\pi $ amplitude through a two-step mechanism $\omega \rightarrow \rho \pi ,$ $\rho
\rightarrow \pi \pi $. The $\omega \rho \pi $ coupling constant can, however, be calculated up to an arbitrary phase only. It is proportional to the $\rho \pi $ transition magnetic moment (with respect to a static magnetic $\omega $-meson field) which is not a diagonal one. The $\omega
\rho \pi $ coupling is proportional to an arbitrary phase difference of the $\rho $- and $\pi $-meson wave functions (as distinct from the $\rho \pi \pi $ diagonal transition where such a phase is identically zero). The relative phase of the quantities $\xi _{\rho }$ and $\xi _{\omega }$ is well defined for identical final states, e.g. for the $\rho \rightarrow $ $\pi \gamma $ and $\omega \rightarrow \pi \gamma $ decays, in which case $\xi _{\rho }/\xi
_{\omega }=+1,$ while the $\omega \rightarrow 3\pi $ phase cannot be fixed with respect to the $\rho \rightarrow \pi \pi $ phase.
Godfrey and Isgur [@Godfrey:1985xj] calculate meson decay amplitudes by considering the pseudoscalar mesons as elementary fields. The overall sign of these amplitudes is not correlated with the amplitudes calculated by considering the vector mesons as elementary fields. It can be seen e.g. from Table V of Ref. [@Godfrey:1985xj], according to which the $\rho
\rightarrow \pi \pi $ amplitude is imaginary, whereas we get above for the $\rho \rightarrow \pi \pi $ decay a real negative amplitude. The reason stems from the fact that the amplitudes of Ref. [@Godfrey:1985xj] are proportional to the coupling constant of the quarks with the pions, whereas the amplitudes of Ref. [@Koniuk:1982ej] are proportional to the coupling constant of the quarks with the vector mesons. The relative signs (phases) of these coupling constants are not correlated.
The model by Koniuk [@Koniuk:1982ej] has, however, a $SU(3)$ symmetric vertex for quarks interacting with the vector mesons. In this model, $1/g_{\rho }\backsim tr\tau ^{3}Q=1$ and $1/g_{\omega }\backsim trQ=1/3$ where $\tau ^{3}$ is the isospin Pauli matrix and $Q=$ diag$(2/3,-1/3)$ is the quark charge matrix. The relative sign of the coupling constants $g_{\rho }$ and $g_{\omega }$ is apparently known. The same relative sign ($+$) is given by Godfrey and Isgur [@Godfrey:1985xj], which is independent on the quark-pion coupling constant.
This knowledge is sufficient to compare predictions of the extended VMD model with the amplitudes of Refs. [@Koniuk:1982ej], which do not include the phases $\xi _{V}$.
Manley and Saleski [@Manley:1992yb] found that the $\pi N$ partial-wave analysis is in good agreement with the results by Koniuk [@Koniuk:1982ej] taken with the opposite sign. We got $\xi _{\rho }=-1$, using the model by Koniuk, so the results by Manley and Saleski for the $\pi \pi N$ waves have apparently the overall sign correlated correctly with the photo- and electroproduction amplitudes, as calculated by Koniuk and Isgur [@Koniuk:1980vy].
We thus perform a fit to the amplitudes for the photo- and electroproduction of the nucleon resonances with the standard phase conventions of PDG [@Groom:2000in] and Koniuk and Isgur [@Koniuk:1980vy]. We fit further the vector meson decay [*amplitudes*]{}. The overall sign of these amplitudes is chosen such as to reproduce in the extended VMD model both the photo- and electroproduction amplitudes and the vector meson amplitudes of the nucleon resonance decays, calculated in the non-relativistic quark model by Koniuk and Isgur [@Koniuk:1980vy] and by Koniuk [@Koniuk:1982ej]. The transition form factors of nucleon resonances determined this way do not depend on the sign convention adopted in the quark model [@Koniuk:1982ej] for the coupling constant, $g$, of quarks with the vector mesons:
In the Koniuk quark model, the amplitudes of the vector meson decays of the nucleon resonances are proportional to the coupling constant $g$. In the extended VMD model, these amplitudes are proportional to the coupling constants $f_{VNN^{*},k}^{(\pm )},$ according to Eq.(\[A1\]). We found earlier $f_{\rho \pi \pi }\backsim g$. The coupling constant $g_{\rho }\ $is also proportional to $g,$ according to Eq.(\[SAC\]). The $SU(3)$ symmetry implies $g_{\omega }\backsim g$. The $g$-dependence drops out from the transition form factors (\[VMDF\]). In agreement with the quark model, Eqs.(\[RESD\]) and (\[RESN\]) give afterwards $f_{VNN^{*},k}^{(\pm )}\backsim g$. This completes the consistency check.
Numerical results
=================
$\;$
The parameters $C_{kj}^{(\pm )}$ of the extended VMD model, entering Eqs.(\[FF\_l\]) and (\[FF\_0\]), are determined from the fit to the photo- and electroproduction data [@Groom:2000in; @Stein:1975yy; @Bartel:1968tw; @Batzner; @Frolov:1999pw] and the vector meson decay amplitudes of the nucleon resonances [@Groom:2000in; @Manley:1992yb; @Longacre:1977ja; @Koniuk:1982ej; @Capstick:1994kb]. We use the minimal $n=0$ extension of the VMD model for all resonances.
The number of the vector mesons required for each isotopic channel to ensure the correct asymptotic behavior depends on the total spin of the nucleon resonance. For spin-$J$ resonances, we need $3 + l$ ($=3 + J - \frac{1}{2}$) excited vector mesons with the same quantum numbers for the minimal extension of the VMD. The nucleon resonances we consider have spins $J$ from $\frac{1}{2}$ to $\frac{7}{2}$. It means that we need at the most 6 excited vector mesons for each isotopic channel. The following masses are used: 0.769, 1.250, 1.450, 1.720, 2.150, 2.350 (in GeV). The numbers appearing on the 1 and 3 - 5 positions are masses of the physical $\rho$-mesons according to PDG [@Groom:2000in]. The possible existence of vector mesons with masses around 1.250 GeV is discussed for a long time. The phenomenology of the nucleon form factors and, in particularly, the scaling laws for the Sachs form factors make the existence of an enhancement in the spectral function of the nucleon form factors at 1.250 GeV very plausible (for details see [@Faessler:2000de]). The results on the dilepton emission do not depend strongly on the exact numerical values of the masses of the excited vector mesons, since the dilepton energy spectrum extends only slightly above 1 GeV for the nucleon resonances with masses about 2 GeV. In the region of the invariant masses $M<1$ GeV, the form factors are smooth functions of the masses of the excited vector mesons. The last mass is set equal to 2.350 GeV for an estimate. We assumed further a degeneracy between the $\rho$ and $\omega$ families. The strange $\phi$ mesons are decoupled in our model from the nucleons due to the OZI rule. The opposite assumption is used in Refs. [@Hohler:1976ax; @Mergell:1996bf]. The widths of the mesons 2 - 6 is set equal to zero, the widths of the ground-state $\rho$- and $\omega$-mesons are taken from PDG.
For the nucleon resonance decays into the vector mesons, we use the data from PDG [@Groom:2000in]. When these data are not available (quite often), we use the Manley and Saleski results (MS) of the multichannel $\pi N$ partial-wave analysis [@Manley:1992yb]. In other cases, we use the quark model predictions by Koniuk [@Koniuk:1982ej] with $50\%$ errors and $0.05$ MeV$^{1/2}$ errors if the values are close to zero. In a few cases, the results [@Longacre:1977ja] of the multichannel $\pi N$ partial-wave analysis of Longacre and Dolbeau (LD) with $50\%$ errors and of Capstick and Roberts (CR) quark model predictions [@Capstick:1994kb] are used, when other results do not agree with the most recent PDG constraints to the total vector meson decay widths. The PDG and MS data are included to the $\chi ^{2}
$ with greater weights. Below we give details of our fitting procedure:
$N(1535)\frac{1}{2}^{-}$: The experimental values for $A_{1/2}$ are from Ref.[@Burk]. The $N\rho$ mode $s_{1/2}$ is taken from PDG. The $N\rho$ mode $d_{3/2}$ is taken from MS. The $\omega$-meson mode $d_{3/2}$ is set equal to zero.
$N(1650)\frac{1}{2}^{-}$: The $N\rho $ mode $s_{1/2}$ is taken from PDG with the negative sign as predicted by K. The MS sign is not well fixed. The mode $d_{3/2}$ is taken from PDG. The $N\omega $ modes are from the Koniuk paper (K) with $0.05$ MeV$^{1/2}$ errors.
$N(1520)\frac{3}{2}^{-}$: The experimental values for $A_{1/2}$ and $A_{3/2}$ are from Ref.[@Burk]. The modes $d_{1/2}$ and $d_{3/2}$ are taken from K with a $0.05$ MeV$^{1/2}$ error. The mode $s_{3/2}$ is taken from PDG.
$N(1700){\frac{3}{2}}^{-}$: The experimental values for $A_{1/2}$ and $A_{3/2}$ are from Ref.[@Burk]. The $N\rho $ modes $d_{1/2}$ and $d_{3/2}$ are taken from K with a $0.05$ MeV$^{1/2}$ error. The mode $s_{3/2}$ is taken from PDG with the negative sign as predicted by MS. The $N\omega $ modes are from K with $0.05$ MeV$^{1/2}$ errors.
$N(1675){\frac{5}{2}}^{-}$: The $N\rho $ mode $d_{1/2}$ is taken from MS, the mode $d_{3/2}$ is taken from PDG. The $g_{3/2}$ modes are from K with a $0.05$ MeV$^{1/2}$ error. The $N\omega$ mode $g_{1/2}$ is set equal to zero.
$N(1440){\frac{1}{2}}^{+}$: The experimental values for $A_{1/2}$ are from Ref.[@Burk]. The mode $p_{1/2}$ is taken from PDG. The sign of the mode which is given by PDG in the absolute value is taken to be positive as predicted by Koniuk. The value of the mode $p_{3/2}$ is taken from K. We set a $50\%$ error for the fit. The coordinate quark wave function of the resonance is known to be symmetric, so the neutron charge radius does vanish. It is proportional to the second derivative of the charge density with respect to the momentum $k$ at $k=0$ or, equivalently, to the first derivative of the helicity amplitude $S_{1/2}$ at pseudothreshold $M^{2}=(m_{*}-m)^{2}$, so we set $2m_{+}F_{1}^{(-)}+m_{-}F_{2}^{(-)}=0$ at $M^{2}=(m_{*}-m)^{2}$.
$N(1710){\frac{1}{2}}^{+}$: The $N\rho $ mode $p_{1/2}$ is taken from PDG. The sign of the mode is taken to be positive, as predicted by MS. The value of the mode $p_{3/2}$ from LD results to $\sqrt{\Gamma
_{N\rho }}=8.9$ MeV$^{1/2}$. This value is too high as compared to the square root of the PDG total $N\rho $ width of $4.3\pm 1.9$ MeV$^{1/2}$. We thus take for the mode $p_{3/2}$ the three times smaller Koniuk quark model value of $\sqrt{B_{N\pi }B_{N\rho }}=0.09$ with a $50\%$ error. The $N\omega
$ modes from K are included to the fit with $0.05$ MeV$^{1/2}$ errors.
$N(1720){\frac{3}{2}}^{+}$: The $N\rho $ mode $p_{1/2}$ from MS and the mode $p_{3/2}$ from LD seems to be overestimated, in view of the PDG value $\sqrt{\Gamma _{N\rho }^{tot}}=11\pm 2$ MeV$^{1/2}$ for the total $N\rho$ width. The modes $p_{1/2}$, $p_{3/2}$ and $f_{3/2}$ are taken from K with $50\%$ errors. The $N\omega $ modes from K are included to the fit with $0.05$ MeV$^{1/2}$ errors.
$N(1900){\frac{3}{2}}^{+}$: The $N\rho $ mode $p_{1/2}$ is from MS. The modes $p_{3/2}$ and $f_{3/2}$ are taken from K with $0.05$ MeV$^{1/2}$ errors. The $N\omega $ modes are from K with $0.05$ MeV$^{1/2}$ errors.
$N(1680){\frac{5}{2}}^{+}$: The experimental values for $A_{1/2}$ and $A_{3/2}$ are from Ref.[@Burk]. The $N\rho $ mode $f_{1/2}$ is from K with a $50\%$ error. The modes $f_{3/2}$ and $p_{3/2}$ are from PDG. The $N\omega $ modes are from K with $0.05$ MeV$^{1/2}$ errors.
$N(2000){\frac{5}{2}}^{+}$: The $N\rho $ mode $f_{1/2}$ is from K with a $50\%$ error. The modes $f_{3/2}$ and $p_{3/2}$ are from MS. The $N\omega $ modes are from K with $0.05$ MeV$^{1/2}$ errors.
$N(1990){\frac{7}{2}}^{+}$: The $N\rho $ modes $f_{1/2}$, $f_{3/2}$, and $h_{3/2}$ are from K with $0.05$ MeV$^{1/2}$ errors. The $N\omega $ modes are from K with $0.05$ MeV$^{1/2}$ errors.
$\Delta (1620){\frac{1}{2}}^{-}$: PDG values are used.
$\Delta (1900){\frac{1}{2}}^{-}$: MS values are used.
$\Delta (1700){\frac{3}{2}}^{-}$: K values are used for the modes $d_{1/2}$ and $d_{3/2}$ with $0.05$ MeV$^{1/2}$ errors. The PDG absolute value is used for the $s_{3/2}$ mode with MS sign.
$\Delta (1940){\frac{3}{2}}^{-}$: CR values are used for the modes $d_{1/2}$ and $d_{3/2}$. MS value is used for the $s_{3/2}$ mode.
$\Delta (1930){\frac{5}{2}}^{-}$: CR values are used for the modes $d_{3/2}$ and $g_{3/2}$. MS value is used for the $d_{1/2}$ mode.
$\Delta (1750){\frac{1}{2}}^{+}$: K values are used with $50\%$ errors.
$\Delta (1910){\frac{1}{2}}^{+}$: K values are used with $50\%$ errors.
$\Delta (1232){\frac{3}{2}}^{+}$: The data in the space-like region on the magnetic transition form factor are from Refs.[@Bartel:1968tw; @Stein:1975yy; @Batzner]. We include into the fit the experimental results of Refs.[@Burkert; @Siddle:1971ug; @Beck:1997ew; @Frolov:1999pw; @AS; @ALDER] for the ratio $G_C/G_M$ and of Refs. [@Frolov:1999pw; @AS; @Siddle:1971ug; @ALDER] for the ratio $G_E/G_M$. The results of Refs.[@Frolov:1999pw] and [@AS] for the ratio $G_E/G_M$ disagree already in sign. We fit well the data [@Frolov:1999pw]. The amplitudes $A_{3/2}$ and $A_{1/2}$ at $M=0$ are given by PDG. The $\Delta(1232)$ magnetic form factor in the space-like region is a smooth, well measured function. It is well reproduced in the VMD model. Using the knowledge of the $G_M$, we translate on Fig. 19, when data are sufficient, the experimental points from plots for the monopole form factors to plots for the helicity amplitudes, and [*v. v.*]{} It is seen that the experimental data for the form factors $G_E$ and $G_C$ are not stable yet. The vector meson decay channels are closed.
$\Delta (1600){\frac{3}{2}}^{+}$: LD values give too high coupling constants $\Delta N\rho $. We use CR predictions which are 5 to 10 times smaller.
$\Delta (1920){\frac{3}{2}}^{+}$: K values are used with $50\%$ errors.
$\Delta (1905){\frac{5}{2}}^{+}$: CR values are used for the $f$-modes and PDG for the $p_{3/2}$-mode.
$\Delta (2000){\frac{5}{2}}^{+}$: K values for the $f$ modes are used with $50\%$ errors. MS value is used for the $p_{3/2}$ mode.
$\Delta (1950){\frac{7}{2}}^{+}$: PDG gives an upper limit of $6$ MeV$^{1/2}$ for the total $N\rho $ width. MS and K results are above this limit, so we use the estimates of CR.
The $S_{1/2}$ amplitudes which we used as an input are from Refs. [@foster; @Gerhardt:1980yg].
In Table I, we show the parameters $C_{kj}^{(\pm )}$ of the extended VMD model for nucleon resonances with masses below $2$ GeV.
In Table II, the coupling constants $g_{T}^{V}\equiv g_{T}^{V}(m_{V})$ of the vector mesons of the magnetic, electric, and Coulomb types with the nucleon resonances are shown. The coupling constants $g_{T}^{V}\ $are defined as in Eqs.(\[RESD\]) and (\[RESN\]) with the selfevident replacements $f_{VNN^{*},k}^{(\pm
)}\rightarrow g_{T}^{V}$ and $F_{k}^{(\pm )}\rightarrow G_{T}^{(\pm )}.$ The coupling constants $f_{VNN^{*},k}^{(\pm )}$ and $g_{T}^{V}$ are related by the same transformation as the covariant form factors $F_{k}^{(\pm )}$ and $G_{T}^{(\pm )}:$$$g_{T}^{V}=\sum_{k}{\rm M}_{Tk}(m_{V}^{2})f_{VNN^{*},k}^{(\pm )}.$$ The matrices ${\rm M}_{Tk}(M^{2})$ are defined by Eqs.(\[FG\_lP\]), (\[FG\_lM\]), and (\[FG\_lM\]), respectively, for the normal- and abnormal-parity $l>0$ resonances, and normal- and abnormal-parity $l=0$ resonances.
The quality of the fit is generally good. It can be controlled with the help of Tables III - VI and Figs. 1 - 25. The minimal $n=0$ extension of the VMD model appears to be sufficient to fit the data. The exception is only the $N^{*}(1520)$-resonance. It has a sharp $M^{2}$-dependence of the helicity amplitude $A_{3/2}^{p}(M^{2})$ at the interval $-4<M^{2}<0$ GeV$^{2}$ (see Fig. 3). The minimal $n=0$ model cannot reproduce it.
In Tables III - VI, we compare the VMD model results for the vector meson decay amplitudes of the nucleon resonances with the results of the $\pi N$ multichannel partial-wave analysis [@Groom:2000in; @Manley:1992yb; @Longacre:1977ja] and the quark models [@Koniuk:1982ej; @Capstick:1994kb; @LeYaouanc:1975mr; @Stassart:1990zt; @Stancu:1993xz; @Bijker:1997tr]. The extended VMD model results should not normally be treated as predictions, since for every nucleon resonance the number of the parameters in the fit is comparable with (being always less than or equal to) the number of points available from the experiment and the quark models. The evident exceptions are the $\Delta (1232)$-resonance, where many data points exist in the space-like region, and a few others. It is seen from the Tables III - VI that the minimal extension $n=0$ of the VMD model describes the vector meson amplitudes in the different partial waves fairly well, with respect to both, the signs and the values.
The vector meson decay widths of the nucleon resonances are calculated using Eqs.(\[GAMMA\_l\]) and (\[GAMMA\_0\]) after substituting $\alpha
\rightarrow 1/(4\pi )$ and $G_{T}(M^{2})\rightarrow g_{T}^{V}(M^{2}).$ The ”running” coupling constant $g_{T}^{V}(M^{2})$ is given by $$g_{T}^{V}(M^{2})=\sum_{kT^{\prime }}{\rm M}_{Tk}(M^{2}){\rm M}_{kT^{\prime
}}^{-1}(m_{V}^{2})g_{T^{\prime }}^{V}(m_{V}^{2}) \label{RUN}$$ The values ${\rm M}_{kT^{\prime }}^{-1}(m_{V}^{2})$ are the inverse transformation matrices given in Sect. 3. The parameters $C_{kj}^{(\pm )}$ and the coupling constants $f_{VNN^{*}}^{(\pm )}$ of Eqs.(\[RESD\]) and (\[RESN\]) are considered as being independent of $M^{2}$, whereas in the multipole basis one should take into account the $M^{2}$-dependence of the transformation matrices ${\rm M}_{Tk}(M^{2})$.
The signs of the vector meson decay amplitudes are defined as follows: We determine first the average vector meson mass $$\bar{M}_{(\pm )}^{2}=\int \varphi ^{(\pm )}(M^{2})M^{2}dW(M^{2})/\int
\varphi ^{(\pm )}(M^{2})dW(M^{2}) \label{MAV}$$ where $\varphi ^{(\pm )}(M^{2})$ are the kinematical parts of the widths (\[GAMMA\_l\]) and (\[GAMMA\_0\]): $$\begin{aligned}
\varphi ^{(\pm )}(M^{2}) &=&(m_{\mp }^{2}-M^{2})^{l+1/2}(m_{\pm
}^{2}-M^{2})^{l-1/2}, \\
\varphi ^{(\pm )}(M^{2}) &=&(m_{\pm }^{2}-M^{2})^{3/2}(m_{\mp
}^{2}-M^{2})^{1/2},\end{aligned}$$ respectively, for $l>0$ and $l=0$ resonances. The integral runs over the Breit-Wigner distribution $dW(M^{2}).$ The sign of the vector meson decay amplitude is given by sign of the quantity $$H_{S,\ell }^{(\pm )}=sign(\xi \xi _{\rho }^{*}\lambda _{l}^{(\pm )}){\frak H}_{S,\ell }^{(\pm )} \label{VERY IMPORTANT}$$ evaluated at $M^{2}=\bar{M}_{(\pm )}^{2}.$ The factor of $\xi _{\rho }$ brings the overall sign to the MS sign conventions [@Manley:1992yb]. The phase of $\xi $ is absorbed by both the photo- and electroproduction amplitudes and the vector meson decay amplitudes of the nucleon resonances. The products $\xi {\frak F}_{\frac{3}{2}}^{(\pm )},$ $\xi {\frak F}_{\frac{1}{2}}^{(\pm )}, $ $\xi {\frak C}_{\frac{1}{2}}^{(\pm )},$ and $\xi {\frak H}_{S,\ell }^{(\pm )}$ are real for all partial waves [@Koniuk:1980vy], so the appearance of the complex conjugate value of $\xi $ in the vector meson decay amplitudes does not change the relative sign, as compared with what we have described.
For some of the resonances, as can be seen from the Figs.1-25, the amplitude changes its sign on the interval $0<M^{2}<1$ GeV.
The plots 1-25 show the magnetic, electric, and Coulomb form factors for the nucleon resonances with masses below $2$ GeV, listed by PDG. We give in the space-like region the ratios between the form factors and the dipole function $$G_{D}(t)=\frac{1}{(1-t/0.71)^{2}} \label{DIP}$$ where $t$ is in GeV$^{2}$ $(t=q^{2})$. The ratios between the helicity amplitudes and the dipole function (\[DIP\]) are also shown, also in the space-like region. Next, we give the partial-wave amplitudes $H_{S,\ell }^{(\pm
)}$ of the vector meson emission in the time-like region. The solid curves stand for the extended VMD model. The data for the $\pi N$ multichannel partial-wave analysis and the quark models used in the fit at $M^{2}>0$ and the experimental data for the transition form factors and the photo- and electroproduction helicity amplitudes used in the fit at $M^{2}<0$ are shown also.
In Table VII, we show the dilepton widths of the nucleon resonances. Figs.26 and 27 show the dilepton $e^{+}e^{-}$ and $\mu ^{+}\mu ^{-}$ spectra from decays of the nucleon resonances.
Conclusion
==========
$\;$
In this work, we derived phenomenological kinematically complete relativistic expressions for the decay rates of nucleon resonances with arbitrary spin and parity into the dilepton pairs in terms of the magnetic, electric, and Coulomb transition form factors. The extended VMD model was used for the description of the transition form factors of the nucleon resonances. The quark counting rules were taken into account to reduce the number of free parameters of the model. The remaining free parameters are fixed by fitting the photo- and electroproduction amplitudes and the decay amplitudes of the nucleon resonances into the vector mesons. The extended VMD model allows to treat the data both in the space- and time-like regions. The transition form factors determined from the fit are used for the calculation of the dilepton widths and dilepton spectra from decays of the nucleon resonances with masses below $2$ GeV.
In many cases, the experimental data we used to fix the form factors are not stable yet. To make the study complete, we used also the quark model predictions. The results of our analysis can be treated as a first quantitative hint to the form factors and the decay amplitudes which they determine. The VMD predictions are useful for planning new experiments for measurements of the transition form factors and vector meson branchings of the nucleon resonances.
We propose thus unified description of the photo- and electroproduction data, the vector meson and dilepton decay amplitudes of the nucleon resonances. The results can be used for modeling the dilepton production in the pion-nucleon, nucleon-nucleon, and heavy-ion collisions.
The authors are grateful to D. M. Manley for valuable comments on the $\pi N$ partial-wave analysis. Two of us (M.I.K. and B.V.M.) are indebted to the Institute for Theoretical Physics of University of Tuebingen for kind hospitality. The work was supported by GSI (Darmstadt) under the contract TÜFÄST, by the Plesler Foundation, and by the Deutsche Forschungsgemeinschaft under the contract No. 436RUS113/367/0(R).
Figures captions:
Figs. 1- 25: Electromagnetic transition form factors of the magnetic, electric, and Coulomb types ($G_M$, $G_E$, and $G_C$), helicity amplitudes ($A_{3/2}$, $A_{1/2}$, and $S_{1/2}$), and partial-wave amplitudes of nucleon resonance decays into the $\rho $- and $\omega$-meson channels ($H_{S,\ell }^{(\pm
)} $) for nucleon resonances listed by PDG with masses below $2$ GeV. The minimal extension $n=0$ of the VMD model is used for all resonances. The value $G_{D}(t)$ is the dipole function of Eq.(\[DIP\]). The photo- and electroproduction experimental data [@Groom:2000in; @Stein:1975yy; @Bartel:1968tw; @Batzner; @Frolov:1999pw] are displayed. The vector meson decay amplitudes of the nucleon resonances [@Groom:2000in; @Manley:1992yb; @Longacre:1977ja; @Koniuk:1982ej; @Capstick:1994kb], that were included into the fit, are also shown.
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Resonance $C_{10}$ $C_{11}$ $C_{20}$ $C_{30}$
----------------------------------- ---------- ---------- ---------- ---------- --
$N^{*} (1535)\frac{1}{2}^-$ 0.979 0.006
1.787 -0.062
$N^{*} (1650)\frac{1}{2}^-$ 0.232 -0.186
-0.394 0.157
$N^{*} (1520)\frac{3}{2}^-$ 2.186 -1.236 -1.976 -0.159
-0.220 1.899 -0.316 -0.249
$N^{*} (1700)\frac{3}{2}^-$ 0.169 0.410 -0.227 -0.454
0.104 1.666 -0.138 -2.281
$N^{*} (1675)\frac{5}{2}^-$ -0.205 0.080 0.371 -3.812
0.579 -0.560 -0.591 3.357
$N^{*} (1440)\frac{1}{2}^+$ 0.863 1.023
0.084 -0.699
$N^{*} (1710)\frac{1}{2}^+$ 0.287 -0.185
-0.382 0.056
$N^{*} (1720)\frac{3}{2}^+$ 0.000 0.608 0.187 -5.312
0.051 -0.304 0.194 1.630
$N^{*} (1900)\frac{3}{2}^+$ 0.024 -0.238 0.028 0.991
-0.054 0.398 0.055 -1.777
$N^{*} (1680)\frac{5}{2}^+$ 2.487 -0.700 -2.116 -0.797
-0.793 4.929 0.735 -6.297
$N^{*} (2000)\frac{5}{2}^+$ 0.201 0.762 -0.267 -1.663
0.049 -0.176 -0.029 0.956
$N^{*} (1990)\frac{7}{2}^+$ -0.199 0.336 0.321 -0.578
1.028 -2.263 -0.796 0.846
$\Delta (1620)\frac{1}{2}^-$ -0.155 -0.081
$\Delta (1900)\frac{1}{2}^-$ 0.123 -0.025
$\Delta (1700)\frac{3}{2}^-$ -0.630 -0.298 1.080 -0.473
$\Delta (1940)\frac{3}{2}^-$ -0.251 0.075 0.252 -0.202
$\Delta (1930)\frac{5}{2}^-$ 0.181 0.845 -0.555 -4.531
$\Delta (1750)\frac{1}{2}^+$ 0.325 0.431
$\Delta (1910)\frac{1}{2}^+$ 0.194 -0.045
$\Delta (1232)\frac{3}{2}^+$ 1.768 0.025 -1.096 -0.926
$\Delta (1600)\frac{3}{2}^+$ 0.086 -0.012 -0.238 1.806
$\Delta (1920)\frac{3}{2}^+$ -0.120 0.187 0.276 -1.386
$\Delta (1905)\frac{5}{2}^+$ -0.209 0.090 0.157 -1.145
$\Delta (2000)\frac{5}{2}^+$ -0.088 -0.388 -0.020 1.299
$\Delta (1950)\frac{7}{2}^+$ 0.867 -1.250 -0.138 1.619
: Residues $C_{jk}^{(\pm )}$ of the extended VMD model, entering Eqs.(\[FF\_l\]) and (\[FF\_0\]), in units GeV$^{-(l+1)}$ where $l=J-\frac{1}{2}.$ The $N^{*}$ residues are shown in two lines for the proton and neutron resonances, respectively. []{data-label="table1"}
Resonance $g^{\rho}_{M}$ $g^{\rho}_{E}$ $g^{\rho}_{C}$ $g^{\omega}_{M}$ $g^{\omega}_{E}$ $g^{\omega}_{C}$
----------------------------------- ---------------- ---------------- ---------------- ------------------ ------------------ ------------------ --
$N^{*} (1535)\frac{1}{2}^-$ 2.21 3.16 -28.03 -42.67
$N^{*} (1650)\frac{1}{2}^-$ -0.26 -0.01 2.01 4.14
$N^{*} (1520)\frac{3}{2}^-$ -0.53 -9.21 -24.62 -7.67 18.16 46.13
$N^{*} (1700)\frac{3}{2}^-$ 0.02 1.31 2.56 -0.17 -1.45 -1.45
$N^{*} (1675)\frac{5}{2}^-$ -9.79 -4.91 -31.08 2.07 -1.61 -10.50
$N^{*} (1440)\frac{1}{2}^+$ -8.21 18.16 -14.14 63.13
$N^{*} (1710)\frac{1}{2}^+$ -0.69 13.35 2.97 -8.13
$N^{*} (1720)\frac{3}{2}^+$ -0.49 -5.72 -25.91 0.14 -8.27 -37.73
$N^{*} (1900)\frac{3}{2}^+$ -1.75 1.71 10.66 1.25 -1.28 -8.85
$N^{*} (1680)\frac{5}{2}^+$ 0.00 5.77 8.01 -1.34 -11.75 -7.98
$N^{*} (2000)\frac{5}{2}^+$ 0.60 -5.19 -26.29 1.72 8.98 5.18
$N^{*} (1990)\frac{7}{2}^+$ 6.13 1.39 7.73 -19.88 -3.73 -28.51
$\Delta (1620)\frac{1}{2}^-$ 1.59 3.32
$\Delta (1900)\frac{1}{2}^-$ -0.32 -1.31
$\Delta (1700)\frac{3}{2}^-$ 0.05 -5.53 -12.08
$\Delta (1940)\frac{3}{2}^-$ -0.37 -2.38 -6.98
$\Delta (1930)\frac{5}{2}^-$ 12.87 -5.38 -48.07
$\Delta (1750)\frac{1}{2}^+$ -6.02 16.69
$\Delta (1910)\frac{1}{2}^+$ -0.44 8.44
$\Delta (1232)\frac{3}{2}^+$ 30.57 0.80 6.56
$\Delta (1600)\frac{3}{2}^+$ 4.92 3.31 12.31
$\Delta (1920)\frac{3}{2}^+$ -1.61 -1.38 -9.88
$\Delta (1905)\frac{5}{2}^+$ -0.19 -15.25 -42.58
$\Delta (2000)\frac{5}{2}^+$ -0.54 2.30 20.56
$\Delta (1950)\frac{7}{2}^+$ 5.82 1.92 22.90
: The $\rho $- and $\omega $-meson coupling constants of the magnetic, electric, and Coulomb types with the nucleon resonances in units GeV$^{-(l-1)}$ where $l=J-\frac{1}{2}.$[]{data-label="table2"}
Resonance Ref. $N\rho$ $N\rho $ $N\rho $ $\sqrt{\Gamma^{tot}_{N\rho}} $ $N\omega$ $N\omega$ $N\omega$ $\sqrt{\Gamma^{tot}_{N\omega}}$
---------------------------- ------ ---------------------- ---------------------- ---------------------- --------------------------------- --------------------- --------------------- ---------------------- ---------------------------------
$s_{1/2}$ $d_{3/2}$ $s_{1/2}$ $d_{3/2}$
$N^{*}(1535)\frac{1}{2}^-$ VMD -2.13 -0.25 2.15 1.43 0.05 1.43
KI -1.7 -6.1 6.3
CR -0.7$\pm0.3$ 0.4$\pm0.1$ 0.8$^{+0.2}_{-0.1}$
SST 1.1
MS -1.7$\pm0.5$ -1.3$\pm0.6$ 2.2$\pm0.6$
PDG -2.0$\pm0.9$ $<2.7$
$N^{*}(1650)\frac{1}{2}^-$ VMD -1.45 1.04 1.78 -0.97 -0.02 0.97
KI -9.7 2.7 10.1 -0.96 0.67 1.2
CR 0.9$^{+0.8}_{-0.6}$ 0.4$\pm0.1$ 1.0$^{+0.3}_{-0.2}$
SST 0.6
MS 0.0$\pm1.6$ 2.2$\pm0.9$ 2.2$\pm0.9$
PDG $\pm1.6\pm1.2$ 3.4$\pm1.0$ 3.6$\pm0.9$
$d_{1/2}$ $d_{3/2}$ $s_{3/2}$ $d_{1/2}$ $d_{3/2}$ $s_{3/2}$
$N^{*}(1520)\frac{3}{2}^-$ VMD -0.37 -0.17 -5.14 5.16 -0.02 0.03 0.28 0.29
KI 0.7 -1.1 -5.0 5.2
CR -0.1$^{+0.1}_{-0.3}$ -0.3$^{+0.2}_{-1.0}$ -2.4$^{+1.9}_{-6.4}$ 2.5$^{+6.5}_{-1.9}$
SST 3.2 -2.7 -1.7 4.6
MS 0 0 -5.1$\pm0.6$ 5.1$\pm0.6$
PDG -4.9$\pm0.6$ 4.9$\pm0.6$
$N^{*}(1700)\frac{3}{2}^-$ VMD -0.94 -2.32 -2.22 3.35 0.21 0.90 1.39 1.67
KI -0.1 -2.7 -4.3 5.1 0.26 0.89 1.4 1.7
CR 0 -0.9$^{+0.3}_{-0.6}$ 0.0$\pm0.1$ 0.9$^{+0.6}_{-0.4}$ 0.0$^{+0.0}_{-0.3}$ 0.0$^{+0.3}_{-0.0}$ 0.0$^{+0.0}_{-16.2}$ $<16.2$
SST 3.7
MS 0 0 -5.6$\pm5.7$ 5.6$\pm5.7$
PDG $\pm2.2\pm2.1$ $<7.4$
$d_{1/2}$ $d_{3/2}$ $g_{3/2}$ $d_{1/2}$ $d_{3/2}$ $g_{3/2}$
$N^{*}(1675)\frac{5}{2}^-$ VMD 0.75 -1.70 0.22 1.87 0.06 0.00 0.00 0.06
KI -1.1 -0.2 0 2.3 0
CR 0.2 -0.4 0 0.5
SST 2.0
MS 0.8$\pm0.4$ -0.5$\pm0.5$ 0 1.0$\pm0.4$
PDG 0.8$\pm0.4$ -1.7$\pm0.6$ $<2.2$
: Predictions of the extended VMD model for the partial widths of the nucleon resonance decays into the $\rho $- and $\omega $-meson channels, inclusive of the sign of the amplitudes. The data quoted by PDG \[24\], the results of the multichannel $\pi N$ partial-wave analysis MS \[25\] and LD \[26\], and predictions of the non-relativistic quark model K \[27\] and the quark-pair creation models CR \[28\] and SST \[53,54\] are given for comparison. The widths are in MeV. This table shows the $\rho$- and $\omega$-meson modes of the negative-parity $N^*$-resonances. []{data-label="table3"}
Resonance Ref. $N\rho$ $N\rho $ $N\rho $ $\sqrt{\Gamma^{tot}_{N\rho}} $ $N\omega$ $N\omega$ $N\omega$ $\sqrt{\Gamma^{tot}_{N\omega}}$
---------------------------- ------ ---------------------- ---------------------- ---------------------- --------------------------------- ---------------------- ---------------------- --------------------- ---------------------------------
$p_{1/2}$ $p_{3/2}$ $p_{1/2}$ $p_{3/2}$
$N^{*}(1440)\frac{1}{2}^+$ VMD -0.29 0.61 0.67 0.00 0.00 0.00
KI 0.3 0.1 0.3
CR -0.3$^{+0.2}_{-0.3}$ -0.5$^{+0.3}_{-0.5}$ 0.6$^{+0.5}_{-0.3}$
SST 1.5
PDG $\pm3.7\pm2.2$ $<6$
$N^{*}(1710)\frac{1}{2}^+$ VMD 2.22 3.30 3.97 0.18 -0.72 0.74
KI 5.5 2.5 6.0 0.6 -0.7 0.9
CR 0.3$\pm0.1$ -3.7$^{+0.9}_{-1.2}$ 3.7$^{+1.2}_{-1.0}$ 0.0$^{+0.0}_{-2.3}$ 0.0$^{+0.0}_{-0.4}$ $<2.3$
SST 4.1 0.03 -0.2 0.2
MS 3.9$\pm4.4$ 0 $<8.3$
PDG $\pm4.0\pm2.0$ 4.3$\pm1.9$
$p_{1/2}$ $p_{3/2}$ $f_{3/2}$ $p_{1/2}$ $p_{3/2}$ $f_{3/2}$
$N^{*}(1720)\frac{3}{2}^+$ VMD 11.03 -2.56 1.02 11.37 5.29 -2.09 0.14 5.69
KI 11.7 -2.6 -3.5 12.5 5.3 -2.1 -0.61 5.7
CR -2.6$^{+0.7}_{-0.8}$ 1.8$^{+0.6}_{-0.5}$ 0.7$^{+0.3}_{-0.2}$ 3.3$^{+1.0}_{-0.8}$ 0.0$^{+0.0}_{-0.2}$ 0.0$^{+1.2}_{-0.0}$ 0.0$^{+0.1}_{-0.0}$ $<1.3$
SST 5.2 0.1 0.2 0.1 0.2
OR -5.7 -2.5 -1.9 6.5
MS 18$\pm5$ 0 0 18$\pm5$
PDG 11$\pm2$
$N^{*}(1900)\frac{3}{2}^+$ VMD -14.82 -1.28 -2.08 15.02 7.97 -0.49 1.25 8.09
KI -0.4 -1.3 -0.5 1.5 9.7 -0.4 -2.0 9.9
CR -1.4$^{+0.9}_{-1.0}$ -1.0$\pm0.6$ 0.2$^{+0.5}_{-0.2}$ 1.8$^{+1.2}_{-1.1}$ 4.4$^{+1.2}_{-4.4}$ 0.6$^{+1.2}_{-0.6}$ $<5.9$
SST 6.1 11.5 -8.2 6.2 15.4
MS -14.7$\pm2.9$ 0 0 14.7$\pm2.9$ 12.3$\pm1.8$ 0 0 12.3$\pm1.8$
PDG
$f_{1/2}$ $f_{3/2}$ $p_{3/2}$ $f_{1/2}$ $f_{3/2}$ $p_{3/2}$
$N^{*}(1680)\frac{5}{2}^+$ VMD -1.35 -1.23 -2.62 3.20 0.09 0.40 0.58 0.71
KI 1.6 -1.3 -4.0 4.5 0.13 -0.19 -1.2 1.2
CR -0.2$\pm0.0$ -0.3$\pm0.1$ -3.0$^{+0.4}_{-0.5}$ 3.0$^{+0.5}_{-0.4}$
SST 3.1 2.7 -1.3 4.3
MS 0 -1.7$\pm0.6$ -2.8$\pm0.7$ 3.3$\pm0.7$
PDG -2.0$\pm0.6$ -2.8$\pm1.4$ 3.4$\pm1.1$
$N^{*}(2000)\frac{5}{2}^+$ VMD 2.50 6.99 -16.02 17.66 0.07 8.19 9.96 12.89
KI -1.7 -4.4 -6.6 8.1 4.0 6.7 10.9 13.4
CR -0.4$\pm0.3$ -0.2$\pm0.1$ -7.8$^{+3.1}_{-0.2}$ 7.8$^{+0.2}_{-3.1}$ -0.3$^{+0.2}_{-0.3}$ -1.6$^{+1.1}_{-1.5}$ 3.1$^{+0.5}_{-0.5}$ 3.5$^{+1.3}_{-0.8}$
MS 0 8.5$\pm5.8$ -17.2$\pm6.2$ 19.2$\pm6.1$
$f_{1/2}$ $f_{3/2}$ $h_{3/2}$ $f_{1/2}$ $f_{3/2}$ $h_{3/2}$
$N^{*}(1990)\frac{7}{2}^+$ -0.96 3.95 0.97 4.18 1.31 -6.90 -0.68 7.06
KI -0.8 4.2 0 4.3 1.3 -7.2 0 7.3
CR 0.6$\pm0.3$ -1.0$^{+0.6}_{-0.5}$ 0 1.2$^{+0.6}_{-0.7}$ -0.8$^{+0.4}_{-0.5}$ 1.4$^{+0.9}_{-0.7}$ 0 1.6$^{+1.0}_{-0.9}$
SST 1.1 2.3 -2.8 0.7 14
: The $\rho$- and $\omega$-meson modes of the positive-parity $N^{*}$-resonances. The notations are the same as in Table III.[]{data-label="table4"}
Resonance Ref. $N\rho$ $N\rho $ $N\rho $ $\sqrt{\Gamma^{tot}_{N\rho}}$
----------------------------- ------ ---------------------- ---------------------- ---------------------- -------------------------------
$s_{1/2}$ $d_{3/2}$
$\Delta(1620)\frac{1}{2}^-$ VMD 4.05 -0.02 4.05
KI 7.8 -1.7 8.0
CR -3.6$^{+1.3}_{-2.5}$ -0.3$^{+0.1}_{-0.2}$ 3.6$^{+2.5}_{-1.3}$
SST 2.5 -3.6 4.4
MS 6.2$\pm0.9$ -2.4$\pm0.2$ 6.6$\pm0.8$
PDG 4.2$\pm1.4$ -2.2$\pm1.5$ 4.9$\pm1.5$
$\Delta(1900)\frac{1}{2}^-$ VMD -5.31 -1.52 5.52
CR 2.5$\pm0.6$ 1.5$^{+0.5}_{-0.3}$ 2.9$^{+0.8}_{-0.6}$
MS -3.5$\pm2.7$ -9.3$\pm1.7$ 9.9$\pm1.9$
$d_{1/2}$ $d_{3/2}$ $s_{3/2}$
$\Delta(1700)\frac{3}{2}^-$ VMD -1.66 0.66 6.67 6.91
KI 4.2 0.9 16.5 17.0
CR -1.2$^{+0.6}_{-1.2}$ 0.5$^{+0.5}_{-0.2}$ 3.4$^{+2.2}_{-1.7}$ 3.6$^{+2.5}_{-1.8}$
SST 4.9
MS 0 0 6.8$\pm2.3$ 6.8$\pm2.3$
PDG $\pm6.7\pm2.4$ 11$\pm3$
$\Delta(1940)\frac{3}{2}^-$ VMD -2.70 1.32 12.83 13.18
CR -3.8$^{+2.3}_{-2.5}$ 1.4$^{+0.9}_{-0.8}$ 1.0$\pm0.3$ 4.2$^{+2.7}_{-2.4}$
MS 0 0 12.7$\pm5.6$ 12.7$\pm5.6$
$d_{1/2}$ $d_{3/2}$ $g_{3/2}$
$\Delta(1930)\frac{5}{2}^-$ VMD -16.90 -2.62 -1.65 17.18
CR 0.1$\pm0.0$ -2.9$^{+0.5}_{-0.8}$ -0.1$^{+0.0}_{-0.1}$ 2.9$^{+0.8}_{-0.5}$
MS -20.8$\pm2.9$ 0 0 20.8$\pm2.9$
: The $\rho$-meson modes of the negative-parity $\Delta$-resonances. The notations are the same as in Table III.[]{data-label="table5"}
Resonance Ref. $N\rho$ $N\rho $ $N\rho $ $\sqrt{\Gamma^{tot}_{N\rho}} $
----------------------------- ------ ---------------------- ---------------------- ---------------------- ---------------------------------
$p_{1/2}$ $p_{3/2}$
$\Delta(1750)\frac{1}{2}^+$ VMD 2.20 -7.65 7.97
KI 2.2 -7.6 7.9
CR -6.5$^{+4.6}_{-4.1}$ 4.7$^{+3.1}_{-3.3}$ 8.0$^{+5.1}_{-5.7}$
SST 17.1
$\Delta(1910)\frac{1}{2}^+$ VMD -2.75 -5.44 6.10
KI -3.7 -4.9 6.1
CR 5.6$^{+0.9}_{-0.4}$ 2.6$^{+0.4}_{-0.2}$ 6.1$^{+1.0}_{-0.5}$
SST 6.9
MS 4.9$\pm1.1$
$p_{1/2}$ $p_{3/2}$ $f_{3/2}$
$\Delta(1600)\frac{3}{2}^+$ VMD 0.56 -1.30 0.13 1.42
KI -1.3 -5.5 -0.4 5.7
CR 0.4$^{+0.7}_{-0.3}$ -0.9$^{+0.6}_{-1.4}$ 0 1.0$^{+1.6}_{-0.6}$
SST 2.9
L 4.5 4.5 0 6.4
PDG $<11$
$\Delta(1920)\frac{3}{2}^+$ VMD -6.19 6.75 -2.05 9.39
KI -8.1 6.2 5.5 11.6
CR 5.3$^{+1.3}_{-0.5}$ 6.6$^{+1.6}_{-0.7}$ -0.7$^{+0.2}_{-0.4}$ 8.5$^{+2.0}_{-0.8}$
SST 5.2
$f_{1/2}$ $f_{3/2}$ $p_{3/2}$
$\Delta(1905)\frac{5}{2}^+$ VMD -1.40 -0.46 17.46 17.53
KI -0.1 -6.4 -2.1 6.7
CR -0.7$\pm0.2$ -0.7$^{+0.1}_{-0.2}$ 6.3$^{+0.8}_{-0.4}$ 6.4$^{+0.8}_{-0.4}$
SST 5.1
OR 0.3 1.3 -6.6 6.7
MS 0 0 16.8$\pm1.3$ 16.8$\pm1.3$
PDG 20$\pm6$ $>17$
$\Delta(2000)\frac{5}{2}^+$ VMD 2.43 5.20 -6.73 8.84
KI 7.2 4.6 17.8 19.7
CR 2.6$^{+2.8}_{-2.1}$ -3.1$^{+2.4}_{-3.2}$ 3.1$\pm1.2$ 5.1$^{+4.2}_{-3.0}$
SST 8.9
MS 0 0 -6.7$\pm2.4$ 6.7$\pm2.4$
$f_{1/2}$ $f_{3/2}$ $h_{3/2}$
$\Delta(1950)\frac{7}{2}^+$ VMD 1.28 -2.38 0.28 2.72
KI -4.7 -8.2 0 9.4
CR 1.3$\pm0.1$ -2.3$\pm0.2$ 0 2.6$\pm0.2$
SST 4.5
OR 1.1 1.9 0 2.2
MS 0 11.4$\pm0.5$ 0 11.4$\pm0.5$
PDG $<6$
: The $\rho$-meson modes of the positive-parity $\Delta$-resonances. The notations are the same as in Table III.[]{data-label="table6"}
----------------------------------- ------------------- -----------------------
Resonance$\;\;$ $\Gamma_{e^+e^-}$ $\Gamma_{\mu^+\mu^-}$
KeV $\;\;$ KeV $\;\;$
$N^{*} (1535)\frac{1}{2}^-$ 2.01 1.87
5.30 4.85
$N^{*} (1650)\frac{1}{2}^-$ 3.23 0.79
2.00 0.31
$N^{*} (1520)\frac{3}{2}^-$ 6.02 0.73
4.42 0.41
$N^{*} (1700)\frac{3}{2}^-$ 0.41 0.32
2.86 2.64
$N^{*} (1675)\frac{5}{2}^-$ 0.21 0.10
1.09 0.22
$N^{*} (1440)\frac{1}{2}^+$ 1.40 0.22
0.56 0.05
$N^{*} (1710)\frac{1}{2}^+$ 0.58 0.31
0.60 0.56
$N^{*} (1720)\frac{3}{2}^+$ 7.93 7.77
3.14 2.77
$N^{*} (1900)\frac{3}{2}^+$ 4.62 4.54
12.22 11.91
$N^{*} (1680)\frac{5}{2}^+$ 2.58 0.43
1.47 1.13
$N^{*} (2000)\frac{5}{2}^+$ 14.17 13.99
20.89 21.56
$N^{*} (1990)\frac{7}{2}^+$ 3.09 2.97
8.24 4.78
$\Delta (1620)\frac{1}{2}^-$ 1.33 0.88
$\Delta (1900)\frac{1}{2}^-$ 1.19 1.09
$\Delta (1700)\frac{3}{2}^-$ 6.10 1.65
$\Delta (1940)\frac{3}{2}^-$ 6.57 6.20
$\Delta (1930)\frac{5}{2}^-$ 9.63 9.16
$\Delta (1750)\frac{1}{2}^+$ 4.61 2.77
$\Delta (1910)\frac{1}{2}^+$ 1.33 1.27
$\Delta (1232)\frac{3}{2}^+$ 5.02 0.04
$\Delta (1600)\frac{3}{2}^+$ 0.24 0.13
$\Delta (1920)\frac{3}{2}^+$ 4.22 3.50
$\Delta (1905)\frac{5}{2}^+$ 10.51 10.36
$\Delta (2000)\frac{5}{2}^+$ 3.25 3.00
$\Delta (1950)\frac{7}{2}^+$ 3.18 0.81
----------------------------------- ------------------- -----------------------
: The decay widths of the nucleon resonances into the $e^{+}e^{-}$ and $\mu ^{+}\mu ^{-}$ pairs. The first line of the $N^{*}$-resonances with $I=1/2$ refers to the proton, the second one to the neutron resonances.[]{data-label="table7"}
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 12cm
= 14cm
= 14cm
= 18cm
= 18cm
= 18cm
= 14cm
= 14cm
= 18cm
= 18cm
= 18cm
= 18cm
= 18cm
= 18cm
= 16cm
= 16cm
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'John Haywood\*'
- Adrian Sescu
- Kevin Adkins
title: |
[under consideration for publication in Wind Energy]{}\
Large eddy simulation study of the humidity variation in the shadow of a large wind farm
---
Introduction {#sec1}
============
With the planned growth of renewable energy and the consequential deployment of a large number of wind turbines in many countries, research was focused on the climatic impact on the atmospheric boundary layer (ABL) [@Roy; @Barrie]. Among other approaches, Large Eddy Simulation (LES) has been used extensively in the last decades to investigate the effect of large wind farm on the ABL (see, for example, Calaf et al. [@Calaf1], Lu & Porté-Agel [@Lu], Yang et al. [@Yang], VerHulst & Meneveau [@VerHulst1], Stevens et al. [@Stevens2], Hayat et al. [@Hayat]). As the kinetic energy extracted from the upstream flow and from above the wind farm (the so-called ’entrainment’, as was found in previous studies, such as Cal et al. [@Cal], Meyers & Meneveau [@Meyers], Abkar & Porté-Agel [@Abkar3], VerHulst & Meneveau [@VerHulst1; @VerHulst2]) is used to generate power, downstream wake recovery from individual turbines or compounded wakes from multiple turbines is an important issue in the development of large wind turbine arrays. LES has shown that higher levels of upstream turbulence intensity aid in the recovery of wakes and moves the location of peak turbulence intensity and turbulent shear stress closer to the turbine [@Yu-Ting].
LES has been used to investigate the influence of atmospheric stability on entrainment, and the results pointed that the stable atmospheric scenarios lead to reduced entrainment [@Abkar1]. Alternatively, simulations point toward a weaker inversion strength or height [@Allaerts], or increased positive buoyancy [@Abkar2], increasing the entrainment rate and shortening the wake recovery. In an LES investigation of Calaf et al. [@Calaf1], it was shown that, by neglecting stratification effects and specifically aiming at determining whether surface scalar fluxes change in the presence of wind turbines, there is an overall increase in scalar fluxes on the order of 10%-15% within a fully developed WTABL. Temperature fields within large wind farms in a stably or unstably stratified ABL have been also investigated using LES (see, for example, Calaf et al. [@Calaf], Sescu & Meneveau [@Sescu2], Ali et al. [@Ali]). Generally, results agree with lower-resolution model studies that show how enhanced vertical mixing lowers the temperature above the rotor turbine top tip height and increases the temperature below the rotor turbine bottom tip height. Very few studies on the application of LES to predict the humidity budget in the ABL exist, however, and this study is an attempt to fill this gap (an example of a previous study is Adkins & Sescu [@Adkins1]). LES numerical experiments have also been utilized to explore the role of large-scale flow structures within the turbulent wake in entrainment (Cal et al. [@Cal], Meyers & Meneveau [@Meyers], VerHulst & Meneveau [@VerHulst1]). Such numerical experiments have also demonstrated how synthetic downward forcing of high velocity flow at upstream wind turbines can enhance kinetic energy entrainment and power extraction [@VerHulst2].
Previous work using both experimental measurements and numerical simulation indicate that wakes generated inside large wind farms can substantially impact the exchanges of sensible heat and humidity within the ABL. More extensive investigations are necessary in order to understand and quantify these exchanges both within and downstream of a wind turbine array boundary layer (WTABL) as associated changes may impact human activities, especially in an agricultural context. For example, such impacts to crop production have been identified previously by Takle [@Takle]. Specific examples include: reduced summer moisture stress, reduced dew duration, spring soil drying, enhanced daytime photosynthesis, enhanced nighttime respiration, and increased moisture loss during drought (see, for example, Mortley et al. [@Mortley], Pareek et al. [@Pareek], Ford and Thorne [@Ford], Tibbitts and Bottenberg [@Tibbitts1], Tibbitts [@Tibbitts2], or Grange and Hand [@Grange]).
In a previous study, Adkins and Sescu [@Adkins1; @Adkins2] used experimental measurements to investigate the impact of a wind turbine on the relative humidity distribution in the near-wake region and numerical simulation to study relative humidity changes within a broader turbine array. Vertical, lateral and longitudinal observations allowed profiles of humidity in a stable ABL to be constructed. Vertical profiles with a decrease in relative humidity below the turbine hub height and an increase above were observed. Within the near-wake region, the relative humidity at the lower turbine tip height quickly decreased and slowly recovered with downstream distance. In the spanwise direction, at the lower turbine tip height, the greatest decreases in humidity were observed on the right-hand side of the wakeÕs centerline. This change was associated with the descending turbine blades on the right-hand side of the turbine disk and the wake interaction with the ground. In Adkins and Sescu [@Adkins1], the agreement between the numerical LES results and the experimental measurements was found to be good although some of the comparisons were qualitative; however, we are confident that the comparisons can serve as a validation of the numerical tool that is being employed in this study, aimed at quantifying the effect of the wind farm shadow (the region that extends in the downstream of the wind farm for several kilometers) on the humidity budget in a stably stratified ABL.
The large-eddy simulation framework, the numerical algorithm employed to solve the equations, and the boundary conditions are outlined in the next section \[sec2\]. In section \[sec3\], numerical results consisting of contour plots of various quantities or distribution of turbulent kinetic energy and relative humidity along different directions are presented and discussed. In the discussion, the focus is on how changes in relative humidity evolve in the shadow region of a wind farm, for both the aligned and staggered layouts.
Large Eddy Simulation framework {#sec2}
===============================
As in Adkins & Sescu [@Adkins2], the LES filtered momentum conservation equations with the Boussinesq approximation, transport equations for potential temperature and specific humidity, as well as the continuity equation are employed,
$$\begin{aligned}
\label{eq1}
\frac{\partial \tilde{u}_{i}}{\partial t}
+ \tilde{u}_{j}\frac{\partial \tilde{u}_{i}}{\partial x_{j}}
=
- \frac{\partial \tilde{p}^{*}}{\partial x_{i}}
- \frac{\partial \tilde{\tau}_{ij}}{\partial x_{j}}
+ \delta_{i3} g \frac{\tilde{\theta} - \langle \tilde{\theta} \rangle }{\theta_{0}}
+ f_c \epsilon_{ij3} (\tilde{u}_{j} - u_{gj})
+ F_{i} + F_{CPM}\end{aligned}$$
$$\begin{aligned}
\label{eq2}
\frac{\partial \tilde{\theta}}{\partial t}
+ \tilde{u}_{j}\frac{\partial \tilde{\theta}}{\partial x_{j}}
=
- \frac{\partial \pi_{j}}{\partial x_{i}} + F^{\theta}_{CPM},
\hspace{12mm}
\frac{\partial \tilde{q_s}}{\partial t}
+ \tilde{u}_{j}\frac{\partial \tilde{q_s}}{\partial x_{j}}
=
- \frac{\partial \pi_{j}^q}{\partial x_{i}}+ F^{q}_{CPM},
\hspace{12mm}
\frac{\partial \tilde{u}_{i}}{\partial x_{i}} = 0 \end{aligned}$$
respectively, where the spatial filtering at scale $\tilde{\Delta}$ is represented by tilde, $\tilde{u}_{i}, i=1,2,3$, are the components of the velocity field corresponding to the axial $x_1$-direction, lateral $x_2$-direction, and vertical $x_3$-direction, respectively, $\tilde{\theta}$ is the resolved potential temperature, $\theta_{0}$ is the reference temperature, $\tilde{q_s}$ is the resolved specific humidity, the angle brackets represent a horizontal average, $g$ is the gravitational acceleration, $f_c$ is the Coriolis parameter, $\delta_{ij}$ is the Kronecker delta, $\epsilon_{ijk}$ is the alternating unit tensor, $\tilde{p}^{*}$ is the effective pressure divided by reference density, $F_{i}$ is a forcing term (here modeling the effect of the wind turbines), and the terms having $CPM$ at the subscript are active in a blending region, where the data from a precursor domain is transferred on to the main domain (more details below). The SGS stress, heat and humidity fluxes are given as $\tau_{ij} = \widetilde{u_i u_j} - \tilde{u}_i \tilde{u}_j$, $\pi_{j} = \widetilde{u_j \theta} - \tilde{u}_j \tilde{\theta}$, and $\pi_{j}^q = \widetilde{u_j q_s} - \tilde{u}_j \tilde{q_s}$. They are modeled using a Lagrangian scale-dependent model as developed by Bou-Zeid et al. [@Bou-Zeid], and extended to scalar transport by Porté-Agel et al. [@Porte-Agel2].
The numerical tool is a pseudo-spectral LES code that solves the filtered Navier-Stokes equations using a pseudo-spectral horizontal discretization and a centered finite difference method in the vertical direction (the grid is uniform in all directions) [@Calaf; @Sescu]. Time marching is performed using a fully-explicit second-order accurate Adams-Bashforth scheme [@Butcher]. The continuity equation is enforced through the solution of the Poisson equation resulting from taking the divergence of the momentum equation. Periodic boundary conditions are imposed along the horizontal directions. The vertical gradients of velocity and the vertical velocity component vanish at the top boundary. The horizontal velocities at the first point away from the wall ($z = \Delta z/2$) are set through the velocity gradients in the vertical direction calculated using the Monin-Obukhov similarity theory, and the vertical velocity at the wall is set to zero.
A concurrent precursor simulation [@Stevens; @Haywood] provides inflow boundary conditions that are introduced at the downstream boundary of the main domain in order to preserve the periodicity in the streamwise direction. It may sound counterintuitive to impose inflow conditions at the downstream boundary of the domain, but since periodic boundary conditions are imposed in the streamwise direction, whatever is imposed at the outflow boundary is automatically copied to the inflow boundary. The precursor and main flow domains considered here are identical, except the wind turbines rotors are added to the main domain. After each time step, a region of the flow data near the outflow boundary of the precursor domain is blended on to the flow data in a region located in proximity to the outflow boundary of the main domain. The blending region ensures that the flow data is smoothly transitioned from the precursor simulation to the main simulation. Assuming that the length of the blending region is $L_{blend}$ and ranges from $x = L_s$ to $x = L_x$, where $L_x$ is the length of entire domain, a generic variable (velocity, temperature, or relative humidity) in the blending region can be penalized using a source term in the governing equations ($F_{CPM}$, $F^{\theta}_{CPM}$, or $F^{q}_{CPM}$) to match the solution from the precursor domain; for example, $$\begin{aligned}
F_{CPM} = w(x) \left[ \left( \tilde{u}_{i} \right)_{main} - \left( \tilde{u}_{i} \right)_{prec} \right]\end{aligned}$$ where $\left( \tilde{u}_{i} \right)_{main}$ and $\left( \tilde{u}_{i} \right)_{prec}$ are flow variables from the main and precursor domains, respectively. The blending function used here is $$\begin{aligned}
w(x) = \sigma
\begin{cases}
\frac{1}{2}\left[1-cos\left(\pi \frac{x-L_s}{L_{pl}-L_s}\right)\right] &;\ L_s \leq x \leq L_{pl} \\
1 & ;\ L_{pl}<x \leq L_x
\end{cases}\end{aligned}$$ where $L_{pl} = L_x - \frac{1}{4} L_{blend}$ (in the simulations included in this study, $L_{blend} = 0.05 L_x$, $L_s = L_x - L_{blend}$, and $L_{pl} = L_x - 0.25 L_{blend}$, where $L_x$ is the length of the flow domain in the streamwise direction) , and $\sigma$ is an amplitude, which is set equal to $0.05$ in this work. Figure \[f1\] illustrates the procedure that is used to impose the inflow condition for the main simulation; the shape of the blending function that smoothly ramps the flow variables of the main simulation to the flow variables of the precursor simulation is also shown.
![The procedure used to impose the inflow condition.[]{data-label="f1"}](inflow2){width="13cm"}
Due to the Coriolis effect, the direction of the wind changes with height in the ABL, subscribing to an Ekman spiral. This presents a challenge in trying to align the geostrophic velocity components to achieve the desired flow direction at hub height. For the simulations, an adjustment to the geostrophic wind direction is accomplished through manipulation of a Coriolis force type source term in the momentum equations (see equation (8) in Sescu & Meneveau [@Sescu]) in order to achieve the desired hub height flow direction. Once the flow direction becomes normal to the rotor disk at the hub elevation, this term is deactivated to avoid unphysical behavior of the flow (more details about this procedure can be found in Sescu & Meneveau [@Sescu]).
An effective top layer of the ABL, isolated from the physically relevant flow within the ABL domain, is specified via a capping inversion created by a temperature gradient (the inversion strength is $0.01$ K/m). To this end, a source or sink of heat is introduced above the top of the ABL within the precursor simulation to enable the desired atmospheric stability (see figure 2 of Sescu & Meneveau [@Sescu]). This is realized by including a source term in the scalar equation (see equation (4) in Sescu & Meneveau [@Sescu]), where the amplitude is set via a PI controller with the input defined as the difference between the actual temperature and the desired temperature (see equation (5) and the following discussion in the same reference [@Sescu]).
An actuator disk method with rotation (ADM-R) similar to that employed by Wu and Porté-Agel [@Wu] is implemented here to model the effect of rotors on the ABL (more details about the implementation of the ADM-R can be found in Adkins and Sescu [@Adkins2]).
Results and Discussion {#sec3}
======================
Two simulations were performed corresponding to an aligned and a staggered configuration, with the lateral spacing between two rotors of $416$ m, the longitudinal spacing of $680$ m, the rotor diameter set to $100$ m, the hub height equal to $80$ m, and a thrust coefficient of $0.6$. The sketch in figure \[f2\] shows the aligned wind farm and the shadow region, the latter having a streamwise length on the order of $13$ km. A stable ABL with a thermal stratification of $2$ K and a geostrophic velocity of $8$ m/s, whose direction is maintained normal to the rotor disk at the hub height, are considered. A positive lapse rate of $0.5$ g/kg in the first 400 m was imposed for the specific humidity (closely resembling the humidity profile found from the experimental measurements of Adkins & Sescu [@Adkins2]), where a constant potential temperature of 300 K and a constant specific humidity flux of $0.01$ g/kg m/s were imposed at the ground level. The distance between the inflow boundary and the first row of the wind farm is approximately $7$ rotor diameters (in the same order as the streamwise rotor spacing), which is sufficiently large to avoid unwanted induction. Simulations were performed within a domain having downstream, lateral, and vertical dimensions of $20$ km x $2.5$ km x $0.5$ km respectively, on a grid consisting of $512$ x $128$ x $96$ points. In the following, the relative humidity will be denoted by $q$, and its time average by $\bar{q}$.
![Wind farm layout with the shadow region.[]{data-label="f2"}](align_layout){width="15cm"}
a\) ![Qualitative representation of turbulence in the ABL via iso-surfaces of $Q$ criterion colored by the streamwise velocity component (the ground is shown in green, and wind turbine rotors are shown in yellow): a) aligned layout; b) staggered layout. The streamwise direction is scaled down by a factor of 10.[]{data-label="f3"}](Q_align4.png "fig:"){width="11cm"}\
b) ![Qualitative representation of turbulence in the ABL via iso-surfaces of $Q$ criterion colored by the streamwise velocity component (the ground is shown in green, and wind turbine rotors are shown in yellow): a) aligned layout; b) staggered layout. The streamwise direction is scaled down by a factor of 10.[]{data-label="f3"}](Q_stag4.png "fig:"){width="11cm"}
![Turbulent kinetic energy (TKE) progression in the streamwise direction (TKE was integrated in the lateral and vertical directions). The blending region is not shown in this plot.[]{data-label="f4"}](tke_comp){width="14cm"}
Results in terms of iso-surfaces of Q-criterion, contour plots of mean relative humidity, and profiles of relative humidity in the longitudinal, lateral and vertical directions are presented and discussed next. The main objective is to analyze the effect of compounding wakes on near-surface relative humidity in the downstream of the wind farm (referred to as ’shadow region’). In figure \[f3\], we show iso-surface of the Q-criterion ($Q = 1/2[|\mathbf{\Omega}|^2 - |\mathbf{S}|^2]$, where $\mathbf{S} = 1/2[\nabla \mathbf{v}+(\nabla \mathbf{v})^T]$ is the rate-of-strain tensor, and $\mathbf{\Omega} = 1/2[\nabla \mathbf{v}-(\nabla \mathbf{v})^T]$ is the vorticity tensor) colored by the streamwise velocity component, where the ground is in green, and wind turbine rotors are in yellow (the streamwise direction is scaled down by a factor of 10). We decided to use Q-criterion as opposed to vorticity magnitude because it is one of the most effective approaches of vortex identification, and it better highlights the turbulence intensity in the wakes, as well as gives a qualitative representation of the level of mixing. The effect of compounding wakes is more prevalent for the aligned layout in figure \[f3\]a than that corresponding to the staggered layout in figure \[f3\]b. Turbulence in the upstream of the wind farm is characteristic of a stable ABL, with small flow structures convected by the base flow - the thermally stratified ABL tends to suppress large flow structures; the appearance of turbulence in the downstream, however, changes since the wind turbine rotors are efficient mixers, generating more intense flow structures, thus increasing the turbulence kinetic energy.
The turbulence kinetic energy (TKE) integrated in the lateral and vertical directions as
$$\begin{aligned}
\label{eq5}
TKE(x) = \frac{1}{z_h(y_2 - y_1)} \int_{0}^{z_h}\int_{y1}^{y2} \left[ \overline{u'^2}(x,y,z) + \overline{v'^2}(x,y,z) + \overline{w'^2}(x,y,z) \right] dy dz,\end{aligned}$$
where $y_1$ and $y_2$ are the coordinates of lateral boundaries, $z_h$ is the height of the domain, and bars represent time average of the fluctuating velocity components $u'$, $v'$ and $w'$, is plotted as a function of the streamwise direction in figure \[f4\]; both curves corresponding to the two farm layouts were superposed to each other. The graph clearly indicates that the integrated TKE reaches a maximum at the end of the wind farm (actually, slightly downstream by roughly a streamwise spacing between two consecutive rotors), after which it experiences a slow decrease in the shadow region. The integrated TKE corresponding to the staggered layout (dashed line) increases at a smaller rate within the farm and decreases at a smaller rate in the shadow region, but reaches a higher level further in the downstream, suggesting a longer recovery region. The reason for which the wakes recover faster in case of the aligned layout is the higher turbulence intensity in the wakes and therefore the creation of a stronger vertical kinetic energy flux. This contrast between the aligned and staggered layouts has been identified in a number of previous studies (see, for example, Chamorro et al. [@Chamorro], VerHulst & Meneveau [@VerHulst1; @VerHulst2], or Stevens et al. [@Stevens3; @Stevens3]). The higher peak in the TKE for the aligned layout is due to the effect of compounding wakes that enhance mixing and turbulence intensity more effectively, because the distance between two consecutive turbines is smaller when compared to the staggered layout.
![Mean relative humidity progression (calculated using equation (\[eq5\]) as a percentage difference) contour plots: aligned layout (top); staggered layout (bottom). The streamwise direction is scaled down by a factor of 12, and the rotors are represented by 8 vertical lines).[]{data-label="f5"}](q_cont_align.png "fig:"){width="15cm"}\
![Mean relative humidity progression (calculated using equation (\[eq5\]) as a percentage difference) contour plots: aligned layout (top); staggered layout (bottom). The streamwise direction is scaled down by a factor of 12, and the rotors are represented by 8 vertical lines).[]{data-label="f5"}](q_cont_stag.png "fig:"){width="15cm"}
Next, the behavior of the relative humidity in the shadow region is studied qualitatively and quantitatively. First, contours of the time-averaged and spanwise-averaged relative humidity progression calculated as
$$\begin{aligned}
\label{eq5}
\Delta q(x,z) = \frac{1}{T(y_2 - y_1)} \int_{0}^{T}\int_{y1}^{y2} \left[ q(x,y,z,t) - q_{up}(y,z,t) \right] dy dt,\end{aligned}$$
where $q_{up}(y,z,t)$ is the humidity two diameters upstream of the first row of turbines, and $T$ being a sufficiently long time window (in the order of $1.5$ hours, which corresponds to approximately two flow through times), are plotted in figure \[f5\] for both the aligned (top) and staggered (bottom) layouts (in this figure, the streamwise coordinate is scaled down by a factor of 12, and the rotors are represented by 8 vertical, black lines). Even with this time window, there where, however, some grid-to-grid oscillations in the reported results that have been eliminated via high-order filters, which do not compromise the validity of the results. In the shadow region, both parts of the figure strongly indicate that there is a reduction of the relative humidity in proximity to the ground, which extends up to the total height of the wind turbines ($\sim130$ m), and that there is an increase above the wind farm. The region of increase gradually moves upward with the streamwise direction reaching altitudes in the order of twice the total wind turbine height ($\sim300$ m). It is interesting to note that the decrease in the relative humidity in the vicinity of the ground continues in the downstream of the wind farm for almost another length of the farm (from $6$ km to $10$ km for the aligned layout and from $6$ km to $11$ km for the staggered layout). The recovery of the relative humidity under the hub (recovery meaning the convergence of the humidity to the upstream levels) seems to be faster for the aligned layout as indicated by the top panel of figure \[f5\].
The variation of the relative humidity in the streamwise direction evaluated as
$$\begin{aligned}
\label{eq6}
\Delta q_{min}(x) = \frac{1}{y_2 - y_1} \int_{y1}^{y2} \min_{0<z<z_h} \left[ \bar{q}(x,y,z) - \bar{q}_{up}(y,z) \right] dy,\end{aligned}$$
$$\begin{aligned}
\label{eq7}
\Delta q_{max}(x) = \frac{1}{y_2 - y_1} \int_{y1}^{y2} \max_{0<z<z_h} \left[ \bar{q}(x,y,z) - \bar{q}_{up}(y,z) \right] dy,\end{aligned}$$
where bar denotes the time average $
\bar{q}(x,y,z) = 1/T \int_{0}^{T} q(x,y,z,t) dt
$, is quantitatively analyzed in figure \[f6\]; the minimum values are representative of the humidity decrease under the hub and the maximum values are representative of the humidity increase above the hub. Apparently, the decrease in relative humidity under the hub (top panel of figure \[f6\]) evolves differently in the streamwise direction for the two layouts: the humidity corresponding to the aligned layout drops at a higher rate between $x=6$ km and $x=8$ km than the rate corresponding to the staggered layout; then, it recovers again at a higher rate in the downstream, while the recovery of the staggered layout is much smaller. This can be correlated with our previous assertion (in the paragraph before figure \[f5\]) about the behavior of the kinetic energy in the wake, which recovers faster for the aligned layout. Above the hub height (bottom panel in figure \[f6\]) the behaviors of the relative humidity variations are not much different between the two farm layouts: both experience a steady decrease in the shadow region, though at a slightly smaller rate for the aligned layout. The results in figure \[f6\] indicate that the observed trends in the wind farm, with respect to the variation of integrated relative humidity both below and above the hub, are similar as in the previous work of the authors (see figure 9 in Adkins & Sescu), except a slight difference in the levels because the turbine spacings and the thrust coefficient are different.
![Mean relative humidity progression in the streamwise direction: decrease - under the hub (top); increase - above the hub (bottom). The blending region is not shown in this plot.[]{data-label="f6"}](q_min_comp "fig:"){width="12cm"}\
![Mean relative humidity progression in the streamwise direction: decrease - under the hub (top); increase - above the hub (bottom). The blending region is not shown in this plot.[]{data-label="f6"}](q_max_comp "fig:"){width="12cm"}
In figure \[f7\], we plot vertical profiles of the time-averaged and spanwise-averaged relative humidity variation (see equation \[eq5\]) at selected streamwise locations (they are in fact a quantitative representation of the contour plots shown in figure \[f5\]). Both panels in figure \[f7\] show that the location of zero humidity variation (where the switching from negative to positive values occurs) gradually moves upward with the increase in the streamwise location, and that this location is well above the hub height: it varies between $130$ m at $x=6$ km and $200$ m at $x=18$ km. Both layouts bring the relative humidity variation to the same level at the ground, with a faster recovery for the aligned layout as previously concluded from the contour plots shown in figure \[f5\]. Comparing the progression of the relative humidity in the upper layers, it seems that the staggered farm configuration yields a higher increase in the immediate proximity to the farm, but then it decreases at a faster rate in the downstream.
The humidity distribution in the vertical direction (i.e., the decrease underneath the hub and increase above the hub) is dictated by the vertical turbulent humidity flux. A parallel can be made with the entrainment of kinetic energy from above layers that was observed in previous studies, except here the humidity is convected in both directions, from and toward the ground. The humidity generated at the ground level (note that we imposed a humidity flux at the bottom boundary) is convected to the upper layers by the rotor wakes, namely by the vertical turbulent humidity flux $\langle w'q' \rangle$. At the same time, the dry air masses (or ’less humid’ air) from above the wind farm is convected downward into the wind farm (this is what we could call ’entrainment’). To support this postulation, in figure \[f8\] we plot contours of the vertical turbulent humidity flux in $(y,z)$ planes at different streamwise locations, starting from 0.1 diameters (in proximity to the rotor) to 2.5 diameters downstream from the rotor disk. It reveals how the humidity is convected from the ground on one side of the rotor, and toward the ground on the other side (the effect of rotation is also captured by these plots). This could be also correlated with the distribution of temperature in a wind farm operating in stable conditions, where it was observed that the rotor wakes slightly heat the air in the region underneath the hub height (while it is known that heating is commonly associated with a decrease in humidity).
![Vertical profiles of the mean relative humidity for different streamwise locations: a) aligned layout; b) staggered layout. $x=6$ km corresponds to the end of the wind farm.[]{data-label="f7"}](q_z_align "fig:"){width="7.cm"} ![Vertical profiles of the mean relative humidity for different streamwise locations: a) aligned layout; b) staggered layout. $x=6$ km corresponds to the end of the wind farm.[]{data-label="f7"}](q_z_stag "fig:"){width="7.cm"}\
a) b)
0.1 D ![Vertical turbulent humidity flux distribution in $(y,z)$ planes at different streamwise locations, in the wakes of a row of rotors. The distance from the rotor is shown on the left, where $D$ is the rotor diameter.[]{data-label="f8"}](wq1 "fig:"){width="12cm"}\
0.5 D ![Vertical turbulent humidity flux distribution in $(y,z)$ planes at different streamwise locations, in the wakes of a row of rotors. The distance from the rotor is shown on the left, where $D$ is the rotor diameter.[]{data-label="f8"}](wq2 "fig:"){width="12cm"}\
1.0 D ![Vertical turbulent humidity flux distribution in $(y,z)$ planes at different streamwise locations, in the wakes of a row of rotors. The distance from the rotor is shown on the left, where $D$ is the rotor diameter.[]{data-label="f8"}](wq3 "fig:"){width="12cm"}\
1.5 D ![Vertical turbulent humidity flux distribution in $(y,z)$ planes at different streamwise locations, in the wakes of a row of rotors. The distance from the rotor is shown on the left, where $D$ is the rotor diameter.[]{data-label="f8"}](wq4 "fig:"){width="12cm"}\
2.0 D ![Vertical turbulent humidity flux distribution in $(y,z)$ planes at different streamwise locations, in the wakes of a row of rotors. The distance from the rotor is shown on the left, where $D$ is the rotor diameter.[]{data-label="f8"}](wq5 "fig:"){width="12cm"}\
2.5 D ![Vertical turbulent humidity flux distribution in $(y,z)$ planes at different streamwise locations, in the wakes of a row of rotors. The distance from the rotor is shown on the left, where $D$ is the rotor diameter.[]{data-label="f8"}](wq6 "fig:"){width="12cm"}\
![Vertical turbulent humidity flux distribution in $(y,z)$ planes at different streamwise locations, in the wakes of a row of rotors. The distance from the rotor is shown on the left, where $D$ is the rotor diameter.[]{data-label="f8"}](legend "fig:"){width="9cm"}
Finally, in figure \[f9\] we analyze lateral profiles of the time-averaged relative humidity progression at four selected streamwise locations to determine the impact of the compounded wakes on the humidity distribution in the spanwise direction. For the aligned layout, the effect of the wakes seems to be visible in the profiles that are taken at $x=6$ km and $x=8$ km, but by the time the flow reaches $x=10$ km, the wakes coalesce and thoroughly mix up the humidity. The mixing from the compounding wakes for the staggered farm configuration is even higher, as expected, as seen from the bottom panel of figure \[f9\]: while the impact of the wakes is predominant at $x=6$ km, by the time the flow reaches $x=8$ km the humidity seems to be well mixed up. The reason for which the wakes’ ’footprints’ in figure \[f9\] (the blue curve of the top plot, for example) do not show periodicity is because there are large flow structures in the atmospheric boundary layer that persist for a long time; these flow structures introduce anisotropy in the time averaging taken along practical time intervals (excessively long time intervals should be taken to eliminate the effect of these structures from the data). Similar profiles for the maximum increase of the mean relative humidity progression above the hub, at the same selected streamwise locations, are illustrated in figure \[f10\]. The above conclusions with respect to the effect of the wakes on the lateral distribution of humidity holds for these profiles as well.
![Mean minimum (close to the ground) relative humidity variation in the lateral direction: aligned layout (top); staggered layout (bottom).[]{data-label="f9"}](q_y_min_align "fig:"){width="13cm"}\
![Mean minimum (close to the ground) relative humidity variation in the lateral direction: aligned layout (top); staggered layout (bottom).[]{data-label="f9"}](q_y_min_stag "fig:"){width="13cm"}
![Mean maximum (above the hub) relative humidity variation in the lateral direction: aligned layout (top); staggered layout (bottom).[]{data-label="f10"}](q_y_max_align "fig:"){width="13cm"}\
![Mean maximum (above the hub) relative humidity variation in the lateral direction: aligned layout (top); staggered layout (bottom).[]{data-label="f10"}](q_y_max_stag "fig:"){width="13cm"}
Conclusions {#sec5}
===========
Changes to near-surface relative humidity were analyzed via LES both within and downstream of a large wind farm in aligned and staggered configurations. In keeping with previous observations and simulations made within the array, mixing brought about by turbines change near-surface relative humidity by reducing humidity adjacent to the ground and increasing it aloft. This investigation shows that this alteration remains well beyond two times the streamwise dimension of the wind farm and, at the surface, is greatest for a staggered layout. Similar to that for TKE, the aligned array configuration allows for a faster recovery of the decrease in relative humidity adjacent to the ground. Along with a faster recovery, the decrease in relative humidity does not extend as far vertically for the aligned configuration but, for both arrangements, extends well above the hub height. While the area of increase in relative humidity aloft rises vertically with downstream distance, the area of greatest increase extends further in the downstream and vertical directions for the aligned array. It was also shown that the humidity distribution in the vertical direction (i.e., the decrease underneath the hub and increase above the hub) is mainly the effect of the vertical turbulent humidity flux, which is positive on one lateral side of the rotor and negative on the other; a consequence of this is a convection of the humidity from and toward the ground, respectively.
Finally, we advise that any changes of concern with respect to humidity within a wind farm must be considered for long distances downstream of the wind farm as well, although more investigations (either numerically or experimentally) in other different conditions are warranted in this respect.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider two dependent Brownian motions with (possibly) different drift, and apply a result by le Gall on cone points of two dimensional Brownian motion to show that with probability one, there will not be a time that is a local maximum for both processes.'
address: |
Delft University of Technology\
DIAM\
Mekelweg 4\
2628 CD Delft\
The Netherlands\
author:
-
title: Local maxima of two dependent Brownian Motions never coincide
---
Introduction
============
In this short note we will consider the following problem: suppose $B_1(t)$ and $B_2(t)$ are independent two-sided standard Brownian motions. Define the processes $$X_1(t) = \phi_1(t) + \sigma B_1(t)\ \ {\rm and}\ \ X_2(t) = \phi_2(t) + \rho_1B_1(t) + \rho_2B_2(t).$$ Here, the drift functions $\phi_1$ and $\phi_2$ are assumed to have an $L^2$ derivative. Furthermore, the constants $\sigma$ and $\rho_2$ are assumed to be non-zero. The processes $X_1$ and $X_2$ are dependent Brownian motions with drift (in fact independent if $\rho_1=0$). We will prove the following statement:
\[thm\] Using the notations introduced above, it holds that with probability one, there does not exist $t\in {{\mathbb R}}$ such that $t$ is a local maximum for $X_1$ and for $X_2$.
An application of this result can be found in [@Lopu] by H.P. Lopuhaä and C. Durot. There the limiting distribution is calculated for a multiple monotone regression testing problem $$H_0:f_1=f_2=\cdots=f_J
\quad\text{ against }\quad
H_1: f_i\ne f_j\text{ for some }i\ne j$$ where all $f_j$’s are decreasing. Think of the $f_j$’s as densities, regression functions or failure rates. Consider the test statistic based on comparing the isotonic estimators $\hat{f}_j$ to the pooled isotonic estimator $\hat{f}_0$ (if $H_0$ is true, all data are generated by the same $f$). Clearly, $\hat{f}_0$ is dependent of each $\hat{f}_j$. When calculating the asymptotic distribution of this test statistic, an important role is played by random variables $V_j$, which are locations of maxima of independent Brownian motions $W_j$ minus a parabola for $1\leq j\leq J$. However, the corresponding “pooled” variable $V_0$ is a similar location of the maximum for a Brownian motion $W_0$ minus a parabola, where $W_0$ is a weighted average of the $W_j$’s. In their analysis, Lopuhaä and Durot need to prove that for ${{\varepsilon }}\to 0$, $${\mathbb P}(|V_0-V_j|\leq {{\varepsilon }}) = o(1).$$ This follows directly from Theorem \[thm\] (see page 28-29 in [@Lopu]).
It seems natural to try and prove Theorem \[thm\] using path properties of one dimensional Brownian motion near a local maximum, of which many are known in the literature. However, it turned out that the most elegant way to prove Theorem \[thm\] is to relate a coinciding local maximum to a path property of two dimensional standard Brownian motion, and apply a result by le Gall.
Proof of main result
====================
We first restrict our time parameter $t$ to the open interval $(-T,T)$. Clearly, if we can prove for all $T>0$ that no simultaneous local maximum can exist in the interval $(-T,T)$, then the theorem follows. On the interval $(-T,T)$, define $\tilde{B}_1(t) = X_1(t)/\sigma$ and $$\tilde{B}_2(t) = \frac{\phi_2(t)}{\rho_2} - \frac{\rho_1\phi_1(t)}{\sigma\rho_2} + B_2(t).$$ Using the Cameron-Martin theorem it is clear that on the time interval $(-T,T)$, the law of $(\tilde{B}_1,\tilde{B}_2)$ is absolutely continuous with respect to the law of $(B_1,B_2)$. Also, $$(X_1(t),X_2(t)) = (\sigma \tilde{B}_1(t), \rho_1\tilde{B}_1(t) + \rho_2\tilde{B}_2(t)).$$ This means that the theorem follows if we can prove that with probability one, the processes $\sigma B_1(t)$ and $\rho_1B_1(t) + \rho_2B_2(t)$ do not have a simultaneous local maximum. Suppose $s\in {{\mathbb R}}$ is such a simultaneous maximum. Then there exists $\eta>0$ such that for all $t\in (s-\eta,s+\eta)$ we would have $$\sigma B_1(t)\leq \sigma B_1(s)\ \ \ \mbox{and}\ \ \ \rho_1B_1(t) + \rho_2B_2(t)\leq \rho_1B_1(s) + \rho_2B_2(s).$$ Define $p=(\sigma B_1(s), \rho_1B_1(s) + \rho_2B_2(s))$. Define $\cal C$ as the intersection of the two half-spaces: $${\cal C} = \{x\in{{\mathbb R}}^2\ :\ \sigma x_1\leq p_1\}\cap\{x\in{{\mathbb R}}^2\ :\ \rho_1x_1+\rho_2x_2\leq p_2\}.$$ Then $\cal C$ is a cone with vertex $p$ and top angle $\alpha$, depending only on $\sigma, \rho_1$ and $\rho_2$, with $\alpha<\pi$, since $\sigma$ and $\rho_2$ are non-zero. Furthermore, the two dimensional Brownian motion $(B_1(t),B_2(t))$ lies inside $\cal C$ for the time interval $(s-\eta,s+\eta)$, and touches $p$ at time $s$. This makes $p$ a two-sided cone point with angle $\alpha<\pi$ for the two dimensional Brownian motion, in the sense of [@leGall]. However, in [@leGall] p.136, it is proven that with probability one, there do not exist any two-sided cone points with angle $\alpha<\pi$.$\Box$
[10]{} Some properties of planar Brownian motion. *Ecole d’été de probabbilités de Saint-Flour XX - 1990.* *Lecture Notes in Mathematics* **1527/1992**, 111–229.
Testing equality of functions under monotonicity constraints. *Submitted*
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Assignment of critical missions to unmanned aerial vehicles (UAV) is bound to widen the grounds for adversarial intentions in the cyber domain, potentially ranging from disruption of command and control links to capture and use of airborne nodes for kinetic attacks. Ensuring the security of electronic and communications in multi-UAV systems is of paramount importance for their safe and reliable integration with military and civilian airspaces. Over the past decade, this active field of research has produced many notable studies and novel proposals for attacks and mitigation techniques in UAV networks. Yet, the generic modeling of such networks as typical MANETs and isolated systems has left various vulnerabilities out of the investigative focus of the research community. This paper aims to emphasize on some of the critical challenges in securing UAV networks against attacks targeting vulnerabilities specific to such systems and their cyber-physical aspects.'
author:
-
title: 'Cyber-Physical Attacks on UAS Networks- Challenges and Open Research Problems '
---
UAV, Cyber-Physical Security, Vulnerabilities
Introduction {#intro}
============
The 21st century is scene to a rapid revolution in our civilization’s approach to interactions. Advancement of communication technologies, combined with an unprecedentedly increasing trust and interest in autonomy, are pushing mankind through an evolutionary jump towards delegation of challenging tasks to non-human agents. From mars rovers to search and rescue robots, we have witnessed this trend of overcoming the limitations inherent to us, through replacement of personnel with cyber-physical systems capable of performing tasks that are risky, repetitive, physically difficult or simply economically infeasible for human actors.
Unmanned Aerial Vehicles, or UAVs, are notable examples of this revolution. Since the early 2000s, military and intelligence theaters have seen an explosive growth in the deployment of tactical UAVs for surveillance, transport and combat operations. In the meantime, civilian use of UAVs has gained traction as the manufacturing and operations costs of small and mid-sized UAVs are undergoing a steady decline. The cheaper cost of such UAVs has also led to a growing interest in collaborative deployment of multiple UAVs to perform specific tasks, such as monitoring the conditions of farms and patrolling national borders. Yet, there are a multitude of challenges associated with this vision, solving which are crucial for safe and reliable employment of such systems in civilian and military scenarios. One such challenge is ensuring the security of systems that comprise UAVs, as their remote operational conditions leave the burden of command and control reliant on the onboard cyber-physical components. The body of literature on this issue has seen an accelerated growth in recent years [@kim2012cyber], which is partly due to major cyber attacks on UAVs [@javaid2012cyber]. The overwhelming number of potential vulnerabilities in UAVs indicates the need for vigorous standards and frameworks for assurance of reliability and resilience to malicious manipulations in all aspects of UAVs, from the mechanical components to the information processing units and communications systems.
In multi-UAV operations, Inter-UAV links are necessary for exchange of situational and operational commands, which are the basis of essential functions such as formation control and task optimization. As for the architecture of these UAV networks, the current consensus in the research community is biased towards decentralized and ad hoc solutions, which allow dynamic deployment of Unmanned Aerial Systems (UAS) with minimal time and financial expenditure on pre-mission preparations.
![Communication Links in a UAS Network[]{data-label="fig:FANET"}](UASNet){width="\linewidth"}
Structure of a typical UAS network is shown in Figure \[fig:FANET\]. By considering the various types of links and interfaces depicted in this figure, it can be deduced that such networks are inherently of a complex nature. Integration of multiple subsystems not only aggregates their individual vulnerabilities, but may result in new ones that are rooted in the interactions between those subsystems. Hence, UAS present the research community with a novel interdisciplinary challenge. The aim of this paper is to emphasize on some of the critical vulnerabilities specific to network and communications aspects of UAVs, and provide the research community with a list of open problems in ensuring the safety and security of this growing technology.
Uniqueness of UAS Networks
==========================
Accurate analysis of vulnerabilities in UAS networks necessitates an understanding of how an airborne network differs from traditional computer networks. Much of recent studies in this area compare UAS networks to Mobile Ad hoc Networks (MANETs) and Wireless Sensor Networks (WSN), as UAS communications and protocols may initially seem similar to those of generic distributed and mobile networks. Yet differences in mobility and mechanical degrees of freedom, as well as their operational conditions, build the grounds for separate classification of UAS networks. One such distinguishing factor is the velocity of airborne vehicles, which may range up to several hundreds of miles per hour. The high mobility of airborne platforms increases the complexity of requirements for the communications subsystem and many aspects of the UAS network. In the link layer, management of links and adaptation of access control has to be fast enough to accommodate tasks such as neighbor discovery and resource allocation in an extremely dynamic environment. Likewise, the network layer must be able to provide fast route discovery and path calculation while preserving the reliability of the information flow.
In the physical layer, not only communications, but the kinetic aspects of the UAS give rise to unique requirements. As the span of a UAS network may vary from close-by clusters to far and sparse distributions, the transmission power of UAV radios must be adjustable for efficient power consumption and sustained communications. Also, since the geography and environment of the mission may vary rapidly, channel availability in UAS links is subject to change. A potential solution is for the UAS to be equipped with Dynamic Spectrum Access (DSA) and adaptive radios to provide the required agility. Furthermore, the conventional antenna arrangement on airborne platforms is such that changes in orientation and attitude of the aircraft affect the gain of onboard radios. This problem is further intensified in unmanned aircraft, as the elimination of risk to human pilot allows longer unconventional maneuvers.
These considerations clarify the demand for a fresh vantage point for analyzing the problem of security in UAS networks. The reliability of today’s mission-critical UAVs need to be studied with models that adopt a more inclusive view of such systems and the impact of seemingly benign deficiencies on the overall vulnerability of UAVs.
Anatomy of a UAV
================
UAVs are cyber-physical systems, meaning that their operations are reliant on the interaction between physical and computational elements of the system. Consequently, security of a UAV is dependent not only on the computation and communications elements and protocols, but also on the physical components of the system [@banerjee2012ensuring]. This heavy entanglement of traditionally independent components requires a thorough framework for analysis of security issues in UAVs to be inclusive of the entire airframe. One obstacle in developing such a framework is the variety of UAV architectures and capabilities which makes the design of a generic model difficult. Yet, the similarity of fundamental requirements of such systems allows for generation of a high level system model for conventional types of UAVs. Figure \[fig:anatomy\] depicts a breakdown of components in a conventional UAV. Most UAVs contain multiple communication antennas, including air to ground (ATG), air to air (ATA), satellite data link and navigation antennas, along with a set of sensors. The positioning and navigation of a UAV is typically consisted of a Global Navigation Satellite System (GNSS) receiver for accurate positioning, and an Inertial Measurement Unit (IMU) for relative positioning based on readings from kinetic sensors. This subsystem can be further extended to include air traffic monitors such as ADS-B and collision avoidance systems.
Inside the fuselage, one or more processors supervise the operation and navigation of the UAV, using the output of various radios and sensors for adjustment of electronic and mechanical parameters. This process is performed by adaptive control mechanisms, many of which are dependent on feedback loops. Each of the elements mentioned in this section may become the subject of malicious exploitation, leading the UAV into undesirable states and critical malfunctions.
![Sensing and Communication Components of a UAV[]{data-label="fig:anatomy"}](Anatomy){width="\linewidth"}
Overview of Potential Attacks {#related}
=============================
Table \[tab:table1\] lists some of the uninvestigated attacks on UAS networks, categorized according to both network functionalities and cyber-physical factors. The table emphasizes on the criticality of the security problem, as the potential for vulnerability exists in every major component, ranging from the outer fuselage and antennas to network layers and application stack. This section provides an overview on the attacks listed in Table \[tab:table1\], and presents preliminary ideas on potential mitigating approaches and areas of research.
[p[0.75in]{}|p[1.75in]{}]{} Component & Attacks\
\
Sensors & Visual Navigation Jamming and Spoofing\
\
Physical Layer & *Adaptive radios:* deceptive attacks on spectrum sensing\
\
& *Antennas:* Disruption and Deception of Direction of Arrival estimator, Beamnull-induced Jamming\
\
& *Orientation:* Self-disruption by Induction of Defensive Maneuvers\
\
Link Layer & Topology Inference, Topological Vulnerability of Formation to Adaptive Jamming, Routing attacks\
\
Network Layer & Traffic Analysis, Disruption of Convergence\
\
Air Traffic Control & ADS-B Spoofing, TCAS-Induced Collisions\
\
Fault Handling & Manipulation of fault detection\
Sensors and Navigation
----------------------
Absence of a human pilot from the airframe of UAVs puts the burden of observing the environment on the set of sensors onboard the aircraft. Whether autonomous or remotely piloted, sensors are the “eyes and ears” of the flight controller and provide the environmental measurements necessary for safe and successful completion of the mission. However, malicious exploitation of sensors in critical cyber-physical systems is widely neglected in vulnerability assessment of such systems. An attacker may manipulate or misuse sensory input or functions to trigger or transfer malware, misguide the processes dependent on such sensors, or simply disable them to cause denial of service attacks and trigger undesired fail-safe mechanisms [@subramanian2014sensory].
For navigational measurements, GNSS and IMU units are traditionally used in tandem to provide accurate positioning of the aircraft. It is well-known that GNSS signals, such as GPS, are highly susceptible to spoofing attacks. The report in [@wesson2013hacking] demonstrates that UAVs that only rely on commercial GPS receivers for positioning are vulnerable to relatively simple jamming and spoofing attacks, which may lead to crash or capture of the UAV by adversaries. Since the establishment of GPS, various countermeasures against GNSS spoofing have been proposed, ranging from exploitation of direction and polarization of the received GPS signal for attack detection to beamforming and statistical signal processing methods for elimination of spoofing signals [@jafarnia2012gps]. However, the speed and spatial freedom of UAVs render many of the basic assumptions and criteria of such techniques inapplicable. In [@humphreys2008assessing], the authors propose the cross-examination of variations in IMU and GPS readings for detection of spoofing attacks from anomalies in fused measurements. While theoretically attractive, practical deployment of this technique requires highly reliable IMUs and adaptive threshold control for an efficient performance, which are economically undesirable for the small UAVs industry. Such practical limitations in accuracy and implementation leave this detection technique ineffective to advanced spoofing attacks, demonstrating the insufficiency of current civilian GNSS technology for mission-critical applications.
Fusion of IMU and GNSS systems with other sensors, such as video camera, may lessen the possibility of spoofing. Yet, vision-based navigation is also subject to attacks, the simplest of which is blinding the camera by saturating its receptive sensors with high intensity laser beams. A more sophisticated attack may aim for deception of the visual navigation system: In smaller areas, homogenizing or periodically modifying the texture of the terrain beneath a camera-equipped UAV may cause miscalculations of movement and orientation. Investigating the effect of such attacks on the control loop of a fused positioning system may determine the feasibility of such attacks and potential mitigation techniques.
Detection of attacks on the navigation subsystem is the basis of reactive countermeasures, such as triggering of hovering or return-to-base mechanisms. However, as the following section demonstrates, fault-handling mechanisms are also potential subjects to malicious manipulation. Robustness of the sensory and navigational subsystem against spoofing attacks may be further improved by implementation of proactive mechanisms through elimination of spoofing signals, applicability of which to UAVs is yet to be investigated.
Fault Handling Mechanisms
-------------------------
Even with the stringent reliability requirements of UAVs, mechanical and electronic subsystems of UAVs remain prone to faults due to physical damage and unpredicted state transitions. Therefore, critical UAV systems must consider the possibility of faults and implement Fault Handling mechanisms to reduce the impact of such events on the system. Typical examples of fault handling mechanisms are entering a hovering pattern when temporary faults occur, return-to-base for persistent faults and self-destruction in the event of fatal faults such as capture or crash. In remotely operated systems, fault handling mechanisms may be triggered automatically once a certain fault is detected. This process adds yet another attack surface to UAS networks, as the fault detection mechanisms may be subject to manipulation [@hartmann2013vulnerability]. For instance, if a temporary disruption of communications triggers the hovering pattern of a UAV, an adversary can jam the link to bind the motion of the aircraft, thus simplify its kinetic destruction or physical capture. A more severe case is when sensory manipulation allows the induction of capture conditions on a tactical UAV, thereby triggering its auto-destruction mechanism.
Air Traffic Control (ATC) and Collision Avoidance
-------------------------------------------------
Integration of unmanned vehicles with national and international airspaces requires guarantees on safety and reliability of UAV operations. One major consideration in the safety of all airborne operations -manned and unmanned- is situation awareness and collision avoidance. Modern manned aircraft in the major civilian airspaces are equipped with secondary surveillance technologies such as Automatic Dependent Surveillance - Broadcast (ADS-B), which allow each aircraft to monitor the air traffic in their vicinity. This information, along with other available means of traffic monitoring, provide situation awareness to the Traffic advisory and Collision Avoidance System (TCAS), which monitors the risk of collision with other aircraft and generates advisories on how to prevent collisions.
With the growing interest in large-scale deployment of UAVs, implementation of similar technologies in UAS is crucial. Recent literature contain several proposals on TCAS and ATC solutions for UAVs, many of which are based on adaptation of ADS-B and commercial TCAS protocols. From a security point of view, this approach suffers from several critical vulnerabilities, rendering it unfeasible for mission-critical UAS applications. Firstly, ADS-B is an insecure protocol by design [@kim2012cyber]. Lack of authentication and the unencrypted broadcast nature of this protocol make room for relatively simple attacks, ranging from eavesdropping to manipulation of air traffic data by jamming or injection of false data. Consequently, a TCAS system relying on ADS-B can produce erroneous results and advisories, leading to unwanted changes in the flight path or in the worst scenario, collisions.
Also, TCAS is shown to be susceptible to a flaw known as “TCAS-Induced Collisions” [@tang2015causal]. Common implementations of TCAS are not equipped with prediction capabilities to foresee the longer-term effect of an advisory that they produce. In dense traffic conditions, certain scenarios may cause the TCAS to generate advisories that lead to a state where avoidance of collision is not possible. Hence, an adversary capable of manipulating the traffic data can intentionally orchestrate conditions leading to TCAS-induced collisions. Authors of [@tang2015causal] provide an example of this flaw for a 4 airplane scenario, as illustrated in Figure \[fig:tcas\]. In this scenario, UAV1 and UAV2 are initially in a collision path, hence the TCAS in each generates a collision avoidance advisory to descend and climb, respectively. At a lower altitude, the same situation holds for UAV3 and UAV4, causing UAV4 to climb, which puts UAV1 and UAV4 on a collision path. Even though that TCAS does not fail to generate new correction advisories in both UAVs, but the advisory is no longer practical as there is not enough time before the collision to implement the new path.
![Example of TCAS-Induced Collision in a 4 Airplane Scenario[]{data-label="fig:tcas"}](TCAS8){width="\linewidth"}
Physical Layer
--------------
Typical UAVs require multiple radio interfaces to retain continuous connectivity with essential links to satellite relays, ground control stations and other UAVs. This degree of complexity, along with the physical and mechanical characteristics of UAVs, widen the scope of potential vulnerabilities and enable multiple attacks that are specific to UAS networks. This section presents a discussion on some of such attacks on the physical layer of UAV nodes.
***1. Adaptive Radios:*** As the operational environment of UAS networks is highly dynamic, sustained and reliable communications necessitates the employment of radios that are capable of adjusting to changes in propagation and links conditions. Depending on the operational requirements, this adaptability may apply to any of the physical layer parameters such as transmit power, frequency, modulation, and configuration of antennas. The procedure responsible for controlling these parameters must essentially rely on environmental inputs, which can be manipulated by adversaries to result in undesirable configurations. This issue is analogous to deceptive attacks on the spectrum sensing process of cognitive radio networks, for which various mitigation techniques have been proposed based on anomaly detection and fusion of distributed measurements [@bhattacharjee2013vulnerabilities]. However, the rapid variation of conditions in a UAS network may lead to situations where determination of a baseline for anomaly detection is not practical. The same consideration also develops a necessity for rapid adjustments, which limits the acceptable amounts of redundancy and overhead. Similarly, deployment of airborne nodes in hostile environments further reduces the feasibility of relying on real-time collaboration between distributed sensors. Therefore, such countermeasures will not be sufficient for agile UAS radios and novel solutions must be tailored according to the unique requirements of airborne networks.
***2. Antennas:*** The current trend in antenna selection for UAV radios is favored towards omnidirectional antennas, defined by their relatively homogeneous reception and transmission in all directions of the horizontal or vertical planes. This feature simplifies communications in mobile nodes, as the homogeneity of gain eliminates the need for considering the direction of transmissions. On the other hand, the indiscriminate nature of omnidirectional antennas extends the attack surface for eavesdroppers and jammers, since they also do not need to tune towards the exact direction of radios to implement their attacks. A countermeasure against this class of attack is the utilization of directional antennas, which can only communicate in certain directions and are “blind” to others. Besides their higher security, other advantages of directional antennas include longer transmission ranges and spatial reuse, thus providing a higher network capacity. One downside associated with this approach is the inevitable escalation of overhead. Maintaining directional communications in highly mobile networks is a complex and costly task, as it requires real-time knowledge of other nodes’ positions, as well as employment of antennas capable of reconfiguring their beam patterns.
To overcome the disadvantages of these two approaches, a midway solution combining the simplicity of omnidirectional radios and spatial selectivity of directional antennas can be actualized in the form of beamforming antenna arrays. Such antennas are capable of detecting the Direction of Arrival (DoA) of individual signals. This measurement, along with other system parameters, are then used to electronically reconfigure the radiation pattern and directionality of the antenna array. Beamforming has been studied as a mitigation technique against jamming attacks, as it allows spatial filtering of the jammer’s signals by adjusting the antenna pattern such that a null is placed towards the direction of the jammer [@bhuniaperformance]. The accuracy and efficiency of this technique depends on correct detection of the jamming signal, as well as the resolution of beamformer’s DoA estimations. An adversary may attack the DoA estimator by shaping its jamming signals to mimic waveforms of a nearby legitimate node, thus avoiding detection or causing false detections.
Another attack scenario exploits the process of beamnulling itself. In an ad hoc UAS network, beamnulling must be implemented in a distributed fashion to allow targeted nodes to retain or regain connectivity with the network independently. Due to lack of coordination, nulls created by one node towards a jammer may also null the direction of legitimate signals. Depending on the mobility model and formation of the network, an adversary may deploy multiple mobile jammers with strategically controlled trajectories to manipulate the DoA measurements, and eventually cause the network to null more of its legitimate links than is necessary. In certain conditions, the adversary can maximize the efficiency of jamming attacks by persistently manipulating the distributed beamnulling mechanism in such a way that its solution converges towards a maximally disconnected state. Analytical studies into feasibility criteria of this attack may produce insights into possible countermeasures and mitigation techniques.
***3. Orientation:*** As depicted in figure \[fig:anatomy\], a conventional UAV employs multiple fixed antennas on different sides, each of which is dedicated to a certain application. Consider the ATG antenna which is placed on the lower side of the UAV. As discussed previously, if the UAV performs a half-roll maneuver or ascends with a steep climb angle, the ATG antenna is no longer capable of communicating with the ground antenna and therefore the ATG link is lost. This issue can be exploited for jamming in UAS Networks that employ the spatial retreat as a mitigation technique. By observing the reaction of the nodes to jamming attacks, an adversary may infer their reformation strategy, and adapt its attack such that the defensive reformation of certain nodes leads to the loss of some links due to the new orientation of antennas.
Link Layer and Formation
------------------------
Similar to generic multihop wireless networks, the topology of a UAS network is determined based on the location of UAVs relative to each other: UAVs closer than a threshold can directly communicate with each other, while those that are farther must utilize relay nodes to reach their destination. Knowledge of the topology of a network allows adversaries to optimize attacks by analyzing the structure of their target and determine the most vulnerable regions by identifying nodes whose disconnection incur the maximum loss of connectivity in the network. Even though the effect of topology on the resilience of the network is widely studied, the proposed mitigation techniques fail to provide practical solutions for UAS networks. A class of such solutions are based on a “security by obscurity” approach, suggesting the employment of covert communications between nodes to hide the topology of the network from adversaries. Besides the undesirable overhead of this approach in terms of decreased network throughput and increased processing costs, it has been shown that the topology of such networks can be estimated with a high degree of accuracy via timing analysis attacks [@Vahid]. Therefore, hiding the topology may not serve as a reliable solution in mission critical scenarios.
An alternative mitigation technique is adaptive control of the topology [@zhu2011attack]. In this approach, detection of a jamming attack triggers a reformation process during which the nodes of a UAS network change their positions to retain connectivity. A fundamental assumption of this approach is the ability of the nodes to detect and localize attacks, which may not always be practical. A promising area of further investigation is the problem of minimizing the topological vulnerability to targeted jamming attacks. Development of real-time and distributed formation control techniques that consider this optimization problem may lead to highly efficient techniques for ensuring dynamic resilience of mission-critical UAS networks.
A mitigation technique against topology inference attacks is randomization of transmission delays. It is expected that introducing randomness in forwarding delays weakens the observed correlation between connected hops, and therefore reduces the accuracy of timing analysis attacks. However, the high mobility of UAS networks and the consequent requirement for minimal latency limit the maximum amount of delay permissible in such networks. This constraint limits the randomness of the forwarding delays, which may neutralize the effect of mitigation technique. A potential alternative for delay randomization is transmission of decoy signals to perturb the adversary’s correlation analysis. This proposal may be extended by incorporating it in topology control, such that the resultant formation is optimized for decoy transmissions in a way that spatial distribution of traffic in the network appears homogeneous to an outside observer, thereby inducing an artificial correlation between all nodes in the network. To the extent of authors’ knowledge, the feasibility, overhead and optimal implementation of this approach are yet to be analytically and experimentally studied.
Network Layer
-------------
The impact of high mobility in UAS networks is greatly accentuated in the network layer. Speed and frequency of changes in the topology of a UAS network give rise to many challenges that are still active subjects of research. Yet, studies on security of routing mechanisms tend to follow the tradition of equating UAS networks with MANETs. Indeed, the unique features of unmanned airborne networks generate a set of challenges in the network layer that do not match the criteria of conventional MANETs. The highly dynamic nature of UAS networks, as well as stringent requirements on latency, necessitate novel routing mechanisms capable of calculating paths in rapidly changing topologies. A survey of the state of the art in this area is presented in [@bekmezci2013flying]. The proposed methods may be prone to potential vulnerabilities, and the demand for a detailed technical analysis and comparison of these proposals in terms of their security is yet to be fulfilled.
Similar to the link layer, the routing layer of UAS networks is also vulnerable to traffic analysis attacks, aiming to infer individual flows, as well as source-destination pairs of end-to-end connections. Various mitigation techniques against such attacks have been proposed [@kong2007identity], many of which rely on traditional approaches such as mixing and decoy transmissions. As such techniques require addition of redundancies and overhead to the UAS networks, a comprehensive feasibility analysis and optimal design of the corresponding defense strategies is vital, but not yet available to the research community.
Mobile routing in UAS networks is a surface for attacks on convergence of the network. As discussed, the topology of unmanned airborne networks is subject to manipulation by adversarial actions such as exploitation of adaptive formation control and jamming attacks. Also, many of the recently proposed routing mechanisms for airborne networks rely on global knowledge of the geographical positions of every node in the network, which may also be prone to manipulation. A sophisticated adversary may be able to design a strategic combination of topological perturbation and sensor manipulations to prevent or slow the convergence of routing in the network. Investigation of this attack in terms of feasibility, as well as potential countermeasures may prove to be valuable for efficient protection of UAS networks operating in hostile environments.
Conclusions {#conclusion}
===========
The cyber-physical nature of UAVs demand an extension to the scope of ordinary vulnerability analysis for such systems. In addition to threats in the electronic and computational components, a largely overlooked class of vulnerabilities is fostered by the interactions between the mechanical elements and the computational subsystems. Pondering on the list of critical attacks presented in this paper, an alarming conclusion can be drawn: serious threats still remain unmitigated not only in every networking component of UAS communications, but also in the interdependency of the network and other components, including sensors and physical elements of UAVs. Considering the seriousness of open issues in the cyber-physical aspects of UAVs, a successful move towards the age of mainstream unmanned aviation cannot be envisioned without remedying the void of effective solutions for such critical challenges.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Schödinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deift’s conjecture [@Deift; @Deift17] for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al [@BDGL; @DGL1; @dgsv; @Go] and Kotani [@Kot97].'
address:
- 'Department of Mathematics, University of Toronto, 40 St George St. Toronto, ON M5S 2E4, Canada'
- ' Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China'
- 'Laboratoire J.A. Dieudonné, Université Côte d’Azur, 06108 Cedex 02 Nice, France'
- 'Department of Mathematics, Nanjing University, Nanjing 210093, China'
author:
- Martin Leguil
- Jiangong You
- Zhiyan Zhao
- Qi Zhou
title: 'Asymptotics of spectral gaps of quasi-periodic Schrödinger operators'
---
Introduction and main results
=============================
We consider one-dimensional discrete Schrödinger operators on $\ell^2({{\mathbb Z}})$:$$\label{schro}
(H_{V, \alpha, \theta} u)_n= u_{n+1}+u_{n-1} + V(\theta + n\alpha) u_n,\quad \forall \ n\in{{\mathbb Z}},$$ where $\theta\in {{\mathbb T}}^d:=({{\mathbb R}}/{{\mathbb Z}})^d$ is the *phase*, $V\colon {{\mathbb T}}^d \to {{\mathbb R}}$ is the *potential*, and $\alpha\in{{\mathbb T}}^d$ is the *frequency*. It is well known that the spectrum of $H_{V, \alpha, \theta}$, denoted by $\Sigma_{V, \alpha}$, is a compact subset of ${{\mathbb R}}$, independent of $\theta$ if $(1,\alpha)$ is rationally independent. The [*integrated density of states*]{} (IDS) $N_{V,\alpha}\colon{{\mathbb R}}\to [0,1]$ of $H_{V,\alpha,\theta}$ is defined as $$N_{V, \alpha}(E):=\int_{{{\mathbb T}}} \mu_{V, \alpha,\theta}(-\infty,E] \, d\theta,$$ where $\mu_{V,\alpha,\theta}$ is the spectral measure of $H_{V,\alpha,\theta}$. Any bounded connected component of ${{\mathbb R}}\backslash\Sigma_{V, \alpha}$ is called a *spectral gap*. By the Gap-Labelling Theorem [@JM], for any spectral gap $G$, there exists a unique $k\in {{\mathbb Z}}^d$ such that $N_{V,\alpha}|_G \equiv \langle k, \alpha\rangle {\rm\ mod} \ {{\mathbb Z}}$. Thus, the gaps in the spectrum of the operator $H_{V,\alpha,\theta}$ can be labelled by integer vectors: we denote by $G_k(V)=(E_k^-,E_k^{+})$ the gap with label $k\neq 0$. When $E^-_k=E^+_k$, we say the gap is [*collapsed*]{}. We also set $\underline{E}:=\inf\Sigma_{V,\alpha}$, $\overline{E}:=\sup\Sigma_{V,\alpha}$, and we let $G_0(V):=(-\infty, \underline{E}) \cup (\overline{E},\infty)$.
Estimates on spectral gaps
--------------------------
In this paper, we will focus on gap estimates for quasi-periodic operators as in $(\ref{schro})$. Before formulating our main results, let us first comment on the importance of gap estimates. From the perspective of physics, $(\ref{schro})$ is a model for quantum Hall effect, and thus has attracted constant interest. In particular, after Von Klitzing’s discovery of quantum Hall effect [@KDP], Thouless and his coauthors [@TKNN], assuming that all gaps are open for almost Mathieu operators, gave a theoretic explanation of the quantization of the Hall conductance by Laughlin’s argument, i.e., the Hall conductance is quantized whenever the Fermi energy lies in an energy gap (Thouless was awarded the 2016 Nobel Prize partly due to this work). From the mathematical point of view, gap estimates are a core problem in the spectral theory of quasi-periodic Schrödinger operators. The question of lower bound estimates on spectral gaps is deeper than the well-known “Dry Ten Martini Problem", while upper bound estimates provide an efficient way for proving the homogeneity of the spectrum, which is a key subject in the study of inverse spectral theory. As we will see, it is also related to Deift’s conjecture [@Deift; @Deift17] on the dynamics of solutions to KdV equation with almost periodic initial data.
We start with the most important example of , namely almost Mathieu operators (AMO), which are defined as $$(H_{\lambda, \alpha, \theta} u)_n= u_{n+1}+u_{n-1} + 2\lambda \cos 2\pi (\theta + n\alpha) u_n,\quad \forall \ n\in{{\mathbb Z}},$$ with $\lambda \in {{\mathbb R}}$ and $\alpha \in {{\mathbb R}}\backslash {{\mathbb Q}}$. For simplicity, we denote by $\Sigma_{\lambda,\alpha}$ the spectrum of $H_{\lambda, \alpha, \theta}$ and by $G_k(\lambda)=(E^-_k, E^+_k)$ the gap with label $k$. Our first result is to establish exponential asymptotics for the spectral gaps of the AMO with Diophantine frequency. Recall that $\alpha \in{{\mathbb R}}^d$ is [*Diophantine*]{} if there exist $\gamma>0$ and $\tau>d-1$ such that $\alpha \in {\rm DC}_d(\gamma,\tau)$, where $$\label{dio}
{\rm DC}_d(\gamma,\tau):=\left\{x \in{{\mathbb R}}^d: \inf_{j \in {{\mathbb Z}}}\left| {\langle}n,x {\rangle}- j \right|
> \frac{\gamma}{|n|^{\tau}},\quad \forall \ n\in{{\mathbb Z}}^d\backslash\{0\} \right\}.$$ Let ${\rm DC}_d:=\bigcup_{\gamma>0,
\, \tau>d-1} {\rm DC}_d(\gamma,\tau)$. In particular, when $d=1$, we simplify the above notations as ${\rm DC}(\gamma,\tau)$ and ${\rm DC}$. Our precise result is the following:
\[thm\_bounds\_Mathieu\] For $\alpha\in{\rm DC}$, and for any $0<\xi<1$, there exist $C=C(\lambda, \alpha,\xi)>0$, $\tilde{C}=\tilde{C}(\lambda,\alpha)$, and a numerical constant $\tilde{\xi}>1$, such that for all $k \in {{\mathbb Z}}\backslash\{0\}$, $$\begin{array}{rccclll}
\tilde{C}\lambda^{\tilde{\xi}|k|} &\leq& |G_k(\lambda)| &\leq& C \lambda^{\xi |k|},&\quad&\text{if} \;\ 0<\lambda<1,\\
\tilde{C}\lambda^{-\tilde{\xi}|k|} &\leq& |G_k(\lambda)| &\leq& C \lambda^{-\xi |k|},&\quad&\text{if} \;\ 1<\lambda<\infty,
\end{array}$$ where $|G_k(\lambda)|$ denotes the length of $G_k(\lambda)$.
Let us review some recent works in connection with the question of gap estimates for quasi-periodic Schrödinger operators. The study of lower bounds dates back to a long-standing conjecture, referred to in the literature as the “Ten Martini Problem" [@Sim], i.e., whether the spectrum of the almost Mathieu operator $H_{\lambda, \alpha, \theta}$ is a Cantor set, in the case where $\lambda\neq0$ and $\alpha$ is irrational. This problem was finally solved by Avila-Jitomirskaya [@AvilaJito1]: readers are invited to consult the history and references therein. The so-called “Dry Ten Martini Problem" is a further elaboration of the “Ten Martini Problem" asking whether for any $\lambda\ne 0$ and irrational $\alpha$, all possible spectral gaps of $H_{\lambda, \alpha, \theta}$ predicted by the Gap-Labelling theorem are non-collapsed. In [@AYZ], Avila-You-Zhou solved this problem for any non-critical coupling constant $\lambda\neq 1$ (consult [@AYZ] for earlier advances on this problem). Note that the “Dry Ten Martini Problem" only concerns the openness of the spectral gaps, without asking any quantitative estimates on their size. After we claimed the result about the Dry Ten Martini Problem in the conference “Almost Periodic and Other Ergodic Problems" in 2015, Goldstein [@Go] asked us whether any quantitative lower bound on the size of the gaps could be obtained. Theorem \[thm\_bounds\_Mathieu\] gives an answer to his question.
Let us move on to upper bounds. The first result in this direction is due to Moser-Pöschel. In [@Moser-Poschel], given an analytic potential $V\colon {{\mathbb T}}^d\to {{\mathbb R}}$, $d\geq 2$, and $\varpi \in {\rm DC}_d$, they consider the continuous quasi-periodic Schrödinger operator on $L^2({{\mathbb R}})$: $$({{{\mathcal L}}}_{V,\varpi} y)(t) = -y''(t)+V(\varpi t)y(t).$$ Thanks to KAM techniques, Moser-Pöschel proved that if $V$ is small enough, then $| G_k(V)|$ is exponentially small with respect to $|k|$ provided that $|k|$ is sufficiently large and ${\langle}k,\varpi{\rangle}$ is not too close to the other ${\langle}m,\varpi{\rangle}$.[^1] Later, Amor [@HA] proved that in the same setting, the spectral gaps have sub-exponential decay for any $k\in {{\mathbb Z}}^d\backslash\{0\}$. Although Amor [@HA] presented the result for discrete Schrödinger operators, her method applies to the continuous case as well. Damanik-Goldstein [@DG] gave a stronger result: $|G_k(V)| \leq \varepsilon e^{-\frac{r_0}{2}|k|}$ if $V\in C_{r_0}^{\omega}({{\mathbb T}}^{d},{{\mathbb R}})$ (i.e., the collection of bounded analytic functions on the strip $\{z\in{{\mathbb C}}:|\Im x|< r_0\}$) and $\varepsilon:=\sup_{|\Im x|< r_0}|V(x)|$ is sufficiently small. We obtain the following upper bound:
\[sharpdecay\] Let $\alpha \in {\rm DC}_d$ and $V\in C_{r_0}^{\omega}({{\mathbb T}}^{d},{{\mathbb R}})$. For any $r\in(0,r_0)$, there exists $\varepsilon_0= \varepsilon_0(V, \alpha, r_0, r)>0$ such that if $\sup_{|\Im x|<r_0} |V(x)| < \varepsilon_0$, then for the discrete operator $H_{V,\alpha,\theta}$, we have $$|G_k(V)| \leq \varepsilon_0^{\frac23} e^{- r |k|}, \quad \forall \ k\in {{\mathbb Z}}^d\backslash\{0\}.$$
We remark that the exponential decay rate of $|G_k(V)| $ can be arbitrarily close to the initial length $r_0$ of the strip.[^2] We emphasize that our proof is based on KAM, which works for both the discrete and the continuous case (cf. Theorem \[sharpcon\] and Corollary \[cor-local\] (1)), while the proof in [@DG] is based on localization arguments, which cannot be directly employed in the discrete case. Finally, the above results are perturbative, in the sense that the smallness of $V$ depends on the Diophantine constants $\gamma$ and $\tau$. This is optimal for the multi-frequency case in view of a counterexample due to Bourgain [@B1]. However, our method does lead to non-perturbative results (i.e., the smallness of $V$ does not depend on $\gamma,\, \tau$) in the one-frequency case. One may consult our Theorem \[thm\_bounds\_Mathieu\], Corollary \[cor-local\](2) for example.
Homogeneous spectrum
--------------------
The exponential decay of the spectral gaps can be used to prove the homogeneity of the spectrum. The concept of homogeneous set was introduced by Carleson [@Car83], and is defined as follows:
\[defi\_homogeneous\] Given $\mu >0$, a closed set ${{{\mathcal S}}}\subset{{\mathbb R}}$ is called $\mu-$homogeneous if for any $0<\epsilon\leq {\rm diam}{{{\mathcal S}}}$ and any $E\in {{{\mathcal S}}}$, we have $$|\mathcal{S}\cap(E-\epsilon,E+\epsilon)| > \mu\epsilon.$$
Homogeneity of the spectrum plays an essential role in the inverse spectral theory of almost periodic potentials (as in the fundamental work of Sodin-Yuditskii [@SY95; @SY97]). Assuming finite total gap length, homogeneity of the spectrum and the condition of being reflectionless (see Subsection \[subsec\_Schrodinger\] for the precise definition), Sodin-Yuditskii [@SY95] proved that the corresponding potential is almost periodic, and Gesztesy-Yuditskii [@GY] proved that the corresponding spectral measure is purely absolutely continuous.
Let us recall recent results on the homogeneity of the spectrum. Building on the localization estimates developed in [@DG], Damanik-Goldstein-Lukic [@DGL1] proved that the spectrum is homogeneous for continuous Schrödinger operators $\mathcal{L}_{V,\varpi}$ with Diophantine $\varpi$ and sufficient small analytic $V$. For the discrete operator $H_{V,\alpha}$ in the positive Lyapunov exponent regime, Damanik-Goldstein-Schlag-Voda [@dgsv] proved that the spectrum is homogeneous for any $\alpha\in {\rm SDC}$[^3] and for some $\alpha\in {\rm DC}_d$ [@GSV]. Inspired by the above results, it is natural to expect a global description of the homogeneity of the spectrum for quasi-periodic Schrödinger operators (see Remark (2) after Theorem 1 in [@dgsv]). In this paper, we will prove the following:
\[theorem\_homo\_spec\] Let $\alpha\in {\rm SDC}$. For a (measure-theoretically) typical analytic potential $V\in C^{\omega}( {{\mathbb T}},{{\mathbb R}})$, the spectrum $\Sigma_{V,\alpha}$ is $\mu-$homogeneous for some $\mu\in(0,1)$.
Let us first explain the meaning of measure-theoretically typical. In infinite-dimensional settings, it is common to replace the notion of [*almost every*]{} by [*prevalence*]{}: we fix some probability measure $\mu$ of compact support (describing a set of admissible perturbations $w$), and declare a property to be [*measure-theoretically typical*]{} if it is satisfied for almost every perturbation $v+w$ of every starting condition $w$. In finite-dimensional vector spaces, prevalence implies full Lebesgue measure.
For any $E \in {{\mathbb R}}$, we define a *Schrödinger cocycle* $(\alpha,S_E^V)$, where $S^{V}_{E}(\theta):=
\begin{pmatrix}
E-V(\theta) & -1\\
1 & 0
\end{pmatrix}$. The energy $E \in \Sigma_{V,\alpha}$ is called [*supercritical*]{} (resp. [*subcritical*]{}), if the associate Lyapunov exponent satisfies $L(\alpha,S_E^V)>0$ (resp. $L(\alpha,S_{E}^V(\cdot+{\rm i}\epsilon))=0$ for any $|\epsilon|<\delta$, with $\delta>0$). By Avila’s global theory of one-frequency quasi-periodic Schrödinger operators [@Aglobal], for a (measure-theoretically) typical analytic potential $V\in C^{\omega}( {{\mathbb T}},{{\mathbb R}})$, any $E \in \Sigma_{V,\alpha}$ is either subcritical or supercritical. More precisely, Avila [@Aglobal] proved that for a (measure-theoretically) typical $V\in C^\omega({{\mathbb T}},{{\mathbb R}})$, there exist some integer $n\geq 1$ and a collection of points $a_1<b_1<\dots<a_n<b_n$ in the spectrum $\Sigma_{V,\alpha}$ such that $\Sigma_{V,\alpha}\subset \bigcup_{j=1}^n[a_j, b_j]$, where energies alternate between supercritical and subcritical along the sequence $(\Sigma_{V,\alpha}\cap [a_j, b_j])_j$. We denote by $I_i:=[a_j, b_j]$ the intervals such that the energies in $\Sigma_{V,\alpha}\cap [a_j, b_j]$ are subcritical, and let $\Sigma^{\mathrm{sub}}_{V,\alpha}:=\bigcup_{i}(\Sigma_{V,\alpha} \cap I_i)$ be the set of subcritical energies. Since Theorem \[theorem\_homo\_spec\] in the supercritical regime has been proved in [@dgsv], we only need to prove the result for energies $E$ in the subcritical part of the spectrum. Let $p_n/q_n$ be the best approximants of $\alpha$ and $\beta(\alpha):=\limsup\limits_{n\rightarrow\infty}\frac{\ln q_{n+1}}{q_n}$. Our precise result is the following:
\[theo calibration gaps bands\] Let $\alpha\in{{\mathbb R}}\backslash {{\mathbb Q}}$ satisfy $\beta(\alpha)=0$. For typical potentials $V\in C^{\omega}({{\mathbb T}},{{\mathbb R}})$, the following assertions hold.
1. There exist constants $C, \vartheta>0$ depending on $V,\alpha$, such that $$|G_k(V)| \leq C e^{-\vartheta |k|}, \quad \forall \ k\in {{\mathbb Z}}\backslash \{0\} \ with \ \overline{G_k(V)} \cap \Sigma_{V,\alpha}^{\rm sub}\neq \emptyset.$$
2. For any $\tilde{\epsilon}>0$, there exists $D=D(V, \alpha,\tilde{\epsilon})>0$ such that $${\rm dist}(G_{k}(V),G_{k'}(V) ) \geq D e^{-\tilde{\epsilon} |k'-k|},$$ if $k\neq k'\in {{\mathbb Z}}$ satisfy $\overline{G_k(V)}\cap I_i \neq \emptyset$ and $\overline{G_{k'}(V)}\cap I_i \neq \emptyset$ for some $ i$.
3. There exists $\mu_0\in (0,1)$ such that $$|\Sigma_{V,\alpha}\cap(E-\epsilon,E+\epsilon)| > \mu_0 \epsilon,\quad \forall \ E\in \Sigma_{V,\alpha}^{\mathrm{sub}}, \quad \forall \ 0<\epsilon\leq {\rm diam}\Sigma_{V, \alpha}.$$
Theorem \[theo calibration gaps bands\] answers an open question raised by Damanik-Goldstein-Schlag-Voda [@dgsv] (Problem 1 of [@dgsv], see also Question 3.1 of [@DGL1]), i.e., whether the spectrum $\Sigma_{\lambda V,\alpha}$ is homogeneous, assuming that $L(\alpha,S_E^{\lambda V})$ vanishes identically on $\Sigma_{\lambda V,\alpha}$ for $0<|\lambda|<\lambda_0$. Actually, Theorem \[theo calibration gaps bands\] gives an even more precise description of the structure of the spectrum.
Let us make a short comment on Damanik-Goldstein-Schlag-Voda’s open problem (Problem 1 of [@dgsv]). Under the assumption that $L(\alpha,S_E^{\lambda V})$ vanishes on the spectrum for $0<|\lambda |<\lambda_0$, they initially asked whether one could find a complete set of Bloch-Floquet eigenfunctions for $0<|\lambda|<\lambda_0$. In fact, this point is an easy consequence of Avila’s Almost Reducibility Conjecture (subcriticality implies almost reducibility)[^4] [@A2; @Aglobal]: one could follow for instance Theorem 4.2 of [@AYZ1] to give a proof. What they really asked was whether the spectrum is homogenous; we refer the reader to Problem 1 of [@dgsv] for more explanations (one may also consult Question 3.1 of [@DGL1]).
For the most important example of AMO, Damanik-Goldstein-Lukic (Question 3.2 of [@DGL1]) asked for which values of $\lambda$ the spectrum of AMO is homogeneous. Back in 1997, Kotani [@Kot97] already asked a similar question: whether the spectrum is homogeneous under the conditions $\lim\limits_{n\rightarrow \infty}q_n^2/q_{n+1}=0$ and $0<\lambda<1$. In this paper we answer their questions as follows:
\[homo-amo\] Assume that $\beta(\alpha)=0$ and $\lambda\neq 1$. The following assertions hold:
1. For any given $r\in (0, \frac{1}{12}|\ln\lambda|)$, there exists $C=C(\lambda,\alpha,r)>0$ such that $$|G_k(\lambda)| \leq C e^{-r |k|}, \quad \forall \ k\in {{\mathbb Z}}\backslash \{0\}.$$
2. For any $\tilde{\epsilon}>0$, there exists a constant $D>0$ depending on $\tilde{\epsilon},\lambda,\alpha$, such that $$\begin{array}{rclll}
\mathrm{dist}(G_k(\lambda),G_{k'}(\lambda)) &\geq& D e^{-\tilde{\epsilon} |k'-k|}, &\ & \forall \ k\neq k'\in {{\mathbb Z}}\backslash\{0\}, \\
|E_k^- - \underline{E}|,\; |E_k^ + - \overline{E}| &\geq& D e^{-\tilde{\epsilon} |k|},&\ & \forall \ k\in {{\mathbb Z}}\backslash\{0\}.
\end{array}$$
3. $\Sigma_{\lambda,\alpha}$ is $\mu-$homogeneous for some $\mu\in (0,1)$.
If $\lambda=1$, one knows that $|\Sigma_{\lambda,\alpha}|=0$ for every $\alpha\in {{\mathbb R}}\backslash {{\mathbb Q}}$ [@AK06; @L], hence the spectrum is not homogeneous. Prior to us, Damanik-Goldstein-Schlag-Voda [@dgsv] proved that if $\alpha \in \mathrm{SDC}$ and $\lambda\neq 1$, then $\Sigma_{\lambda,\alpha}$ is homogeneous. Compared to their result, not only we weaken the condition $\alpha \in \mathrm{SDC}$ to $\beta(\alpha)=0$, but more importantly, we establish a calibration between the gaps and the bands of the operator (see (1) and (2) in the above statements). Indeed, as pointed out by Damanik-Goldstein-Schlag-Voda [@dgsv]: “This feature was not known for the almost Mathieu operator even in the regime of small coupling". In their work, they established the following weaker estimate: there exists $N_0(\alpha,\lambda)\geq 0$ such that if $N\geq N_0$ and if $G_{k}(\lambda),G_{k'}(\lambda)$ are two gaps with $|G_{k}(\lambda)|,|G_{k'}(\lambda)| > e^{-N^{1-}}$, then $\mathrm{dist}(G_{k}(\lambda),G_{k'}(\lambda))>e^{-(\log N)^{C_0}}$ for some constant $C_0=C_0(\lambda,\alpha)>0$.
The arithmetic property $\beta(\alpha)=0$ is essential to the homogeneity of the spectrum. After this work, Avila-Last-Shamis-Zhou [@ALSZ] proved that if $\beta>0$ and $e^{-\beta/2}<\lambda<e^{\beta/2}$, then $\Sigma_{\lambda,\alpha}$ is not homogeneous.
Deift’s conjecture
------------------
As an application of homogeneity of the spectrum, we can prove the discrete version of Deift’s conjecture for some almost periodic initial datum (not neccessarily small). Recall that Deift’s conjecture (Problem 1 of [@Deift; @Deift17]) asks whether for almost periodic initial datum, the solutions to the KdV equation are almost periodic in the time variable.
The calibration estimates between the gaps and the bands of Schrödinger operators (similar to items $(1)-(3)$ in Theorem \[theo calibration gaps bands\]) played an important role in the proof of Deift’s conjecture for small analytic quasi-periodic data [@BDGL; @DG2016; @DGL2]. Let us make a short review of the recent developments on this important conjecture. Tsugawa [@Ts] proved local existence and uniqueness of solutions to the KdV equation when the frequency is Diophantine and the Fourier coefficients of the potential decay at a sufficiently fast polynomial rate. Damanik-Goldstein [@DG2016] then proved global existence and uniqueness for a Diophantine frequency and small quasi-periodic analytic initial datum. Recently, Binder-Damanik-Goldstein-Lukic [@BDGL] showed that in the same setting, the solution is in fact almost periodic in time, thus proving Deift’s conjecture in this special case. In this paper, we consider the discrete version of Deift’s conjecture, namely that for almost periodic initial data, the Toda flow is almost periodic in the time variable.
The Toda flow is defined to be any solution of the Toda lattice equation $$\label{Toda}
\left\{ \begin{array}{rcl}
\displaystyle a'_n(t)&=&a_n(t)\left(b_{n+1}(t)-b_n(t)\right),\\[2mm]
\displaystyle b'_n(t)&=&2(a^2_{n}(t)-a^2_{n-1}(t)),
\end{array} \right. \quad n\in{{\mathbb Z}}.$$ In view of Theorem 12.6 in [@Teschl], with initial condition $(a(0),b(0))\in \ell^{\infty}({{\mathbb Z}})\times\ell^{\infty}({{\mathbb Z}})$, there is a unique solution $(a,b)\in{{{\mathcal C}}}^{\infty}({{\mathbb R}}, \ell^{\infty}({{\mathbb Z}})\times\ell^{\infty}({{\mathbb Z}}))$ to . If we identify $(a(t), b(t))$ with a doubly infinite Jacobi matrix $J(t)$, i.e., $$\label{Jacobi}
(J(t) u)_n=a_{n-1}(t)\, u_{n-1}+b_n(t) \,u_n+ a_{n}(t) \, u_{n+1},$$ then (\[Toda\]) can be expressed equivalently as a Lax pair: $$\label{Toda_Lax}
\frac{d}{dt} J(t)=P(t)J(t)-J(t)P(t)$$ where $P(t)$ is an operator defined as $$( P(t) u)_n:=-a_{n-1}(t)\, u_{n-1}+a_n(t)\, u_{n+1}.$$
Now, we take the almost periodic initial condition $(a_n(0), b_n(0))=(1,V(\theta+n\alpha))$, $n\in{{\mathbb Z}}$, with $V\in C^{\omega}({{\mathbb T}},{{\mathbb R}})$ and $\alpha\in {{\mathbb R}}\backslash {{\mathbb Q}}$, i.e., $J(0)=H_{V,\alpha,\theta}$, and consider the almost periodicity of the solution $(a(t), b(t))$. In fact, Binder-Damanik-Goldstein-Lukic [@BDGL] asked whether one could generalize their result to Avila’s subcritical regime: in particular, for the most important example of almost Mathieu operators, whether the result holds for $0<\lambda<1$. For partial advance on this problem, one can consult [@BDLV]. In this paper, we give an affirmative answer to their question as follows:
\[thm\_deift\_conjecture\] Let $\alpha \in {{\mathbb R}}\backslash{{\mathbb Q}}$ with $\beta(\alpha)=0$. Given a potential $V\in C^{\omega}({{\mathbb T}}, {{\mathbb R}})$ which is subcritical[^5], we consider the Toda flow (\[Toda\]) with initial condition $(a_n,b_n)(0)=(1,V(\theta+n\alpha))$. We then have:
1. For any $\theta\in{{\mathbb T}}$, (\[Toda\]) admits a unique solution $(a(t), b(t))$ defined for all $t \in {{\mathbb R}}$.
2. For every $t$, the Jacobi matrix $J(t)$ given by (\[Jacobi\]) is almost periodic with constant spectrum $\Sigma_{V,\alpha}$.
3. The solution $(a(t),b(t))$ is almost periodic in $t$ in the following sense: there exists a continuous map ${{{\mathcal M}}}\colon{{\mathbb T}}^{{{\mathbb Z}}} \to \ell^{\infty}({{\mathbb Z}})\times\ell^{\infty}({{\mathbb Z}})$, a point $\varphi\in {{\mathbb T}}^{{{\mathbb Z}}}$ and a direction $\varpi\in{{\mathbb R}}^{{{\mathbb Z}}}$, such that $(a(t),b(t))={{{\mathcal M}}}(\varphi+\varpi t)$.
In particular, the above conclusion holds for $V=2\lambda\cos2\pi(\cdot)$ with $0<\lambda < 1$.
Ideas of the proofs
-------------------
While we answer a series of open problems posed in [@BDGL; @DGL1; @dgsv; @Go], we used a totally different approach compared to these papers. Our approach is from the perspective of dynamical systems, and is based on quantitative almost reducibility. The philosophy is that nice quantitative almost reducibility should induce nice spectral applications. This approach has been proved to be very fruitful [@A2; @AvilaJito; @AYZ1; @AYZ].
As for the upper and lower bounds on the size of spectral gaps, we need to analyze the behavior of Schrödinger cocycles at the edge points of the spectral gaps. At the edge points, the cocycles are reducible to constant parabolic cocycles. The crucial points for us are the exponential decay of the off-diagonal element of the parabolic matrix and the control of the growth of the conjugacy with respect to the label $k$. Furthermore, in order to obtain uniformity of the decay rate with respect to the label $k$, we need some strong almost reducibility result, namely that the cocycle is almost reducible in a fixed band.
Now, we distinguish between two cases in the proof. If the frequency is Diophantine, we will develop a new KAM scheme to prove the almost reducibility with nice estimates (which works for multifrequencies, and for both continuous and discrete cocycles). Moreover, in order to get a sharp decay of the spectral gaps (Theorem \[sharpdecay\]), we prove almost reducibility of the cocycle in a fixed band, arbitrarily close to the initial band. We remark that although Chavaudret [@Chavaudret] developed some kind of strong almost reducibility result, the estimates there are not sufficient to yield good spectral applications. On the other hand, if the frequency $\alpha$ satisfies $\beta(\alpha)=0$, we need the almost localization argument (via Aubry duality) given by Avila [@A1] (initially developed by Avila-Jitomirskaya [@AvilaJito]): as we can see from the proof, Corona estimates are the key ingredient for these nice almost reducibility estimates.
However, each of these two approaches leads to local results. In order to deal with the global regime, we need Avila’s global theory of analytic ${\rm SL}(2,{{\mathbb R}})-$cocycles [@Aglobal], especially his proof of the Almost Reducibility Conjecture [@A2; @Aglobal]. Moreover, in order to have a uniform control on the conjugacies with respect to $E\in \Sigma_{V,\alpha}$ (which ultimately yields uniform decay rate with respect to the label $k$), we shall perform some compactness argument. Here, the key point still follows from Avila’s global theory, namely openness of the almost reducibility property, and compactness of the subcritical spectrum. We should also point out that for AMO, the strategy is different, since the proof is based on Avila-Jitomirskaya’s almost localization technique [@AvilaJito] instead of Avila’s Almost Reducibility Conjecture. In particular, it is the main reason why we can get sharp decay of the spectral gaps for noncritical AMO (Theorem \[thm\_bounds\_Mathieu\]). Avila’s Almost Reducibility Conjecture does not allow us to get this sharp result since the analytic strip has to be shrinked greatly in his proof.
In Section \[Sec\_bounds\], we will prove a criterion (Theorem \[thm\_upperbound\]) to get quantitative upper and lower bounds on the size of the gaps, building on quantitative reducibility results which even work for Liouvillean frequencies. Although the method developed by Moser-Pöschel [@Moser-Poschel] and Amor [@HA] can be used to obtain some decay of the upper bounds for small potentials, yet, when dealing with large potentials, their approach does not work since their estimates need explicit dependence on the parameters. In fact, when reducing the global potential to local regimes by Avila’s global theory, the explicit dependence of the parameters is lost. However, our method is purely dynamical, which means that we only need the information for the fixed cocycle, and it is the main reason why we can deal with all subcritical regimes. We also emphasize that our estimates on the lower bounds of the gaps of AMO crucially depend on a key proposition of [@AYZ] which was initially used by the authors to prove the non-critical “Dry Ten Martini Problem" for Liouvillean frequencies.
Homogeneity of the spectrum in the subcritical regime is derived from the upper bounds on the size of spectral gaps, together with Hölder continuity of the IDS. Theorem \[theorem\_homo\_spec\] is proved by combining this with previous work of Damanik-Goldstein-Schlag-Voda [@dgsv] and Avila’s global theory of one-frequency Schrödinger operators [@Aglobal]. As a consequence of homogeneity (Theorem \[theo calibration gaps bands\]) and purely absolutely continuous spectrum of subcritical Schrödinger operators [@A2], we are then able to prove the discrete version of Deift’s conjecture for such initial data, building on an previous result of Vinnikov-Yuditskii [@VY].
Preliminaries
=============
For a function $f$ defined on a strip $\{|\Im z|<h\}$, we define $|f|_{h}:=\sup_{|\Im z|<h} |f(z)|$. Analogously, for $f$ defined on ${{\mathbb T}}$, we set $|f|_{{{\mathbb T}}}:=\sup_{x \in {{\mathbb T}}} |f(x)|$. For any $f\colon{{\mathbb T}}^d\to {{\mathbb C}}$, we let $[f]:=\int_{{{\mathbb T}}^d} f(\theta) d\theta$. When $\theta \in {{\mathbb R}}$, we also set $\|\theta\|_{{{\mathbb T}}}:=\inf_{j \in {{\mathbb Z}}} |\theta-j|$.
Continued Fraction Expansion {#sec:2.1}
----------------------------
Let $\alpha \in (0,1)\backslash {{\mathbb Q}}$, $ a_0:=0$ and $\alpha_{0}:=\alpha.$ Inductively, for $k\geq 1$, we define $$a_k:=\left\lfloor\alpha_{k-1}^{-1}\right\rfloor,\qquad \alpha_k:=\alpha_{k-1}^{-1}-a_k.$$ Let $p_0:=0$, $p_1:=1$, $q_0:=1$, $q_1:=a_1$. We define inductively $p_k:=a_k p_{k-1}+p_{k-2}$, $q_k:=a_kq_{k-1}+q_{k-2}$. Then $(q_n)_n$ is the sequence of denominators of the best rational approximations of $\alpha$, since we have $\|k\alpha\|_{{{\mathbb T}}} \geq \|q_{n-1}\alpha\|_{{{\mathbb T}}}$, $\forall \ 1
\leq k < q_n$, and $${1 \over 2q_{n+1}}\leq \|q_n \alpha \|_{{{\mathbb T}}} \leq {1 \over q_{n+1}}.$$ Let $\displaystyle \beta(\alpha):=\limsup_{n\rightarrow\infty}\frac{\ln q_{n+1}}{q_n}$. Equivalently, we have $$\label{equibeta}
\beta(\alpha)=\limsup_{k\rightarrow \infty} \frac{1}{|k|} \ln \frac{1}{ \|k\alpha\|_{{{\mathbb T}}}} .$$
Schrödinger operators {#subsec_Schrodinger}
---------------------
Given $V\in C^\omega({{\mathbb T}}^d,{{\mathbb R}})$ and $\alpha\in{{\mathbb R}}^d$, we define the Schrödinger operator $H_{V,\alpha,\theta}$ as in (\[schro\]). For any $\psi\in\ell^2({{\mathbb Z}})$, we let $\mu_{V,\alpha,\theta}^\psi$ be the spectral measure of $H_{V,\alpha,\theta}$ corresponding to $\psi$: $${\langle}(H_{V,\alpha,\theta}-E)^{-1}\psi, \psi {\rangle}= \int_{{\mathbb R}}\frac{1}{E-E'}d\mu_{V,\alpha,\theta}^\psi(E'), \quad \forall \ E\in {{\mathbb C}}\backslash\Sigma_{V,\alpha}.$$ We denote $\mu_{V,\alpha,\theta}:=\mu_{V,\alpha,\theta}^{e_{-1}}+\mu_{V,\alpha,\theta}^{e_{0}}$, where $\{e_n\}_{n\in{{\mathbb Z}}}$ is the canonical basis of $\ell^2({{\mathbb Z}})$. More generally, we consider the self-adjoint Jacobi matrices $J$: $$(Ju)_n=a_{n-1}u_{n-1}+b_n u_n+a_n u_{n+1},\quad n\in{{\mathbb Z}}.$$ Let $\Sigma\subset{{\mathbb R}}$ be the spectrum of $J$. Given any $z\not\in\Sigma$, the Green’s function of $J$ is the integral kernel of $(J-z)^{-1}$: $$G_J(m, n; z) := {\langle}e_n, (J -z)^{-1} e_m {\rangle}.$$
A Jacobi operator $J$ is said to be [*reflectionless*]{} on $\Sigma$ if $\Re(G_J(0,0; E + i0)) = 0$ for Lebesgue$-$a.e. $E \in \Sigma$.
Given any $z\in{{\mathbb H}}:=\{z\in{{\mathbb C}}: \Im z>0\}$, the difference equation $Ju=zu$ has two solutions $u^{\pm}$ (defined up to normalization) with $u_0^{\pm}\neq 0$, which are in $\ell^2({{\mathbb Z}}_{\pm})$ respectively. Let $m_J^{\pm}:=\mp\frac{u^{\pm}_{\pm1}}{a_0 u^{\pm}_{0}}$. Then $m_J^+$ and $m_J^-$ are Herglotz functions, i.e., they map ${{\mathbb H}}$ holomorphically into itself. For almost every $E\in{{\mathbb R}}$, the non-tangential limits $\lim\limits_{\epsilon\rightarrow0^+}m_J^{\pm}(E+{\rm i}\epsilon)$ exist. Note that we have $$\label{green_m+-}
G_J(0,0; z)=\frac{1}{a_0^2(m_J^+(z)+m_J^-(z))},\quad z\in{{\mathbb H}}.$$
Quasiperiodic cocycles {#subslyap}
----------------------
Given $A \in C^\omega({{\mathbb T}}^d,{\rm SL}(2,{{\mathbb C}}))$ and $\alpha\in{{\mathbb R}}^d$ rationally independent, we define the quasi-periodic *cocycle* $(\alpha,A)$: $$(\alpha,A)\colon \left\{
\begin{array}{rcl}
{{\mathbb T}}^d \times {{\mathbb C}}^2 &\to& {{\mathbb T}}^d \times {{\mathbb C}}^2\\[1mm]
(x,v) &\mapsto& (x+\alpha,A(x)\cdot v)
\end{array}
\right. .$$ The iterates of $(\alpha,A)$ are of the form $(\alpha,A)^n=(n\alpha, \mathcal{A}_n)$, where $$\mathcal{A}_n(x):=
\left\{\begin{array}{l l}
A(x+(n-1)\alpha) \cdots A(x+\alpha) A(x), & n\geq 0\\[1mm]
A^{-1}(x+n\alpha) A^{-1}(x+(n+1)\alpha) \cdots A^{-1}(x-\alpha), & n <0
\end{array}\right. .$$ The [*Lyapunov exponent*]{} is defined by $\displaystyle
L(\alpha,A):=\lim\limits_{n\to \infty} \frac{1}{n} \int_{{{\mathbb T}}^d} \ln |\mathcal{A}_n(x)| dx
$.
The cocycle $(\alpha,A)$ is [*uniformly hyperbolic*]{} if, for every $x \in {{\mathbb T}}^d$, there exists a continuous splitting ${{\mathbb C}}^2=E^s(x)\oplus E^u(x)$ such that for every $n \geq 0$, $$\begin{array}{rl}
|A_n(x) \, v| \leq C e^{-cn}|v|, & v \in E^s(x),\\[1mm]
|A_n(x)^{-1} v| \leq C e^{-cn}|v|, & v \in E^u(x+n\alpha),
\end{array}$$ for some constants $C,c>0$. This splitting is invariant by the dynamics, i.e., $$A(x) E^{*}(x)=E^{*}(x+\alpha), \quad *=``s"\;\ {\rm or} \;\ ``u", \quad \forall \ x \in {{\mathbb T}}^d.$$
Assume that $A \in C ({{\mathbb T}}^d, {\rm SL}(2, {{\mathbb R}}))$ is homotopic to the identity. It induces the projective skew-product $F_A\colon {{\mathbb T}}^d \times \mathbb{S}^1 \to {{\mathbb T}}^d \times \mathbb{S}^1$ with $$F_A(x,w):=\left(x+\a,\, \frac{A(x) \cdot w}{|A(x) \cdot w|}\right),$$ which is also homotopic to the identity. Thus we can lift $F_A$ to a map $\tF_A\colon {{\mathbb T}}^d \times {{\mathbb R}}\to {{\mathbb T}}^d \times {{\mathbb R}}$ of the form $\tF_A(x,y)=(x+\alpha,y+\psi_x(y))$, where for every $x \in {{\mathbb T}}^d$, $\psi_x$ is ${{\mathbb Z}}$-periodic. The map $\psi\colon{{\mathbb T}}^d \times {{\mathbb T}}\to {{\mathbb R}}$ is called a [*lift*]{} of $A$. Let $\mu$ be any probability measure on ${{\mathbb T}}^d \times {{\mathbb R}}$ which is invariant by $\widetilde{F}_A$, and whose projection on the first coordinate is given by Lebesgue measure. The number $$\rho(\alpha,A):=\int_{{{\mathbb T}}^d \times {{\mathbb R}}} \psi_x(y)\ d\mu(x,y) \ {\rm mod} \ {{\mathbb Z}}$$ depends neither on the lift $\psi$ nor on the measure $\mu$, and is called the *fibered rotation number* of $(\alpha,A)$ (see [@H; @JM] for more details).
Given $\theta\in{{\mathbb T}}^d$, let $
R_\theta:=
\begin{pmatrix}
\cos2 \pi\theta & -\sin2\pi\theta\\
\sin2\pi\theta & \cos2\pi\theta
\end{pmatrix}$. If $A\colon {{\mathbb T}}^d\to{\rm PSL}(2,{{\mathbb R}})$ is homotopic to $\theta \mapsto R_{\frac{{\langle}n, \theta{\rangle}}{2}}$ for some $n\in{{\mathbb Z}}^d$, then we call $n$ the [*degree*]{} of $A$ and denote it by $\deg A$. The fibered rotation number is invariant under real conjugacies which are homotopic to the identity. More generally, if $(\alpha,A_1)$ is conjugated to $(\alpha, A_2)$, i.e., $B(\cdot+\alpha)^{-1}A_1(\cdot)B(\cdot)=A_2(\cdot)$, for some $B \colon {{\mathbb T}}^d\to{\rm PSL}(2,{{\mathbb R}})$ with ${\rm deg} B=n$, then $$\label{rot-conj}
\rho(\alpha, A_1)= \rho(\alpha, A_2)+ \frac{{\langle}n,\alpha {\rangle}}2.$$ Moreover, it follows immediately from the definition of rotation number that
\[esti\_rot\_num\] If $A \colon {{\mathbb T}}^d\to{\rm SL}(2,{{\mathbb R}})$ is homotopic to the identity, then $$|\rho(\alpha, A)-\theta|< |A-R_{\theta}|_{{{\mathbb T}}^d}.$$
A typical example is given by the so-called *Schrödinger cocycles* $(\alpha,S_E^V)$, with $$S^V_{E}(\cdot):=
\begin{pmatrix}
E-V(\cdot) & -1\\
1 & 0
\end{pmatrix}, \quad E\in{{\mathbb R}}.$$ Those cocycles were introduced because in connection with the eigenvalue equation $H_{V, \alpha, \theta}u=E u$: indeed, any formal solution $u=(u_n)_{n \in {{\mathbb Z}}}$ of $H_{V, \alpha, \theta}u=E u$ satisfies $$\begin{pmatrix}
u_{n+1}\\
u_n
\end{pmatrix}
= S_E^V(\theta+n\alpha) \begin{pmatrix}
u_{n}\\
u_{n-1}
\end{pmatrix},\quad \forall \ n \in {{\mathbb Z}}.$$ The spectral properties of $H_{V,\alpha,\theta}$ and the dynamics of $(\alpha,S_E^V)$ are closely related by the well-known fact: $E\in \Sigma_{V,\alpha}$ if and only if $(\alpha,S^V_{E})$ is *not* uniformly hyperbolic.
For any fixed $E \in {{\mathbb R}}$, the map $x \mapsto S_E^V(x)$ is homotopic to the identity, hence the rotation number $\rho(\alpha,S_E^V)$ is well defined. Moreover, $\rho(\alpha,S_E^V)\in [0,\frac12]$ relates to the integrated density of states $N=N_{V,\alpha}$ as follows: $$N_{V,\alpha}(E)=1-2 \rho(\alpha,S_E^V).$$ By *Thouless formula*, we also have the following relation between the integrated density of states $N$ and the Lyapunov exponent $L$: $$L(\a, S^V_{E}) = \int \ln |E-E'| \, dN_{V,\alpha}(E').$$
Aubry duality and almost localization {#subs_duality}
-------------------------------------
Let $\theta\in {{\mathbb T}}$, $V\in C^\omega({{\mathbb T}},{{\mathbb R}})$, and denote by $(\widehat v_l)_{l \in {{\mathbb Z}}}$ the Fourier coefficients of $V$. The *dual Schrödinger operator* $\widehat H_{V,\alpha,\theta}$ is defined on $\ell^2({{\mathbb Z}})$ by: $$\left(\widehat H_{V,\alpha,\theta} u\right)_j:=\sum\limits_{l\in {{\mathbb Z}}} \widehat v_l u_{j-l} + 2 \cos 2\pi (\theta + j \alpha) u_j, \quad \forall \ j \in {{\mathbb Z}}.$$ *Aubry duality* involves an algebraic relation between the families of operators $\{H_{V,\alpha,\theta}\}_{\theta \in {{\mathbb T}}}$ and $\{\widehat H_{V,\alpha,\theta}\}_{\theta \in {{\mathbb T}}}$: given an eigenvector of $\widehat H_{V,\alpha,\theta}$ whose coefficients decay exponentially, one can construct an analytic *Bloch wave* for the dual operator $H_{V,\alpha,\theta}$. However, if one wants to obtain information for all energies $E$, one cannot expect that all the eigenfunctions decay exponentially. The weaker notion of almost localization proved to be very useful in this context.
Fix $\epsilon_0>0$ and $\theta\in {{\mathbb T}}$. An integer $k\in {{\mathbb Z}}$ is called an *$\epsilon_0-$resonance* of $\theta$ if $\|2 \theta - k\alpha\|_{{{\mathbb T}}}\leq e^{-\epsilon_0 |k|}$ and $\|2 \theta - k\alpha\|_{{{\mathbb T}}}=\min_{|l|\leq |k|} \|2 \theta - l\alpha\|_{{{\mathbb T}}}$. We denote by $\{n_l\}_l$ the set of $\epsilon_0-$resonances of $\theta$, ordered in such a way that $|n_1|\leq |n_2|\leq \dots$. We say that $\theta$ is $\epsilon_0-$*resonant* if the set $\{n_l\}_l$ is infinite.
The family $\{\widehat H_{V,\alpha,\theta}\}_{\theta\in{{\mathbb T}}}$ is said to be *almost localized* if there exist constants $C_0, C_1,\epsilon_0, \epsilon_1>0$ such that for all $\theta\in{{\mathbb T}}$, any generalized solution $u=(u_k)_{k \in {{\mathbb Z}}}$ to the eigenvalue problem $\widehat H_{V,\alpha,\theta} u = E u$ with $u_0=1$ and $|u_k| \leq 1+|k|$ satisfies $$\label{almost_localization}
|u_k| \leq C_1 e^{-\epsilon_1 |k|},\quad \forall \ C_0 |n_j| \leq |k| \leq C_0^{-1} |n_{j+1}|,$$ where $\{n_l\}_l$ is the set of $\epsilon_0-$resonances of $\theta$.
The basic fact for us is the following result of Avila and Jitomirskaya [@AvilaJito]:
[@AvilaJito]\[almostredth\] Let $\alpha\in {{\mathbb R}}\backslash {{\mathbb Q}}$ satisfy $\beta(\alpha)=0$. There exists an absolute constant $c_0>0$ such that for any given $0<r_0<1$, $C_0>1$, there exist $\epsilon_0=\epsilon_0(r_0)>0$, $\epsilon_1=\epsilon_1(r_0,C_0)\in(0,r_0)$ and $C_1=C_1(\alpha, r_0,C_0)>0$ such that the following is true: given any $V \in C^\omega({{\mathbb T}},{{\mathbb R}})$ satisfying $|V|_{r_0}\leq c_0 r_0^3$, the family $\{\widehat H_{V,\alpha,\theta}\}_{\theta\in {{\mathbb T}}}$ is almost localized with parameters $C_0,\, C_1,\, \epsilon_0,\, \epsilon_1$ as in (\[almost\_localization\]).
If we restrict ourselves to almost Mathieu operators, then we expect the decay rate of the eigenfunction to be $\ln\lambda$, which is the content of the following result.
\[thm\_almost\_almost-1\] Let $\alpha \in {{\mathbb R}}\backslash {{\mathbb Q}}$ satisfy $\beta(\alpha)=0$. If $\lambda>1$, then $\{H_{\lambda,\alpha,\theta}\}_{\theta}$ is almost localized. Moreover, for any $\delta\in (0,\ln \lambda)$, any $C_0>1$, there exists $\epsilon_0=\epsilon_0(\lambda,C_0,\delta)>0$ such that the following holds. Let $H_{\lambda,\alpha,\theta} u =E u$ for some $E \in \Sigma_{\lambda,\alpha}$, with $|u_j|\leq 1$ for all $j \in {{\mathbb Z}}$.
1. If $\theta$ is not $\epsilon_0-$resonant, then $|u_{j}|\leq e^{-(\ln \lambda -\delta)|j|}$ for $|j|$ large enough.
2. Else, let $\{n_l\}_l$ be the set of $\epsilon_0-$resonances of $\theta$. Given any $\eta>0$, $$|u_{j}|\leq e^{-(\ln \lambda -\delta)|j|},\quad \forall \ 2C_0 |n_l|+\eta|n_{l+1}| < |j| < (2C_0)^{-1} |n_{l+1}|,$$ provided that $|j|$ is large enough.
If $\alpha \in \mathrm{DC}$, then the above result is shown in [@Jito1999] for a full measure set of $\theta\in{{\mathbb T}}$. As for the case $\beta(\alpha)=0$, we could not find a reference in the literature. For completeness, we give a proof in Appendix \[Appendix\_A\].\
As a direct corollary of Theorem \[almostredth\], one can see that the dual cocycle has subexponential growth on the strip $\{|\Im z|<\frac{\epsilon_1}{2\pi}\}$, which was first realized by Avila (one may consult footnote 5 of [@A1]). We sketch the proof here for completeness.
\[subexp groowth\] Let $\alpha\in {{\mathbb R}}$ satisfy $\beta(\alpha)=0$ and $V \in C^\omega({{\mathbb T}},{{\mathbb R}})$ satisfy $|V|_{r_0}\leq c_0 r_0^3$ for the absolute constant $c_0$ in Theorem \[almostredth\]. For any $\delta>0$, there exists a constant $C_2= C_2(\alpha,\epsilon_1, \delta)>0$ such that for any $E \in \Sigma_{V,\alpha}$, the iterates of the cocycle $(\alpha,S_{E}^V)$ satisfy $$\label{subpolgrowth}
\sup_{ |\Im z|<\frac{\epsilon_1}{2\pi}}|\mathcal{A}_m (z)| \leq C_2 e^{\delta |m|},\quad \forall \ m \in \mathbb{N}.$$
We first show that for any $E \in \Sigma_{V,\alpha}$, $$\label{eq vanish lyap exp}
L(\alpha, S_{E}^V(\cdot+{\rm i} y))=0, \quad \forall \ |y| <\frac{\epsilon_1}{2\pi}.$$
As above, given any $E \in \Sigma_{V,\alpha}$, there exist $\theta=\theta(E)\in {{\mathbb R}}$ and $\widehat u$ such that $\widehat H_{V,\alpha,\theta} \widehat u= E \widehat u$, with $\widehat u_0=1$ and $|\widehat u_j|\leq 1$ for all $j \in {{\mathbb Z}}$. We claim that it is enough to show at energies $E \in \Sigma_{V,\alpha}$ such that $\theta=\theta(E)$ is not $\epsilon_0-$resonant. Indeed, by Theorem 4.2 in [@AvilaJito], there exists $c>0$ depending on $V, \alpha$ such that if $\theta$ is $\epsilon_0-$resonant, then $\rho(E)$ is $c-$resonant. Then, it follows from a standard Borel-Cantelli argument that the set of $E \in \Sigma_{V,\alpha}$ such that $\theta$ is $\epsilon_0-$resonant has zero Lebesgue measure (indeed, it has zero Hausdorff dimension, as shown by Avila [@A1]). On the other hand, by [@BJ; @JKS], we know that given any $|y|< \frac{\epsilon_1}{2\pi} $, the map $E \mapsto L(\alpha,S_{E}^V(\cdot+{\rm i} y))$ is continuous, and thus, the claim is proved.
Now, fix $E \in \Sigma_{V,\alpha}$ such that $\theta$ is not $\epsilon_0-$resonant and let $\widehat u$ be as above. By , $|\widehat u_j| \leq C_1 e^{-\epsilon_1 |j|}$ for all sufficiently large $|j|$. Therefore, the function $u\colon z \mapsto \sum_{j \in {{\mathbb Z}}} \widehat u_j e^{2 \pi {\rm i} j z}$ is well-defined on $\{|\Im z|<\frac{\epsilon_1}{2\pi}\}$, and ${{{\mathcal U}}}\colon z \mapsto
\begin{pmatrix}
e^{2 \pi {\rm i}\theta} u(z)\\
u(z-\a)
\end{pmatrix}$ satisfies $S_{E}^V(z) \, {{{\mathcal U}}}(z) = e^{2\pi {\rm i}\theta} {{{\mathcal U}}}(z+\alpha)$. Set $Z:=(\mathcal{U},\, \frac{1}{\|{{\mathcal U}}\|^2}R_{\frac14}{{\mathcal U}})$, where $R_{\frac14}$ denotes the rotation of angle $\frac{\pi}{2}$. Then $Z$ is defined on $\{|\Im z|<\frac{\epsilon_1}{2\pi}\}$ and conjugates $(\alpha,S_{E}^V)$ to $(\alpha,B)$, with $B(z)=\begin{pmatrix}
e^{2 \pi \mathrm{i} \theta} & \kappa(z)\\[1mm]
0 & e^{-2 \pi \mathrm{i} \theta}
\end{pmatrix}$ for some continuous function $\kappa$ on $\{|\Im z|<\frac{\epsilon_1}{2\pi}\}$. Thus, $L(\alpha,S_{E}^V)$ vanishes identically on the strip $\{|\Im z|<\frac{\epsilon_1}{2\pi}\}$. Combining this fact with the previous claim, this concludes the proof of .
Let $(m\alpha, \mathcal{A}_m^y)$ be the $m^{\mathrm{th}}$ iterate of $(\alpha, S_{E}^V(\cdot+{\rm i} y))$. Following Remark 2.1 in [@AvilaJito] (see also Lemma 3.1 of [@AYZ1]), by subadditivity and compactness of $\Sigma$, we see that for any $\delta>0$, there exists $C=C(\alpha,\epsilon_1, \delta)>0$ such that for any $m \geq 0$, $$\sup_{E\in \Sigma_{V,\alpha}}\sup_{ |y|<\frac{\epsilon_1}{2\pi}}|\mathcal{A}_m^y|_{{\mathbb T}}\leq C+ m\left( \sup_{E\in \Sigma_{V,\alpha}}\sup_{ |y|<\frac{\epsilon_1}{2\pi}}L(\alpha, S_{E}^V(\cdot+{\rm i} y))+\delta\right)=C+m \delta,$$ which concludes the proof of Lemma \[subexp groowth\].
Global theory of one-frequency Schrödinger operators {#ARCGlobal}
----------------------------------------------------
Let us make a short review of Avila’s global theory of one-frequency ${\rm SL}(2,{{\mathbb R}})-$cocycles [@Aglobal]. Suppose that $A\in C^\omega({{\mathbb T}},{\rm SL}(2,{{\mathbb R}}))$ admits a holomorphic extension to $\{|\Im z|<h\}$. Then for $|\epsilon|<h$, we define $A_\epsilon \in
C^\omega({{\mathbb T}},{\rm SL}(2,{{\mathbb C}}))$ by $A_\epsilon(\cdot)=A(\cdot+i \epsilon)$. The cocycles which are not uniformly hyperbolic are classified into three classes: subcritical, critical, and supercritical. In particular, $(\alpha, A)$ is said to be subcritical if there exists $h>0$ such that $L(\alpha,A_{\epsilon})=0$ for $|\epsilon|<h.$
One main result of Avila’s global theory is the following:
\[global-red\] Given any $\alpha\in{{\mathbb R}}\backslash{{\mathbb Q}}$, for a (measure-theoretically) typical $V\in C^\omega({{\mathbb T}},{{\mathbb R}})$, there exist $n\geq 1$ and a collection of points $a_1<b_1<\dots<a_n<b_n$ in the spectrum $\Sigma_{V,\alpha}$ such that $\Sigma_{V,\alpha}\subset \bigcup_{i=1}^n[a_i, b_i]$, and energies alternate between supercritical and subcritical along the sequence $\{\Sigma_{V,\alpha}\cap [a_i, b_i]\}_i$. Moreover, for any $i=1,\, \dots,\, n$, the set $\Sigma_{V,\alpha}\cap [a_i, b_i]$ is compact, and it depends continuously (in the Hausdorff topology) on $(\alpha,V).$
A cornerstone in Avila’s global theory is the Almost Reducibility Conjecture(ARC), which says that $(\alpha,A)$ is almost reducible if it is subcritical. Recall that the cocycle $(\alpha, A)$ is said to be [*reducible*]{} if it can be conjugated to a constant cocycle, i.e., there exist $Z\in C^\omega({{\mathbb T}}^d,{\rm PSL}(2,{{\mathbb R}}))$ and $B\in {\rm SL}(2,{{\mathbb R}})$ such that $$Z(\cdot+\alpha)^{-1}A(\cdot)Z(\cdot)=B.$$ Moreover, $(\alpha, A)$ is (analytically) [*almost reducible*]{} if the closure of its analytic conjugates contains a constant. The complete solution of ARC was recently given by Avila in [@A3; @A2]. In the case where $\beta(\alpha)=0$, it is the following:
\[arc-conjecture\] Given $\alpha\in{{\mathbb R}}\backslash{{\mathbb Q}}$ with $\beta(\alpha)=0$, and $A\in C^\omega({{\mathbb T}},{\rm SL}(2,{{\mathbb R}}))$, if $(\alpha,A)$ is subcritical, then it is almost reducible.
Quantitative KAM scheme {#sec_Quantitative}
=======================
In this section, we present a new KAM scheme and give a quantitative almost reducibility result for the local quasi-periodic linear system $$\label{linear_system}
\left\{
\begin{array}{l}
\dot{x}=(A_0+F_0(\theta))x\\
\dot{\theta}= \varpi
\end{array}
\right. ,$$ where $A_0\in {\rm sl}(2,{{\mathbb R}})$, and $F_0(\theta)$ is a perturbation. We also abbreviate (\[linear\_system\]) as $(\varpi,A_0+F_0)$. There is a parallel result for the quasi-periodic cocycle introduced in Subsection \[subslyap\]. The reason why we chose to present the detailed proof in the continuous case is simply for comparing our Theorem \[sharpdecay\] with the result of Damanik-Goldstein [@DG].
Given any $A_1, A_2\in {{{\mathcal B}}}_{r_0}:=C^\omega_{r_0}({{\mathbb T}}^{d}, {\rm sl}(2,{{\mathbb R}}))$ and $W\in C^{\omega}( {{\mathbb T}}^d,{\rm PSL}(2,{{\mathbb R}}))$, we say that $(\varpi,A_1)$ is [*conjugated*]{} to $(\varpi,A_2)$ by $W$ if $\partial_\varpi W=A_1 W- W A_2$, where $ \partial_\varpi W:=\langle \varpi, \nabla W\rangle$. The system $(\varpi,A_1)$ is called [*reducible*]{} if it is conjugated to a constant system $(\varpi,B)$ with $B\in {\rm sl}(2,{{\mathbb R}})$. It is called [*almost reducible*]{} if the closure of its analytical conjugates contains a constant system $(\varpi,B)$.
Iteration
---------
Suppose that $A\in {\rm sl}(2,{{\mathbb R}})$ and $F\in{{{\mathcal B}}}_r$ with $|F|_{r}\leq \varepsilon$ for some $r,\varepsilon>0$. For any given $r_+\in(0,r)$, the aim of the following argument is to find $\hat W\in C^{\omega}_{r_+}({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$, $A_{+}\in{\rm sl}(2,{{\mathbb R}})$ and $F_{+}\in{{{\mathcal B}}}_{r_{+}}$ with $|F_{+}|_{r_{+}}\ll \varepsilon$ such that $(\varpi, A+F)$ is conjugated to $(\varpi, A_++F_+)$ by $\hat W(\theta)$.
\[prop\_iteration\] Let $\varpi\in{\rm DC}_d(\gamma,\tau)$. Given any $r_+\in(0,r)$, there is a constant $D_0=D_0(\gamma,\tau,d)>0$ such that if $\varepsilon$ satisfies $$\label{smallness_condition}
\varepsilon\leq D_0 \left(1+|A|^{120d(1+\frac{1}{\tau})}\right)(rr_+-r_+^2)^{800d(\tau+1)},$$ then there exist $F_{+}\in {{{\mathcal B}}}_{r_{+}}$, $\hat W\in C_{r_{+}}^{\omega}({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$ and $A_{+}\in {\rm sl}(2,{{\mathbb R}})$ such that $(\varpi, A+F )$ is conjugated to $(\varpi, A_+ +F_+ )$ by $\hat W$. Let $N:=\frac{2|\ln\varepsilon|}{r-r_+}$, and $\pm2\pi{\rm i}\xi$ be the two eigenvalues of $A$. Then we have the following:
- (Non-resonant case) Assume that $$\label{non-resonant_condition}
|2\xi-{\langle}n,\varpi {\rangle}|\geq \varepsilon^{\frac{1}{15}}, \quad \forall \ n\in{{\mathbb Z}}^d \;\ with \;\ 0<|n|\leq N.$$ In this case, we have the estimates: $$|F_{+}|_{r_{+}}\leq \varepsilon^{2},\quad |\hat W- {\rm Id}|_{r_{+}}\leq 2 \varepsilon^{\frac12}, \quad |A_+-A|\leq \varepsilon^{\frac12}.$$
- (Resonant case) If there exists $n_*\in{{\mathbb Z}}^{d}$ with $0<|n_*|\leq N$ such that $$\label{resonant_condition}
|2\xi-{\langle}n_*,\varpi{\rangle}|< \varepsilon^{\frac{1}{15}},$$ then $|F_{+}|_{r_{+}}\leq \varepsilon e^{-r_+\varepsilon^{-\frac{1}{18\tau}}}$, ${\rm deg} \hat W=n_*$, with the estimate $$|\hat W|_{r''}\leq \frac{4\sqrt{|A|}}{\sqrt{\gamma}}|n_*|^{\frac\tau2} e^{\pi r''|n_*|}, \quad \forall \ 0< r''\leq r_+,$$ and $A_+$ satisfies $|A_+|\leq \varepsilon^{\frac{1}{16}}$ with two eigenvalues $\pm 2\pi {\rm i}\xi_+$ satisfying $|\xi_+|\leq \varepsilon^{\frac{1}{16}}$. Moreover, for $M:=\frac{1}{1+{\rm i}}\left(\begin{matrix}
1 & -{\rm i}\\
1 & {\rm i}
\end{matrix}\right)$, we have $$\label{A+}
A_{+}=2\pi M^{-1}\begin{pmatrix}
{\rm i}(\xi-\frac{{\langle}n_*,\varpi{\rangle}}2+ g_0) & g_{*} \\[2mm]
\overline{g_{*}} & -{\rm i}(\xi-\frac{{\langle}n_*,\varpi{\rangle}}2+ g_0)
\end{pmatrix} M$$ for some $g_0\in{{\mathbb R}}$, $g_{*}\in {{\mathbb C}}$ with $|g_0|\leq \varepsilon^{\frac{15}{16}}$, $|g_{*}|\leq \varepsilon^{\frac{15}{16}} e^{-2\pi r|n_*|}$.
Before giving the proof, we present a decomposition for the space ${{{\mathcal B}}}_h$, $h>0$. Given any $\varpi\in {{\mathbb R}}^d$, for $\eta>0$ and $\tilde A\in {\rm sl}(2,{{\mathbb R}})$, we decompose ${{{\mathcal B}}}_h={{{\mathcal B}}}_h^{(\rm nre)}(\eta)\oplus {{{\mathcal B}}}_h^{(\rm re)}(\eta)$ in such a way that for any $Y\in {{{\mathcal B}}}_h^{(\rm nre)}(\eta)$, $$\label{non_resonant_condition}
\partial_\varpi Y, \;\ [\tilde A,Y]\in {{{\mathcal B}}}_h^{(\rm nre)}(\eta), \quad |\partial_\varpi Y-[\tilde A,Y]|_h\geq \eta |Y|_h.$$ Moreover, we let ${{\mathbb P}}_{\rm nre}^{\eta}Y$ and ${{\mathbb P}}_{\rm re}^{\eta}Y$ be the standard projections from ${{{\mathcal B}}}_h$ onto ${{{\mathcal B}}}_h^{(\rm nre)}(\eta)$ and ${{{\mathcal B}}}_h^{(\rm re)}(\eta)$ respectively.
Associated with this decomposition, we have
\[lemma\_HouYou\] Given $\tilde F\in{{{\mathcal B}}}_h$ with $|\tilde F|_h\leq \tilde \varepsilon$, assume that $\tilde \varepsilon\in (0, 10^{-8})$ and $\eta\geq \tilde \varepsilon^{\frac14}$. There exist $Y\in {{{\mathcal B}}}_{h}$, $G \in {{{\mathcal B}}}^{({\rm re})}_{h}(\eta)$, with the estimates $|Y|_{h}\leq \tilde \varepsilon^{\frac12}$, $|G|_{h}\leq 2\tilde \varepsilon$, such that $(\varpi,\tilde A+\tilde F)$ is conjugated to $(\varpi,\tilde A+G)$ by $e^{Y(\theta)}$.
### Non-resonant case
Consider the linear system $
(\varpi, A+F)$, where $\varpi\in {\rm DC}_d(\gamma,\tau)$, $A\in {\rm sl}(2,{{\mathbb R}})$ has two eigenvalues $\pm2\pi{\rm i}\xi$, and $F\in{{{\mathcal B}}}_r$ satisfies $|F|_{r}\leq \varepsilon$. For $\tilde A=A$ and $\eta=2\varepsilon^{\frac14}$, we focus on the decomposition ${{{\mathcal B}}}_r={{{\mathcal B}}}_r^{(\rm nre)}(\eta)\oplus {{{\mathcal B}}}_r^{(\rm re)}(\eta)$. The key observation is the following:
\[lemma\_nonresonant\_diophantine\] Assume that $(\ref{non-resonant_condition})$ holds. For any $G\in {{{\mathcal B}}}^{({\rm re})}_r(\eta)$, we have $\widehat G(n)=0$ if $0<|n|\leq N$.
Since $\varpi \in{\rm DC}_d(\gamma,\tau)$, we have $$\label{diophantine_diagonal}
|{\langle}n, \varpi {\rangle}|\geq \frac{\gamma}{N^\tau} \geq\varepsilon^{\frac1{15}},\quad \forall \ n\in{{\mathbb Z}}^d \ {\rm with} \ 0<|n|\leq N,$$ if $\varepsilon$ satisfies $(\ref{smallness_condition})$. Combining $(\ref{diophantine_diagonal})$ with the non-resonant condition $(\ref{non-resonant_condition})$, it is easy to check that $$|\partial_\varpi G_N-[A,G_N]|_h\geq \varepsilon^{\frac{1}{5}} |G_N|_h,$$ holds for any $G_N:=\sum_{0<|n|\leq N}\widehat G(n)e^{2\pi {\rm i}{\langle}n, \theta {\rangle}}$. One can consult Lemma 1 of [@E92] for details. Hence $G_N\in {{{\mathcal B}}}^{({\rm nre})}_r(\eta)$, which means $$\widehat{{{\mathbb P}}_{\rm re}^{\eta} G }(n)=0 \ {\rm if} \ 0<|n|\leq N.$$ This finishes the proof.
Applying Lemma \[lemma\_HouYou\], we know that $(\varpi, A+F)$ is congugated to a cocycle $(\varpi, A+G)$ with $G\in {{{\mathcal B}}}^{(\rm re)}_r(\eta)$. By Lemma \[lemma\_nonresonant\_diophantine\], we know that the only non-vanishing Fourier mode $\widehat G(n)$ with $|n|\leq N$ is $\widehat G(0)$. Recalling that $N=\frac{2|\ln\varepsilon|}{r-r_+}$, for $\varepsilon$ small enough, we have $$|G-\widehat G(0)|_{r_+}\leq \sum_{|n|>N}|\widehat G(n)| e^{2\pi r_+ |n| } \leq |G|_r \, e^{-2\pi N(r-r_+)} \sum_{j=0}^{d-1}\frac{(d-1)!}{j!} \frac{N^j}{(r-r_+)^{d-j}} \leq \varepsilon^{2}.$$ Let $\hat W:=e^Y$, $A_+:=A+\widehat G(0)$ and $F_+:=G- \widehat G(0)$. This concludes the proof of Proposition \[prop\_iteration\] for the non-resonant case.
### Resonant case
Recall that $A\in {\rm sl}(2,{{\mathbb R}})$ has two eigenvalues $\pm 2\pi{\rm i}\xi$ with $\xi \in {{\mathbb R}}\cup {\rm i}{{\mathbb R}}$. In view of the Diophantine property (\[dio\]) of $\varpi\in{{\mathbb R}}^d$, it is easy to see that the resonant case of $A$ does not occur unless $A$ is of elliptic type, i.e., $\xi\in {{\mathbb R}}\backslash\{0\}$. Moreover, the resonant condition (\[resonant\_condition\]) implies that $$|\xi|\geq \frac12(|{\langle}n_*, \varpi{\rangle}|-\varepsilon^{\frac1{15}})\geq \frac{\gamma}{2|n_*|^{\tau}}-\frac12\varepsilon^{\frac1{15}} \geq \frac{\gamma}{3|n_*|^{\tau}}.$$ In view of Lemma 8.1 in [@HouYou], there is $C_{A}\in {\rm SL}(2,{{\mathbb R}})$ with $$|C_A|\leq 2\sqrt{\frac{|A|}{|\xi|}} \leq \frac{2\sqrt{3}\sqrt{|A|}}{\sqrt{\gamma}}|n_*|^{\frac\tau2}$$ such that $A=C_{A} \begin{pmatrix}
0 & 2\pi\xi \\
-2\pi\xi & 0
\end{pmatrix} C_{A}^{-1}$. Let $\tilde F:=C_{A}^{-1}F C_{A}$. Then we have $$|\tilde F|_r\leq \frac{12|A|}{\gamma}|n_*|^\tau\varepsilon=:\tilde\varepsilon.$$ Let $\tilde A:=\begin{pmatrix}
0 & 2\pi\xi \\
-2\pi\xi & 0
\end{pmatrix}$ and $\eta:=2\tilde\varepsilon^{\frac14}$. Applying Lemma \[lemma\_HouYou\], one can conjugate $(\varpi, \tilde A+\tilde F)$ to $(\varpi, \tilde A+G)$ with $G\in {{{\mathcal B}}}^{(\rm re)}_r(\eta)$. Let us now characterize the precise structure of $G\in {{{\mathcal B}}}_r^{(\rm re)}(\eta)$.
\[lemma\_structrue\_resonant\] Assume that (\[resonant\_condition\]) holds. For any $G\in{{{\mathcal B}}}^{({\rm re})}_{r}(\eta)$ with $|G|_r\leq 2\tilde\varepsilon$, there exist $g_0\in{{\mathbb R}}$, $g_{*}\in {{\mathbb C}}$ and $P\in C^{\omega}_{r}({{\mathbb T}}^d, {\rm su}(1,1))$ satisfying $$|g_0|\leq \varepsilon^{\frac{15}{16}}, \quad |g_{*}|\leq\varepsilon^{\frac{15}{16}} e^{-2\pi r |n_*|};\qquad |P|_{r_+}\leq \varepsilon e^{-r_+\varepsilon^{-\frac{1}{16\tau}}}, \quad \forall \ 0<r_+<r,$$ such that $M G(\theta) M^{-1}= 2\pi\begin{pmatrix}
{\rm i} g_0 & g_{*}e^{2\pi{\rm i}{\langle}n_*,\theta{\rangle}} \\[1mm]
\overline{g_{*}}e^{-2\pi{\rm i}{\langle}n_*,\theta{\rangle}} & -{\rm i} g_0
\end{pmatrix} + P(\theta)$.
First we note that $M=\frac{1}{1+{\rm i}}\left(\begin{matrix}
1 & -{\rm i}\\
1 & {\rm i}
\end{matrix}\right)$ induces an isomorphism from ${\rm sl}(2,{{\mathbb R}})$ to ${\rm su}(1,1)$, which is the group of matrices of the form $\left(
\begin{array}{ccc}
{\rm i} t & \nu\\
\bar{ \nu} & -{\rm i} t
\end{array}\right)$ with $t\in {{\mathbb R}}$, $\nu\in {{\mathbb C}}$. Thus for any $\widehat G(n)\in {\rm sl}(2,{{\mathbb R}})$, there exist $g_{11}(n)\in{{\mathbb R}}$, $g_{12}(n)\in{{\mathbb C}}$ such that $$M \widehat G(n) M^{-1}=\begin{pmatrix}
{\rm i}g_{11}(n) & g_{12}(n) \\[1mm]
\overline{g_{12}(n)} & -{\rm i}g_{11}(n)
\end{pmatrix}.$$ By the decay property of Fourier coefficients, we get $$\label{esti_g}
|g_{11}(n)|,\, |g_{12}(n)|\leq 4 \tilde\varepsilon e^{-2\pi r |n|}.$$ Since $\tilde A=M^{-1} \begin{pmatrix}
2\pi {\rm i}\xi & 0 \\
0 & -2\pi {\rm i}\xi
\end{pmatrix} M$, a direct computation shows that $$M (\partial_\varpi G-[\tilde A,G]) M^{-1}=2\pi{\rm i}\sum_{n\in{{\mathbb Z}}^d}
\begin{pmatrix}
{\rm i} {\langle}n,\varpi {\rangle}g_{11}(n) & ({\langle}n,\varpi {\rangle}+2\xi)g_{12}(n)\\[1mm]
({\langle}n,\varpi {\rangle}-2\xi)\overline{g_{12}(n)} & -{\rm i} {\langle}n,\varpi {\rangle}g_{11}(n)
\end{pmatrix} e^{2\pi{\rm i}{\langle}n,\theta{\rangle}}.$$ By the definition of ${{{\mathcal B}}}_r^{({\rm nre})}(\eta)$ in (\[non\_resonant\_condition\]), for any $G\in {{{\mathcal B}}}^{({\rm nre})}_{r}(\eta)$, $M G(\theta) M^{-1}$ equals $$\sum_{n\notin\Lambda_1} \begin{pmatrix}
{\rm i} g_{11}(n) & 0 \\[1mm]
0 & -{\rm i} g_{11}(n)
\end{pmatrix} e^{2\pi{\rm i}{\langle}n,\theta{\rangle}}
+ \sum_{n\notin\Lambda_2}\begin{pmatrix}
0 & g_{12}(-n)e^{-2\pi{\rm i}{\langle}n,\theta{\rangle}} \\[1mm]
\overline{g_{12}(n)} e^{2\pi{\rm i}{\langle}n,\theta{\rangle}} & 0
\end{pmatrix}$$ with $\Lambda_1:=\{n\in{{\mathbb Z}}^{d}:|{\langle}n,\varpi {\rangle}|< 2\tilde\varepsilon^{\frac{1}{4}}\}$, $\Lambda_2:=\{n\in{{\mathbb Z}}^{d}:|2\xi-{\langle}n,\varpi {\rangle}|< 2\tilde\varepsilon^{\frac{1}{4}}\}$. Hence $G\in {{{\mathcal B}}}^{({\rm re})}_{r}(\eta)$ means $M G(\theta) M^{-1}$ has the form $$\label{form_resonant_elliptic}
\sum_{n\in\Lambda_1} \begin{pmatrix}
{\rm i} g_{11}(n) & 0 \\[1mm]
0 & - {\rm i} g_{11}(n)
\end{pmatrix} e^{2\pi{\rm i}{\langle}n,\theta{\rangle}}
+ \sum_{n\in \Lambda_2}\begin{pmatrix}
0 & g_{12}(-n)e^{-2\pi{\rm i}{\langle}n,\theta{\rangle}} \\[1mm]
\overline{g_{12}(n)} e^{2\pi{\rm i}{\langle}n,\theta{\rangle}} & 0
\end{pmatrix}$$
We have the following observations: $$\begin{aligned}
\Lambda_1\cap \{n\in{{\mathbb Z}}^{d}: |n| \leq \gamma^{\frac1\tau}\varepsilon^{-\frac{1}{15\tau}}\}&=\{0\},\label{resonant_site_1}\\
\Lambda_2\cap \{n\in{{\mathbb Z}}^{d}: |n| \leq 2^{-\frac1\tau} \gamma^{\frac1\tau}\varepsilon^{-\frac{1}{15\tau}}-N\}&=\{n_*\}.\label{resonant_site_2}\end{aligned}$$
Indeed, given any $n\in\Lambda_1$ and $n\neq0$, we have $$\frac{\gamma}{|n|^\tau}< |{\langle}n,\varpi {\rangle}|<2\tilde\varepsilon^{\frac{1}{4}}< \varepsilon^{\frac{1}{15}}.$$ Therefore, $|n|> \gamma^{\frac{1}{\tau}}\varepsilon^{-\frac{1}{15\tau}}$, which gives (\[resonant\_site\_1\]).
For any $n'_*\neq n_*$ with $|2\xi-{\langle}n'_*,\varpi{\rangle}|< \varepsilon^{\frac{1}{15}}$, since $\varpi\in {\rm DC}_d(\gamma,\tau)$, we have $$\frac{\gamma}{|n'_*-n_*|^{\tau}}\leq|{\langle}n'_*-n_*,\varpi {\rangle}|< 2\varepsilon^{\frac{1}{15}},$$ which implies $|n'_*|> 2^{-\frac1\tau} \gamma^{\frac1\tau}\varepsilon^{-\frac{1}{15\tau}}-N > N$ under the hypothesis (\[smallness\_condition\]), and thus $(\ref{resonant_site_2})$ follows.
Let ${{{\mathcal N}}}_1:= \gamma^{\frac1\tau}\varepsilon^{-\frac{1}{15\tau}}$ and ${{{\mathcal N}}}_2:=2^{-\frac1\tau} \gamma^{\frac1\tau}\varepsilon^{-\frac{1}{15\tau}}-N$. In view of (\[resonant\_site\_1\]) and (\[resonant\_site\_2\]), the two parts of $M G(\theta) M^{-1}$ given in (\[form\_resonant\_elliptic\]) can be decomposed as $$\begin{aligned}
& \sum_{n\in\Lambda_1} \begin{pmatrix}
{\rm i}g_{11}(n) & 0 \\[1mm]
0 & -{\rm i}g_{11}(n)
\end{pmatrix} e^{2\pi {\rm i}{\langle}n,\theta{\rangle}}\\
=&\begin{pmatrix}
{\rm i}g_{11}(0) & 0 \\[1mm]
0 & -{\rm i}g_{11}(0)
\end{pmatrix}+\sum_{n\in \Lambda_1\atop{|n|>{{{\mathcal N}}}_1}} \begin{pmatrix}
{\rm i}g_{11}(n) & 0 \\[1mm]
0 & -{\rm i}g_{11}(n)
\end{pmatrix} e^{2\pi {\rm i}{\langle}n,\theta{\rangle}},\\
& \sum_{n\in\Lambda_2}\begin{pmatrix}
0 & g_{12}(n)e^{2\pi {\rm i}{\langle}n,\theta{\rangle}} \\[1mm]
\overline{g_{12}(n)} e^{-2\pi {\rm i}{\langle}n,\theta{\rangle}} & 0
\end{pmatrix}\\
=& \begin{pmatrix}
0 & g_{12}(n_*)e^{2\pi {\rm i}{\langle}n_*,\theta{\rangle}} \\[1mm]
\overline{g_{12}(n_*)}e^{-2\pi {\rm i}{\langle}n_*,\theta{\rangle}} & 0
\end{pmatrix} + \sum_{n\in\Lambda_2\atop{|n|> {{{\mathcal N}}}_2}}\begin{pmatrix}
0 & g_{12}(n)e^{2\pi {\rm i}{\langle}n,\theta{\rangle}} \\[1mm]
\overline{g_{12}(n)} e^{-2\pi {\rm i}{\langle}n,\theta{\rangle}} & 0
\end{pmatrix}.\end{aligned}$$ Let $g_0:=\frac{1}{2\pi}g_{11}(0)$, $g_{*}:=\frac{1}{2\pi} g_{12}(n_*)$ and let $P(\theta)$ be $$\sum_{n\in \Lambda_1\atop{|n|> {{{\mathcal N}}}_1}} \begin{pmatrix}
{\rm i} g_{11}(n) & 0 \\[1mm]
0 & -{\rm i} g_{11}(n)
\end{pmatrix} e^{2\pi {\rm i}{\langle}n,\theta{\rangle}}+ \sum_{n\in\Lambda_2\atop{|n|> {{{\mathcal N}}}_2}}\begin{pmatrix}
0 & g_{12}(-n)e^{-2\pi {\rm i}{\langle}n,\theta{\rangle}} \\[1mm]
\overline{g_{12}(n)} e^{2\pi {\rm i}{\langle}n,\theta{\rangle}} & 0
\end{pmatrix}.$$ By (\[esti\_g\]), and noting that ${{{\mathcal N}}}_2<{{{\mathcal N}}}_1$, we have $$|P|_{r_+}\leq (d-1)! \left({{{\mathcal N}}}_2+\frac{1}{r-r_+}\right)^d \cdot 4 \tilde\varepsilon e^{-2\pi (r-r_+){{{\mathcal N}}}_2}\leq \varepsilon e^{-r_+\varepsilon^{-\frac{1}{16\tau}}}.$$
Now we define $$Z(\theta):=e^{-\frac{{\langle}n_*,\theta{\rangle}}{2\xi}\tilde A}= M^{-1}\begin{pmatrix}
e^{-\pi {\rm i}{\langle}n_*,\theta{\rangle}} & 0\\
0 & e^{\pi {\rm i}{\langle}n_*,\theta{\rangle}}
\end{pmatrix} M .$$ Obviously, $Z\in C^\omega_{r}({{\mathbb T}}^d,{\rm PSL}(2,{{\mathbb R}}))$, and for any $r''\in (0,r)$, $|Z|_{r''}\leq 2e^{\pi r''|n_*|}$. Given any $G\in{{{\mathcal B}}}^{({\rm re})}_{r}(\eta)$, we have thus $$\partial_\varpi Z= (\tilde A + G) Z-Z\left[\left(1-\frac{{\langle}n_*,\varpi{\rangle}}{2\xi}\right)\tilde A + Z^{-1} G Z\right].$$ By a direct calculation, we get $$Z(\theta)^{-1}G(\theta)Z(\theta)=2\pi M^{-1}\begin{pmatrix}
{\rm i} g_0 & g_{*} \\[2mm]
\overline{g_{*}} & - {\rm i} g_0
\end{pmatrix}M + Z(\theta)^{-1}P(\theta)Z(\theta).$$ Let $F_{+}:=Z^{-1}PZ$ and $$A_{+}:=\left(1-\frac{{\langle}n_*,\varpi{\rangle}}{2\xi}\right)\tilde A + 2\pi M^{-1}\begin{pmatrix}
{\rm i} g_0 & g_{*} \\[2mm]
\overline{g_{*}} & - {\rm i} g_0
\end{pmatrix}M.$$ Therefore the system $(\varpi,\tilde A + G)$ is conjugated to $(\varpi,A_{+} + F_{+})$ by $Z$, with estimates $$\begin{aligned}
&&|A_+|\leq 2\pi \varepsilon^{\frac{1}{15}}+4\pi \varepsilon^{\frac{15}{16}}(1+e^{-2\pi r |n_*|})\leq \varepsilon^{\frac{1}{16}},\\
&&|F_+|_{r_{+}}\leq
4 e^{2\pi r_+|n_*|}\cdot
\varepsilon e^{-r_+\varepsilon^{-\frac{1}{16\tau}}}
\leq \varepsilon e^{-r_+\varepsilon^{-\frac{1}{18\tau}}},\quad \forall \ 0<r_+<r.\end{aligned}$$
Let $\hat W:=C_A \cdot e^{Y} \cdot Z$ with $e^Y$ obtained in Lemma \[lemma\_HouYou\]. Obviously, ${\rm deg} \hat W=n_*$. Then we finish the proof for the resonant case of Proposition \[prop\_iteration\].
Reducibility of quasi-periodic linear systems
---------------------------------------------
Consider the quasi-periodic linear system $(\varpi, A_0 + F_0)$. Denote by $\rho(\varpi,A_0+F_0)$ its rotation number (we refer to [@E92; @JM] for the detailed definition). In the same way as in [@E92], one can prove that if $\rho(\varpi,A_0+F_0)$ is Diophantine or rational with respect to $\varpi$, then the system $(\varpi, A_0 + F_0)$ is reducible. For our purpose, in this paper we will specially focus our attention on the quantitative reducibility in the case where $\rho(\varpi, A_0+F_0)=\frac{{\langle}k, \varpi{\rangle}}2$ for some $k\in{{\mathbb Z}}^{d}\backslash\{0\}$.
\[thm\_gap\_edge\_algebra\] Assume that $\varpi\in{\rm DC}_d(\gamma,\tau)$, $d\geq 2$. Given any $r\in (0,r_0)$, there is $\varepsilon_*=\varepsilon_*(|A_0|, \gamma, \tau, r_0, r, d)>0$ such that if $|F_0|_{r_0}=\varepsilon_0 <\varepsilon_*$, then the following holds.
1. The system $(\varpi,A_0+F_0)$ is almost reducible in the strip $|\Im \theta|<r$.
2. If $\rho(\varpi, A_0+F_0)=\frac{{\langle}k,\varpi{\rangle}}{2}$ for $k\in{{\mathbb Z}}^{d}\backslash\{0\}$, and $(\varpi,A_0+F_0)$ is not uniformly hyperbolic, then there exists $W\in C_{r}^{\omega}({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$ such that $$\partial_\varpi W=(A_0+F_0)W - W \begin{pmatrix}
0 & \kappa \\
0 & 0
\end{pmatrix}$$ with $|\kappa|\leq \varepsilon_0^{\frac34} e^{-2\pi r|k|}$. Moreover, for any $r''\in (0, r]$, $|W|_{r''}\leq D_1 e^{\frac{3\pi r''}{2} |k|}$ with $D_1=D_1(\gamma, \tau, |A_0|, r_0, d)>0$.
The result can be proved by applying Proposition \[prop\_iteration\] iteratively. Take $\varepsilon_0$, $r_0$ and $r$ as above. Assume that we are at the $(j+1)^{\rm th}$ KAM step, where we have $A_{j}\in {\rm sl}(2,{{\mathbb R}})$ with eigenvalues $\pm2\pi {\rm i}\xi_{j}$ and $F_j\in {{{\mathcal B}}}_{r_j}$ satisfying $|F_j|_{r_j}\leq \varepsilon_j$ for some $\varepsilon_j\leq \varepsilon_0$. Let $\tilde r=\frac{r_0+r}{2}$. Then we define $$\label{sequences}
r_j - r_{j+1}:=\frac{r_0-\tilde r}{4^{j+1}} ,\quad N_j:=\frac{ 2|\ln\varepsilon_j|}{r_j - r_{j+1}}=\frac{2\cdot 4^{j+1} |\ln\varepsilon_j|}{r_0-\tilde r}.$$ If $\varepsilon_j$ is sufficiently small such that the condition (\[smallness\_condition\]) is satisfied for $\varepsilon=\varepsilon_j$, $r=r_j$, $r_+=r_{j+1}$ and $A=A_j$, then, by Proposition \[prop\_iteration\], we can construct $$\hat W_{j}\in C_{r_{j+1}}^{\omega}({{\mathbb T}}^d, \mathrm{PSL}(2,{{\mathbb R}}) ),\;\ A_{j+1}\in {\rm sl}(2,{{\mathbb R}}), \;\ F_{j+1}\in {{{\mathcal B}}}_{r_{j+1}},$$ such that $(\varpi, A_j + F_j)$ is conjugated to $(\varpi, A_{j+1} + F_{j+1})$ by $\hat W_{j}(\theta)$. Moreover,
- if for any $n\in{{\mathbb Z}}^{d}$ with $0<|n|\leq N_j$, we have $|2\xi_j-{\langle}n,\varpi{\rangle}|\geq \varepsilon_j^{\frac{1}{15}}$, then $$\label{esti_non-resonant}
|A_{j+1}-A_j|\leq \varepsilon_j^{\frac12},\;\ |\hat W_j- {\rm Id}|_{r_{j+1}}\leq 2\varepsilon_j^{\frac12},\;\ |F_{j+1}|_{r_{j+1}}\leq \varepsilon_{j+1}:= \varepsilon_j^{2};$$
- if there is $n_j\in{{\mathbb Z}}^{d}$ with $0<|n_j|\leq N_j$ such that $|2\xi_j-{\langle}n_j,\varpi {\rangle}|< \varepsilon_j^{\frac{1}{15}}$, then $$\label{esti_resonant}
|A_{j+1}|\leq \varepsilon_j^{\frac{1}{16}},\quad |F_{j+1}|_{r_{j+1}}\leq \varepsilon_{j+1}:= \varepsilon_j e^{-r_{j+1} \varepsilon_j^{-\frac{1}{18\tau}}}, \quad {\rm deg}\hat W_j=n_j$$ and for any $r''\in(0,r_{j+1}]$, $$\label{esti_resonant_W}
|\hat W_j|_{r''}\leq 4\sqrt\frac{|A_j|}{\gamma}|n_j|^{\frac\tau2} e^{\pi r'' |n_j|}.$$
In view of (\[esti\_non-resonant\]) and (\[esti\_resonant\]), one sees that $\varepsilon_j\leq \varepsilon_0^{2^j}$ and $|A_j|\leq 2|A_0|$ for any $j\geq 0$. So, if $\varepsilon_0$ is sufficiently small (depending on $|A_0|, \gamma, \tau, r_0, r, d$) such that (\[smallness\_condition\]) holds, then Proposition \[prop\_iteration\] can be applied iteratively. Indeed, $\varepsilon_j$ on the left side of the inequality (\[smallness\_condition\]) decays at least super-exponentially with $j$, while $(r_j-r_{j+1})^{800d(\tau+1)}$ on the right side decays exponentially with $j$. Hence $(\varpi,A_0+F_0)$ is almost reducible.
Assume that there are at least two resonant steps in the above almost reducibility precedure. Let us focus on two consecutive resonant steps, say the $(j_i+1)^{\rm th}$ and $(j_{i+1}+1)^{\rm th}$. At the $(j_{i+1}+1)^{\rm th}-$step, the resonance condition implies $\left|\xi_{j_{i+1}}- \frac{{\langle}n_{j_{i+1}},\varpi{\rangle}}2\right|\leq \frac12\varepsilon_{j_{i+1}}^{\frac{1}{15}}$, hence $|\xi_{j_{i+1}}|>\frac{\gamma}{3|n_{j_{i+1}}|^{\tau}}$. On the other hand, according to Proposition \[prop\_iteration\], after the $(j_{i}+1)^{\rm th}-$step, $|\xi_{j_{i}+1}|\leq \varepsilon_{j_i}^\frac{1}{16}$. By (\[esti\_non-resonant\]), $|\xi_{j_{i+1}}|\leq 2\varepsilon^{\frac{1}{16}}_{j_{i}}\leq \frac{\varepsilon^{\frac{1}{18}}_{j_{i}}\gamma}{3|n_{j_i}|^{\tau}}.$ Thus $$\label{skip_of_resonances}
|n_{j_{i+1}}|\geq \varepsilon_{j_i}^{-\frac{1}{18\tau}} |n_{j_{i}}|.$$ Recall that ${\rm deg}\hat W_{j+1}=n_{j}$ if the $(j+1)^{\rm th}-$step is resonant. In view of (\[rot-conj\]) and (\[skip\_of\_resonances\]), we deduce that there are at most finitely many resonant steps in the above almost reducibility procedure under the hypothesis $\rho_{(\varpi, A_0+F_0)}=\frac{{\langle}k,\, \varpi{\rangle}}2$. This means that we can find a sequence $(\hat W_{l})_{l\in{{\mathbb N}}}$ with $\hat W_{l}\in C^\omega_{r_{l+1}}({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$, in which the resonant case occurs only finitely many times. By the estimate of $\hat W_j$ in and the sequence $(r_j)_{j\in{{\mathbb N}}}$ given in (\[sequences\]), we see that the product $\prod_{l=0}^j\hat W_{l+1}$ converges to some $W\in C^\omega_{\tilde r}({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$ such that $\partial_\varpi W=(A_0+F_0)W-W B$ for some $B\in {\rm sl}(2,{{\mathbb R}})$ with $\rho_{(\varpi,B)}=0$.
Assuming that there are $s+1$ resonant steps, associated with integers vectors $$n_{j_0},\dots , n_{j_s}\in {{\mathbb Z}}^{d}, \qquad 0<|n_{j_i}|\leq N_{j_i}, \;\ i=0,1,\dots,s,$$ then $k=n_{j_0}+ \dots + n_{j_s}$. In view of the inequalities (\[skip\_of\_resonances\]) and the fact that $$|n_{j_s}|-\sum_{i=0}^{s-1}|n_{j_i}|\leq |k| \leq |n_{j_s}|+\sum_{i=0}^{s-1}|n_{j_i}|,$$ we get $(1-2\varepsilon_{0}^{\frac{1}{18\tau}})|n_{j_s}|\leq |k| \leq (1+2\varepsilon_{0}^{\frac{1}{18\tau}})|n_{j_s}|$. By (\[esti\_resonant\_W\]), for any $r''\in (0,r]$, $$|W|_{r''}
\leq 2|\hat W_{j_0+1}|_{r''}\cdots|\hat W_{j_s+1}|_{r''}
\leq \frac{2^{2s+3}}{\sqrt{\gamma^{s+1}}}\prod_{i=0}^s |A_{j_i}| |n_{j_i}|^{\frac\tau2} e^{\pi r'' |n_{j_i}|}
\leq D_1 e^{\frac{3\pi r''}{2} |k|}$$ for some $D_1=D_1(\gamma, \tau, |A_0|, r_0, d)>0$.
Now we estimate the constant matrix $B$. Since we have assumed that the initial system $(\varpi,A_0+F_0)$ is not uniformly hyperbolic, one concludes that $B$ can not be a hyperbolic matrix. As we have proved, $\rho(\varpi,B)=0$, thus ${\rm det}B=0$. Assume that $B=
\begin{pmatrix}
B_{11} & B_{12} \\
B_{21} & -B_{11}
\end{pmatrix}.$ Then there exists $\phi\in{{\mathbb T}}$ such that $R_{-\phi}B R_{\phi}=
\begin{pmatrix}
0 & B_{21}-B_{12} \\
0 & 0
\end{pmatrix}$. By taking $W R_{\phi}$ instead of $W$ and $\kappa=B_{21}-B_{12}$, we know that $(\varpi, A_0+F_0)$ is conjugated to $\tilde B=\begin{pmatrix}
0 & \kappa \\
0 & 0
\end{pmatrix}$.
To estimate $|\kappa|$, let us focus on $(\varpi, A_{j_s+1}+F_{j_s+1})$, i.e., the system just after the last resonant step. In view of (\[A+\]), we have $$A_{j_s+1}=2\pi M^{-1}\begin{pmatrix}
{\rm i}\left(\xi_{j_s}-\frac{{\langle}n_{j_s},\, \varpi{\rangle}}2+ q_{0}\right) & q_{j_s} \\[2mm]
\overline{q_{j_s}} & -{\rm i}\left(\xi_{j_s}-\frac{{\langle}n_{j_s},\, \varpi{\rangle}}2 + q_{0}\right)
\end{pmatrix} M,$$ with $q_{0}\in {{\mathbb R}}, \, q_{j_s}\in {{\mathbb C}}$ satisfying $$\label{off-diagonal_js+1}
\left|\xi_{j_s}-\frac{{\langle}n_{j_s},\, \varpi{\rangle}}2 + q_{0}\right|\leq \varepsilon^{\frac{1}{15}}_{j_s}+\varepsilon^{\frac{15}{16}}_{j_s}\leq 2\varepsilon^{\frac{1}{15}}_{j_s},\quad |q_{j_s}|\leq\varepsilon^{\frac{15}{16}}_{j_s} e^{-2\pi r_{j_s}|n_{j_s}|}.$$ Since $(j_s+1)^{\rm th}-$step is the last resonant step, we have $|A_{l+1}-A_l|\leq \varepsilon_{l}^{\frac12}$, $l\geq j_s+1$. Hence, noting that $\varepsilon_{j_s+1}=\varepsilon_{j_s} e^{-r_{j_s+1} \varepsilon_{j_s}^{-\frac{1}{18\tau}}}$, we get $$|A_{j_s+1}-B|\leq \sum_{l=j_s+1}^{\infty} |A_{l+1}-A_l|\leq 2\varepsilon^{\frac12}_{j_s} e^{-\frac{r_{j_s+1}}{2}\varepsilon_{j_s}^{-\frac{1}{18\tau}}}.$$ Rewrite $B$ as $B=M^{-1}\begin{pmatrix}
{\rm i} \beta_{11} & \beta_{12} \\[1mm]
\overline{\beta_{12}} & -{\rm i} \beta_{11}
\end{pmatrix} M$ with $\beta_{11} \in {{\mathbb R}}$, $\beta_{12} \in{{\mathbb C}}$. In view of (\[off-diagonal\_js+1\]), we have $$|\beta_{12}| \leq
2\pi \varepsilon^{\frac{15}{16}}_{j_s} e^{-2\pi r_{j_s} |n_{j_s}|} +4\varepsilon^{\frac12}_{j_s} e^{-\frac{r_{j_s+1}}{2}\varepsilon_{j_s}^{-\frac{1}{18\tau}}}
\leq \varepsilon^{\frac78}_{j_s}e^{-2\pi r_{j_s+1}|n_{j_s}|}.$$ Then we have $|\beta_{11}| \leq \varepsilon^{\frac78}_{j_s}e^{-2\pi r_{j_s+1}|n_{j_s}|}$ since ${\rm det}B=0$. So $$|B_{12} |, \, |B_{21} |\leq 2 \varepsilon^{\frac78}_{j_s}e^{-2\pi r_{j_s+1}|n_{j_s}|} \leq \frac12\varepsilon^{\frac34}_{j_s}e^{-2\pi r_{j_s+1}|n_{j_s}|}.$$ Hence, in view of the fact $|k|\leq (1+2\varepsilon_{0}^{\frac{1}{18\tau}})|n_{j_s}|$, $$|\kappa|=|B_{21}-B_{12}|\leq \varepsilon^{\frac34}_{j_s}e^{-2\pi r_{j_s+1}|n_{j_s}|}\leq \varepsilon^{\frac34}_{j_s}e^{-\frac{2\pi \tilde r |k|}{1+2\varepsilon_{0}^{1/18\tau}}}\leq \varepsilon^{\frac34}_{j_s}e^{-2\pi r|k|}.$$
Reducibility of quasi-periodic cocycles
---------------------------------------
In analogy with Theorem \[thm\_gap\_edge\_algebra\] for quasi-periodic linear systems, we obtain a similar result for quasi-periodic cocycles $$\begin{pmatrix}
u_{n+1} \\
u_n
\end{pmatrix}=(A_0+F_0(\theta+n\alpha))\begin{pmatrix}
u_n \\
u_{n-1}
\end{pmatrix}.$$
\[thm\_gap\_edge\_SL\] Let $\alpha\in{\rm DC}_d(\gamma,\tau)$ and $A_0\in {\rm SL}(2,{{\mathbb R}})$. Given $r\in (0,r_0)$, there is $\varepsilon_*=\varepsilon_*(|A_0|, \gamma, \tau, r_0, r, d)>0$ such that if $|F_0|_{r_0}=\varepsilon_0 <\varepsilon_*$, then the following holds:
1. The quasi-periodic ${\rm SL}(2,{{\mathbb R}})$ cocycle $(\alpha,A_0+F_0)$ is almost reducible in the strip $|\Im \theta|<r$.
2. If $2\rho(\alpha,\, A_0+F_0)-{\langle}k,\alpha{\rangle}\in{{\mathbb Z}}$ for $k\in{{\mathbb Z}}^{d}\backslash\{0\}$, and $(\alpha,A_0+F_0(\cdot))$ is not uniformly hyperbolic, then there exists $W\in C_{r}^{\omega}({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$, such that $$W(\cdot +\alpha)^{-1}(A_0+F_0(\cdot)) W(\cdot)=B=\begin{pmatrix}
1 & \kappa \\
0 & 1
\end{pmatrix},$$ with $|\kappa|\leq \varepsilon_0^{\frac34} e^{-2\pi r|k|}$. Moreover, for any $r''\in(0,r]$, $|W|_{r''}\leq D_1 e^{\frac{3\pi r''}{2} |k|}$ with $D_1=D_1(\gamma, \tau, |A_0|,r_0, d)>0$.
\[uniformcons\] Let $\gamma,\tau>0$ be fixed. If $A_0$ varies in ${\rm SO}(2,{{\mathbb R}})$, then $\varepsilon_*=\varepsilon_*( \gamma, \tau, r_0, r, d)>0$ can be taken uniform with respect to $A_0$.
Almost localization and duality argument
========================================
While in the previous part, we considered the case of a Diophantine vector of frequencies, here, we let $\alpha\in {{\mathbb R}}$ be a frequency satisfying $\beta(\alpha)=0$, and we study the reducibility of associate quasi-periodic Schrödinger cocycles by non-perturbative methods. The quantitative statement we prove is based on two importants ingredients: quantitative almost localization properties of dual Schrödinger operators and quantitative Aubry duality.
From now on, in the formulations and proofs of various assertions about quasi-periodic Schrödinger cocycles and Schrödinger operators, we shall encounter several positive constants depending on the potential $V$, the frequency $\alpha$, etc. For the convenience of notation, we denote by $C$ constants depending only on $V$, $\alpha$ (or only on $\lambda$, $\alpha$ for the almost Mathieu case). And we use other notations ($c_1$, $c_2$, $\cdots$, $C_1'$, $C_2'$ $\cdots$, $C_3$, $C_4$, $\cdots$) to denote absolute constants or constants depending on some other quantities (e.g., the given radius of analyticity).
Quantitative reducibility – general analytic potential
------------------------------------------------------
Assume that the family $\{\widehat H_{V,\alpha,\theta}\}_{
\theta}$ is almost localized. Given $E\in \Sigma_{V,\alpha}$ on the boundary of a spectral gap, it is well-known that the cocycle $(\alpha, S_E^V)$ can be reduced to a constant parabolic cocycle. The main goal of this section is to show that the off-diagonal coefficient of the parabolic matrix is exponentially small in terms of the label of the spectral gap. We stress that the exponential decay rate of the off-diagonal element is directly related to the exponential decay rate of the spectral gaps (c.f. Theorem \[thm\_upperbound\]). If one just wants to show that the off-diagonal element is exponentially small, one may consult [@A1] where a more concise proof is given. However, if one wants to explicitly estimate the decay rate as in our paper, it is more technically involved (consult Remark \[compare\] for more discussions).
As in [@A1], we use truncations to obtain lower bounds on the Bloch waves involved in the definition of conjugacies. As was first realized by Avila [@A1], a crucial fact to obtain sharp estimates in the non-perturbative regime is the Corona Theorem (with the Uchiyama estimates), whose statement we now recall.
\[corona theorem\] Let $M\in C^{\omega}({{\mathbb T}}, {{\mathbb C}}^2)$. Assume that for some constants $a,\delta_1,\delta_2>0$, we have $\delta_1\leq |M(z)| \leq \delta_2^{-1}$ for $|\Im z|<a$. Then there exists $Z \in C^\omega( \mathbb{T} , \mathrm{SL}(2,{{\mathbb C}}))$ with first column $M$ and such that $|Z|_a \lesssim \delta_1^{-2} \delta_2^{-1} (1 - \ln (\delta_1 \delta_2))$.
We first use the parametrization by some auxiliary phase $\theta(E)\in {{\mathbb R}}$, and the estimates are expressed in terms of its last resonance. Then we show how they can be translated in terms of the label of the spectral gap. A key fact is that the constants in the following statements are independent of the spectral gap we focus on. Indeed, our proof is based on almost localization, which provides constants that are uniform with respect to the energy. Our main statement is as follows.
\[prop\_duality\_para\] Let $\alpha\in{{\mathbb R}}\backslash{{\mathbb Q}}$ satisfy $\beta(\alpha)=0$. Given $r_0\in (0,1)$, let $V \in C_{r_0}^\omega({{\mathbb T}},{{\mathbb R}})$ with $|V|_{r_0}\leq c_0 r_0^{3}$ and take $\epsilon_1=\epsilon_1(r_0)\in(0,r_0)$ as in Theorem \[almostredth\]. For any $r\in (0,\frac{\epsilon_1}{2\pi})$, there exists $k_1=k_1(\alpha, r_0, r)>0$ such that for any $E\in \Sigma_{V,\alpha}$ satisfying $2\rho{(\alpha, S_{E}^V)}- k\alpha \in {{\mathbb Z}}$ with $|k|\geq k_1$, there exist $U\in C_{r}^\omega(\mathbb{T}, \mathrm{PSL}(2,{{\mathbb R}}))$, $\varphi\in{{\mathbb R}}$ and $n=n(k) \in {{\mathbb Z}}$ satisfying $|n|\geq\frac{|k|}{4}$ such that $$\label{eqrota}
U(\cdot +\a)^{-1} S_{E}^{V}(\cdot) U(\cdot)=
\begin{pmatrix}
1 & \varphi \\[1mm]
0 & 1
\end{pmatrix}$$ with $|\varphi| \leq C_3 e^{- \frac{ \pi r}{5} |n|}$ for some $C_3=C_3(\alpha,r_0,r)>0$. Moreover, for any $r''\in(0, r]$, $|U|_{r''} \leq C_4 e^{22 \pi r'' |n|}$ for some $C_4=C_4(\alpha,r_0,r'')>0$.
If $2\rho{(\alpha, S_{E}^V)}- k\alpha \in {{\mathbb Z}}$, then by Theorem 3.3 in [@AvilaJito], for some phase $\theta=\theta(E) \in {{\mathbb R}}$, there exists a solution $\widehat u$ to $\widehat H_{V,\alpha,\theta} \widehat u= E \widehat u$ with $\widehat u_0=1$ and $|\widehat u_j|\leq 1$ for every $j \in {{\mathbb Z}}$. By Theorems 2.5 (also Theorem 4.2) in [@AvilaJito], $\theta=\pm \rho{(\alpha, S_E^V)}+\frac{l \alpha}2$ for some $l \in {{\mathbb Z}}$. Set $n=n(k):=\pm k +l \in \mathbb{Z}$. Since $|V|_{r_0}\leq c_0 r_0^{3}$, by Theorem \[almostredth\], $\{\widehat H_{V,\alpha,\theta'}\}_{\theta'}$ is almost localized: let $\epsilon_1:=\epsilon_1(r_0,2)>0$, $C_1:=C_1(\alpha,r_0,2)>0$ be the constants defined in Theorem \[almostredth\]. Then it follows from that $$\label{localize_duality}
|\widehat u_j| \leq C_1 e^{-\epsilon_1 |j|}, \quad \forall \ |j|\geq 2 |n|.$$ Therefore, the function $u\colon z \mapsto \sum_{j \in {{\mathbb Z}}} \widehat u_j e^{2 \pi {\rm i} j z}$ is analytic on the strip $\{|\Im z|<\frac{\epsilon_1}{2 \pi}\}$, and the analytic Bloch wave ${{{\mathcal U}}}\colon z \mapsto
\begin{pmatrix}
e^{2 \pi {\rm i}\theta} u(z)\\
u(z-\a)
\end{pmatrix}$ satisfies $$S_{E}^{V}(z) \, {{{\mathcal U}}}(z) = e^{2\pi {\rm i}\theta} {{{\mathcal U}}}(z+\alpha),\quad \forall \ z\in {{\mathbb C}}/{{\mathbb Z}}\ {\rm with} \ |\Im z|<\frac{\epsilon_1}{2 \pi}. $$ In particular, by the minimality of $x \mapsto x+\alpha$, and the fact that $\widehat u_0=1$, we see that $\mathcal{U}$ does not vanish. Define ${{{\mathcal U}}}^{(1)}(z):=e^{ \pi {\rm i} n z} {{{\mathcal U}}}(z)\in {{\mathbb C}}^2\backslash\{0\}$. Since $2\theta -n\alpha \in {{\mathbb Z}}$, we get $$\label{eq widedtiel u}
S_{E}^{V}(\cdot) \, {{{\mathcal U}}}^{(1)}(\cdot) = e^{\pi {\rm i}(2\theta -n\alpha)} \, {{{\mathcal U}}}^{(1)}(\cdot+\alpha)= \pm \, {{{\mathcal U}}}^{(1)}(\cdot+ \alpha). $$ Without loss of generality, we assume that $S_{E}^{V}(\cdot) \, {{{\mathcal U}}}^{(1)} (\cdot)= {{{\mathcal U}}}^{(1)}(\cdot+ \alpha)$. Set $\mathcal{S}:=\Re({{{\mathcal U}}}^{(1)})$ and $\mathcal{T}:=\Im( {{{\mathcal U}}}^{(1)})$. Then, by the minimality of $x \mapsto x+\alpha$, the map $x \mapsto \mathrm{det}(\mathcal{S}(x),\mathcal{T}(x))$ is constant on ${{\mathbb T}}$, equal to $\pm d_0$ for some $d_0 \geq 0$.
If $d_0>0$, let $\sigma=\pm1$ be chosen such that $d_0^{-1/2} (\mathcal{S}, \sigma \mathcal{T})\colon {{\mathbb T}}\to \mathrm{PSL}(2,{{\mathbb R}})$. Note that in this case, $d_0^{-1/2} (\mathcal{S}, \sigma \mathcal{T})$ conjugates $(\alpha,S_{E}^{V})$ to $(\alpha,\mathrm{Id})$. Otherwise, there exist $\psi\colon {{\mathbb T}}\to {{\mathbb C}}$ with $|\psi|=1$ and $\mathcal{V}\colon {{\mathbb T}}\to {{\mathbb R}}^2\backslash\{0\}$ such that ${\mathcal{U}^{(1)}}=\psi \, \mathcal{V}$ on ${{\mathbb T}}$. By , we have $$S_{E}^{V} (x) \, \mathcal{V}(x)=\frac{\psi(x+\alpha)}{\psi(x)} \mathcal{V}(x+\alpha),\quad \forall \ x \in {{\mathbb T}},$$ hence $\frac{\psi(x+\alpha)}{\psi(x)}\in {{\mathbb R}}$. By the minimality of $x \mapsto x+\alpha$, we deduce that $\psi|_{{\mathbb T}}\equiv e^{2 \pi \mathrm{i} \theta_0}\in {{\mathbb C}}$ for some $\theta_0\in {{\mathbb R}}$. The map $$\label{defv}\mathcal{V}\colon z \mapsto e^{ \pi \mathrm{i}(n z- 2 \theta_0)}\mathcal{U}(z)$$ is analytic on $\{|\Im z|<\frac{\epsilon_1}{2\pi}\}$ and satisfies $$\label{invarireim2}
S_{E}^{V}(z) \, \mathcal{V}(z) = \mathcal{V}(z+ \alpha),\quad \forall \ z\in {{\mathbb C}}/{{\mathbb Z}}\ {\rm with} \ |\Im z|<\frac{\epsilon_1}{2\pi}.$$
\[eq1estimee\] For any $r\in (0,\frac{\epsilon_1}{2\pi})$, there is $k_1=k_1(\alpha, r_0, r)>0$ such that for any $E\in \Sigma_{V,\alpha}$ satisfying $2\rho{(\alpha, S_{E}^V)}- k\alpha \in {{\mathbb Z}}$ with $|k|\geq k_1$, there exist $U\in C_{r}^\omega( {{\mathbb T}}, \mathrm{PSL}(2,{{\mathbb R}}))$, $\varphi\in{{\mathbb R}}$ and $n=n(k) \in {{\mathbb Z}}$ with $|n|\geq\frac{|k|}{4}$ such that $$\label{eqrota}
U(\cdot+\a)^{-1} S_{E}^{V}(\cdot) U(\cdot)=
\begin{pmatrix}
1 & \varphi \\[1mm]
0 & 1
\end{pmatrix}.$$ Moreover, for any $r''\in(0, r]$, $|U|_{r''} \leq C_4 e^{22 \pi r'' |n|}$ for some $C_4=C_4(\alpha,r_0,r'')>0$.
Fix $r\in (0,\frac{\epsilon_1}{2\pi})$, choose some $\delta\in(0,\frac{\epsilon_1}{2}-\pi r)$, and set $h:=r+\frac{\delta}{2\pi}$. Recall that $|\widehat u_j|\leq 1$, for all $j \in {{\mathbb Z}}$. In view of and , for any $r''
\in(0, h]$, there exists $C'_1=C'_1(\alpha,r_0,r'')>0$ such that for $|n|$ sufficiently large, we have $$\begin{aligned}
|\mathcal{V}|_{r''} &\leq 4|n|e^{5 \pi r''|n|}+4 C_1 e^{\pi r'' |n|} \sum_{j \geq 2|n|} e^{-(\epsilon_1- 2\pi r'') j}\leq C'_1 e^{5 \pi (r''+\delta) |n|}.\label{upper bound on v 2}\end{aligned}$$
Let us now show the lower bounds on $\mathcal{V}$. Set $I:=[-2|n|+1,2|n|-1]$ and consider the trigonometric polynomial $u^I\colon z\mapsto \sum_{j \in I} \widehat u_j e^{2 \pi {\rm i} j z}$. We define $\mathcal{U}^I, \mathcal{V}^I$ accordingly, for $u^I$ in place of $u$. By , for any $r''\in (0,h]$, we have $$\label{diff U and truncat}
|\mathcal{U}- \mathcal{U}^I|_{r''} \leq \frac{4 C_1}{1-e^{-(\epsilon_1- 2 \pi r'')}}e^{-2(\epsilon_1- 2 \pi r'')|n|}.
$$
Given any analytic function $f$ defined on the strip $\{|\Im z|<\frac{\epsilon_1}{2\pi}\}$, any $|y| \leq h$, let $f_y\colon x \mapsto f(x+\mathrm{i}y)$, $x\in{{\mathbb T}}$. Recall that $\int_{{\mathbb T}}u_y (x) dx =\widehat u_0=1$. By , we thus get $$\label{lowww bounddd cui}
\int_{{{\mathbb T}}} \left| {{\mathcal U}}_y^I(x) \right|dx\geq \left| \int_{{{\mathbb T}}} {{\mathcal U}}_y(x) dx \right|-\frac{4 C_1}{1-e^{-(\epsilon_1- 2 \pi h)}}e^{-2(\epsilon_1- 2 \pi h)|n|}\geq \frac{3}{4}$$ for $|n|\geq n_1(\alpha,\epsilon_1,h)$ large enough. Let us denote by $(\widehat{v}_j)_{j \in \mathbb{Z}}$ the Fourier coefficients of $V$, and let $\chi_{I}$ be the characteristic function of $I$. We get $$\label{equation inv first bloch wave}
S_{E}^{V}(\cdot)\, \mathcal{U}^I(\cdot)= e^{2 \pi \mathrm{i} \theta} \mathcal{U}^I(\cdot+\alpha)+e^{2 \pi\mathrm{i}\theta} \begin{pmatrix}
g(\cdot)\\
0
\end{pmatrix}$$ for some function $g\in C^\omega({{\mathbb T}},{{\mathbb C}})$ whose Fourier coefficients $(\widehat{g}_j)_{j \in \mathbb{Z}}$ satisfy $$\widehat g_j=\chi_{I}(j) \left(E-2 \cos2 \pi (\theta+j\a)\right) \widehat u_j - \sum_{l\in {{\mathbb Z}}} \chi_{I}(j-l) \widehat u_{j-l} \widehat v_l.$$ Since $\widehat H \widehat u=E \widehat u$, we also have $$\label{equation coeffc gg}
\widehat g_j=-\chi_{{{\mathbb Z}}\backslash I}(j) \left(E-2 \cos2 \pi (\theta+j\a)\right) \widehat u_j + \sum_{l\in {{\mathbb Z}}} \chi_{{{\mathbb Z}}\backslash I}(j-l) \widehat u_{j-l} \widehat v_l.$$ If $E \in \Sigma_{V,\alpha}$, then $|E|\leq 2+ |V|_{{\mathbb T}}\leq 2+ c_0 r_0^3$. By and , we therefore obtain $$\begin{aligned}
|g|_{r''} \leq& \ \sum_{j\in {{\mathbb Z}}} \chi_{{{\mathbb Z}}\backslash I}(j) (4+ c_0 r_0^3) \, C_1 e^{-(\epsilon_1-2 \pi r'')|j|}\\
&\ + \sum_{j,l\in {{\mathbb Z}}} \chi_{{{\mathbb Z}}\backslash I}(j-l) c_0 r_0^3 \, C_1 e^{-(\epsilon_1-2 \pi r'')|j-l|} e^{- 2\pi (r_0-r'')|l|}\\
\leq&\ C_1(4+ c_0 r_0^3)\left(1+ \sum_{l\in {{\mathbb Z}}} e^{- 2\pi (r_0-r'')|l|}\right)\sum_{|j|\geq 2|n|} e^{-(\epsilon_1- 2 \pi r'')|j|}\\
\leq&\ C'_2(\alpha,r_0,r'') e^{-2(\epsilon_1- 2 \pi r'')|n|}.$$ Together with , this implies that for all $z\in {{\mathbb C}}/{{\mathbb Z}}$ with $|\Im z| \leq r''\leq h$, all $m \geq 1$: $$\label{est2cw}
|\mathcal{U}^I(z+m\alpha)| \leq |\mathcal{A}_m(z)|\, |\mathcal{U}^I(z)|+ \sum_{j=1}^{m}|\mathcal{A}_{m-j}(z+j\alpha)|\cdot C'_2 e^{-2(\epsilon_1- 2 \pi r'')|n|}. $$
To get a lower bound on $\mathcal{V}$, we will use the following result of Avila-Jitomirskaya.
\[theo lagrang interp\] Let $\ell\geq 1$ and $1 \leq p \leq \lfloor q_{\ell+1}/q_\ell\rfloor$. If $P$ has essential degree[^6] at most $p q_\ell -1$ and $x_0 \in {{\mathbb T}}$, then for some absolute constant $K_0>0$, $$|P|_{{{\mathbb T}}} \leq K_0 q_{\ell+1}^{K_0 p} \sup\limits_{0 \leq m \leq p q_\ell -1}|P(x_0+m\alpha)|.$$
Choose $\ell\geq 1$ and $1 \leq p \leq \lfloor q_{\ell+1}/q_\ell\rfloor$ such that $(p-1) q_\ell -1 \leq 4 | n|< p q_\ell -1 \leq q_{\ell+1}$. In particular, under the assumption $\beta(\alpha)=0$, it implies that $$|P|_{{{\mathbb T}}} \leq K_0 e^{o(|n|)} \sup\limits_{0 \leq m \leq 4 | n|+q_l}|P(x+m\alpha)|\leq K_0 e^{o(|n|)} \sup\limits_{0 \leq m \leq 8 | n|}|P(x+m\alpha)|.$$ Since $\mathcal{U}^I$ has essential degree at most $4| n|$, by Lemma \[theo lagrang interp\], for any $x \in {{\mathbb T}}$, we have $$\label{ineqq}
|\mathcal{U}^I_y|_{{{\mathbb T}}} \leq K_0 e^{o(|n|)} \sup\limits_{0 \leq m \leq 8 | n|}|\mathcal{U}^I_y(x+m\alpha)|,\quad \forall \ |y|\leq r'' .$$
Let us show that for any $\delta>0$, there exists $C'_3=C'_3(\alpha,r_0,r'',\delta)>0$ such that $$\label{lowboundu}
\inf_{|\Im z|\leq r''} |\mathcal{U}^I(z)| \geq 2 C'_3 e^{-2\pi \delta |n|}.$$ Else for some $\delta'\in (0,\frac{\epsilon_1}{2\pi}-r'')$, we would have $|\mathcal{U}_y^I(x)| \leq e^{- 4\pi \delta' |n|}$ for $|n|$ arbitrarily large, and $y=y(n)\in[-r'',r'']$. We deduce from Corollary \[subexp groowth\], and that $|\mathcal{U}_{y(n)}^I|_{{\mathbb T}}\leq e^{-2\pi\delta'|n|}\leq \frac{1}{2}$ for $|n|$ large enough, which contradicts . Combining with and , for $\delta>0$ arbitrarily small, we get $$\label{subexpo}
\inf_{|\Im z|\leq r''} |{\mathcal{V}}(z)| \geq C'_3 e^{-\pi(\delta+r'') |n|},\quad \forall \ r''\in(0,h].$$
Applying Theorem \[corona theorem\] to $\mathcal{V}$, combining with and , we deduce that there exists $U_1=(\mathcal{V}, \mathcal{W}) \in C_{h}^\omega(\mathbb{T}, \mathrm{PSL}(2,{{\mathbb C}}))$ such that for all $r''\in(0,h]$, $$\label{rstim U1bisss}
|U_1|_{r''} \leq C'_4(\alpha,r_0,r'',\delta) e^{7\pi(r''+\delta) |n|}.$$ Indeed, one can choose $U_1\in C_{h}^\omega(\mathbb{T}, \mathrm{PSL}(2,{{\mathbb R}}))$, since $\mathcal{V}|_{\mathbb{T}}$ takes values in ${{\mathbb R}}^2\backslash\{0\}$, then one only need to replace $U_1$ by $U_1=(\mathcal{V},\widetilde{\mathcal{W}})$, with $\widetilde{\mathcal{W}}\colon z \mapsto \frac{1}{2}(\mathcal{W}(z)+\overline{\mathcal{W}(\overline{z}}))$.
By , there exists $\varphi^{(1)}\ \in C_{h}^\omega( \mathbb{T}, {{\mathbb R}})$ such that $$\label{eqatori}
U_1(\cdot+\a)^{-1} S_{E}^{V}(\cdot) U_1(\cdot)=
\begin{pmatrix}
1 & \varphi^{(1)}(\cdot)\\[1mm]
0 & 1
\end{pmatrix}.$$ By and , we have: $$|\varphi^{(1)}|_{h} \leq (4+c_0 r_0^3) (C_4')^2 e^{14 \pi (h+\delta)|n|}.$$ Since $\beta(\a)=0$, we can solve the cohomological equation $$\label{cohomolllfognri}
\phi(z+\a) - \phi(z) =\varphi^{(1)}(z)-\int_{{\mathbb T}}\varphi^{(1)}(x) \, dx,$$ with $\int_{{\mathbb T}}\phi(x)dx=0$. Moreover, $\phi\ \in C_{r}^\omega( \mathbb{T}, {{\mathbb R}})$, and for any $r''\in(0,r]$, one has $$\label{esti_phi}
|\phi|_{r''} \leq C_5'(\alpha,r_0,r'',\delta) e^{14\pi(r''+\delta) |n|}.$$ Let $U:=U_1
\begin{pmatrix}
1 & \phi\\
0 & 1
\end{pmatrix}$ and $\varphi:= \int_{{\mathbb T}}\varphi^{(1)}(x) \, dx$. and implies that $$\label{equation conjugaison finale}
U(\cdot+\a)^{-1}S_{E}^{V}(\cdot) U(\cdot)=
\begin{pmatrix}
1 & \varphi\\
0 & 1
\end{pmatrix} .$$ Obviously, $U\in C_{r}^\omega(\mathbb{T}, \mathrm{PSL}(2,{{\mathbb R}}))$. By and , we get the estimate $$|U|_{r''} \leq C_4 (\alpha,h_0,r'') e^{22\pi r''|n|},\quad \forall \ r''\in(0, r].$$
To find a relation between $k$ and $n=n(k)$, we estimate the topological degree of the conjugacy map $U$. Recall that $2\rho{(\alpha,S_E^V)} - k\alpha \in{{\mathbb Z}}$, and then, by , we have $|k|=|\mathrm{deg} U|$. Since $x \mapsto
\begin{pmatrix}
1 & \phi(x)\\
0 & 1
\end{pmatrix}$ is homotopic to the identity, we also know that $|k|=|\mathrm{deg} U_1|$. Thus, it remains to estimate $\mathrm{deg}U_{1}$. For this purpose, we look at the degree of the first column ${\mathcal{V}}\colon {{\mathbb T}}\rightarrow {{\mathbb R}}^2\backslash\{0\}$ of $U_1$. By , for any $\delta>0$, $$\label{estttttttti}
\inf\limits_{x \in {{\mathbb T}}} |{\mathcal{V}}(x)| \geq e^{-\pi \delta |n|}$$ for all sufficiently large $|n|$. Consider the truncated vector $\mathcal{V}^I$ as defined above. Since $\mathcal{V}(z)= e^{ \pi \mathrm{i}(n z- 2 \theta_0)}\mathcal{U}(z)$, we deduce from that $$|\mathcal{V}-\mathcal{V}^I|_{{\mathbb T}}\leq \frac{4 C_1}{1-e^{-\epsilon_1}} e^{-2\epsilon_1|n|}.$$ Comparing with , for any sufficiently large $|n|$, we obtain $$|\mathcal{V}(x)-\mathcal{V}^I(x)| \leq |{\mathcal{V}}(x)|, \quad \forall \ x\in {{\mathbb T}}.$$ By Rouché’s theorem, we deduce that $\mathrm{deg}{\mathcal{V}}=\mathrm{deg}\mathcal{V}^I$. Consider a coordinate of $\mathcal{V}^I$ which is not identically vanishing. It is a trigonometric polynomial of degree less than $4|n|$, so it has at most $4|n|$ zeros in ${{\mathbb T}}$, and we get $|\mathrm{deg} {\mathcal{V}}| \leq 4 |n|$. Therefore, for $|k|$ sufficiently large, we conclude that $|k| = |\mathrm{deg} U_1| \leq 4 |n|$.
Let us now estimate the size of $\varphi$. We will first need the following.
\[lemma first conjugaison\] Let $\{n_{l}\}_{l}$ be the set of resonances of $\theta$. For any $\delta\in(0,\frac{\epsilon_1}{20 \pi})$, there exist constants $C'_i=C'_i(\alpha,r_0,\delta)>0$, $i=6,7,8$, such that the following holds. There exists $B\in C^{\omega}({{\mathbb T}}, \mathrm{PSL}(2,{{\mathbb C}}))$ with $|B|_{\frac{\epsilon_1}{20 \pi}} \leq C'_6 e^{\delta |n_l|}$ such that $$\label{conj par complexe bis}
B(\cdot+\a)^{-1} S_E^V(\cdot) B(\cdot)=\begin{pmatrix}
e^{2 \pi \mathrm{i} \theta} & 0\\[1mm]
0 & e^{-2 \pi \mathrm{i} \theta}
\end{pmatrix}+\begin{pmatrix}
\beta_1(\cdot) & \beta(\cdot)\\[1mm]
\beta_3(\cdot) & \beta_4(\cdot)
\end{pmatrix},$$ with $|\beta_j|_{\frac{\epsilon_1}{20 \pi}} \leq C'_7 e^{-(\frac{\epsilon_1}{10}-2\pi\delta) |n_l|} $ for $j=1,3,4$ and $|\beta|_{{{\mathbb T}}} \leq C'_8 e^{-(\frac{\epsilon_1}{10}-2\pi \delta) |n_l|}$.
Without loss of generality, we assume in the following that $n_l\geq 0$. Let $u^J\colon z \mapsto \sum_{j \in J} \widehat u_j e^{2 \pi \mathrm{i} j z}$ and ${{{\mathcal U}}}^J\colon z \mapsto
\begin{pmatrix}
e^{2 \pi {\rm i}\theta} u^J(z)\\
u^J(z-\a)
\end{pmatrix}$, where $J:=[-\lfloor\frac{n_l}{4}\rfloor, \lfloor\frac{n_l}{4} \rfloor]$. Consequently, we have $$\label{equation inv first bloch wave bis bis}
S_E^V(\cdot)\, \mathcal{U}^J(\cdot)= e^{2 \pi \mathrm{i} \theta} \mathcal{U}^J(\cdot+\alpha)+e^{2 \pi\mathrm{i}\theta} \begin{pmatrix}
g^*(\cdot)\\
0
\end{pmatrix}$$ for some analytic function $g^*\in C^\omega({{\mathbb T}},{{\mathbb C}})$ whose Fourier coefficients $(\widehat g^*_j)_{j\in {{\mathbb Z}}}$ satisfy $$\label{fourier coeff gj bis 1}
\widehat g^*_j=\chi_{J}(j) \left(E-2 \cos2 \pi (\theta+j\a)\right) \widehat u_j - \sum_{m\in {{\mathbb Z}}} \chi_{J}(j-m) \widehat u_{j-m} \widehat v_m. $$ Since $\widehat H \widehat u=E \widehat u$, we also have $$\label{fourier coeff gj bis 2}
\widehat g^*_j=-\chi_{{{\mathbb Z}}\backslash J}(j) \left(E-2 \cos2 \pi (\theta+j\a)\right) \widehat u_j + \sum_{m\in {{\mathbb Z}}} \chi_{{{\mathbb Z}}\backslash J}(j-m) \widehat u_{j-m} \widehat v_m. $$ Let us assume that $j\notin J$, i.e., $|j|> \lfloor\frac{n_l}{4}\rfloor$. By , and since $2 n_{l-1}=o(\frac{n_l}{8})$ we have $|\widehat u_{j-m}| \leq C_1 e^{-\epsilon_1 |j-m|}$ for $\frac{n_l}{8}<|j-m|< \frac{n_l}{2}$, while $|\widehat u_{j-m}|\leq 1$ in other cases. Besides, $|\widehat v_m| \leq c_0 r_0^3 e^{-2 \pi r_0 |m|}$ for all $m \in {{\mathbb Z}}$. Thus, we deduce from that $$\begin{aligned}
|\widehat g^*_j| &\leq \sum_{ |j-m| \leq \frac{n_l}{8}} |\widehat u_{j-m}| |\widehat v_m| + \sum_{\frac{n_l}{8} < |j-m| \leq \frac{n_l}{4}} |\widehat u_{j-m}| |\widehat v_m| \\
&\leq c_0 r_0^3\left(\sum_{ |m| \geq |j|-\frac{n_l}{8}} e^{-2 \pi r_0 |m|} + \sum_{\frac{n_l}{8}< |j-m| \leq \frac{n_l}{4}} C_1 e^{-\epsilon_1 |j-m|} e^{-2 \pi r_0 |m|}\right) \\ &\leq c_0 r_0^3\left(\sum_{|m| \geq \frac{|j|}{2}} e^{-2 \pi r_0 |m|} + \frac{n_l}{4} C_1 e^{-\epsilon_1 |j|}\right)\\
&\leq C'_9 e^{- \frac{\epsilon_1}2 |j|}\end{aligned}$$ for some constant $C'_9=C'_9(\alpha,r_0)>0$. Similarly, if $j \in J$, by we get $$\begin{aligned}
|\widehat g^*_j| &\leq \sum_{\frac{n_l}{4} \leq |j-m|< \frac{n_l}{2}} |\widehat u_{j-m}| |\widehat v_m| + \sum_{|j-m| \geq \frac{n_l}{2}} |\widehat u_{j-m}| |\widehat v_m| \\ &\leq c_0 r_0^3\left(\frac{n_l}{2} C_1 e^{-\frac{\epsilon_1}{4}n_l}+ \sum_{|m|\geq \frac{n_l}{4}} e^{-2 \pi r_0 |m|}\right)\\
&\leq C'_{10} e^{-\frac{\epsilon_1}{4}n_l}\end{aligned}$$ for some constant $C'_{10}=C'_{10}(\alpha,r_0)>0$. We thus obtain $$\label{eq applica a bloch}
\sup_{|\Im z|\leq \frac{\epsilon_1}{20\pi}}|S_E^V(z)\, \mathcal{U}^J(z)- e^{2 \pi \mathrm{i} \theta} \mathcal{U}^J(z+\alpha)|=|g^*|_{\frac{\epsilon_1}{20\pi}} \leq C'_{11}(\alpha,r_0) e^{- \frac{\epsilon_1}{10} n_l}.$$ Arguing as in , we deduce that for any $\delta>0$, one has $$\label{uestimeeef bisbis}
\inf_{|\Im z|\leq \frac{\epsilon_1}{20\pi}} |\mathcal{U}^J(z)| \geq C'_{12}(\alpha,r_0,\delta) e^{-\delta n_l}.$$ On the other hand, since $|\widehat u_j|\leq C_1 e^{-\epsilon_1 |j|}$ for $2 n_{l-1} \leq |j| \leq \frac{1}{2} n_l$[^7], $|\widehat u_j| \leq 1$ in other cases, and since $n_{l-1}=o(n_l)$, we also have $$\label{uestimeeef bisbister}
\sup_{|\Im z|\leq \frac{\epsilon_1}{20\pi}} |\mathcal{U}^J(z)| \leq C'_{13}(\alpha,r_0,\delta) e^{\delta n_l}.$$ Combining $-$, and by Theorem \[corona theorem\], we can define $U_2 \in C_{ \frac{\epsilon_1}{20\pi}}^{\omega}({{\mathbb T}},\mathrm{PSL}(2,{{\mathbb C}}))$ with $\mathcal{U}^J$ as first column which satisfies $$\label{conj par complexe bis}
U_2(\cdot+\a)^{-1} S_E^{V}(\cdot) U_2(\cdot)=\begin{pmatrix}
e^{2 \pi \mathrm{i} \theta} & 0\\[1mm]
0 & e^{-2 \pi \mathrm{i} \theta}
\end{pmatrix}+\begin{pmatrix}
\widetilde{\beta}_1(\cdot) & \varphi^{(2)}(\cdot)\\[1mm]
\widetilde{\beta}_3(\cdot) & \widetilde{\beta}_4(\cdot)
\end{pmatrix}.$$ Besides, we have $|U_2|_{\frac{\epsilon_1}{20\pi}} \leq C'_{14}(\alpha,r_0,\delta) e^{\delta n_l}$, $|\varphi^{(2)}|_{ \frac{\epsilon_1}{20\pi}} \leq C'_{15}(\alpha,r_0,\delta) e^{2\delta n_l}$, and for $j=1,3,4$, $|\widetilde{\beta}_j|_{ \frac{\epsilon_1}{20\pi}} \leq C'_{16}(\alpha,r_0) e^{- \frac{\epsilon_1}{10} n_l}$.
Let us write $\varphi^{(2)}(z)=\sum_j\widehat \varphi_j e^{2 \pi \mathrm{i} j z}$, and let $\tau$ satisfy $$\varphi^{(2)} (z)-e^{-2 \pi \mathrm{i} \theta}\tau(z+\alpha)+e^{2 \pi \mathrm{i} \theta} \tau(z)= \sum_{|j| \geq n_l} \widehat \varphi_j e^{2 \pi \mathrm{i} j z}.$$ We have $\tau(z)=\sum_{|j| < n_l} \widehat \tau_j e^{2 \pi \mathrm{i} j z}$, where $
\widehat\tau_j:=\frac{-\widehat \varphi_j e^{-2 \pi \mathrm{i} \theta}}{1-e^{-2 \pi \mathrm{i} (2\theta-j\alpha)}}
$. By the assumption $\beta(\alpha)=0$, and the definition of resonances, for $j \neq n_l$, we have $$\|2\theta- j \alpha\|_{{{\mathbb T}}} \geq \|(j-n_l)\alpha\|_{{{\mathbb T}}}- \|2\theta- n_l\alpha\|_{{{\mathbb T}}} \geq e^{-o(|j-n_l|)}-e^{-\epsilon_0n_l }\geq \frac{1}{2}e^{-o(|j-n_l|)}.$$ Therefore, we deduce that $|\tau|_{h} \leq C'_{17}(\alpha,r_0,\delta) e^{2\delta n_l}$.
Let $B:=U_2
\begin{pmatrix}
1 & \tau\\
0 & 1
\end{pmatrix}$ conjugate the initial cocycle to the following: $$B(\cdot+\a)^{-1} S_E^V(\cdot) B(\cdot)=\begin{pmatrix}
e^{2 \pi \mathrm{i} \theta} & 0 \\[1mm]
0 & e^{-2 \pi \mathrm{i} \theta}
\end{pmatrix}+
\begin{pmatrix}
\beta_1(\cdot) & \beta_2(\cdot)+\varsigma(\cdot)\\[1mm]
\beta_3(\cdot) & \beta_4(\cdot)
\end{pmatrix}$$ with $\varsigma:z\mapsto \sum_{|j| \geq n_l} \widehat \varphi_{j} e^{2 \pi \mathrm{i} j z}$, $\beta_1(\cdot):=\widetilde{\beta}_1(\cdot)-\widetilde{\beta}_3(\cdot)\tau(\cdot+\alpha)$, $\beta_3(\cdot):=\widetilde{\beta}_3(\cdot)$, $\beta_4(\cdot):=\widetilde{\beta}_4(\cdot)+\widetilde{\beta}_3(\cdot)\tau(\cdot)$ and $$\beta_2(\cdot):=\widetilde{\beta}_1(\cdot) \tau(\cdot)- \widetilde{\beta}_4(\cdot)\tau(\cdot+\alpha)+\widetilde{\beta}_3(\cdot)\tau(\cdot)\tau(\cdot+\alpha).$$ By the estimates on $\widetilde{\beta}_1,\widetilde{\beta}_3,\widetilde{\beta}_4$, we have $|\beta_j|_{\frac{\epsilon_1}{20\pi}}\leq C'_{18}(\alpha,r_0,\delta) e^{-(\frac{\epsilon_1}{10} -4 \delta) n_l}$, for all $j=1,2,3,4$. On the other hand, $$\big|\varsigma\big|_{{{\mathbb T}}}\leq \sum_{|j| \geq n_l} |\varphi^{(2)}|_{h}e^{-\frac{\epsilon_1}{10} |j|} \leq C'_{19}(\alpha,r_0,\delta) e^{-(\frac{\epsilon_1}{10}-2 \delta) n_l},$$ which concludes.
Let us apply the previous result for $n_l=n$ when $2 \theta-n\alpha \in {{\mathbb Z}}$. According to Proposition \[lemma first conjugaison\], for any $\delta>0$, there exists $C'_{20}=C'_{20}(\alpha,r_0,\delta)>0$ such that $$\label{eqgorwth}
\sup_{0 \leq l \leq e^{\frac{\epsilon_1|n|}{10}}}|{{\mathcal A}}_l|_{{{\mathbb T}}} \leq C'_{20} e^{\delta |n|}.$$ On the other hand, for any $l \in \mathbb{N}$, we obtain by iterating : $$\begin{pmatrix}
1 & l \varphi\\[1mm]
0 & 1
\end{pmatrix}=U(\cdot+l \alpha)^{-1} {{\mathcal A}}_l(\cdot) \, U(\cdot).$$ Take $l:=\left\lfloor e^{\frac{\epsilon_1|n|}{10}}\right\rfloor$. By Proposition \[eq1estimee\], $|U|_{{{\mathbb T}}},|U^{-1}|_{{{\mathbb T}}} \leq C'_{21} e^{\delta |n|}$ for some $C'_{21}=C'_{21}(\alpha,r_0,\delta)>0$, and by , $|{{\mathcal A}}_{l}|_{{{\mathbb T}}} \leq C'_{20} e^{\delta |n|}$. We conclude that $e^{\frac{\epsilon_1|n|}{10}} |\varphi| \leq C'_{20}(C'_{21})^2 e^{3 \delta |n|}$, and the desired estimate on $\varphi$ follows.
\[compare\] The readers should compare our Proposition \[lemma first conjugaison\] with Theorem 3.8 in [@A1]. In Theorem 3.8 in [@A1], we only know that $\beta(z)$ has exponential decay, while the decay rate is very small. However, in our result, $\beta(z)$ has a large and explicit decay, which allows us to show that $\varphi$ has large exponential decay.
Quantitative almost reducibility – almost Mathieu case
------------------------------------------------------
For almost Mathieu operators, we have the following improved result.
\[thm\_almost\_almost-2\] Let $\alpha \in {{\mathbb R}}$ satisfy $\beta(\alpha)=0$. If $0<\lambda<1$, then for any $r\in (0,-\frac{1}{2\pi}\ln \lambda)$, $E \in \Sigma_{\lambda,\alpha}$, the following holds on $\{|\Im z|<r\}$:
1. either $(\alpha,S_E^\lambda)$ is almost reducible to $(\alpha,R_{\theta})$ for some $\theta=\theta(E)\in {{\mathbb R}}$: for any $\varepsilon>0$, there exists $U\in C^\omega_{r}({{\mathbb T}},\mathrm{PSL}(2,{{\mathbb R}}))$ such that $$|U(\cdot+\alpha)^{-1} S_E^\lambda(\cdot) U(\cdot)-R_{ \theta}|_{r} < \varepsilon;$$
2. or $(\alpha,S_E^\lambda)$ is reducible:
1. if $2 \rho(\alpha,S_E^\lambda)-j\alpha \notin {{\mathbb Z}}$ for any $j \in {{\mathbb Z}}$, then $(\alpha,S_E^\lambda)$ is reducible to $(\alpha,R_\theta)$ for some $\theta=\theta(E)\in {{\mathbb R}}$;
2. if $2 \rho(\alpha,S_E^\lambda)-k\alpha \in {{\mathbb Z}}$ for some $k \in {{\mathbb Z}}$, then there is $k_2=k_2(\lambda,\alpha, r)>0$ such that if $|k|\geq k_2$, then there exist $\varphi \in {{\mathbb R}}\backslash\{0\}$ and $U \in C_{r}^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ such that $$\label{reduci_parabis}
U(\cdot+\a)^{-1} S_{E}^{\lambda}(\cdot) U(\cdot) =
\begin{pmatrix}
1 & \varphi \\
0 & 1
\end{pmatrix},$$ and there is $n=n(k) \in {{\mathbb Z}}$ with $|n|\geq\frac{|k|}{4}$, such that $|\varphi| \leq C_5 e^{- \frac{2 \pi r}{3} |n|}$ for some $C_5=C_5(\lambda,\alpha,r)>0$ and for any $0<r'' \leq r$, $|U|_{r''} \leq C_6 e^{22 \pi r'' |n|}$ for some $C_6=C_6(\lambda,\alpha,r'')>0$.
Fix $0< r<-\frac{\ln \lambda}{2\pi}$, and set $\tilde r:=\frac12(-\frac{\ln\lambda}{2\pi}+r)$. Hence $-\frac{\ln\lambda}{2\pi}-\tilde r=\tilde r-r$. The proof that follows is similar to those of Proposition \[eq1estimee\] and Proposition \[lemma first conjugaison\], but with the improved estimates obtained in Theorem \[thm\_almost\_almost-1\] for almost Mathieu operators. For any $E \in \Sigma_{\lambda,\alpha}$, there exist $\theta=\theta(E)\in {{\mathbb R}}$ and $(\widehat u_j)_{j \in {{\mathbb Z}}}$ satisfying $\widehat H_{\lambda,\alpha,\theta} \widehat u= E \widehat u$ with $\widehat u_0=1$ and $|\widehat u_j|\leq 1$ for every $j \in {{\mathbb Z}}$. Let us denote by $\{n_{l}\}_{l}$ the set of $\epsilon_0-$resonances of $\theta$. We fix $\eta\in(0, \frac{\pi(\tilde r-r)}{-4\ln\lambda})$ sufficiently small and let $\delta\in(-\eta\ln\lambda, \frac{\pi(\tilde r-r)}{4})$. By Theorem \[thm\_almost\_almost-1\], there exists $N_0\geq 0$ such that for $|n_{l}| \geq N_0$, we have $$\label{decay expllc}
|\widehat u_j| \leq e^{-2 \pi r |j|},\quad \forall \ 2 |n_{l-1}|+\eta |n_{l}| < |j| < \frac12 |n_{l}|.$$
Let $n_l\geq N_0$ and set $J:=[-\lfloor\frac{n_l}{2}\rfloor+2, \lfloor\frac{n_l}{2}\rfloor-2]$. We define $u^J\colon z \mapsto \sum_{j \in J} \widehat u_j e^{2 \pi \mathrm{i} j z}$ and ${{{\mathcal U}}}^J\colon z \mapsto
\begin{pmatrix}
e^{2 \pi {\rm i}\theta} u^J(z)\\
u^J(z-\a)
\end{pmatrix}$. Then $$S_E^{\lambda}(\cdot)\, \mathcal{U}^J(\cdot)= e^{2 \pi \mathrm{i} \theta} \mathcal{U}^J(\cdot+\alpha)+e^{2 \pi\mathrm{i}\theta} \begin{pmatrix}
g^*(\cdot)\\
0
\end{pmatrix}$$ for some analytic function $g^*$ whose Fourier coefficients $(\widehat g^*_j)_{j\in {{\mathbb Z}}}$ satisfy $$\label{fourier coeff gj ter 11}
\widehat g^*_j=\chi_{J}(j) \left(E-2 \cos2 \pi (\theta+j\a)\right) \widehat u_j - \lambda \sum_{m=\pm1} \chi_{J}(j-m) \widehat u_{j-m}. $$ Since $\widehat H \widehat u=E \widehat u$, we also have $$\label{fourier coeff gj ter 22}
\widehat g^*_j=-\chi_{{{\mathbb Z}}\backslash J}(j) \left(E-2 \cos2 \pi (\theta+j\a)\right) \widehat u_j + \lambda \sum_{m=\pm1} \chi_{{{\mathbb Z}}\backslash J}(j-m) \widehat u_{j-m} . $$ Applying either or , we see that $\widehat{g}_j^*=0$ if $|j|\not\in(\lfloor\frac{n_l}{2}\rfloor-4,\lfloor\frac{n_l}{2}\rfloor)$. Then by , there exists a constant $c'_{1}=c'_{1}(\lambda,\alpha,\tilde r)>0$ such that $$\label{def sup g' on r'}
\sup_{|\Im z|\leq \tilde r}|S_E^\lambda(z)\, \mathcal{U}^J(z)- e^{2 \pi \mathrm{i} \theta} \mathcal{U}^J(z+\alpha)|=|g^*|_{\tilde r} \leq c'_{1} e^{-\pi (-\frac{\ln\lambda}{2\pi}-\tilde r) n_l}=c'_{1} e^{-\pi (\tilde r-r) n_l}.$$ Similar to (\[uestimeeef bisbis\]) and (\[uestimeeef bisbister\]), we deduce that $$c'_{2}(\alpha,\lambda,\delta,\tilde r) e^{-\delta n_l} \leq \inf_{|\Im z|\leq \tilde r} |\mathcal{U}^J(z)| \leq \sup_{|\Im z|\leq \tilde r} |\mathcal{U}^J(z)| \leq c'_{3}(\alpha,\lambda,\delta,\tilde r) e^{\delta n_l}.$$ As previously in , we can define $U_2 \in C^{\omega}_{\tilde r}({{\mathbb T}},\mathrm{PSL}(2,{{\mathbb C}}))$ with first column $\mathcal{U}^J$ such that $$U_2(\cdot+\a)^{-1} S_E^\lambda(\cdot) U_2(\cdot)=\begin{pmatrix}
e^{2 \pi \mathrm{i} \theta} & 0\\[1mm]
0 & e^{-2 \pi \mathrm{i} \theta}
\end{pmatrix}+\begin{pmatrix}
\widetilde{\beta}_1(\cdot) & \varphi^{(2)}(\cdot)\\[1mm]
\widetilde{\beta}_3(\cdot) & \widetilde{\beta}_4(\cdot)
\end{pmatrix}$$ for some function $\varphi^{(2)}\colon z \mapsto\sum_j\widehat \varphi_j e^{2 \pi \mathrm{i} j z}$. Besides, $|U_2|_{\tilde r} \leq c'_4(\alpha,\lambda,\delta,\tilde r) e^{\delta n_l}$, $|\varphi^{(2)}|_{\tilde r} \leq c'_5(\alpha,\lambda,\delta,\tilde r) e^{2\delta n_l}$, and for $j=1,3,4$, $|\widetilde{\beta}_j|_{\tilde r} \leq c'_6(\alpha,\lambda,\tilde r) e^{- \pi (\tilde r-r) n_l}$.
Consequently, one can define $B \in C^{\omega}_{\tilde r}({{\mathbb T}},\mathrm{PSL}(2,{{\mathbb C}}))$ such that $$\label{con-2}
B(\cdot+\a)^{-1} S_E^{\lambda}(\cdot) B(\cdot)=\begin{pmatrix}
e^{2 \pi \mathrm{i} \theta} & 0 \\[1mm]
0 & e^{-2 \pi \mathrm{i} \theta}
\end{pmatrix}+
\begin{pmatrix}
\beta_1(\cdot) & \beta_2(\cdot)+\varsigma(\cdot)\\[1mm]
\beta_3(\cdot) & \beta_4(\cdot)
\end{pmatrix}$$ with $|B|_{\tilde r} \leq c'_7(\alpha,\lambda,\delta,\tilde r) e^{2\delta n_l},$ and $|\beta_j|_{\tilde r}\leq c'_8(\alpha,\lambda,\delta,\tilde r) e^{-( \pi(\tilde r-r) -4 \delta) n_l}$, for all $j=1,2,3,4$. Moreover, $\varsigma(z):=\sum_{|j| \geq n_l} \widehat \varphi_{j} e^{2 \pi \mathrm{i} j z}$. Therefore, we have $$\left|\varsigma\right|_{r}\leq \sum_{|j| \geq n_l} |\varphi^{(2)}|_{\tilde r}\, e^{-2 \pi (\tilde r-r) |j|} \leq c'_{5} (\alpha,\lambda,\delta,\tilde r)e^{-( \pi(\tilde r-r)- 4\delta)n_l}.$$
If $\theta$ is $\epsilon_0-$resonant, which means that the collection $\{n_l\}$ is infinite, then we set $U:=\frac{1}{1+ \mathrm{i}}B \begin{pmatrix}
\mathrm{i} & -1\\
\mathrm{i} & 1
\end{pmatrix}\in C^\omega_{r}({{\mathbb T}}, \mathrm{PSL}(2,{{\mathbb R}}))$, so that $$|U(\cdot+\alpha)^{-1} S_E^{\lambda}(\cdot) U(\cdot)- R_\theta |_{r} \leq c'_9(\alpha,\lambda,\delta,\tilde r) e^{-( \pi(\tilde r-r) -4 \delta) n_l}.$$ This concludes the proof of the almost reducibility statement in (1).
Assume now that $\theta$ is not $\epsilon_0-$resonant, then $\theta=\pm \rho{(\alpha, S_E^{\lambda})}+\frac{k' \alpha}2$ for some $k' \in {{\mathbb Z}}$ (see Remark 4.2 in [@AvilaJito]). If $2 \rho{(\alpha, S_E^\lambda)} - j\alpha \notin {{\mathbb Z}}$ for all $j\in {{\mathbb Z}}$, then $2 \theta - j\alpha \notin {{\mathbb Z}}$ for all $j \in {{\mathbb Z}}$. Theorem \[thm\_almost\_almost-2\] (a) actually follows from Theorem 2.5 of [@AvilaJito]. Now if $2 \rho{(\alpha, S_E^\lambda)} - k\alpha \in {{\mathbb Z}}$ for some $k \in {{\mathbb Z}}$, we thus have $2 \theta-n\alpha \in {{\mathbb Z}}$ for some $n=n(k) \in {{\mathbb Z}}$. As in Proposition \[eq1estimee\], there exist $U \in C^\omega_{\tilde r} ( {{\mathbb T}},\mathrm{PSL}(2,{{\mathbb C}}))$, $\varphi \in {{\mathbb R}}$ such that $$U(\cdot+\a)^{-1} S_E^\lambda(\cdot) \, U(\cdot)=
\begin{pmatrix}
1 & \varphi \\[1mm]
0 & 1
\end{pmatrix}.$$ Note that the case $\varphi=0$ cannot happen. Otherwise $\widehat H_{\lambda,\alpha,\theta}=\lambda H_{ \lambda^{-1},\alpha,\theta}$ would have an eigenvalue with two linearly independent eigenvectors in $\ell^2({{\mathbb Z}})$, which is impossible by the limit-point character of Schrödinger operators. The estimate on $U$ and the relation between $k$ and $n$ are obtained as in Proposition \[eq1estimee\]. To estimate $\varphi$, we argue as in the proof of Theorem \[prop\_duality\_para\], but using the following improved version of Proposition \[lemma first conjugaison\] in the case of almost Mathieu operators.
Assume that $2\theta-n\alpha\in{{\mathbb Z}}$ for some $n \in {{\mathbb Z}}$. For any $\delta \in(0, -\frac{\ln \lambda}{6 \pi})$, there exist $c'_i=c'_i(\alpha,\lambda,\delta)>0$, $i=10,11,12$, and there exists $B\in C^{\omega}({{\mathbb T}}, \mathrm{PSL}(2,{{\mathbb C}}))$ with $|B|_{-\frac{\ln \lambda}{6 \pi}} \leq c'_{10} e^{\delta |n|}$ such that $$\label{conj par complexe bis-2}
B(\cdot+\a)^{-1} S_E^\lambda(\cdot) B(\cdot)=\begin{pmatrix}
e^{2 \pi \mathrm{i} \theta} & 0\\[1mm]
0 & e^{-2 \pi \mathrm{i} \theta}
\end{pmatrix}+\begin{pmatrix}
\beta_1(\cdot) & \beta(\cdot)\\[1mm]
\beta_3(\cdot) & \beta_4(\cdot)
\end{pmatrix},$$ with $|\beta_j|_{-\frac{\ln \lambda}{6 \pi}}\leq c'_{11} e^{- (-\frac{\ln \lambda}{3}-2 \pi\delta) |n|} $ and $|\beta|_{{{\mathbb T}}} \leq c'_{12} e^{-(-\frac{\ln \lambda}{3}-2 \pi\delta) |n|}$.
Set $h:=-\frac{\ln \lambda}{6 \pi}$ and let $\delta >0$ be taken arbitrarily small. With the same notations as above, for $|n|$ which is large enough, becomes now $$|g^*|_{h} \leq c'_1(\alpha,\lambda,\delta) e^{-2 \pi (h-\delta) |n|}.$$ Consequently, holds with $|B|_{h} \leq c'_7 (\alpha,\lambda,\delta) e^{\delta |n|}$, $|\beta_j|_{h}\leq c'_8(\alpha,\lambda,\delta) e^{-2 \pi( h -\delta) |n|}$, for $j=1,2,3,4$, while $$|\varsigma|_{{{\mathbb T}}}\leq \sum_{|j| \geq |n|} |\varphi^{(2)}|_{h}e^{-2 \pi h |j|} \leq c'_{5}(\alpha,\lambda,\delta) e^{-2 \pi( h- \delta) |n|},$$ which concludes.
Global to local reduction {#subsectlocalglobal}
=========================
In this section, we extend the reducibility results for Schrödinger cocycles, which were obtained previously for small potentials, to the global subcritical regime.
General subcritical potential
-----------------------------
We first consider typical $V\in C^\omega({{\mathbb T}},{{\mathbb R}})$ such that $\Sigma_{V,\alpha}$ presents the structure given in Theorem \[global-red\]. We denote by $\{I_i\}_{1\leq i \leq m}$ the intervals such that the energies in $\Sigma_{V,\alpha}\cap I_i$ are subcritical, and by $\{J_i\}_{1\leq i \leq m'}$ (the number $m'$ of intervals can be $m-1$, $m$ or $m+1$) the intervals such that the energies in $\Sigma_{V,\alpha}\cap J_i$ are supercritical. Let $\Sigma^{\mathrm{sup}}_{V,\alpha}:=\bigcup_{i}(\Sigma_{V,\alpha} \cap J_i)$ and $\Sigma^{\rm sub}_{V,\alpha}:=\bigcup_{i}(\Sigma_{V,\alpha} \cap I_i) $.
Let us focus on the case where $E\in\Sigma^{\rm sub}_{V,\alpha}$. By Theorem \[arc-conjecture\], the Schrödinger cocycle $(\alpha, S^V_E)$ is almost reducible. While almost reducibility allows one to conjugate the dynamics of a cocycle close to constant, it is convenient to have the conjugated cocycle in Schrödinger form, since many results (particularly those depending on Aubry duality) are obtained only in this setting.
\[prop\_ar\_1\] Let $\alpha\in{{\mathbb R}}\backslash{{\mathbb Q}}$ satisfy $\beta(\alpha)=0$. There exists $h_1=h_1(V,\alpha)>0$ such that for any $\eta>0$, $E\in \Sigma_{V,\alpha}^{\rm sub}$, one can find $\Phi_E\in C^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ with $|\Phi_E|_{h_1}<\Lambda$ for some $\Lambda=\Lambda(V,\alpha, \eta,h_1)>0$, $E_*=E_*(E)$ locally constant (as a function of $E$), and $V_{*}=V_*(E) \in C^\omega_{h_1}({{\mathbb T}},{{\mathbb R}})$, $|V_*|_{h_1}<\eta$, such that $$\label{conjPhi}
\Phi_E(\cdot+\alpha)^{-1} S_E^V(\cdot) \Phi_E(\cdot)=S_{E_*}^{V_*}(\cdot).$$
The crucial fact in this proposition is that we can choose $h_1$ to be independent of $E$ and $\eta$, and choose $\Gamma$ to be independent of $E$.
The key ingredient for us is the following:
\[lemma conjugaaison socrhd\] Let $\alpha\in{{\mathbb R}}\backslash{{\mathbb Q}}$ and $A \in C_{h_*}^\omega({{\mathbb T}},{\rm SL}(2,{{\mathbb R}}))$ for some $h_*>0$, such that $(\alpha,A)$ is almost reducible. There exists $h_0\in(0,h_*)$ such that for any $\eta>0$, one can find $V \in C_{h_0}^\omega({{\mathbb T}},{{\mathbb R}})$ with $|V|_{h_0} < \eta$, $E \in {{\mathbb R}}$, and $Z \in C_{h_0}^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ such that $$Z(\cdot+\alpha)^{-1}A(\cdot)Z(\cdot)=S_E^V(\cdot).$$ Moreover, for every $0< h\leq h_0$, there is $\delta>0$ such that if $A' \in C_{h}^\omega({{\mathbb T}},{\rm SL}(2,{{\mathbb R}}))$ satisfies $|A-A'|_{h} < \delta$, then there exist $ V' \in C_{h}^\omega({{\mathbb T}},{{\mathbb R}})$ with $|V'|_{h} < \eta$ and $Z' \in C_{h}^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ such that $|Z-Z'|_{h} < \eta$ and $$Z'(\cdot+\alpha)^{-1}A'(\cdot) Z'(\cdot)=S_E^{V'}(\cdot).$$
For any $E_0\in\Sigma^{\rm sub}_{V,\alpha}$, the cocycle $(\alpha, S_{E_0}^V)$ is subcritical, hence almost reducible by Theorem \[arc-conjecture\]. Fix $\eta>0$. By Lemma \[lemma conjugaaison socrhd\], there is $h_0= h_0(E_0,V,\alpha)>0$, such that one can find $V_*(E_0) \in C_{h_0}^\omega({{\mathbb T}},{{\mathbb R}})$ with $|V_*(E_0)|_{h_0}\leq \eta$, $E_*= E_*(E_0)\in{{\mathbb R}}$ and $\Psi_{E_0}\in C_{h_0}^\omega({{\mathbb T}},\mathrm{PSL}(2,{{\mathbb R}}))$ with $|\Psi_{E_0}|_{h_0}\leq\tilde\Lambda$ for $\tilde\Lambda=\tilde\Lambda(V,\alpha, \eta, h_0,E_0)>0$ such that $$\Psi_{E_0}(\cdot+\alpha)^{-1} S_{E_0}^V(\cdot) \Psi_{E_0}(\cdot)= S_{ E_*}^{V_*(E_0)}(\cdot).$$ Moreover, for every $0<h \leq h_0$, there exists $\delta > 0$ such that for any $E\in (E_0-\delta, E_0 +\delta)$, one can find $\Psi_{E}\in C_{h}^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ with $|\Psi_{E}-\Psi_{E_0}|_{h}\leq \eta$, and $V'_*(E)\in C_{h}^\omega({{\mathbb T}},{{\mathbb R}})$ with $|V'_*(E)|_{h}\leq \eta$ satisfying $$\Psi_{E}(\cdot + \alpha)^{-1} S_{E}^V(\cdot) \Psi_{E} (\cdot) = S_{ E_*}^{ V'_*(E)}(\cdot).$$
By Theorem \[global-red\], $\Sigma^{\rm sub}_{V,\alpha}$ is compact, by compactness argument, we obtain $h_1=h_1(V,\alpha)$ and $\Lambda=\Lambda(V,\alpha,\eta,h_1)$, both independent of $E$, such that for any $E \in \Sigma^{\rm sub}_{V,\alpha}$, there exist $\Phi_E\in C_{h_1}^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ with $|\Phi_E|_{h_1} < \Lambda$, and $V_*=V_*(E) \in C^\omega_{h_1}({{\mathbb T}},{{\mathbb R}})$ with $|V_*(E)|_{h_1}<\eta$, such that $(\ref{conjPhi})$ holds.
\[prop\_duality\_global\] Let $\alpha\in{{\mathbb R}}\backslash{{\mathbb Q}}$ satisfy $\beta(\alpha)=0$. There exist $h_1=h_1(V,\alpha)>0$, $0<c=c(V,\alpha)<h_1$, $\tilde{k}=\tilde{k}(V,\alpha)>0$, such that for any $E \in \Sigma^{\rm sub}_{V,\alpha}$ satisfying $2\rho(\alpha, S_E^{V})- k\alpha\in {{\mathbb Z}}$ with $|k| \geq \tilde{k}$, there exist $Y \in C^\omega_{\frac{c}{2\pi}}({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ and $\varphi\in {{\mathbb R}}$ s.t. $$Y(\cdot+\a)^{-1} S_{E}^{V}(\cdot) Y(\cdot) =
\begin{pmatrix}
1 & \varphi\\
0 & 1
\end{pmatrix}.$$ Moreover, there is $n=n(k)\in {{\mathbb Z}}$ satisfying $ |n|\geq \frac{|k|}{5}$ such that $|\varphi| \leq C e^{-\frac{c}{10}|n|}$ and $$|Y|_{r''} \leq C_7(V,\alpha,r'') e^{22\pi r''|n|},\quad \forall \ r''\in(0,\frac{c}{2\pi}).$$
Let $(\eta_n)_n$ be a sequence of positive numbers going to zero. For any $E \in \Sigma^{\rm sub}_{V,\alpha}$, we can apply Proposition \[prop\_ar\_1\] and get $h_1=h_1(V,\alpha)>0$, $\Lambda_n=\Lambda_n(V,\alpha,\eta_n,h_1)>0$, $\Phi_E^{n}\in C^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$, $E^n=E^n(E)\in{{\mathbb R}}$ and $V^n=V^n(E) \in C^\omega_{h_1}({{\mathbb T}},{{\mathbb R}})$ such that $$\Phi_E^{n}(\cdot+\alpha)^{-1} S_E^V(\cdot) \Phi_E^{n}(\cdot)=S_{E^{n}}^{V^n}(\cdot),$$ with $|\Phi_E^{n}|_{h_1}<\Lambda_n$ and $|V^n|_{h_1}<\eta_n$. Note that since $h_1(V,\alpha)$ is fixed, one can always find $N_*=N_*(h_1)$ large enough such that $\eta_{N_*} \leq c_0 h_1^3$, where $c_0>0$ is the absolute constant given by Theorem \[almostredth\]. It follows that $$|\Phi_E^{N_*}|_{h_1} \leq \Lambda = \Lambda(V,\alpha, c_0h^3_1, h_1).$$ By footnote 5 of [@A3], $|{\rm deg}\Phi_E^{N_*}|\leq C|\ln \Lambda|$ with $C=C(V,\alpha)$ independent of $E$.
Assume that $2\rho{(\alpha, S_E^{V})}-k\alpha\in{{\mathbb Z}}$. We abbreviate $E_*=E^{N_*}(E)$, $V_*=V^{N_*}(E)$, and $k_*={\rm deg}\Phi_E^{N_*}$. Then $2\rho{(\alpha, S_{ E_*}^{ V_{*}})}-( k -k_*)\alpha\in{{\mathbb Z}}$. Clearly, $ E_* \in \Sigma_{ V_{*}, \alpha}$ since uniform hyperbolicity is invariant under conjugacy, and then $(\alpha,\, S^{V_{*}}_{ E_*})$ is not uniformly hyperbolic. Applying Theorem \[prop\_duality\_para\], we thus get $c=c(h_1)>0$, $k_1=k_1(\alpha, h_1, \frac{c}{2\pi})$, $\varphi\in {{\mathbb R}}$ and $U \in C^\omega_{\frac{c}{2\pi}}(2{{\mathbb T}}, {\rm SL}(2,{{\mathbb R}}))$ such that $$U(\cdot+\alpha)^{-1}S^{V_{*}}_{E_*}(\cdot)U(\cdot)=\begin{pmatrix}
1 & \varphi \\
0 & 1
\end{pmatrix}.$$ Moreover, if $|k-k_*|\geq k_1$, then we have $$|\varphi| \leq C_3(\alpha,h_1, \frac{c}{2\pi}) e^{-\frac{c}{10}|n|}< C(V,\alpha) e^{-\frac{c}{10}|n|}$$ and for any $r''\in(0, \frac{c}{2\pi})$, $$|U|_{r''} \leq C_4 (\alpha, h_1, r'') e^{22\pi r'' |n|}$$ for some $n$ satisfying $|{\rm deg}U|=|k-k_*| \leq 4|n|$. Set $Y := \Phi^{N_*}_{E_*} U$. Thus, if $$|k|\geq k_1+C |\ln\Lambda|:=\tilde k(V,\alpha),$$ and hence $|k-k_*| \geq k_1$, then $$|{\rm deg}Y|=|k| \leq |k_*|+|k-k_*|\leq C |\ln\Lambda| + 4 |n|\leq 5 |n|.$$ Furthermore, $$|Y|_{r''}\leq \Lambda(V,\alpha, c_0h^3_1, h_1)
C_4(\alpha,h_1,r'') e^{22\pi r'' |n|} \leq C_7(V,\alpha,r'') e^{22\pi r'' |n|}.$$ Finally, let us emphasize that the constants $c=c(h_1)$, $C_3=C_3(\alpha, h_1,\epsilon)$, which come from almost localization estimates, only depend on the sizes of the strip, but not on the potential $V_*$, thus not on our choice of $E$. It is the main reason why the estimates we get are uniform with respect to $k$.
Almost Mathieu operator
-----------------------
Now we focus on the subcritical almost Mathieu operator. Compared with Corollary \[prop\_duality\_global\], we obtain even stronger results for $\alpha \in {\rm DC}$:
\[redu amo case\] Let $\alpha\in {\rm DC}$. Given $0<\lambda<1$, we consider the operator $H_{\lambda, \alpha,\theta}$. For any $E\in \Sigma_{\lambda,\alpha}$ satisfying $2\rho{(\alpha, S_E^\lambda)} - k \alpha\in {{\mathbb Z}}$ with $k\in {{\mathbb Z}}\backslash\{0\}$, for any $0< r<\frac{1}{2\pi}|\ln \lambda|$, there exist $U \in C_{r}^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ and $\varphi \in {{\mathbb R}}\backslash\{0\}$ such that $$\label{reduci_para}
U(\cdot+\a)^{-1} S_{E}^{\lambda}(\cdot) U(\cdot) =
\begin{pmatrix}
1 & \varphi \\
0 & 1
\end{pmatrix}.$$ Moreover, there exist $C_8, C_9>0$, depending on $\lambda,\alpha,r$, such that $|\varphi| \leq C_{8} e^{- 2\pi r |k|}$ and for any $r''\in (0,r]$, $|U|_{r''} \leq C_{9} e^{\frac{3 \pi r''}2 |k|}$.
Assume that $\alpha\in {\rm DC}(\gamma,\tau)$. Fix $r\in (0,\frac{1}{2\pi}|\ln \lambda|)$. Let $\tilde r:=\frac12(-\frac{\ln \lambda}{2\pi}+r)$. By Theorem \[thm\_almost\_almost-2\], for any sequence of positive numbers $(\eta_n)_n$ going to zero, there are $\Phi_E^{n}\in C_{\tilde r}^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$, $F_n\in C_{\tilde r}^\omega({{\mathbb T}},{\rm gl}(2,{{\mathbb R}}))$ and $\phi_n=\phi_n(E)\in{{\mathbb T}}$ such that $$\Phi_E^{n}(\cdot+\alpha)^{-1} S_E^{\lambda }(\cdot) \Phi_E^{n}(\cdot)=R_{\phi_n}+F_n(\cdot),$$ with $|F_n|_{\tilde r}< \eta_n/2$ and $|\Phi_E^{n}|_{\tilde r}<\Gamma_n$ for some $\Gamma_n=\Gamma_n(\lambda,\alpha, \eta_n, \tilde r,E)>0$. As a consequence, for any $E'\in {{\mathbb R}}$, one has $$\left|\Phi_E^{n}(\cdot+\alpha)^{-1} S_{E'}^{\lambda}(\cdot) \Phi_E^{n}(\cdot) - R_{\phi_n}\right|_{\tilde r} < \frac{\eta_n}{2}+ |E-E'| \, | \Phi_E^{n}|_{\tilde r}^2.$$ It follows that with the same $\Phi_E^{n}$, we have $|\Phi_E^{n}(x+\alpha)^{-1} S_{E'}^{\lambda}(x) \Phi_E^{n}(x) - R_{\phi_n}|_{ \tilde r} < \eta_n$ for any energy $E'$ in a neighborhood $\mathcal{U}(E)$ of $E$.
One can always take $N_*$ large enough such that $\eta_{N_*} \leq \varepsilon_*(\gamma, \tau, \tilde r, r, 1)$, where $\varepsilon_*(\gamma, \tau, \tilde r, r, 1)$ is define in Theorem \[thm\_gap\_edge\_SL\] (see also Remark \[uniformcons\]). It follows that $$\label{norm}|\Phi_E^{N_*}|_r \leq \Gamma:=\Gamma_{N_*}(\lambda,\alpha, \eta_{N_*},\tilde r,E).$$ By the compactness of $\Sigma_{\lambda,\alpha}$, $\Gamma>0$ can be chosen independently of $E$.
Let $k_*:={\rm deg}\Phi_E^{N_*}$, the assumption $2\rho{(\alpha, S_E^{\lambda})}-k\alpha\in{{\mathbb Z}}$ implies $2\rho{(\alpha,R_{\phi_{N_*}}+F_{N_*})}- (k-k_*)\alpha\in {{\mathbb Z}}$. Since we have chosen $\eta_{N_*} \leq \varepsilon_*(\gamma, \tau, \tilde r, r, 1)$, then by Theorem \[thm\_gap\_edge\_SL\], we get $\phi\in{{\mathbb R}}$ and $W\in C_{r}^\omega({{\mathbb T}}, {\rm PSL}(2,{{\mathbb R}}))$ such that $$W(\cdot+\alpha)^{-1} (R_{\phi_{N_*}}+F_{N_*}(\cdot)) W(\cdot)=\begin{pmatrix}
1 & \phi \\
0 & 1
\end{pmatrix}.$$ Letting $U:=\Phi_E^{N_*} W\in C^\omega_{r}({{\mathbb T}}, {\rm PSL}(2,{{\mathbb R}}))$, we have (\[reduci\_para\]). Moreover, $$\begin{aligned}
&& |\varphi|\leq \varepsilon_*^{\frac34}e^{2\pi r|k_*|}e^{-2\pi r|k|} \leq C_{8}(\lambda,\alpha,r) e^{-2\pi r|k|},\\
&& |U|_{r''} \leq \Gamma \cdot D_1 e^{\frac{3\pi r}{2}|k_*|}e^{\frac{3\pi r''}{2}|k|}\leq C_{9}(\lambda,\alpha,r) e^{\frac{3\pi r''}{2}|k|},\quad \forall \ r''\in(0,r].\end{aligned}$$ The above inequalities follows since by $(\ref{norm})$ and footnote 5 of [@A3], we have $|{\rm deg}\Phi_E^{N_*}|\leq C |\ln \Gamma|$.
Gap estimates via Moser-Pöschel argument {#Sec_bounds}
========================================
We consider the quasi-periodic Schrödinger operator on $\ell^2({{\mathbb Z}})$: $$(H_{V,\alpha,\theta} u)_n= u_{n+1}+u_{n-1} + V( \theta+n\alpha) u_n,$$ with $\alpha\in{{\mathbb T}}^d$ such that $(1,\alpha)$ is rationally independent, and $V\in C^\omega({{\mathbb T}}^d, {{\mathbb R}})$ non-constant. Based on Moser-Pöschel argument [@Moser-Poschel], we will estimate the size of the spectral gap $G_k(V)=(E_k^-, E_k^+)$ via quantitative reducibility of corresponding Schrödinger cocycle at its edge points.
Now we assume that the cocycle $(\alpha,S_{E_k^+}^{V})$ is reducible, i.e., there exist $X \in C_R^{\omega}({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$ for some $0<R<1$ and a constant matrix $B$, such that $$X(\cdot+\alpha)^{-1}S_{E_k^+}^{V}(\cdot)X(\cdot)=B.$$ Since $ E_k^+\in \Sigma_{V,\alpha}$ is a right edge point of a gap, $(\alpha,S_{E_k^+}^{V})$ is reduced to a constant parabolic cocycle $B=
\begin{pmatrix}
1 & \zeta \\
0 & 1
\end{pmatrix}$ with $0\leq \zeta< \frac12$. Recall that $\zeta=0$ if and only if the corresponding gap is collapsed. We will show that the size of gap is determined by $X$ and $\zeta$.
For any $0<\delta<1$, a direct calculation yields $$X(\cdot+\alpha)^{-1}S_{E_k^+ -\delta}^{V}(\cdot)X(\cdot)= B-\delta P(\cdot)$$ with $$P(\cdot):=
\begin{pmatrix}
X_{11}(\cdot) X_{12}(\cdot) - \zeta X_{11}^2(\cdot) & -\zeta X_{11}(\cdot) X_{12}(\cdot) + X_{12}^2(\cdot) \\[1mm]
- X_{11}^2(\cdot) & - X_{11}(\cdot) X_{12}(\cdot)
\end{pmatrix}.$$ Obviously, $$\label{esti_PX_on_T}
|P|_{r''}\leq (1+\zeta)|X|^2_{r''}< 2|X|^2_{r''},\quad \forall \ r''\in (0, R].$$ In fact, moving the energy $E$ from the right end of the gap $E_k^+$ to $E_k^+-\delta$, we can determine the other edge point of the spectral gap according to the variation of the rotation number $\rho{(\alpha, X(\cdot+\alpha)^{-1}S^{V}_{E_k^+-\delta}(\cdot)X(\cdot))}$. Note that the rotation number of the constant cocycle $(\alpha, B)$ vanishes since $B$ is parabolic. Then, as shown symbolically in Figure \[f.graph\], we have the following:
- If the rotation number of $(\alpha, X(\cdot+\alpha)^{-1}S^{V}_{E_k^+-\delta_1}(\cdot)X(\cdot))$ is positive, then $E_k^+-\delta_1$ is beyond the left edge of $G_k(V)$, thus $|G_k(V)|\leq \delta_1$.
- If the rotation number of $(\alpha, X(\cdot+\alpha)^{-1}S^{V}_{E_k^+-\delta_2}(\cdot)X(\cdot))$ vanishes, then $E_k^+-\delta_2$ is still in $\overline{G_k(V)}$ and hence $|G_k(V)|\geq \delta_2$.
Of course, one can estimate the size of spectral gap $G_k(V)$ similarly by starting from the left edge point $E_k^-$.
(-0.5,0) – (6.5,0); (0,-0.8) – (0,1.5); plot (, [0.49+0.7\*sqrt(1.3-)]{}); plot (, [0.49]{}); plot (, [0.42+0.7\*sqrt(1.48-)]{}); plot (, [0.42]{}); plot (, [0.28+0.7\*sqrt(1.56-)]{}); plot (, [0.28]{}); plot (, [0.14+0.7\*sqrt(1.9-)]{}); plot (, [0.14]{}); plot (, [0.7\*sqrt(2-)]{}); plot (, [0]{}); plot (, [-0.7\*sqrt(-3.5)]{}); plot (, [-0.14]{}); plot (, [-0.14-0.7\*sqrt(-3.6)]{}); plot (, [-0.28]{}); plot (, [-0.28-0.7\*sqrt(-3.94)]{}); plot (, [-0.42]{}); plot (, [-0.42-0.7\*sqrt(-4.02)]{}); plot (, [-0.49]{}); plot (, [-0.49-0.7\*sqrt(-4.2)]{}); (1.9,0) circle \[radius=0.025\]; (3.5,0) circle \[radius=0.025\]; (2.1,0) circle \[radius=0.025\]; at (8,1.3) [$E':=E_k^+-\delta_1$]{}; at (8,0.9) [$E'':=E_k^+-\delta_2$]{}; at (0,1.5) [$\rho$]{}; at (-0.15,0) [$0$]{}; at (6.5,0) [$E$]{}; at (3.5,0) [$E_k^+$]{}; at (1.85,-0.01) [$E'$]{}; at (2.3,0) [$E''$]{};
Although we focus on the case of a Diophantine frequency, our approach also works for a Liouvillean frequency. For any rationally independent $\alpha\in{{\mathbb T}}^d$, we set $$\beta=\beta(\alpha):=\limsup_{k\rightarrow \infty} \frac{1}{|k|} \ln \frac{1}{\|{\langle}k,\alpha{\rangle}\|_{{{\mathbb T}}}},$$ which is a generalization of $(\ref{equibeta})$ to the multi-frequency case. For convenience, we let $$\label{D1}
D_{\alpha,R}:= 2+40\sum_{n\in{{\mathbb Z}}^d} \frac{e^{-(R+3\beta)|n|/2}}{|e^{{\rm i}{\langle}n,\alpha{\rangle}}-1|^3},$$ which is finite if $R>3\beta$. For $\tau>d-1$, let $$\label{D2}
D_{\tau}:= 2^{4\tau+9} \, \Gamma(4\tau+2).$$
In the following, we first apply one standard KAM step to the cocycle $(\alpha, B-\delta P(\cdot))$, which is the starting point of our estimate on the size of the gap.
\[prop\_ave\] Given $\alpha \in {{\mathbb T}}^d$ with $R>3\beta(\alpha)\geq 0$. We have the following:
1. If $0<\delta< D_{\alpha,R}^{-1}|X|^{-2}_R$, then there exist $\tilde X\in C_{\frac{R-3\beta}2}^{\omega}({{\mathbb T}}^d, {\rm SL}(2, {{\mathbb R}}))$ and $P_1\in C_{\frac{R-3\beta}2}^\omega({{\mathbb T}}^d, {\rm gl}(2, {{\mathbb R}}))$ such that $$\label{first_ave}
\tilde X(\cdot+\alpha)^{-1}(B-\delta P(\cdot))\tilde X(\cdot)
=e^{b_0-\delta b_1}+ \delta^2 P_1(\cdot),$$ where $b_0:=\begin{pmatrix}
0 & \zeta \\[1mm]
0 & 0
\end{pmatrix}$ and $$b_1:=
\begin{pmatrix}
[X_{11} X_{12}] - \frac{\zeta}{2} [X_{11}^2] & -\zeta [X_{11} X_{12}] + [X_{12}^2] \\[1mm]
-[X_{11}^2] & - [X_{11} X_{12}] + \frac{\zeta}{2} [X_{11}^2]
\end{pmatrix},$$ with the estimates $$\label{first_ave_esti}
|\tilde X- {\rm Id}|_{\frac{R-3\beta}2}\leq 2 D_{\alpha,R} \, \delta| X|^2_{R},
\quad |P_1|_{\frac{R-3\beta}2} \leq 2 D_{\alpha,R}^2 \left| X\right|^4_{R}.$$
2. In particular, for $\alpha\in{\rm DC}_d(\gamma,\tau)$, if $0<\delta< D^{-1}_{\tau} \gamma^{3} R^{4\tau+1}|X|^{-2}_R$, then (\[first\_ave\]) holds with $$\label{first_ave_esti_DC}
|\tilde X- {\rm Id}|_{\frac{R}2}\leq 2 D_{\tau}\gamma^{-3} R^{-(4\tau+1)}\delta|X|^2_{R},
\quad |P_1|_{\frac{R}2} \leq 2 D^2_{\tau}\gamma^{-6} R^{-2(4\tau+1)} |X|^4_{R}.$$
Let $G:=-\delta B^{-1} P$. Noting that $B^{-1}=\begin{pmatrix}
1 & -\zeta \\
0 & 1
\end{pmatrix}$, we can see that ${\rm tr}(B^{-1} P)=0$, hence $G\in {{{\mathcal B}}}_R$. By a standard KAM step, we can construct $Y\in {{{\mathcal B}}}_{\frac{R-3\beta}{2}}$ such that $$\label{eq_homo}
Y(\cdot+\alpha)B-BY(\cdot)=B(G(\cdot)-[G]).$$ Indeed, by identifying the Fourier coefficients of the two sides of (\[eq\_homo\]), we have $$\label{homo_Y}
\left \{\begin{array}{l}
\widehat Y_{21}(n)=\frac{\widehat G_{21}(n)}{e^{{\rm i}{\langle}n,\alpha {\rangle}}-1}\\[1mm]
\widehat Y_{11}(n)=\frac{\widehat G_{11}(n)+\zeta\widehat Y_{21}(n)}{e^{{\rm i}{\langle}n,\alpha {\rangle}}-1}\\[1mm]
\widehat Y_{12}(n)=\frac{\widehat G_{12}(n)-\zeta(1+e^{{\rm i}{\langle}n,\alpha {\rangle}} )\widehat Y_{11}(n)}{e^{{\rm i}{\langle}n,\alpha {\rangle}}-1}
\end{array}\right. , \quad \forall \ n\in{{\mathbb Z}}^d\backslash\{0\}.$$ Hence, by the decay property of the Fourier coefficient $\widehat G(n)$, we have $$|Y|_{\frac{R-3\beta}{2}}= \sum_{n\in{{\mathbb Z}}^d} |\widehat Y(n)| e^{\frac{R-3\beta}{2}|n|} \leq \frac12(D_{\alpha,R}-2) \, \delta |P|_{R}.$$
In the same manner as in Proposition 2 of [@HA], for $\tilde X:=e^Y$, we have $$\tilde X(\cdot+\alpha)^{-1}(B-\delta P(\cdot))\tilde X(\cdot)=Be^{[G]}+ \tilde P(\cdot),$$ where $$\begin{aligned}
\tilde P(\cdot)&:=BY(\cdot)-Y(\cdot+\alpha)B-B [G]- \delta P(\cdot) +\sum_{m+n\geq 2}\frac{1}{m!}(-Y(\cdot+\alpha))^m B \frac{1}{n!}Y(\cdot)^n\\
&+ \delta\sum_{m+n\geq 1} \frac{1}{m!}(-Y(\cdot+\alpha))^m P(\cdot) \frac{1}{n!}Y(\cdot)^n +B \sum_{n\geq 2}\frac{1}{n!} [G]^n.\end{aligned}$$ Obviously, $|\tilde X-{\rm Id}|_{\frac{R-3\beta}{2}}\leq 2|Y|_{\frac{R-3\beta}{2}}\leq D_{\alpha,R} \delta |P|_{R}.$ Since $\sum_{m+n=k}\frac{k!}{m!n!}=2^k$ and $|G|_R\leq \delta|P|_R$, we get $$\begin{aligned}
\left|\sum_{m+n\geq 2}\frac{1}{m!}(-Y(\cdot+\alpha))^m B \frac{1}{n!}Y(\cdot)^n\right|_{\frac{R-3\beta}{2}}&\leq& (D_{\alpha,R}-2)^2 \, \delta^2 |P|^2_{R},\\
\left|\delta\sum_{m+n\geq 1} \frac{1}{m!}(-Y(\cdot+\alpha))^m P(\cdot) \frac{1}{n!}Y(\cdot)^n\right|_{\frac{R-3\beta}{2}}&\leq&(D_{\alpha,R}-2) \, \delta^2 |P|^2_{R},\\
\left|B \sum_{n\geq 2}\frac{1}{n!} [G]^n\right|_{\frac{R-3\beta}{2}}&\leq& \delta^2 |P|^2_{R}.\end{aligned}$$ Note that (\[eq\_homo\]) implies $BY(\cdot)-Y(\cdot+\alpha)B-B [G]- \delta P(\cdot)=0$. We thus get $$|\tilde P|_{\frac{R-3\beta}{2}}\leq D_{\alpha,R}^2 \, \delta^2 |P|^2_{R}.$$ With $\tilde P_1:=\delta^{-2} \tilde P+\sum_{j\geq2}\frac{(-\delta)^{j-2}}{j!} B[B^{-1}P]^j$, we have $$Be^{[G]}+\tilde P(\cdot)=B-\delta [P] +\delta^2 \tilde P_1(\cdot).$$ By a direct calculation, we can see that $$B-\delta [P] ={\rm Id}+(b_0-\delta b_1)-\frac{\delta}{2}(b_0b_1+b_1b_0).$$ Then, with $P_1:=\tilde P_1-\frac12b_1^2-\delta^{-2}\sum_{j\geq 3} \frac{1}{j!}(b_0-\delta b_1)^j$, we obtain (\[first\_ave\]). Note that $b_0$ is nilpotent. Thus, combining with (\[esti\_PX\_on\_T\]), we get (\[first\_ave\_esti\]).
If $\alpha\in {\rm DC}_d(\gamma, \tau)$, then, by (\[homo\_Y\]), we have $$\begin{aligned}
|Y|_{\frac{R}{2}}
&\leq&10 \delta |P|_{R} \sum_{n\in{{\mathbb Z}}^d} \frac{e^{-\frac{R}{2}|n|}}{|e^{{\rm i}{\langle}n,\alpha{\rangle}}-1|^3}\\
&\leq& 20 \gamma^{-3}\delta |P|_{R} \sum_{n\in{{\mathbb Z}}^d} |n|^{3\tau}e^{-\frac{R}{2}|n|} \\
&\leq& 40 \gamma^{-3}\delta |P|_{R} \int_{0}^{+\infty} x^{d-1}x^{3\tau}e^{-\frac{R}{2}x} dx,\end{aligned}$$ where the above integral can be estimated as $$\int_{0}^{+\infty} x^{d-1}x^{3\tau}e^{-\frac{R}{2}x} dx\leq 2+ \int_{0}^{+\infty} x^{4\tau}e^{-\frac{R}{2}x} dx\leq 2^{4\tau+2}\, \Gamma(4\tau+2)\cdot R^{-(4\tau+1)}.$$ The rest proof of (\[first\_ave\_esti\_DC\]) is similar to that of (\[first\_ave\_esti\]).
Since $\tilde X$ is homotopic to identity by construction, we have $$\rho(\alpha, B-\delta P(\cdot))=\rho(\alpha, e^{b_0-\delta b_1}+ \delta^2 P_1(\cdot)).$$ Let $d(\delta):={\rm det} (b_0-\delta b_1)$. By a direct calculation, we get $$\label{d_delta}
d(\delta)=-\delta[X^2_{11}]\zeta+\delta^2\left([X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2\right).$$ As we will see, $d(\delta)$ is the key quantity in our estimates on the size of the gaps.
Criterion for quantitative bounds of spectral gaps
--------------------------------------------------
In this subsection, we give a criterion to obtain bounds on the size of the gaps in terms of the information provided by quantitative reducibility. With this criterion at our disposal, the exponential decay of the spectral gaps in various settings follows at once.
\[thm\_upperbound\] Let $\alpha\in {{\mathbb T}}^d$ with $R> 3\beta(\alpha)\geq 0$, $\kappa\in(0,\frac{1}{4})$, and $V\in C^{\omega}({{\mathbb T}}^d, {{\mathbb R}})$ be a non-constant function. Let $E$ be an edge point of the spectral gap $G(V)$. Assume that there are $\zeta\in(0,\frac12)$ and $X\in C_R^\omega({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$ such that $$\label{reduce_to_parabolic}
X(\cdot+\alpha)^{-1}S_E^{V}(\cdot)X(\cdot)=
\begin{pmatrix}
1 & \zeta \\
0 & 1
\end{pmatrix}.$$ Then the following holds:
1. If $$\label{smallness_zeta_beta}
|X|_R^{14} \zeta^\kappa\leq 10^{-5}D^{-4}_{\alpha,R},$$ then $\zeta^{1+\kappa}\leq |G(V)|\leq \zeta^{1-\kappa}$, where $D_{\alpha, R}$ is the constant defined in (\[D1\]).
2. In particular, for $\alpha\in {\rm DC}_d(\gamma,\tau)$, if $$\label{smallness_zeta_diophantine}
|X|_R^{14} \zeta^\kappa\leq 10^{-5}D_{\tau}^{-4} \gamma^{12} R^{4(4\tau+1)},$$ then $\zeta^{1+\kappa}\leq |G(V)|\leq \zeta^{1-\kappa}$, where $D_{\tau}$ is the constant defined in (\[D2\]).
\[remark\_frequency\] We remark that the optimal condition for reducibility at the edge points of spectral gaps was assumed to be $R>2\beta(\alpha)$, which was first conjectured by Avila-Jitomirskaya [@AvilaJito1]. For technical reasons, we have to require $R>3\beta(\alpha)$ in this approach (due to Lemma \[prop\_ave\]).
Before giving the proof of Theorem \[thm\_upperbound\], we first make some technical preparations:
\[z1-estimate\] For any $X\in C^\omega({{\mathbb T}}^d,{\rm PSL}(2,{{\mathbb R}}))$, $[X_{11}^2]\geq (2|X|_{{{\mathbb T}}^d})^{-2}$.
The proof is essentially contained in Lemma 4.2 of [@AYZ1], we include the proof here for completeness. Let $$u_1(\theta):=\begin{pmatrix}
X_{11}(\theta)\\
X_{21}(\theta)
\end{pmatrix},
\quad u_2(\theta):=\begin{pmatrix}
X_{12}(\theta)\\
X_{22}(\theta)
\end{pmatrix}
.$$ Since $|{\rm det}X(\theta)|=1$, we have $\|u_1\|_{L^2({{\mathbb T}}^d)} \|u_2\|_{L^2({{\mathbb T}}^d)}> 1$, which implies that $$\|X_{11}\|_{L^2({{\mathbb T}}^d)}+\|X_{21}\|_{L^2({{\mathbb T}}^d)}=\|u_1\|_{L^2({{\mathbb T}}^d)}>\|u_2\|_{L^2({{\mathbb T}}^d)}^{-1}>(|X|_{{{\mathbb T}}^d})^{-1}.$$ By (\[reduce\_to\_parabolic\]), we know $X_{21}(\cdot+\alpha)=X_{11}(\cdot)$. So $[X_{11}^2]=\|X_{11}\|_{L^2({{\mathbb T}}^d)}^2\geq (2|X|_{{{\mathbb T}}^d})^{-2}$.
Once we have Lemma \[z1-estimate\], then we have the following key observation for the transformation $X(\cdot)$.
\[esti\_X11\_X12\] For any $\kappa\in (0,\frac1{4})$, if $$\label{x2} |X|_{R}\, \zeta^{\frac\kappa2}\leq \frac{1}{4},$$ then the following holds: $$\begin{aligned}
\label{ineqzz}
0< \frac{[X^2_{11}]}{[X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2} &\leq \frac12\zeta^{-\kappa}, \\ \label{ineqzz-1}
[X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2 &\geq 8\zeta^{2\kappa}.\end{aligned}$$
Assume by contradiction that $$\frac{[X^2_{11}]}{[X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2}> \frac12 \zeta^{-\kappa}.$$ The quadratic polynomial $$Q(z):=[(X_{12}-zX_{11})^2]=[X_{11}^2]z^2-2[X_{11}X_{12}]z+[X_{12}^2]$$ attains its minimum when $z=\frac{[X_{11}X_{12}]}{[X_{11}^2]}$, and we have $$Q\left(\frac{[X_{11}X_{12}]}{[X_{11}^2]}\right)=\left[\left(X_{12}- \frac{[X_{11}X_{12}]}{[X_{11}^2]} X_{11}\right)^2\right]=\frac{[X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2}{[X^2_{11}]}< 2\zeta^{\kappa}.$$ Hence, $X_{12}=\frac{[X_{11}X_{12}]}{[X_{11}^2]}X_{11}+\sigma$ for some $\sigma: {{\mathbb T}}^d \to {{\mathbb R}}$ with $[\sigma^2]< 2\zeta^{\kappa}$.
By (\[reduce\_to\_parabolic\]), we can check that $$X_{11}(\cdot+\alpha)X_{12}(\cdot)-X_{11}(\cdot)X_{12}(\cdot+\alpha)=1+\zeta X_{11}(\cdot+\alpha)X_{11}(\cdot).$$ Hence, we obtain $$X_{11}(\cdot+\alpha)\sigma(\cdot)-X_{11}(\cdot)\sigma(\cdot+\alpha)=1+\zeta X_{11}(\cdot+\alpha)X_{11}(\cdot).$$ By Cauchy-Schwarz inequality and $(\ref{x2})$, we have $$\label{contradiction_1}
\left|[X_{11}(\cdot+\alpha)\sigma(\cdot)-X_{11}(\cdot)\sigma(\cdot+\alpha)]\right|\leq \frac{\sqrt2}2.$$ On the other hand, $\zeta |X_{11}(\cdot+\alpha)X_{11}(\cdot)|_{{{\mathbb T}}^d}\leq \frac{1}{16}\zeta^{1-\kappa}$, which implies $$\label{contradiction_2}
\left|[1+\zeta X_{11}(\cdot+\alpha)X_{11}(\cdot)]\right|>1-\frac{1}{16}\zeta^{1-\kappa}.$$ By (\[contradiction\_1\]) and (\[contradiction\_2\]), we reach a contradiction.
Combining with Lemma \[z1-estimate\], we get $[X_{11}^2]\geq \frac14 |X|_{{{\mathbb T}}^d}^{-2}\geq 4 \zeta^{\kappa}$, which implies $(\ref{ineqzz-1})$.
By (\[d\_delta\]), the quantity $d(\delta)={\rm det}(b_0-\delta b_1)$ satisfies $$\begin{aligned}
d(\delta) &= -\delta [X^2_{11}]\zeta+ \delta^2([X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2) \\
&= \delta([X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2)\left(\delta-\frac{[X^2_{11}]\zeta}{[X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2}\right).\end{aligned}$$
Fix $\kappa\in(0,\frac{1}{4})$, and let $\delta_1=\zeta^{1-\kappa}$. If $\zeta>0$ satisfies (\[smallness\_zeta\_beta\]), then it is obvious that $0<\delta_1\leq D^{-1}_{\alpha,R}|X|_R^{-2}$. In particular, for $\alpha\in {\rm DC}_d(\gamma,\tau)$, (\[smallness\_zeta\_diophantine\]) implies that $0<\delta_1\leq D^{-1}_{\tau} \gamma^3 R^{4\tau+1} |X|_R^{-2}$. Hence, we can apply Lemma \[prop\_ave\], and conjugate the system to the cocycle $(\alpha, e^{b_0-\delta_1 b_1}+\delta_1^2 P_1)$.
As shown symbolically in Figure \[f.graph\], in order to show $|G(V)|\leq \delta_1$, it is sufficient to show that $\rho(\alpha, e^{b_0-\delta_1 b_1}+\delta_1^2 P_1)>0$. By $(\ref{smallness_zeta_beta})$ or $(\ref{smallness_zeta_diophantine})$, one has $|X|_{R} \zeta^{\frac\kappa2} \leq \frac{1}{4}$. Then we can apply Lemma \[esti\_X11\_X12\], and get $$\frac{[X^2_{11}]\zeta}{[X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2} \leq \frac{1}{2}\delta_1.$$ Hence, for $d(\delta_1)={\rm det}(b_0-\delta_1 b_1)$, we have $$\label{lower_determinant}
d(\delta_1) \geq \zeta^{1-\kappa} \cdot 8 \zeta^{2\kappa}\cdot \frac12 \zeta^{1-\kappa}
=4 \zeta^{2}.$$ Following the expressions of $b_0$ and $b_1$ in Lemma \[prop\_ave\], we have $$\label{Mdelta1}
|b_0-\delta_1 b_1|\leq \zeta + \delta_1 (1+\zeta) |X|_{{{\mathbb T}}^d}^2 \leq 2 \, \zeta^{1-\kappa} |X|_{R}^2.$$ In view of Lemma 8.1 in [@HouYou], there exists ${{{\mathcal P}}}\in {\rm SL}(2,{{\mathbb R}})$, with $|{{{\mathcal P}}}|\leq 2\left(\frac{|b_0-\delta_1 b_1|}{\sqrt{d(\delta_1)}}\right)^{\frac12}$ such that $${\mathcal P}^{-1} e^{b_0-\delta_1 b_1} {{{\mathcal P}}}= R_{\sqrt{d(\delta_1)}}.$$ Combining and , we have $$\frac{|b_0-\delta_1 b_1|}{\sqrt{d(\delta_1)}}\leq \frac{2\, \zeta^{1-\kappa} |X|_{R}^2 }{\sqrt{4 \zeta^{2}}}= |X|_{R}^2 \zeta^{-\kappa}.$$ Then, according to Lemma \[esti\_rot\_num\] and Lemma \[prop\_ave\], $$|\rho(\alpha, e^{b_0-\delta_1 b_1}+\delta_1^2 P_1)-\sqrt{d(\delta_1)}|\leq \delta_1^2 |{\mathcal P}|^2 |P_1|_{{{\mathbb T}}^d}
\leq 8 D^2_{\alpha,R} |X|_{R}^6 \zeta^{2-3\kappa}.$$ Under the assumption (\[smallness\_zeta\_beta\]), combining with (\[lower\_determinant\]), we have $$4 D^2_{\alpha,R} |X|_{R}^6 \zeta^{1-3\kappa}< 1,$$ which implies that $$\rho{(\alpha, e^{b_0-\delta_1 b_1}+\delta_1^2 P_1)}\geq \sqrt{d(\delta_1)}-|\rho{(\alpha, e^{b_0-\delta_1 b_1}+\delta_1^2 P_1)}-\sqrt{d(\delta_1)}|>0.$$ In particular, when $\alpha\in {\rm DC}_d(\gamma,\tau)$, in view of (\[first\_ave\_esti\_DC\]), we have $$|\rho{(\alpha, e^{b_0-\delta_1 b_1}+\delta_1^2 P_1)}-\sqrt{d(\delta_1)}|\leq 8 D^2_{\tau} \gamma^{-6} R^{-2(4\tau+1)} |X|_{R}^6 \zeta^{2-3\kappa}.$$ Since (\[smallness\_zeta\_diophantine\]) implies that $$4 D^2_{\tau} \gamma^{-6} R^{-2(4\tau+1)} |X|_{R}^6 \zeta^{1-3\kappa}< 1,$$ and we get $\rho{(\alpha, e^{b_0-\delta_1 b_1}+\delta_1^2 P_1)}>0$. This concludes the proof of the upper bound estimate.
Let us now consider the lower bound estimate on the size of the gap. Let $\delta_2:=\zeta^{1+\kappa}$. We are going to show that $|G(V)| \geq \delta_2$. We first note that $$\delta_2^2\left|[X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2\right|\leq 2 \zeta^{2+2\kappa}|X|_{R}^{4},$$ and, by Lemma \[z1-estimate\], one has $\delta_2[X^2_{11}]\zeta\geq \frac{1}{4} \zeta^{2+\kappa} |X|_{R}^{-2}$. Thus, if $\zeta$ is small enough such that $
|X|_{R}^{6}\zeta^{\kappa}\leq \frac1{40}$ (which can be deduced from (\[smallness\_zeta\_beta\]) or (\[smallness\_zeta\_diophantine\])), then $$d(\delta_2)=-\delta_2[X^2_{11}]\zeta+\delta_2^2\left([X^2_{11}][X^2_{12}]-[X_{11}X_{12}]^2\right)<-\frac{1}{5}\zeta^{2+\kappa} |X|_{R}^{-2},$$ and hence $$\label{upper_determinant} \sqrt{-d(\delta_2)}> \frac{1}{\sqrt{5}}\zeta^{1+\frac\kappa2} |X|_{R}^{-1}.$$ In view of Proposition 18 of [@Puig06], there exists ${{{\mathcal P}}}\in {\rm SL}(2,{{\mathbb R}})$, with $|{{{\mathcal P}}}|\leq 2\left(\frac{|b_0-\delta_2 b_1|}{\sqrt{-d(\delta_2)}}\right)^{\frac12}$ such that $${{{\mathcal P}}}^{-1} e^{b_0-\delta_2 b_1} \, {{{\mathcal P}}}=\begin{pmatrix}
e^{\sqrt{-d(\delta_2)} }& 0 \\[1mm]
0 & e^{-\sqrt{-d(\delta_2)}}
\end{pmatrix}.$$ Since $|X|_{R}^{6} \zeta^{\kappa}\leq \frac{1}{8}$, we have $$|b_0-\delta_2 b_1|\leq \zeta + \zeta^{1+\kappa} (1+\zeta) |X|_{{{\mathbb T}}^d}^2 \leq 2 \zeta,$$ and then, by $(\ref{upper_determinant})$, one has $$\frac{|b_0-\delta_2 b_1|}{\sqrt{-d(\delta_2)}}\leq \frac{\sqrt{5}\cdot 2 \zeta}{\zeta^{1+\frac\kappa2} |X|_{R}^{-1}}= 2\sqrt{5}|X|_{R} \zeta^{-\frac\kappa2}.$$ By $(\ref{first_ave_esti})$ of Lemma \[prop\_ave\], we have $${{{\mathcal P}}}^{-1} \delta_2^2|P_1|_{(R-3\beta)/2}{{\mathcal P}}\leq 16 \sqrt{5} D^2_{\alpha,R} \zeta^{2+\frac{3\kappa}2} |X|^5_{R}.$$ Then, under the condition (\[smallness\_zeta\_beta\]), we have $${{{\mathcal P}}}^{-1} \delta_2^2|P_1|_{(R-3\beta)/2}{{\mathcal P}}\leq -d(\delta_2),$$ consequently, the cocycle $(\alpha, e^{b_0-\delta_2 b_1}+\delta_2^2 P_1)$ is uniformly hyperbolic, and $E-\delta_2 \not \in \Sigma_{V,\alpha}$, which means that $|G(V)|\geq \zeta^{1+\kappa}$. In particular, if $\alpha\in {\rm DC}_{d}(\gamma, \tau)$, by $(\ref{first_ave_esti_DC})$ of Lemma \[prop\_ave\], we have $${{{\mathcal P}}}^{-1} \delta_2^2|P_1|_{R/2}{{\mathcal P}}\leq 16\sqrt{5} D^2_{\tau} R^{-2(4\tau+1)} \zeta^{2+\frac{3\kappa}2} |X|^5_{R}.$$ Similarly as above, under the condition (\[smallness\_zeta\_diophantine\]), we have $|G(V)|\geq \zeta^{1+\kappa}$.
Applications of the criterion – upper bound
-------------------------------------------
As the first application of Theorem \[thm\_upperbound\], for discrete quasi-periodic Schrödinger operator with small potential, we get exponentially decaying upper bounds on the size of spectral gaps. As we mentioned before, the result is perturbative for a multifrequency. However, it is non-perturbative in the case of a one-dimensional frequency.
\[cor-local\] Consider the operator $H_{V,\alpha,\theta}$ with $V\in C_{r_0}^{\omega}({{\mathbb T}}^d,{{\mathbb R}})$ non-constant.
1. If $\alpha\in {\rm DC}_d(\gamma,\tau)$, then for any $r\in(0,r_0)$, there exists $\varepsilon_*=\varepsilon_*(\gamma,\tau, r_0, r, d )>0$ such that if $|V|_{r_0}=\varepsilon_0<\varepsilon_*$, then $$|G_k(V)|\leq \varepsilon_0^{\frac23} e^{-2\pi r |k|},\quad \forall \ k\in{{\mathbb Z}}^d\backslash\{0\}.$$
2. If $d=1$, $\beta(\alpha)=0$, and $|V|_{r_0}\leq c_0 r_0^3$ with $c_0$ the absolute constant in Theorem \[almostredth\], then there are $C_{10}=C_{10}(r_0,\alpha)>0$ and $\vartheta=\vartheta(r_0) \in (0,r_0)$ such that $$|G_k(V)|\leq C_{10} e^{-\vartheta |k|},\quad \forall \ k\in{{\mathbb Z}}\backslash\{0\}.$$
We fist consider the case $\alpha\in {\rm DC}_d(\gamma,\tau)$. Fix $r\in(0,r_0)$ and set $\tilde r:=\frac{r_0+r}{2}$. Write the Schrödinger cocycle $(\alpha, S_{E}^{V}(\cdot))$ as $(\alpha,A_E+F_0(\cdot))$, where $$A_E=\begin{pmatrix}
E & -1 \\
1 & 0
\end{pmatrix},\quad F_0(\cdot)=\begin{pmatrix}
-V(\cdot) & 0 \\
0 & 0
\end{pmatrix}.$$ Since we consider the case where $V$ is small, we have $$E\in \Sigma_{V,\alpha} \subset [-2-\inf V(\theta), 2+\sup V(\theta)]\subset [-3,3].$$ Then the norm of $A_E$ is bounded uniformly with respect to $E$. Hence one can apply Theorem \[thm\_gap\_edge\_SL\] to obtain a uniform $\varepsilon_*=\varepsilon_*(\gamma,\tau, r_0, \tilde r, d )>0$ which is independent of $E$, such that if $|V|_{r_0}=\varepsilon_0<\varepsilon_*$, then $(\alpha,A_E+F_0(\cdot))$ is almost reducible. Moreover, since $2\rho(\alpha, S_{E_k^+}^{V}) -{\langle}k,\alpha {\rangle}\in{{\mathbb Z}}$, by Theorem \[thm\_gap\_edge\_SL\], we have $$X(\cdot+\alpha)^{-1}S_{E_k^+}^{V}(\cdot)X(\cdot)=
\begin{pmatrix}
1 & \zeta \\
0 & 1
\end{pmatrix},$$ with $\zeta\leq \varepsilon^{\frac34}_{0} e^{-2\pi \tilde r|k|}$ and $ |X|_{r''}\leq D_1(\gamma,\tau, r_0, d)e^{\frac32\pi r'' |k|}$ for any $r''\in (0,\tilde r)$.
Let $\kappa:=\frac{\tilde r-r}{9\tilde r}$ and $R:=\varepsilon_0^{\frac{\tilde r -r}{60\tilde r(4\tau+1)}}$. Then for any $k\in{{\mathbb Z}}^d\backslash\{0\}$, we have $$|X|_{R}^{14} \zeta^\kappa\leq D_1^{14} e^{21\pi R |k|} \cdot
\varepsilon_0^{\frac{\tilde r- r}{12\tilde r}} e^{-\frac{2\pi(\tilde r- r)}{9}|k|}\leq 10^{-5} D_{\tau}^{-4}\gamma^{12} R^{4(4\tau+1)}.$$ The above inequality is possible since $\varepsilon_0$ is sufficiently small (the smallness only depend on $\gamma$, $\tau$, $r_0$, $\tilde r$, $d$). Hence, by Theorem \[thm\_upperbound\], we have $$|G_k(V)|\leq \zeta^{1-\kappa}\leq \varepsilon_0^{\frac{8\tilde r+r}{12\tilde r}} e^{-\frac{2\pi}{9}(8\tilde r+r)|k|}\leq \varepsilon_0^{\frac23} e^{-2\pi r|k|},\quad \forall \ k\in{{\mathbb Z}}^d\backslash\{0\}.$$ This concludes the proof of the first statement.
Now we consider the case where $d=1$, $\beta(\alpha)=0$ and $|V|_{r_0}\leq c_0 r_0^3$. By Theorem \[prop\_duality\_para\], there exists $r_1=r_1(r_0)\in (0,r_0)$, such that for any $r\in(0,r_1)$, one has $X\in C^{\omega}_{r}({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ and $\zeta\in {{\mathbb R}}$ such that $$X(\cdot+\alpha)^{-1}S_{E_k^+}^{V}(\cdot)X(\cdot)=
\begin{pmatrix}
1 & \zeta \\
0 & 1
\end{pmatrix}.$$ Moreover, there exists $k_1=k_1(\alpha,r_0,r)>0$ such that if $|k| \geq k_1$, then for some $n=n(k)\in{{\mathbb Z}}$ with $|n|\geq \frac{|k|}{4}$, one has $\zeta\leq C_3(\alpha,r_0,r) e^{-\frac{\pi r}{5} |n|}$, and for any $r''\in (0,r]$, $|X|_{r''}\leq C_4(\alpha, r_0, r'') e^{22\pi r'' |n|}$.
Fix any $\kappa\in (0,\frac{1}{4})$ and let $R:=\frac{\kappa r_1}{2100}$, $r'=\frac56 r_1$. By a direct calculation, if $|k|$ is large enough (hence $|n|$ is large enough), then $$|X|_{R}^{14} \zeta^\kappa\leq C_4^ {14} C_3^{\kappa} e^{-\pi\kappa r_1 (\frac{1}{6}-\frac{11}{75})|n|} \leq 10^{-5}D_{\alpha,R}^{-4}.$$ Thus, by Theorem \[thm\_upperbound\], we have $$|G_k(V)|\leq \zeta^{1-\kappa}\leq C_{10}(\alpha, r_0)e^{-\frac{\pi r_1}{6}(1-\kappa)|n|}\leq C_{10}(\alpha, r_0) e^{-\frac{3\pi r_1}{24}|k|}.$$ Modifying the constant coefficient $C_{10}$, we get the exponential upper bound for all $k\in{{\mathbb Z}}\backslash\{0\}$. This thus concludes the whole proof.
As the second application of Theorem \[thm\_upperbound\], we get exponential decay of the upper bounds of the spectral gaps for subcritical quasi-periodic Schrödinger operators.
\[cor\_global\] Consider the Schrödinger operator $H_{V,\alpha,\theta}$ with $\beta(\alpha)=0$. For a typical potential $V\in C^{\omega} ({{\mathbb T}},{{\mathbb R}})$, there exist constants $C,\vartheta>0$ depending only on $V$ and $\alpha$, such that $$|G_k(V)| \leq C e^{-\vartheta |k|}, \quad \forall \ k\in {{\mathbb Z}}\backslash \{0\} \; { with } \; \overline{G_k(V)} \cap \Sigma_{V,\alpha}^{\mathrm{sub}}\neq \emptyset.$$
The proof is the same as that of Corollary \[cor-local\] (2). One only needs to replace Proposition \[prop\_duality\_para\] with Corollary \[prop\_duality\_global\].
If we restrict ourselves to subcritical almost Mathieu operators, we obtain much better estimates:
\[cor\_mathieu\_upperbound\] Consider the almost Mathieu operator $H_{\lambda,\alpha,\theta}$ with $0<\lambda<1$. For any $0<\xi<1$, the following assertions hold.
1. For $\alpha\in {{\mathbb R}}\backslash{{\mathbb Q}}$ with $\beta(\alpha)=0$, there exists $C_{11}=C_{11}(\lambda,\alpha,\xi)>0$ such that $$|G_k(\lambda)|\leq C_{11}(\lambda,\alpha,\xi) \lambda^{ \frac{\xi}{12} |k|}, \quad \forall \ k\in {{\mathbb Z}}\backslash\{0\}.$$
2. For $\alpha\in{\rm DC}$, there exists $C_{12}=C_{12}(\lambda,\alpha,\xi)>0$ such that $$|G_k(\lambda)|\leq C_{12}(\lambda,\alpha,\xi) \lambda^{\xi |k|}, \quad \forall \ k\in {{\mathbb Z}}\backslash\{0\}.$$
We first consider the case $\beta(\alpha)=0$. If $2\rho{(\alpha,S_E^{\lambda})}-k\alpha \in{{\mathbb Z}}$, then by Theorem \[thm\_almost\_almost-2\], for any $0<\tilde r<-\frac{1}{2\pi}\ln\lambda$, there exist $X\in C_{\tilde r}^{\omega}({{\mathbb T}}, {\rm PSL}(2,{{\mathbb R}}))$, $k_2=k_2(\lambda, \alpha, \tilde r)>0$ such that if $|k|\geq k_2$, then we have $$X(\cdot+\alpha)^{-1}S_{E_k^+}^{\lambda}(\cdot)X(\cdot)=
\begin{pmatrix}
1 & \zeta \\
0 & 1
\end{pmatrix}$$ with $\zeta \leq C_5(\lambda,\alpha, \tilde r) e^{-\frac{2\pi \tilde r}{3} |n|}$ and $|X|_{r''}\leq C_6(\lambda,\alpha, r'') e^{22\pi r'' |n|}$ for any $r''\in(0,\tilde r]$, where $n=n(k)\in {{\mathbb Z}}$ satisfies $|n|\geq \frac{|k|}{4}$.
For any $\xi\in(0,1)$, let $r:=-\frac{\xi\ln \lambda}{2\pi}$, $\tilde r:=\frac12(r-\frac{\ln \lambda}{2\pi})$, $\kappa:=\frac{\tilde r-r}{10\tilde r}$ and $R:=\frac{\kappa \tilde r}{700}$. By a direct calculation, if $k$ is large enough (thus $n$ is large enough too), then we have $$|X|_{R}^{14} \zeta^\kappa\leq C_6^ {14} C_5^{\kappa} e^{-\pi\kappa\tilde r(\frac{2}{3}-\frac{11}{25})|n|}\leq 10^{-5} D_{\alpha,R}^{-4}.$$ Hence, by Theorem \[thm\_upperbound\], we have $$|G_k(\lambda)|\leq \zeta^{1-\kappa}\leq C_5^{1-\kappa} e^{-\frac{\pi}{15}(9\tilde r+r)|n|}\leq C_{11}(\lambda,\alpha,\xi) \lambda^{ \frac{\xi}{12} |k|},$$ since $|n|\geq \frac{|k|}{4}$. Modifying the constant coefficient $C_{11}$, we get the exponential upper bound for all $k\in{{\mathbb Z}}\backslash\{0\}$.
If $\alpha\in{\rm DC}$, the proof is similar to that of Corollary \[cor-local\] (1). One only needs to replace Theorem \[thm\_gap\_edge\_SL\] with Proposition \[redu amo case\] and corresponding arguments. Therefore, for any $0<r <-\frac{ \ln \lambda}{2\pi}$, one has $|G_k(\lambda)|\leq C_8(\lambda ,\alpha, r) e^{-2\pi r|k|}$. Now for any $0<\xi<1$, let $r:=-\frac{\xi \ln \lambda}{2\pi}$, which gives the desired result.
In the same way, we can also derive a criterion to obtain quantitative upper bounds on the size of spectral gaps for continuous quasi-periodic Schrödinger operators on $L^2({{\mathbb R}})$: $$({{{\mathcal L}}}_{V,\varpi}\psi)(x)=-\psi''(x)+V(\varpi x)\psi(x)$$ with $V\in C^{\omega}({{\mathbb T}}^{d},{{\mathbb R}})$ sufficiently small and $\varpi\in {\rm DC}_{d}$, $d\geq 2$. As the proof is the same as Theorem \[thm\_upperbound\], we state the result without proof. In fact, as the reader can see, our result is based on Moser-Pöschel argument [@Moser-Poschel], which was first stated in the case of a continuous operator.
\[thm\_upperbound\_continuous\] Consider the operator ${{{\mathcal L}}}_{V,\varpi}$ with $V\in C^{\omega}({{\mathbb T}}^{d},{{\mathbb R}})$ non-constant and $\varpi\in {\rm DC}_{d}(\gamma, \tau)$, $d\geq 2$. Assume that there are $\zeta>0$ and $X\in C_{R}^\omega({{\mathbb T}}^d, {\rm PSL}(2,{{\mathbb R}}))$ for some $R>0$ such that $$\partial_{\varpi}X=\begin{pmatrix}
0 & 1\\
V(\theta) -E & 0
\end{pmatrix}X-X\begin{pmatrix}
0 & \zeta\\
0 & 0
\end{pmatrix}.$$ Fix $\kappa\in (0, \frac{1}{4})$. If $ |X|_{R}^{14} \zeta^\kappa\leq 10^{-5} D^{-4}_{\tau} \gamma^{12} R^{4(4\tau+1)} $, then $|G(V)|\leq \zeta^{1-\kappa}$.
\[sharpcon\] Let $\varpi \in {\rm DC}_d$ and $V\in C_{r_0}^{\omega}({{\mathbb T}}^{d},{{\mathbb R}})$. For any $r\in(0,r_0)$, there exists $\varepsilon_0= \varepsilon_0(V, \varpi, r_0, r)>0$ such that if $|V|_{r_0} < \varepsilon_0$, then for the operator ${{{\mathcal L}}}_{V,\varpi}$, $$|G_k(V)| \leq \varepsilon_0^{\frac23} e^{- r |k|}, \quad \forall \ k\in {{\mathbb Z}}^d\backslash\{0\}.$$
The proof is exactly the same as that of Corollary \[cor-local\] (1). One only needs to replace Theorem \[thm\_gap\_edge\_SL\] with Theorem \[thm\_gap\_edge\_algebra\], and replace Theorem \[thm\_upperbound\] with Theorem \[thm\_upperbound\_continuous\].
Applications of the criterion – lower bound {#subsec_lower_bound}
-------------------------------------------
For general Schrödinger operators, the spectral gaps may collapse since the corresponding off-diagonal element $\zeta$ may vanish. However, this is not true for non-critical almost Mathieu operators $H_{ \lambda,\alpha,\theta}$ [@AvilaJito; @AYZ]. Now we further derive exponentially decaying lower bounds on the size of the gaps $G_k(\lambda)$.
\[thm\_lowerbound\] Consider the almost Mathieu operator $H_{\lambda,\alpha,\theta}$ with $0<\lambda<1$, $\alpha\in {\rm DC}(\gamma,\tau)$. There exists an absolute constant $\tilde\xi>1$ such that $$|G_k(\lambda)| \geq C(\lambda,\alpha) \lambda^{\tilde\xi |k|}, \quad \forall \ k\in{{\mathbb Z}}\backslash\{0\}.$$
The following proposition plays a key role in Avila-You-Zhou’s proof in solving the non-critical “Dry Ten Martini Problem” [@AYZ]. We point out that it works for all irrational frequencies while in [@AYZ] the authors mainly deal with Liouvillean frequencies.
\[estimate-kappa\] Let $\alpha\in{{\mathbb R}}\backslash {{\mathbb Q}}$, $0<\lambda<1$, $E\in\Sigma_{\lambda,\alpha}$ and $0<R<-\frac{1}{2\pi}\ln \lambda$. There exists $T=T(R,\lambda)>0$ such that for $\varepsilon>0$ sufficiently small, there is no $Z \in C_{R}^\omega({{\mathbb T}}, {\rm PSL}(2,{{\mathbb R}}))$ satisfying $$\label{almost}
Z(\cdot+\alpha)^{-1}S_{E}^{\lambda}(\cdot)Z(\cdot)=\mathrm{Id}+F(\cdot),$$ with $|Z|_R \leq \varepsilon^{-1}$, $|F|_R\leq\varepsilon^T$.
\[remark\_T\] If one checks the argument in [@AYZ], it gives us $T(R,\lambda)=C' \left(\frac{\ln \lambda}{2\pi R}\right)^2$ where $C'$ is a large absolute constant. If (\[almost\]) in the above proposition can be promoted to reducibility, i.e., $$Z(\cdot+\alpha)^{-1}S_{E}^{\lambda}(\cdot)Z(\cdot)={\rm Id}+F$$ for some constant $F$, then one can actually obtain more precise estimates on $C'$.
\[Proof of Corollary \[thm\_lowerbound\]\] By Proposition \[redu amo case\] (2), for $r=-\frac{1}{4\pi}\ln\lambda$, and any $k\in{{\mathbb Z}}\backslash\{0\}$, there exist $\zeta\in{{\mathbb R}}$ and $X\in C^\omega_r({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ such that $$X(\cdot+\alpha)^{-1}S_{E_k^+}^{\lambda}(\cdot)X(\cdot)
= \begin{pmatrix}
1 & \zeta \\
0 & 1 \\
\end{pmatrix}$$ with $\zeta\leq C_{8}(\lambda, \alpha) e^{-2\pi r |k|}$ and $|X|_{r''} \leq C_{9}(\lambda, \alpha) e^{\frac{3\pi r''}{2}|k|}$ for any $0<r''< r$. Then for $R:=\frac{r}{210}$ and $\kappa:=\frac{1}{10}$, it is easy to see that for $|k|$ large enough, $$|X|_{R}^{14} \zeta^\kappa \leq C_{9}^{14} C_{8}^{\frac{1}{10}} e^{-\frac{\pi r}{10} |k|}\leq 10^{-5} D^{-4}_{\tau} \gamma^{12} R^{4(4\tau+1)}.$$ Hence, by Theorem \[thm\_upperbound\], we get $|G_k(\lambda)|\geq \zeta^{\frac{11}{10}}$.
Now it suffices to obtain a lower bound on $\zeta$. By Proposition \[estimate-kappa\] (see also Remark \[remark\_T\]), for $|k|$ large enough, we have $$\zeta> C_{9}^{-T(R,\lambda)} e^{-\frac{\pi T(R,\lambda) }{140} r |k|} >C(\lambda,\alpha)e^{- \frac{40\pi}{11}\tilde{\xi} r|k|},$$ where $\tilde{\xi}>1$ is an absolute constant. As a consequence, one can conclude that $$|G_k(\lambda)|>C(\lambda,\alpha) \lambda^{\tilde{\xi}|k|}.$$ By modifying the constant $C$, we get the above lower bound for every $k\in{{\mathbb Z}}\backslash\{0\}$. This finishes the proof of Corollary \[thm\_lowerbound\].
By Aubry duality, it is enough to consider the case where $0<\lambda<1$. Then the result follows from the assertion (2) of Corollary \[cor\_mathieu\_upperbound\] and Corollary \[thm\_lowerbound\].
Homogeneous spectrum {#sec_homo}
====================
Criterion for homogeneity of spectrum
-------------------------------------
We first present a general criterion for establishing the homogeneous spectrum via gap estimates. The idea first appeared in Corollary 3 of [@HA] and then Theorem H of [@DGL1]. The philosophy is that the Hölder continuity together with some decay of the spectral gaps should yield homogeneity of the spectrum. In the following, contrary to [@DGL1; @HA], we give a criterion which works for a large potential and a Liouvillean frequency. We emphasize that the homogenous spectrum for $\beta(\alpha)=0$ can be obtained if the exponential decay of the spectral gaps is established.
\[thm\_homo\_spec\] Let $\alpha\in{{\mathbb T}}^d$ with $\beta=\beta(\alpha)\geq 0$, and let $V\in C^{\omega}({{\mathbb T}}^d,{{\mathbb R}})$ be non-constant. Assume that
- $N=N_{V,\alpha}$ is $\sigma-$Hölder continuous on $[a,b]$ with $0<\sigma<1$.
Then for any $\tilde\epsilon>0$, there exists $C_{12}=C_{12}(V,\alpha,\sigma,\tilde\epsilon)>0$ such that for any two spectral gaps $G_k(V)$ and $G_{k'}(V)$ with $$\overline{G_k(V)}\cap [a,b], \;\ \overline{G_{k'}(V)}\cap [a,b] \neq \emptyset,$$ we have $$\begin{aligned}
\mathrm{dist}(G_k(V),G_{k'}(V)) &\geq C_{12} e^{-(\frac{\beta}{\sigma}+\tilde\epsilon) |k-k'|}, \;\ if \ k\neq k',\label{distance_1}\\
|E_k^--\underline{E}|&\geq C_{12} e^{-(\frac{\beta}{\sigma}+\tilde\epsilon) |k|},\;\ if \ a=\underline{E}, \label{distance_01}\\
|E_k^+-\overline{E}| &\geq C_{12} e^{-(\frac{\beta}{\sigma}+\tilde\epsilon) |k|}, \;\ if \ b=\overline{E}. \label{distance_02}\end{aligned}$$ Furthermore, for an interval $[a, b]$ with $a=\underline{E}$ or $E_{m}^+$ for some $m\in{{\mathbb Z}}^d\backslash\{0\}$ and $b=\overline{E}$ or $E_{n}^-$ for some $n\in{{\mathbb Z}}^d\backslash\{0\}$, if
- there exist $C$, $\vartheta>0$, which only depend on $V,\alpha$, such that $|G_k(V)|\leq C e^{-\vartheta |k|}$ if $\overline{G_k(V)}\cap [a,b] \neq \emptyset$,
- $ \beta=\beta(\alpha)\leq \frac{\sigma \vartheta}{2}$,
hold, then there exists $\mu=\mu(a,b,V, \alpha, \sigma, C, \vartheta, d)\in (0,1)$, such that $$| (E-\epsilon,E+\epsilon) \cap \Sigma_{V, \alpha}| > \mu\epsilon,\quad \forall \ E\in \Sigma_{V, \alpha}\cap [a,b], \;\ \forall \ 0<\epsilon\leq {\rm diam}\Sigma_{V, \alpha}.$$
Since the decay rate $\vartheta$ is related to the analytic radius of the potential, ${\rm (H3)}$ means that the radius should be relatively large compared to $\beta$. This kind of condition is necessary for the homogeneity of the spectrum, see counterexamples of Avila-Last-Shamis-Zhou [@ALSZ].
For two different gaps $G_k(V)$, $G_{k'}(V)$ with $\overline{G_k(V)}\cap [a,b]\neq \emptyset$ and $\overline{G_{k'}(V)}\cap [a,b]\neq \emptyset$, without loss of generality, we assume that $E_{k}^+\leq E_{k'}^-$. Hence $$\mathrm{dist}(G_k(V),G_{k'}(V) )=E_{k'}^--E_k^+.$$ On the one hand, the $\sigma-$Hölder continuity of $N$ on $[a,b]$ implies $$|N(E_{k}^-)-N(E_{k'}^+)|\leq c' (E_k^--E_{k'}^+)^{\sigma}$$ for some constant $c'>0$ independent of $E$. On the other hand, the definition of $\beta=\beta(\alpha)$ means that for any $\tilde\epsilon>0$, there exists $\tilde c=\tilde c(\alpha, \sigma\tilde\epsilon)>0$ such that $$|N(E_{k}^-)-N(E_{k'}^+)| \geq \|{\langle}k-k', \alpha {\rangle}\|_{{{\mathbb T}}} \geq \tilde c \, e^{-(\beta+\sigma\tilde\epsilon)|k-k'|}.$$ Combining the above estimates, we conclude that $$\mathrm{dist}(G_k(V),G_{k'}(V) ) \geq \left(\frac{\tilde c}{c'}\right)^{\frac1\sigma} e^{-(\frac{\beta}{\sigma}+\tilde\epsilon) |k-k'|},$$ which gives (\[distance\_1\]). The proof of (\[distance\_01\]) and (\[distance\_02\]) is similar, we omit the details.
By assumption (H3), we have $D:=\frac{\vartheta\sigma+2\beta}{2\beta}>2$. Given any $E\in \Sigma_{V, \alpha}\cap [a,b]$ and any $\epsilon>0$, let $${{{\mathcal N}}}={{{\mathcal N}}}(E,\epsilon):=\{k \in {{\mathbb Z}}^d \backslash\{0\}: G_k(V) \cap (E-\epsilon, E+\epsilon)\neq \emptyset, \; \overline{G_k(V)} \cap [a,b] \neq \emptyset \},$$ and let $k_0\in {{{\mathcal N}}}$ be such that $|k_0|=\min_{k\in {{{\mathcal N}}}}|k|$. By (\[distance\_1\])–(\[distance\_02\]), there exists a constant $C_{12}=C_{12}(V,\alpha,\sigma,\frac{\vartheta}{D}-\frac{\beta}{\sigma})>0$ such that $$\begin{aligned}
&&{\rm dist}(G_k(V),G_{k_0}(V) )\geq C_{12} e^{-\frac{\vartheta}{D} |k-k_0|}\geq C_{12} e^{-\frac{2\vartheta}{D} |k|},\ \forall \ k\in{{{\mathcal N}}}\backslash\{k_0\},\label{dist_1}\\
&&|E_k^--\underline{E}|\geq C_{12} e^{-\frac{\vartheta}{D} |k|},\ \forall \ k\in{{{\mathcal N}}}, \quad {\rm if} \ a=\underline{E},\label{dist_01}\\
&&|E_k^+-\overline{E}| \geq C_{12} e^{-\frac{\vartheta}{D} |k|},\ \forall \ k\in{{{\mathcal N}}}, \quad {\rm if} \ b=\overline{E}.\label{dist_02}\end{aligned}$$ Since $E \in \Sigma_{V,\alpha}$, it is easy to see that $$\begin{aligned}
&|G_{k_0}(V) \cap (E-\epsilon,E+\epsilon)|\leq \epsilon,\\
&|(-\infty, \underline{E})\cap(E-\epsilon, E+\epsilon)|\leq \epsilon, \quad {\rm if} \;\ a=\underline{E}, \\
&|(\overline{E}, +\infty)\cap(E-\epsilon, E+\epsilon)|\leq \epsilon,\quad {\rm if} \;\ b=\overline{E}.\end{aligned}$$
Without loss of generality, assume that $\epsilon< \frac{b-a}{2}$. Then $[a,b] \not\subset (E-\epsilon, E+\epsilon)$. We consider the following three cases.
**Case 1. $(E-\epsilon, E+\epsilon)\subset [a,b]$.** By the definition of ${{{\mathcal N}}}$, we have $${\rm dist}(G_k(V),G_{k_0}(V) )\leq 2\epsilon,\quad \forall \ k\in {{{\mathcal N}}}.$$ Combining with (\[dist\_1\]), we get $|k|\geq \frac{D}{2\vartheta}\left|\ln\frac{2\epsilon}{C_{12}}\right|$ for any $k\in{{{\mathcal N}}}\backslash\{k_0\}$. Thus, $$\sum\limits_{k \in {{{\mathcal N}}}\backslash \{k_0\}} |G_k(V) \cap (E-\epsilon,E+\epsilon)|
\leq C \sum_{|k|\geq \frac{D}{2\vartheta}\left|\ln\frac{2\epsilon}{C_{12}}\right|} e^{-\vartheta|k|}\leq \epsilon^{\frac{D+2}4},$$ provided that $0<\epsilon\leq\epsilon_1$ for some $\epsilon_1=\epsilon_1(V, \alpha, \sigma, C, \vartheta, d)>0$ (but independent of the choice of $E$). So we have $$\begin{aligned}
& |(E-\epsilon,E+\epsilon)\cap \Sigma_{V,\alpha}|\nonumber\\
\geq&\ 2 \epsilon - |G_{k_0}(V)\cap(E-\epsilon, E+\epsilon)|
-\, \sum_{k \in {{{\mathcal N}}}\backslash\{k_0\} } |G_k(V) \cap (E-\epsilon,E+\epsilon)|\nonumber\\
\geq&\ 2\epsilon - \epsilon - \epsilon^{\frac{D+2}4}\nonumber\\
\geq&\ \frac34 \epsilon, \quad \forall \ 0<\epsilon\leq \epsilon_1.\label{local_homo_0}\end{aligned}$$
**Case 2. $(E-\epsilon, E+\epsilon)\cap (-\infty,a)\neq\emptyset$.** In this case, one has $$|E_{k}^--a|\leq 2\epsilon,\quad \forall \ k\in {{{\mathcal N}}}.$$ We need to distinguish two cases: if $a=\underline{E}$, then by $(\ref{dist_01})$, we get $|k|\geq \frac{D}{\vartheta}\left|\ln\frac{2\epsilon}{C_{12}}\right|$. If $a=E^{+}_{m}$, then by (\[dist\_1\]), if $\epsilon$ is small enough (the smallness depends on $m$), we have $$|k|\geq \frac{D}{\vartheta}\left|\ln\frac{2\epsilon}{C_{12}}\right|-|m| \geq \frac{D}{2\vartheta}\left|\ln\frac{2\epsilon}{C_{12}}\right|.$$ Hence, if $0<\epsilon\leq \epsilon_2$ for some $\epsilon_2=\epsilon_2(a, V, \alpha, \sigma, C, \vartheta, d)$, then we have $$\sum\limits_{k \in {{{\mathcal N}}}} |G_k(V) \cap (E-\epsilon,E+\epsilon)|
\leq C \sum_{k \in {{{\mathcal N}}}} e^{-\vartheta|k|}\leq \epsilon^{\frac{D+2}4}.$$ So we have $$\begin{aligned}
& |(E-\epsilon,E+\epsilon)\cap \Sigma_{V,\alpha}|\nonumber\\
\geq &\ |(E-\epsilon,E+\epsilon)\cap \Sigma_{V,\alpha}\cap [a,b]|\nonumber\\
\geq&\ 2 \epsilon - |(-\infty, a)\cap(E-\epsilon, E+\epsilon)|
-\, \sum_{k \in {{{\mathcal N}}} } |G_k(V) \cap (E-\epsilon,E+\epsilon)|\nonumber\\
\geq&\ 2\epsilon - \epsilon - \epsilon^{\frac{D+2}4}\nonumber\\
\geq&\ \frac34 \epsilon,\quad \forall \ 0<\epsilon\leq \epsilon_2.\label{local_homo_1}\end{aligned}$$
**Case 3. $(E-\epsilon, E+\epsilon)\cap (b,+\infty)\neq\emptyset$.** Similarly to the above case, there exists $\epsilon_3=\epsilon_3(b, V, \alpha, \sigma, C, \vartheta, d)>0$ such that $$\label{local_homo_2}
|(E-\epsilon,E+\epsilon)\cap \Sigma_{V,\alpha}|\geq \frac34 \epsilon, \quad \forall \ 0<\epsilon\leq \epsilon_3.$$
Let $\epsilon_0:=\min\{\frac{b-a}{2}, \epsilon_1,\epsilon_2, \epsilon_3\}$. By (\[local\_homo\_0\]) – (\[local\_homo\_2\]), for any $E\in\Sigma_{V,\alpha}\cap [a,b]$, we have $$|(E-\epsilon,E+\epsilon)\cap \Sigma_{V,\alpha}|\geq \frac34 \epsilon, \quad \forall \ 0<\epsilon\leq \epsilon_0.$$ As for the case $\epsilon\in (\epsilon_0, {\rm diam}\Sigma_{V,\alpha})$, we have $$|(E-\epsilon,E+\epsilon)\cap \Sigma_{V,\alpha}|\geq |(E-\epsilon_0,E+\epsilon_0)\cap \Sigma_{V,\alpha}|\geq \frac34 \epsilon_0\geq \frac{3\epsilon_0}{4\, {\rm diam}\Sigma_{V,\alpha}} \cdot \epsilon,$$ which completes the proof.
Applications of the criterion
-----------------------------
Let us consider the spectrum of $H_{V,\alpha,\theta}$ with $\alpha\in{{\mathbb R}}$ satisfying $\beta(\alpha)=0$, and $V\in C^\omega({{\mathbb T}},{{\mathbb R}})$. Split $\Sigma_{V,\alpha}$ into $\Sigma^{\rm sup}_{V,\alpha}\cup\Sigma^{\rm sub}_{V,\alpha}$ as presented in Theorem \[global-red\]. As an application of Theorem \[thm\_homo\_spec\], we can get the homogeneity of $\Sigma_{V, \alpha}^{\rm sub}=\bigcup_{i} (\Sigma_{V, \alpha}\cap I_i)$ after showing the $\frac12-$Hölder continuity of the integrated density of states $N=N_{V,\alpha}$ in the global subcritical regime.
\[thm\_holder\] If $\beta(\alpha)=0$, then IDS is $\frac{1}{2}-$Hölder continuous on $I_i$, $1\leq i \leq m$.
In the non-perturbative regime considered in Section \[subs\_duality\], the above result has been shown in [@A1] and [@AvilaJito]. In the global subcritical regime, the result was assumed to be known, however, we could not find a reference in the literature. For completeness, we give a proof in Appendix \[Appendix\_B\].
Let us focus on $E\in \Sigma_{V,\alpha}\cap I_i$ such that the corresponding cocycle is subcritical. Set $I_i:=[a'_i,b'_i]$. By Theorem \[global-red\], $a'_i=\underline{E}$ or $E_{m_i}^+$ for some $m_i\in{{\mathbb Z}}\backslash\{0\}$ and $b'_i=\overline{E}$ or $E_{n_i}^-$ for some $n_i\in{{\mathbb Z}}\backslash\{0\}$.
Assertion (1) in Theorem \[theo calibration gaps bands\] was already shown in Corollary \[cor\_global\]. Since $\alpha\in {{\mathbb R}}\backslash {{\mathbb Q}}$ with $\beta(\alpha)=0$, hypothesis (H2) holds with $C$ and $\vartheta$ given by Corollary \[cor\_global\]. Moreover, (H1) follows from Proposition \[thm\_holder\] with $\sigma=\frac12$, while (H3) holds automatically, since $\vartheta> 0=\beta(\alpha)$. Hence, by applying Theorem \[thm\_homo\_spec\], we get the assertion (2) of Theorem \[theo calibration gaps bands\], and there exists $\mu_i=\mu_i(a'_i,b'_i,V, \alpha)\in (0,1)$, such that $$| (E-\epsilon,E+\epsilon) \cap \Sigma_{V, \alpha}| > \mu_i\epsilon,\quad \forall \ E\in \Sigma_{V, \alpha}\cap [a'_i,b'_i], \;\ \forall \ 0<\epsilon\leq {\rm diam}\Sigma_{V, \alpha}.$$ Taking $\mu_0:=\min_{1\leq i\leq m}\{\mu_i\}$, we can show assertion (3) of Theorem \[theo calibration gaps bands\]. Note that the structure of $\Sigma_{V,\alpha}$ is uniquely determined by $V$ and $\alpha$, thus $\mu_0$ only depends on $V$ and $\alpha$.
For $\alpha \in \mathrm{SDC}$ as defined in , we have that $\Sigma_{V,\alpha}^{\rm sup}$ is $\mu_1-$homogeneous for some $\mu_1\in(0,1)$ in view of Theorem H in [@dgsv]. In particular, $\alpha \in \mathrm{SDC}$ implies that $\beta(\alpha)=0$. Combining with the homogeneity of $\Sigma_{V,\alpha}^{\rm sub}$ (Theorem \[theo calibration gaps bands\] (3)), we can prove directly that $\Sigma_{V,\alpha}=\Sigma_{V,\alpha}^{\rm sub}\cup\Sigma_{V,\alpha}^{\rm sup}$ is $\mu-$homogeneous for some $0<\mu\leq \min\{\mu_0, \, \mu_1\}$.
As another application of the criterion, we prove the homogeneity of spectrum for the noncritical almost Mathieu operator $H_{\lambda,\alpha,\theta}$.
By Aubry duality, it is enough for us to consider the case where $0<\lambda<1$. If $\beta(\alpha)=0$, assertion (1) was shown in Corollary \[cor\_mathieu\_upperbound\], which implies the hypothesis (H2) for $[a,b]=[\underline{E},\overline{E}]$ with $\vartheta= -\frac{\ln \lambda}{4}$, while the hypothesis (H3) holds automatically since $\vartheta>0=\beta(\alpha)$. Moreover, the $\frac12-$Hölder continuity of the integrated density of states on $[\underline{E},\overline{E}]$ was shown in Corollary 3.10 of [@A1]. So we can apply Theorem \[thm\_homo\_spec\] and get assertions (2) and (3).
Deift’s conjecture – Proof of Theorem \[thm\_deift\_conjecture\]
================================================================
To consider Toda lattice equation (\[Toda\]) or equivalently the Lax pair (\[Toda\_Lax\]), let us recall some basic notions and results about the almost periodic Jacobi matrix in the work of Sodin-Yuditskii [@SY95; @SY97].
Consider a self-adjoint almost periodic Jacobi matrix $J$: $$\label{Jacobi_matrix}
(Ju)_n=a_{n-1}u_{n-1}+b_nu_n+a_n u_{n-1},$$ with a compact spectrum $\Sigma=[\inf\Sigma,\sup\Sigma]\setminus\bigcup_{k\in{{\mathbb Z}}}(E_k^-,E_k^+).$ Assume that $\Sigma$ is homogeneous, and let ${{{\mathcal J}}}_\Sigma$ be the class of reflectionless Jacobi matrices with spectrum $\Sigma$. Let $\pi_1({{\mathbb C}}\backslash\Sigma)$ be the fundamental group of ${{\mathbb C}}\backslash\Sigma$. This is a free group admitting a set of generators $\{c_k\}_{k\in{{\mathbb Z}}}$, where $c_k$ is a counterclockwise simple loop intersecting ${{\mathbb R}}$ at $\inf\Sigma-1$ and $\frac12(E_k^{+}+E_k^{-})$. Then, consider the Abelian compact group $\pi_*({{\mathbb C}}\backslash\Sigma)$ of unimodular characters on $\pi_1({{\mathbb C}}\backslash\Sigma)$. Here a character means a function ${{{\mathcal K}}}\colon\pi_1({{\mathbb C}}\backslash\Sigma)\rightarrow{{\mathbb T}}$ satisfying $${{{\mathcal K}}}(\gamma_1 \gamma_2)={{{\mathcal K}}}(\gamma_1)\, {{{\mathcal K}}}(\gamma_2), \quad \gamma_1, \, \gamma_2 \in \pi_1({{\mathbb C}}\backslash\Sigma).$$ An element ${{{\mathcal K}}}\in \pi_*({{\mathbb C}}\backslash\Sigma)$ is uniquely determined by its action on loops $c_k$, so we can write ${{{\mathcal K}}} =({{{\mathcal K}}}(c_k))_{k\in{{\mathbb Z}}}= (e^{2\pi i \tilde{K}_k})_{k\in{{\mathbb Z}}}$.
\[theorem\_SY97\] There is a continuous one-to-one correspondence between almost periodic Jacobi matrices $J\in {{{\mathcal J}}}_{\Sigma}$ and characters ${{{\mathcal K}}}\in\pi_*({{\mathbb C}}\backslash \Sigma)$.
If we identify the Jacobi matrix $J$ given in (\[Jacobi\_matrix\]) with $(a,b)\in\ell^\infty({{\mathbb Z}})\times\ell^\infty({{\mathbb Z}})$, then by Theorem \[theorem\_SY97\], there exists a continuous map ${{{\mathcal H}}}\colon{{\mathbb T}}^{{{\mathbb Z}}}\to \ell^\infty({{\mathbb Z}})\times\ell^\infty({{\mathbb Z}})$ such that for any $J\in {{{\mathcal J}}}_{\Sigma}$ given as in (\[Jacobi\_matrix\]), one can find a unique ${{{\mathcal K}}}\in\pi_*({{\mathbb C}}\backslash \Sigma)$ such that $$\label{one-to-one_map}
(a,b)={{{\mathcal H}}}\left(\left({{{\mathcal K}}}(c_k)\right)_{k\in{{\mathbb Z}}}\right)={{{\mathcal H}}}\left(\left(e^{-{2\pi{\rm i}\tilde{K}_k}}\right)_{k\in{{\mathbb Z}}}\right).$$
Now we consider the Lax pair (\[Toda\_Lax\]) in a more general form. Given any $f\in L^\infty(X,{{\mathbb R}})$ with $\Sigma\subset X$, we define the infinite-dimensional matrix $f(J)$ in the sense of standard functional calculus and decompose it into $
f^+(J)+f^-(J)$, the sum of an upper triangular matrix $f^+(J)$ and a lower triangular matrix $f^-(J)$. We also set $
M_f(J):=f^+(J)-f^-(J).
$ Then, given an almost periodic Jacobi matrix $J_0\in{{{\mathcal J}}}_{\Sigma}$, we define the Lax pair $$\label{Toda_Lax_proof}
\frac{d}{dt}J(t)=[M_f(J(t)),J(t)], \quad J(0)=J_0.$$
\[theorem\_VY\] Assume that $\Sigma$ is homogeneous and the almost periodic Jacobi matrix $J_0\in{{{\mathcal J}}}_\Sigma$ has purely absolutely continuous spectrum. Given $f \in L^\infty(X,{{\mathbb R}})$ with $\Sigma \subset X$, the following holds.
1. There exists a unique solution $J=J(t)$ of (\[Toda\_Lax\_proof\]), well-defined for all $t \in {{\mathbb R}}$. Moreover, for every $t$, $J(t)$ is an almost periodic Jacobi matrix with constant spectrum $\Sigma$.
2. For $t\in{{\mathbb R}}$, let ${{{\mathcal K}}}^t\in \pi_*({{\mathbb C}}\backslash\Sigma)$ be the character corresponding to $J(t)$. There exists a homomorphism $\xi\colon\pi_1({{\mathbb C}}\backslash\Sigma)\rightarrow {{\mathbb R}}$, depending on $f$, such that ${{{\mathcal K}}}^t(c_k)={{{\mathcal K}}}^0(c_k) \, e^{-2\pi {\rm i} t\xi(c_k)}$.
Obviously, with $f(x)=x$ and assuming that all the diagonal elements of $M_f(J)$ vanish, we get the Lax pair (\[Toda\_Lax\]), which is equivalent to the Toda flow (\[Toda\]). By the assertion (2) of Theorem \[theorem\_VY\], combining with (\[one-to-one\_map\]), we get $$(a(t),b(t))={{{\mathcal H}}}\left(\left( e^{-2\pi {\rm i} \tilde{K}_k^t} \right)_{k\in{{\mathbb Z}}}\right)={{{\mathcal H}}}\left(\left( e^{-2\pi {\rm i} \left[\tilde{K}_k^0 + t \xi(c_k)\right]} \right)_{k\in{{\mathbb Z}}}\right),$$ which implies the time almost periodicity of solutions of (\[Toda\]).
Now we are going to prove Theorem \[thm\_deift\_conjecture\]. Let $V\in C^{\omega}({{\mathbb T}},{{\mathbb R}})$ be subcritical and $\alpha\in{{\mathbb R}}\backslash{{\mathbb Q}}$ with $\beta(\alpha)=0$. By Kotani’s theory [@Kot84], for almost every $\theta\in{{\mathbb T}}$, for almost every $E$ such that $L_{V,\alpha}(E) = 0$, we have $m^+_{H_{V,\alpha,\theta}}(E) = -\overline{m^-_{H_{V,\alpha,\theta}}}(E)$. It was later improved in Theorem 2.2 of [@A1], where it is shown that the above assertion is true for every $\theta\in{{\mathbb T}}$. By (\[green\_m+-\]), we have that $H_{V,\alpha,\theta}$ is reflectionless for every $\theta\in{{\mathbb T}}$. Moreover, it follows from Theorem \[theo calibration gaps bands\] that $\Sigma_{V,\alpha}$ is homogeneous. Thus, by Theorem \[theorem\_VY\], it is sufficient to verify the purely absolute continuity of spectrum.
\[thm\_suncritical\_ac\] If $\beta(\alpha)=0$ and $V\in C^{\omega}({{\mathbb T}},{{\mathbb R}})$ is subcritical, then the spectrum of the operator $H_{V,\alpha,\theta}$ is purely absolutely continuous.
Theorem \[thm\_suncritical\_ac\] was proved from the viewpoint of dynamics. Roughly speaking, in view of Theorem \[global-red\], we can transform the corresponding Schrödinger cocycle into the “non-perturbative regime" (Proposition \[prop\_ar\_1\]), for which the purely absolute continuity has been shown in [@A1].
Theorem \[thm\_suncritical\_ac\] can also be shown by inverse spectral theory. Assuming finite total gap length, homogeneity of the spectrum together with the reflectionless condition, Gesztesy-Yuditskii [@GY] have shown that the corresponding spectral measure is purely absolutely continuous. Then, Theorem \[thm\_suncritical\_ac\] follows from Theorem \[theo calibration gaps bands\].
Proof of Theorem \[thm\_almost\_almost-1\] {#Appendix_A}
==========================================
Given $\theta \in {{\mathbb R}}$ and $\epsilon_0>0$, we denote by $\{n_l\}_l$ the set of $\epsilon_0-$resonances of $\theta$, i.e., $$\|2 \theta - n_l \alpha\|_{{\mathbb T}}\leq e^{-\epsilon_0|n_l|},\quad \text{and}\quad \|2 \theta - n_l \alpha\|_{{\mathbb T}}=\min_{|m|\leq |n_l|} \|2 \theta - m \alpha\|_{{\mathbb T}}.$$
Let $\lambda>1$. By [@AvilaJito], the family $\{H_{\lambda,\alpha,\theta}\}_\theta$ is almost localized. Fix $\theta \in {{\mathbb R}}$, and let $u=(u_j)_{j\in{{\mathbb Z}}}$ be a generalized solution to $ H_{\lambda,\alpha,\theta} u= E u$, with $ u_0=1$ and $| u_{j}|\leq 1$ for all $j \in {{\mathbb Z}}$. Given an interval $I=[i_1,i_2] \subset {{\mathbb Z}}$ of length $N\geq 0$, we denote by $G_I$ the Green’s function $(x,y)\mapsto (H_{ \lambda,\alpha,\theta}-E)^{-1}(x,y)$ restricted to $I$ with zero boundary conditions at $i_1-1$ and $i_2+1$. Then for any $j \in I$, we have $$\label{exrpe green}
u_j=-G_I(i_1,j) u_{i_1-1}-G_I(j,i_2) u_{i_2+1}.$$ Let us denote by $P_m(\theta)$ the upper-left coefficient of the $m^{th}$ iterate $(m\alpha,\mathcal{A}_{m}(E))$ of the cocycle $(\alpha,S_E^\lambda)$. Then by Cramer’s rule, we have $$\label{cramerrule}
|G_I(i_1,j)|=\left| \frac{P_{i_2-j}(\theta+(j+1)\alpha)}{P_N(\theta+ i_1 \alpha)}\right|,\quad |G_I(j,i_2)|=\left| \frac{P_{j-i_1}(\theta+i_1 \alpha)}{P_N(\theta+ i_1 \alpha)}\right|.$$ Given $\xi > 0$ and $m\in {{\mathbb N}}$, we say that $y \in {{\mathbb Z}}$ is $(\xi,m)-$*regular* if there exists an interval $J=[x_1,x_2]\subset {{\mathbb Z}}$ of length $m$ such that $y \in J$ and $$|G_{J}(y,x_i)| < e^{-\xi |y-x_i|},\quad |y-x_i| \geq \frac{1}{7}m,\quad i=1,2.$$ Recall that $L(\alpha,S_E^\lambda)=\ln \lambda$ for any energy $E\in \Sigma_{\lambda,\alpha}$. By subadditivity, for any $\eta>0$, any $E' \in \Sigma_{\lambda,\alpha}$, and for $m \geq 0$ large enough, we have $|\mathcal{A}_{m}(E')|_{{\mathbb T}}\leq e^{ (\ln \lambda- \eta)m}$. In particular, $P_m(\theta) \leq e^{ (\ln \lambda- \eta)m}$.
Let $(q_i)_{i \geq 1}$ be the sequence of denominators of best approximants of $\alpha$. We associate with any integer $C_0 |n_l| < |j| < C_0^{-1} |n_{l+1}|$ scales $\ell \geq 0$ and $s\geq 1$ so that $$2 s q_\ell \leq\zeta j < \min(2(s+1)q_\ell,2q_{\ell+1}),$$ where $\zeta:=\frac{1}{32}$ if $2|n_l|< j < 2^{-1} |n_{l+1}|$, and $\zeta:=\frac{C_0-1}{16C_0}$ otherwise. We set
- $I_1:=[-2s q_\ell+1,0]$ and $I_2:=[j-2 s q_\ell +1,j+2 sq_\ell]$ if $j<|n_{l+1}|/3$, $n_{l}\geq 0$.
- $I_1:=[1,2s q_\ell]$ and $I_2:=[j-2 s q_\ell +1,j+2 sq_\ell]$ if $j<|n_{l+1}|/3$ and $n_{l}<0$.
- $I_1:=[-2s q_\ell+1,2s q_\ell]$ and $I_2:=[j-2 s q_\ell +1,j]$ if $|n_{l+1}|/3\leq j< |n_{l+1}|/2$.
- $I_1:=[-2s q_\ell+1,2s q_\ell]$ and $I_2:=[j+1,j+2 sq_\ell]$ if $j\geq |n_{l+1}|/2$.
In particular, the total number of elements in $I_1 \cup I_2$ is $6 sq_\ell$. Fix $\delta>0$ arbitrary. If $\epsilon_0>0$ is chosen sufficiently small, then in view of $\beta(\alpha)=0$, Lemma 5.8 in [@AvilaJito] implies that there exists an integer $j_0=j_0(C_0,\alpha,n,\delta)>0$ such that for $j> j_0$, the set $\{\theta_m:=\theta+m \alpha\}_{m \in I_1 \cup I_2}$ is $\delta-$*uniform*, i.e., $$\max\limits_{z \in [-1,1]}\max\limits_{m \in I_1 \cup I_2} \prod\limits_{m\neq p\in I_1 \cup I_2} \frac{|z-\cos (2 \pi \theta_p)|}{|\cos (2 \pi \theta_m)-\cos(2 \pi \theta_p)|}< e^{(6sq_\ell-1)\delta}.$$ Following the proof of Lemma 5.4 in [@AvilaJito], we conclude that for any $\eta>0$, there exists $j_1=j_1(C_0,\alpha,\lambda,\eta)>0$ such that any $j> j_1$ is $(\ln \lambda-\eta,6 s q_\ell-1)-$regular.
We consider the case that $\theta$ is $\epsilon_0-$resonant. We will show that the sequence $( u_j)_j$ decays exponentially in some suitable interval between two consecutive resonances, with a rate close to the Lyapunov exponent $L(\alpha,S_E^{\lambda})=\ln\lambda$. By the condition $\beta(\alpha)=0$, we know that $|n_l|=o(|n_{l+1}|)$. Let us fix some small $\eta>0$. Given $l>0$ sufficiently large, take $\ell>0$ such that $2q_\ell \leq \zeta(2C_0 |n_l|+1)<2q_{\ell+1}$, and let $2C_0 |n_l|+\eta|n_{l+1}| \leq |j| \leq (2C_0)^{-1} |n_{l+1}|$. We set $b_l:=2 C_0 |n_l|+1$. Then for any $y \in [b_l,2j]$, there exists an interval $I(y)=[x_1,x_2]\subset {{\mathbb Z}}$ with $y \in I(y)$ and $$\mathrm{dist}(y,\partial I(y))\geq \frac{1}{7}|I(y)|\geq \frac{6 q_{\ell}-1}{7}\geq \frac{q_{\ell}}{2},$$ where $\partial I(y):=\{x_1,x_2\}$, and such that $$|G_{I(y)}(y,x_i)|\leq e^{-(\ln \lambda-\eta)|y-x_i|}\leq e^{-(\ln \lambda-\eta)\frac{q_{\ell}}{2}},\quad i=1,2.$$ For $z \in \partial I(y)$, we denote by $z'$ the neighbour of $z$ not belonging to $I(y)$. If $x_2+1<2 j$ or $x_1-1 >b_l$, we can expand $ u_{x_2+1}$ or $u_{x_1-1}$ following , with $I=I(x_2+1)$ or $I=I(x_1-1)$. We continue to expand each term until we arrive to $\widetilde z$ such that either $\widetilde z\leq b_l$, or $\widetilde z> 2j$, or the number of $G_I$ terms in the following product becomes $\lfloor \frac{2j}{q_{\ell}}\rfloor$, whichever comes first: $$u_j=\sum\limits_{r,\ z_{i+1} \in \partial I(z_i')} G_{I(j)}(j,z_1)G_{I(z_1')}(z_1',z_2)\dots G_{I(z_r')}(z_r',z_{r+1}) u_{z_{r+1}'}.$$ In the first two cases, we estimate $$\begin{aligned}
&|G_{I(j)}(j,z_1)G_{I(z_1')}(z_1',z_2)\dots G_{I(z_r')}(z_r',z_{r+1}) u_{z_{r+1}'}|\\ \leq\ &e^{-(\ln \lambda-\eta)(|j-z_{1}|+\sum_{i=1}^r |z_i'-z_{i+1}|)}\\
\leq\ &e^{-(\ln \lambda-\eta)(|j-z_{r+1}|-(r+1))} \\
\leq\ &\max\big(e^{-(\ln \lambda-\eta)(j-b_l-\frac{2j}{q_{\ell}})},e^{-(\ln \lambda-\eta)(2j-j-\frac{2j}{q_{\ell}})}\big),\\
\leq\ &e^{-(\ln \lambda-\eta)(j+o(j))},\end{aligned}$$ where we have used that $|b_l| = o(|j|)$, while in the third case, we have $$|G_{I(j)}(j,z_1)G_{I(z_1')}(z_1',z_2)\dots G_{I(z_r')}(z_r',z_{r+1}) u_{z_{r+1}'}|\leq e^{-(\ln \lambda-\eta)\frac{q_{\ell}}{2}\lceil \frac{2j}{q_{\ell}}\rceil}.$$ Fix $\delta>0$ arbitrarily small. By taking $|j|$ to be sufficiently large, resp. $\eta$ small enough in the previous expression, we conclude that $| u_j| \leq e^{-(\ln \lambda-\delta) |j|}$ for $|j|$ large enough with $2C_0 |n_l|+\eta|n_{l+1}| \leq |j| \leq (2C_0)^{-1} |n_{l+1}|$.
We consider the other case, i.e., when $\theta$ is not $\epsilon_0-$resonant. Denote by $n$ its last $\epsilon_0-$resonance, set $b:=2 C_0 |n|+1$ and let $|j | \geq b$. Let us fix some small $\eta>0$. Then for any $y \in [b,2j]$, there exists an interval $I(y)=[x_1,x_2]\subset {{\mathbb Z}}$ with $y \in I(y)$ and $$\mathrm{dist}(y,\partial I(y))\geq \frac{1}{7}|I(y)|\geq \frac{6 q_{\ell}-1}{7}\geq \frac{q_{\ell}}{2},$$ where $\partial I(y):=\{x_1,x_2\}$, and such that $$|G_{I(y)}(y,x_i)|\leq e^{-(\ln \lambda-\eta)|y-x_i|}\leq e^{-(\ln \lambda-\eta)\frac{q_{\ell}}{2}},\quad i=1,2.$$ As previously, we can expand $ u_{x_2+1}$ or $u_{x_1-1}$ following , with $I=I(x_2+1)$ or $I=I(x_1-1)$. We continue to expand each term until we arrive to $\widetilde z$ such that either $\widetilde z\leq b$, or $\widetilde z> 2j$, or the number of $G_I$ terms in the following product becomes $\lfloor \frac{2j}{q_{\ell}}\rfloor$, whichever comes first: $$u_j=\sum\limits_{r,\ z_{i+1} \in \partial I(z_i')} G_{I(j)}(j,z_1)G_{I(z_1')}(z_1',z_2)\dots G_{I(z_r')}(z_r',z_{r+1}) u_{z_{r+1}'}.$$ In the first two cases, we estimate $$\begin{aligned}
&|G_{I(j)}(j,z_1)G_{I(z_1')}(z_1',z_2)\dots G_{I(z_r')}(z_r',z_{r+1}) u_{z_{r+1}'}|\\ \leq\ &e^{-(\ln \lambda-\eta)(|j-z_{1}|+\sum_{i=1}^r |z_i'-z_{i+1}|)}\\
\leq\ &e^{-(\ln \lambda-\eta)(|j-z_{r+1}|-(r+1))} \\
\leq\ &\max\left(e^{-(\ln \lambda-\eta)(j-b-\frac{2j}{q_{\ell}})},e^{-(\ln \lambda-\eta)(2j-j-\frac{2j}{q_{\ell}})}\right),\\
\leq\ &e^{-(\ln \lambda-\eta)(j+o(j))},\end{aligned}$$ while in the third case, we have $$|G_{I(j)}(j,z_1)G_{I(z_1')}(z_1',z_2)\dots G_{I(z_r')}(z_r',z_{r+1}) u_{z_{r+1}'}|\leq e^{-(\ln \lambda-\eta)\frac{q_{\ell}}{2}\lfloor \frac{2j}{q_{\ell}}\rfloor}.$$ Fix $\delta>0$ arbitrarily small. By taking $|j|$ be sufficiently large, resp. $\eta$ small enough in the previous expression, we conclude that $| u_j| \leq e^{-(\ln \lambda-\delta) |j|}$ for $|j|$ large enough.
Proof of Proposition \[thm\_holder\] {#Appendix_B}
====================================
The proof follows Theorem 1.6 of [@AvilaJito] (see also Corollary 3.10 of [@A1]), the key points are the quantitative almost reducibility results and Thouless formula.
If $\beta(\alpha)=0,$ then by Proposition \[prop\_ar\_1\] (see also Corollary \[prop\_duality\_global\]), there exists $0<h_1=h_1(V,\alpha)<1$, such that for any $E\in\Sigma_{V,\alpha}\cap I_i$, $1\leq i \leq m$, there exists $\Phi_E\in C^\omega({{\mathbb T}},{\rm PSL}(2,{{\mathbb R}}))$ with $|\Phi_E|_{h_1}<\Lambda=\Lambda(V,\alpha, c_0 h_1^{3}, h_1)$, $E_*=E_*(E)$ locally constant, and $V_{*}=V_*(E) \in C^\omega_{h_1}({{\mathbb T}},{{\mathbb R}})$, $|V_*|_{h_1}<c_0 h_1^{3}$, such that $$\Phi_E(\cdot+\alpha)^{-1} S_E^V(\cdot) \Phi_E(\cdot)=S_{E_*}^{V_*}(\cdot),$$ where $c_0> 0$ is the absolute constant given in Theorem \[almostredth\]. In particular, the family $\{\widehat H_{V_*,\alpha,\theta}\}_{\theta\in {{\mathbb T}}}$ is almost localized.
Therefore, by Theorem 3.8 of [@A1], there exist a phase $\theta'=\theta'(E) \in {{\mathbb T}}$ and positive constants $C=C(\alpha,h_1)$, $c=c(\alpha,h_1)$, $\epsilon_0=\epsilon_0(h_1)$ such that the following is true. Let $\{n_j\}_{j}$ be the set of $\epsilon_0-$resonances of $\theta'$, ordered in such a way that $|n_j| \leq |n_{j+1}|$. For any small $\varepsilon>0$, take $j$ such that $e^{-c{{{\mathcal N}}}}\leq\varepsilon\leq e^{-o(n)}$, with $n:=|n_j|+1$ and ${{{\mathcal N}}}:=|n_{j+1}|$ (if defined, otherwise ${{{\mathcal N}}}:=+\infty$). By composing $\Phi_E$ with the conjugacy $B$ given by Theorem 3.8 of [@A1], and noting that $\Phi_E$ is uniformly bounded, we get $\Psi:=\Phi_E B\in C_c^\omega({{\mathbb T}}, {\rm PSL}(2,{{\mathbb C}}))$ satisfying $|\Psi|_{c}\leq e^{o(n)}$, such that $$\Psi(\cdot+\alpha)^{-1} S_{E}^{V}(\cdot) \Psi(\cdot)=
\begin{pmatrix}
e^{2 \pi{\rm i}\theta'} & 0 \\
0 & e^{-2 \pi{\rm i}\theta'}
\end{pmatrix}
+
\begin{pmatrix}
q_1(\cdot) & q(\cdot) \\[1mm]
q_3(\cdot) & q_4(\cdot)
\end{pmatrix},$$ with $|q_1|_{c},\, |q_3|_{c},\, |q_4|_{c}\leq Ce^{-c{{{\mathcal N}}}}$ and $|q|_{c}\leq Ce^{-cn}$. Let $D:=\begin{pmatrix}
d^{-1} & 0\\
0 & d
\end{pmatrix}$ with $d:= \varepsilon^{\frac14} |\Psi|_c$, and set $W:=\Psi D\in C_c^{\omega}(2{{\mathbb T}}, {\rm SL}(2,{{\mathbb C}}))$. It follows from the bounds on $\Psi$ and $\varepsilon$ that $|W|_{c}\leq C'\varepsilon^{-\frac14}$ for some uniform constant $C'>0$. Hence, for $$U_\varepsilon(\cdot):=W(\cdot+\alpha)^{-1} S_{E+\mathrm{i}\varepsilon}^{V}W(\cdot)=W(\cdot+\alpha)^{-1} \left[S_{E}^{V}(\cdot)+ \begin{pmatrix}
\mathrm{i} \varepsilon & 0\\
0 & 0
\end{pmatrix}\right ]W(\cdot),$$ we get $|U_\varepsilon|_{c}\leq 1+ C''\varepsilon^{\frac12}$ for some uniform constant $C''>0$. As a result, we obtain the following estimate on the Lyapunov exponent: $$\label{distance_subcritical}
L(\a, S^V_{E+i \varepsilon})= L(\alpha,U_\varepsilon)\leq \ln |U_\varepsilon|_c \leq C'' \varepsilon^{\frac12}.$$ The above conclusions are similar to Theorem 4.4 and Corollary 4.6 in [@AvilaJito], and we refer to them for more details.
On the other hand, by Thouless formula, there exists a constant $c'>0$ such that for any $\varepsilon>0$, $$\begin{aligned}
L(\a, S^V_{E+i \varepsilon})&= L(\a, S^V_{E+i \varepsilon})-L(\a, S^V_{E})\\ &=\frac12 \int \ln \left(1+\frac{\varepsilon^2}{(E-E')^2}\right)dN_{V,\alpha}(E')\\ &\geq c'(N_{V,\alpha}(E+\varepsilon)-N_{V,\alpha}(E-\varepsilon)).\end{aligned}$$ Combining the last estimate with , we deduce that $$N_{V,\alpha}(E+\varepsilon)-N_{V,\alpha}(E-\varepsilon)\leq C'' c'^{-1}\varepsilon^{\frac12}$$ for $E\in \Sigma_{V,\alpha}\cap I_i$ and $0<\varepsilon<1$ such that $[E-\varepsilon,E+\varepsilon]\subset I_i$. Since $N_{V,\alpha}$ is locally constant on the complement of $\Sigma_{V,\alpha}$, we have that $N_{V,\alpha}$ is $\frac12-$Hölder on $I_i$.
Acknowledgements
================
We would like to thank A. Avila, M. Goldstein and S. Jitomirskaya for useful discussions. During his PhD, M. Leguil was supported by IMJ-PRG/Université Paris 6/7 and by “ANR-15-CE40-0001-03" for the project “BEKAM". J. You was partially supported by NSFC grant (11471155) and 973 projects of China (2014CB340701). Z. Zhao was supported by ANR grant “ANR-15-CE40-0001-03" for the project “BEKAM“. Q. Zhou was partially supported by ”Deng Feng Scholar Program B" of Nanjing University, Specially-appointed professor programe of Jiangsu province and NSFC grant (11671192).
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[^1]: More precisely, ${\langle}k,\varpi{\rangle}\in \mathcal{R}(k)$, where $$\mathcal{R}(k):=\left\{ {\langle}k,\varpi {\rangle}\in{{\mathbb R}}: \inf_{j \in {{\mathbb Z}}}\left| {\langle}m-k,\varpi {\rangle}- j \right| \geq \frac{\gamma}{|m|^{\tau}},\quad \forall \ m \in {{\mathbb Z}}^d\setminus\{k\} \right\}.$$
[^2]: After this work was completed, D. Damanik and M. Goldstein mentioned to us that in their setting it is also possible to get the sharp decay based on their method.
[^3]: We say $\alpha\in{{\mathbb R}}$ is strong Diophantine if there exist $\gamma,\tau>0$ such that $$\label{strongdiophantinecondition}
\inf_{j\in{{\mathbb Z}}}|n\alpha- j| \geq \frac{\gamma}{|n| (\log |n|)^\tau}, \quad \forall \ n \in {{\mathbb Z}}\backslash\{0\}.$$ For fixed $\gamma, \tau$, let ${\rm SDC}(\gamma,\tau)$ be the set of numbers satisfying , and let $\mathrm{SDC}:= \bigcup\limits_{\gamma,\tau>0}\ {\rm SDC}(\gamma,\tau)$.
[^4]: Consult Section \[ARCGlobal\] for details.
[^5]: i.e., for any $E \in \Sigma_{V,\alpha}$, the cocycle $(\alpha,S_E^V)$ is subcritical.
[^6]: Recall that a trigonometric polynomial $P_0\colon {{\mathbb T}}\to {{\mathbb C}}$ has *essential degree* at most $d\geq 1$ if its Fourier coefficients outside an interval of length $d$ are vanishing.
[^7]: In the case of an almost Mathieu operator $H_{\lambda,\alpha,\theta}$ with $\lambda<1$, then by Theorem \[thm\_almost\_almost-1\], for all $r \in(0,-\frac{\ln \lambda}{2 \pi})$ and for $\eta>0$ arbitrarily small, we have $|\widehat u_j|\leq C_1 e^{-2 \pi r |j|}$ for $2 n_{l-1}+\eta n_l \leq |j| \leq \frac{1}{2} n_l$, which is also sufficient for our purpose.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Lucas Gren\
Chalmers and the University of Gothenburg\
Gothenburg, Sweden 412–92\
[email protected]\
bibliography:
- 'references.bib'
nocite: '[@ex1; @ex2]'
title: Using the Agile Adoption Framework to Assess Agility and Guide Improvements
---
Introduction {#ooo}
============
Agile software development has become an extremely popular way of managing projects, and there have also been studies showing the effectiveness of such approaches [@serrador]. Many different measurement models for agility have been proposed but according to [@ozcan] Sidky’s Agile Adoption Framework [@sidkyphd] is the most complete to date. Many companies have implemented an agile approach but lack a way to evaluate the level of implementation, and need contextually specific tools [@xp20141]. However, getting some quantitative feedback from the teams could be a useful indication of agility that can be used to study (and improve) agile teams and also to compare them. This paper presents preliminary results from such a tool.
Sidky’s [@sidkyphd] framework assesses the level of agility an organization is ready to implement and recommends what methods these should be. There are two main differences between Sidky’s [@sidkyphd] tool and how we use it here. First, we measure the present level of agility and not agile potential. In order to measure agility level, the items were altered in time in our tool so they ask about the current situation. Second, it lets the group members fill out the survey and allows a statistical confidence interval to the result. One validation study has been conducted testing the tool in such a way [@grenjss]. The authors report that the feedback was considered useful by the team participating in the pretest, but that the measurement itself shows problems with validity. Despite that, and since the tool was considered useful by the team itself in the focus group, we believe our tool can be used to guide improvement efforts. Sidky [@sidkyphd], also validated the content in the tool by letting practitioners evaluate the items and their connection to what they think agility is.
Sidky’s framework is divided into “agile levels”, “principles”, “practices and concepts”, and “items or indicators”.
To assess the results the Agile Adoption Framework includes 4 steps: First, calculating a weight for each item (the weight of 1 is divided by the number of items if all items are considered equally important), second, computing weighted intervals. These intervals are created by taking the answer of each item and calculating a pessimistic and optimistic result for each one, and the Likert scale is then divided into a percentage.
To clarify, this means that the practices “Reflect and tune process”, “Collaborative planning”, “Collaborative teams”, “Empowered and motivated teams”, “Working standards/procedures”, “Knowledge sharing tools”, “Task volunteering”, and “Customer commitment” are assessed in the tool. The actual items (or indicators) are published in [@grenjss]. The tool consists of one survey for managers and one survey for developers. To assess an agile practice the analysis method proposed uses answers from both surveys. Some of the items are also used to assess more than one practice.
Below is the description of what the agile characteristics set out to determine. This list is needed to find what the different scores mean in the feedback table presented later.
\(1) Whether or not a collaborative or a command-control relation exists between managers and subordinates. The management style is an indication of whether or not management trusts the developers and vice versa. (2) Whether or not management is supportive of or resistive to having a collaborative environment. (3) Whether or not management can be open with customers and developers, i.e., no politics and secrets. (4) Whether or not people are intimidated/afraid to give honest feedback and participation in the presence of their managers. (5) Whether or not the developers are willing to plan in a collaborative environment. (6) Whether or not the organization does basic planning for its projects. (7) Whether or not any levels of interaction exist between people thus laying a foundation for more team work. (8) Whether or not people believe in group work and helping others or are just concerned about themselves. (9) Whether or not people are willing to work in teams. (10) Whether or not people recognize that their input is valuable in group work. (11) Whether or not the developers see the benefit and are willing to apply coding standards. (12) Whether or not developers believe in and can see the benefits of having project information communicated to the whole team. (13) Whether or not managers believe in and can see the benefits of having project information communicated to the whole team. (14) Whether or not management will be willing to buy into and can see benefits from employees volunteering for tasks instead of being assigned. (15) Whether or not developers are willing to see the benefits from volunteering for tasks. (16) Whether or not management empowers teams with decision making authority. (17) Whether or not people are treated in a way that motivates them. (18) Whether or not managers trust and believe in the technical team in order to truly empower them. (19) Whether or not developers are willing to commit to reflecting about and tuning the process after each iteration or release. (20) Whether or not management is willing to commit to reflecting about and tuning the process after each iteration or release. (21) Whether or not the organization can handle process change in the middle of the project.
Feedback to Companies {#sub:feedback}
=====================
So far, we have tested the tool on seven teams in three multinational companies consisting of two US-based and one European-based. Below is the result and recommendations that were given to each team based on the survey result. Only one result and feedback from the one team at Company A is shown in this paper as an example, but all teams were given feedback the same way.
#### Results and Recommendations for Company A — One team
Company A could focus on the following aspects to improve their agility (they are explained in more detail below): (1) Collaborative Planning with regard to the Power Distance between management and group members. (2) Task volunteering. (3) Management’s buy-in of reflecting and tuning the process after each iteration or release.
Table \[fig:Company A\] shows the results for the team. Collaborative planning and its aspect of power distance got a lower score, which means the teams would benefit from having a flatter and less hierarchical organizational structure when planning projects. Having management plan and set up the projects without participation from team members, will result in less accurate and suboptimal goals. The team members are experts on how much work the team can do under a given period of time.
At Team A, “task volunteering” got a lower score from both mangers and developers. Task volunteering is also a key to success, since developers need to take full responsibility for their current chores and deliverables together with the team as a whole. Team members are often also better at knowing their capability and making time estimates on tasks.
The last aspect of Team A concerned manager’s buy-in of “Reflect and Tune Process”. This means that the management is not willing to reflect and tune the process after each iteration or release. Continuous improvement is a key to achieving as high performance as possible. Management of Team A would benefit from having a workshop on how agile principles are supposed to work in teams, especially how the process needs fine-tuning and adaptation to each team/or context. A practice that works well in one team might need to be adapted to a new context but with the reason behind it kept. I.e. the agile principle behind an implementation can be kept, but the actual implementation must be tuned to, and reflected on, in the new context. Hopefully this will guide them into accepting and seeing the benefits of reflecting and tuning the processes and task volunteering.
Conclusions and Future Work {#sec:conclusions_and_future_work}
===========================
This paper has presented how the Agile Adoption Framework can be used to assess agility and pinpoint focus areas for companies that want to improve. Management found it useful in general to get data on possible focus points for improvement. Agility implies a set of principles that need to be followed in order to have the proposed responsiveness to change. We believe our tool could be useful as one step in the agile process assessment.
|
{
"pile_set_name": "ArXiv"
}
|
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